# Exterior products of operators and superoptimal analytic approximation

Dimitrios Chiotis  School of Mathematics, Statistics and Physics, Newcastle
University, Newcastle upon Tyne NE1 7RU, U.K<EMAIL_ADDRESS>,  Zinaida
A. Lykova  School of Mathematics, Statistics and Physics, Newcastle
University, Newcastle upon Tyne NE1 7RU, U.K<EMAIL_ADDRESS>and
N. J. Young  School of Mathematics, Statistics and Physics, Newcastle
University, Newcastle upon Tyne NE1 7RU, U.K. and School of Mathematics, Leeds
University, Leeds LS2 9JT, U.K<EMAIL_ADDRESS>
(Date: 5th August 2021)

###### Abstract.

We give a new algorithm for the construction of the unique superoptimal
analytic approximant of a given continuous matrix-valued function on the unit
circle, making use of exterior powers of operators in preference to spectral
or Wiener-Masani factorizations.

###### Key words and phrases:

Analytic approximation matrix functions, Nehari problem, singular values

###### 2020 Mathematics Subject Classification:

47B35, 30-08, 30J99, 15A75

Lykova and Young were supported by the Engineering and Physical Sciences
Research Council grant EP/N03242X/1. Chiotis was supported by a PhD
studentship from the School of Mathematics, Statistics and Physics of
Newcastle University

## Contents

1. Introduction Page 1

2. History and recent work 2

3. Exterior powers of Hilbert spaces 3

3.1. Exterior powers 3.1

3.2. Pointwise wedge product and pointwise creation operators 3.2

4. Superoptimal analytic approximation 4

4.1. Known results 4.1

4.2. Algorithm for superoptimal analytic approximation 4.2

5. Pointwise orthonormality of $\\{\xi_{i}\\}_{i=1}^{j}$ and $\\{\bar{\eta}_{i}\\}_{i=1}^{j}$ almost everywhere on $\mathbb{T}$ 5

6. The closed subspace $X_{j+1}$ of $H^{2}(\mathbb{D},\wedge^{j+2}\mathbb{C}^{n})$ 6

7. The closed subspace $Y_{j+1}$ of $H^{2}(\mathbb{D},\wedge^{j+2}\mathbb{C}^{m})^{\perp}$ 7

8. $T_{j}$ is a well-defined operator 8

9. Compactness of the operators $T_{1}$ and $T_{2}$ 9

10. Compactness of the operator $T_{j}$ 10

11. Application of the algorithm 11

References References

##  1\. Introduction

In this paper we put forward a new algorithm for the computation of the
superoptimal analytic approximation of a continuous matrix-valued function on
the circle, a notion that arises naturally in the context of the classical
“Nehari problem”, and also in the “robust stabilization problem” in control
engineering.

To explain the term “superoptimal”, let us start from the elementary
observation that a measure of the “size” of a compact operator $T$ between
Hilbert spaces is provided by the operator norm $\|T\|$ of $T$. However, a
single number can only ever provide a coarse measure of the size of a multi-
dimensional object, and there is a well-developed classical theory [12] of
‘s-numbers’ or ‘singular values’ of an operator or matrix, which provides much
more refined information about an operator than the operator norm. Consider
Hilbert spaces $\mathcal{H},\mathcal{K}$ and an operator
$T:\mathcal{H}\to\mathcal{K}$, and let $j\geq 0$. The quantity $s_{j}(T)$ is
defined to be the distance, with respect to the operator norm, of $T$ from the
set of operators of rank at most $j$:

$s_{j}(T)\stackrel{{\scriptstyle\rm
def}}{{=}}\inf\\{\|T-R\|:R\in\mathcal{L}(\mathcal{H},\mathcal{K}),\operatorname{rank}R\leq
j\\}.$

Here, for Hilbert spaces $H,K$, we define by $\mathcal{L}(H,K)$ the Banach
space of bounded linear operators from $H$ to $K$ with the operator norm. We
denote by $\mathcal{K}(H,K)$ the Banach space of compact linear operators from
$H$ to $K$ with the operator norm. In the setting of matrices $T$ (that is, in
the case that $\mathcal{H}$ and $\mathcal{K}$ are finite-dimensional),
$s_{j}(T)$ is often called the $j$th singular value of $T$. In this setting
one can show that the singular values of $T$ are precisely the eigenvalues of
$\sqrt{T^{*}T}$. The largest singular value of $T$ is the spectral radius of
$\sqrt{T^{*}T}$, that is, $\|T\|$, and so clearly the set of all singular
values of $T$ contains much more information than the norm $\|T\|$ alone. The
use of $s$-numbers immediately gives rise to a measure of the error in an
approximation of an operator- or matrix-valued function. Consider, for
example, an $m\times n$-matrix-valued function $G$ on the unit circle
$\mathbb{T}$, and suppose we wish to approximate $G$ by a matrix-valued
function $Q$ of a specified form (such as a rational function of a prescribed
McMillan degree). It is natural to regard the difference $G-Q$ as the “error”
in the approximation, and to regard the quantities

$s_{j}^{\infty}(G-Q)\stackrel{{\scriptstyle\rm
def}}{{=}}\operatorname{ess\;sup}_{z\in\mathbb{T}}s_{j}(G(z)-Q(z))$

for $j\geq 0$ as measures of how good an approximation $Q$ is to $G$. We set

$s^{\infty}(G-Q)\stackrel{{\scriptstyle\rm
def}}{{=}}(s_{0}^{\infty}(G-Q),s_{1}^{\infty}(G-Q),\dots,s_{j}^{\infty}(G-Q),\dots),$

and say that $\mathcal{A}G$ is a superoptimal approximation of $G$ in a given
class $\mathcal{F}$ of functions if $s^{\infty}(G-Q)$ attains its minimum with
respect to the lexicographic ordering of the set of sequences of non-negative
real numbers over $\mathcal{F}$ when $Q=\mathcal{A}G$.

The notion of superoptimality pertains to matricial or operator-valued
functions, and is therefore particularly relevant to control engineering and
electrical networks more generally, since in these fields one must analyse
engineering constructs whose mathematical representations are typically
matrix-valued functions on the circle or the real line. In particular, a
primary application is to the problem of designing automatic controllers for
linear time-invariant plants with multiple inputs and outputs. Such design
problems are often formulated in the frequency domain, that is, in terms of
the Laplace or $z-$transform of signals. By this means the problem becomes to
construct an analytic matrix-valued function in a disc or half-plane, subject
to various constraints. An important requirement is usually to minimize, or at
least to bound, some cost- or penalty-function. In practical engineering
problems a wide variety of constraints and cost functions arise, and the
engineer must take account of many complications, such as the physical
limitations of devices and the imprecision of models. Engineers have developed
numerous ways to cope with these complications [11, 7]. One of them, developed
in the 1980s, is $H^{\infty}$ control theory [10]. It is a wide-ranging
theory, that makes pleasing contact with some problems and results of
classical analysis; a seminal role is played by Nehari’s theorem on the best
approximation of a bounded function on the circle by an analytic function in
the disc. Also important in the development of the theory was a series of deep
papers by Adamyan, Arov and Krein [1],[2] which greatly extend Nehari’s
theorem and which apply to matrix-valued functions.

In this context the notion of a superoptimal analytic approximation arose very
naturally. Simple diagonal examples of a $2\times 2$-matrix-valued function
$G$ on $\mathbb{T}$ show that the set of best analytic approximants to $G$ in
the $L^{\infty}$ norm typically comprises an entire infinite-dimensional ball
of functions, and so one is driven to ask for a stronger optimality criterion,
and preferably one which will provide a unique optimum. The very term
“superoptimal” was coined by engineers even before its existence had been
proved in generality. The paper [26] proved that the superoptimal approximant
does indeed exist, and moreover is unique, as long as the approximand $G$ is
the sum of a continuous function and an $H^{\infty}$ function on the circle.
In engineering examples $G$ is usually rational and so continuous on the
circle.

Let us first provide some preliminary definitions and then formulate the
problem. Throughout the paper, $\mathbb{C}^{m\times n}$ denotes the space of
$m\times n$ complex matrices with the operator norm and
$\mathbb{D},\mathbb{T}$ denote the unit disc and the unit circle respectively.

######  Definition 1.1.

Let $E$ be a Banach space.

$H^{\infty}(\mathbb{D},E)$ denotes the space of bounded analytic $E$-valued
functions on the unit disk with supremum norm:

$\|Q\|_{H^{\infty}}\stackrel{{\scriptstyle\emph{def}}}{{=}}\|Q\|_{\infty}\stackrel{{\scriptstyle\emph{def}}}{{=}}\sup\limits_{z\in\mathbb{D}}\|Q(z)\|_{E}.$

$L^{\infty}(\mathbb{T},\mathbb{C}^{m\times n})$ is the space of essentially
bounded weakly measurable $E$-valued functions on the unit circle with
essential supremum norm

$\|f\|_{L^{\infty}}=\mathrm{ess}\sup\limits_{|z|=1}\|f(z)\|_{E},$

and with functions equal almost everywhere identified.

Also, $C(\mathbb{T},E)$ is the space of continuous $E$-valued functions from
$\mathbb{T}$ to $E.$

Naturally engineers need to be able to compute the superoptimal approximant of
$G$.

######  Problem 1.2 (The superoptimal analytic approximation problem).

Given a function $G\in L^{\infty}(\mathbb{T},\mathbb{C}^{m\times n}),$ find a
function $Q\in H^{\infty}(\mathbb{D},\mathbb{C}^{m\times n})$ such that the
sequence $s^{\infty}(G-Q)$ is minimized with respect to the lexicographic
ordering.

In general, the superoptimal analytic approximant may not be unique. However,
it has been proved that if the given function $G$ belongs to
$H^{\infty}(\mathbb{D},\mathbb{C}^{m\times
n})+C(\mathbb{T},\mathbb{C}^{m\times n}),$ then Problem 1.2 has a unique
solution. The following theorem, which was proved by V.V. Peller and N.J.
Young in [26], asserts what we have just stated.

######  Theorem 1.3 ([26], p. 303).

Let $G\in H^{\infty}(\mathbb{D},\mathbb{C}^{m\times
n})+C(\mathbb{T},\mathbb{C}^{m\times n}).$ Then the minimum with respect to
the lexicographic ordering of $s^{\infty}(G-Q)$ over all $Q\in
H^{\infty}(\mathbb{D},\mathbb{C}^{m\times n})$ is attained at a unique
function $\mathcal{A}G.$ Moreover, the singular values
$s_{j}(G(z)-\mathcal{A}G(z))$ are constant almost everywhere on $\mathbb{T}$
for $j\geq 0$ .

The topic of this paper is not the existence and uniqueness of the function
$\mathcal{A}G$ described in Theorem 1.3, but rather the construction of
$\mathcal{A}G$. In the proof of the validity of our construction, we have no
compunction in making any use of results proved in [26], such as the existence
of some special matrix functions. For example, to justify our algorithm we
shall prove, using results of [26], that certain operators that we introduce
are unitarily equivalent to block Hankel operators, which fact enables us to
make use of general properties of Schmidt vectors of Hankel operators, without
the need to calculate the symbols of those Hankel operators.

The existence proof in [26] can in principle be turned into an algorithm, but
into a very computationally intensive one. The construction is recursive, and
at each step of the recursion one must augment a column-matrix function to a
unitary matrix-valued function on the circle with some special properties.
Computationally this step requires a spectral factorization of a positive
semi-definite matrix-valued function on the circle. There are indeed
algorithms for this step, but they involve an iteration which may be slow to
converge and badly conditioned, especially if some function values have
eigenvalues on or close to the unit circle.

It is certainly desirable to avoid the matricial spectral factorization step
if it is possible to do so. Our aim in this project was to devise an algorithm
in which the iterative procedures are as few and as well-conditioned as
possible. Iteration cannot be completely avoided; even in the scalar case,
optimal error is the norm of a certain operator, and the best approximant is
given by a simple formula involving the corresponding Schmidt vectors. Thus
one has to perform a singular value decomposition. In the case that the
approximand $G$ is of type $m\times n$ one must expect to solve $\min(m,n)$
successive singular value problems. However, from the point of view of
numerical linear algebra, singular value decomposition is regarded as a fast,
accurate and well-behaved operation. In this paper we describe an algorithm
that is, in a sense, parallel to the construction of [27] and that in addition
to the spectral factorisation of _scalar_ functions, requires only rational
arithmetic and singular-value decompositions. Several engineers have developed
alternative approaches [16],[30] based on state-space methods. These too are
computationally intensive.

For practical purposes, before even looking for an algorithm for the
construction of $\mathcal{A}G$, we need to know that the problem of
superoptimal analytic approximation is well posed, in sense that arbitrarily
small perturbations of $G$ do not result in large fluctuations in
$\mathcal{A}G$. This issue arises even for scalar $G$, and in fact it is known
[23] that, for general continuous functions $G$, $\mathcal{A}G$ does not
depend continuously on $G$. However, Peller and Khruschev have shown in [23]
that, for $G$ in suitable subspaces $X$ of the continuous functions on
$\mathbb{T}$, the best analytic approximation operator is continuous for
$\|\cdot\|_{X}$, and so it makes sense to compute it. A similar assertion
holds for matrix-valued functions $G$, as was shown by Peller and Young in
[29].

We believe that the present method, which makes use of exterior powers of
Hilbert spaces and operators, provides a conceptual approach to the
construction of superoptimal approximants which is a promising basis for
computation. The theoretical justification of the algorithm we present in this
paper is lengthy and elaborate. However, the implementation of the algorithm
should be straightforward. It will be very interesting to see whether it leads
to an efficient numerical method in the future.

For vector-valued $L^{p}$ spaces we use the terminology of [17].

######  Definition 1.4.

Let $E$ be a separable Hilbert space and let $1\leq p<\infty.$ Define

  1. (i)

$L^{p}(\mathbb{T},E)$ to be the normed space of measurable (weakly or
strongly, which amounts to the same thing, in view of the separability of $E$)
$E$-valued maps $f\colon\mathbb{T}\to E$ such that

$\|f\|_{p}=\left(\displaystyle\frac{1}{2\pi}\int_{0}^{2\pi}\|f(e^{i\theta})\|_{E}^{p}d\theta\right)^{1/p}<\infty;$

  2. (ii)

$H^{p}(\mathbb{D},E)$ to be the normed space of analytic $E$-valued maps
$f\colon\mathbb{D}\to E$ such that

$\|f\|_{p}=\sup\limits_{0<r<1}\left(\frac{1}{2\pi}\int_{0}^{2\pi}\|f(re^{i\theta})\|_{E}^{p}d\theta\right)^{1/p}<\infty,$

the left hand side of this inequality defining a norm on
$H^{p}(\mathbb{D},E)$.

Our algorithm provides a solution $\mathcal{A}G$ to Problem 1.2. By computing
the value of each $t_{k}$ at every step, we obtain each term
$s_{k}^{\infty}(G-\mathcal{A}G)$ of the sequence $s^{\infty}(G-\mathcal{A}G).$
First we need the notion of a Hankel operator and the definitions of some
long-established standard function spaces; for a more detailed account of
these spaces see [17, Chapter V].

If $E$ is a separable Hilbert space, then every function $f\in
H^{2}(\mathbb{D},E)$ has a radial limit at almost every point of $\mathbb{T}$,
by a theorem of Fatou [17, Chapter V], and the map that takes a function $f\in
H^{2}(\mathbb{D},E)$ to its radial limit function embeds $H^{2}(\mathbb{D},E)$
isometrically in $L^{2}(\mathbb{T},E)$. In this paper we shall only envisage
the case that $E$ is separable, and so we can always regard
$H^{2}(\mathbb{D},E)$ as a closed subspace of $L^{2}(\mathbb{T},E)$. The
operators $P_{+},P_{-}$ on $L^{2}(\mathbb{T},E)$ are the operators of
orthogonal projection onto the closed subspaces $H^{2}(\mathbb{D},E)$ and
${H^{2}(\mathbb{D},E)}^{\perp}\stackrel{{\scriptstyle\rm
def}}{{=}}L^{2}(\mathbb{T},E)\ominus H^{2}(\mathbb{D},E)$.

######  Definition 1.5.

Let $E$ be a separable Hilbert space, and let $\varphi$ be an essentially
bounded measurable $\mathcal{L}(E)$-valued function on $\mathbb{T}$; then the
Hankel operator $H_{\varphi}$ is the operator from $H^{2}(\mathbb{D},E)$ to
${H^{2}(\mathbb{D},E)}^{\perp}$ given by

$H_{\varphi}x=P_{-}(\varphi x)\quad\mbox{ for }x\in
H^{2}(\mathbb{D},E),\;\text{ where}\;(\varphi x)(z)=\varphi(z)x(z)\mbox{ for
}\;z\in\mathbb{T}.$

######  Remark 1.6.

In this paper we call an operator $U\colon H\to K$ between Hilbert spaces
$H,K$ a _unitary operator_ if $U$ is both isometric and surjective. Some
authors restrict the name ”unitary operator” to the case that $H=K.$ Such
authors would use a terminology like ”isometric isomorphism” for our ”unitary
operator” in the case that $H\neq K.$

For any vector $x$ in a Hilbert space $E$, we denote by $x^{*}$ the linear
functional $\langle\cdot,x\rangle_{E}$ on $E$. For an $E$-valued function $x$
on a set $S\subset\mathbb{C}$, we define $E^{*}$-valued function $x^{*}$ on
$S$ by $x^{*}(z)=x(z)^{*}$ for all $z\in S$. We observe that if $x\in
L^{p}(\mathbb{T},E)$, where $1\leq p\leq\infty$, then $x^{*}\in
L^{p}(\mathbb{T},E)$ and $\|x^{*}\|_{p}=\|x\|_{p}$.

If $x,y\in E$, then $xy^{*}$ denotes the operator of rank one on $E$ defined
by $xy^{*}(u)=\langle u,y\rangle_{E}x$ for all $u\in E$. This operator is
sometimes denoted by $x\otimes y$ (see, for example, [3, equation (1.17)]). If
$x,y$ are $E$-valued functions on a set $S\subset\mathbb{C}$, then $xy^{*}$ is
the function from $S$ to ${\mathcal{L}}(E)$ given by $xy^{*}(z)=x(z)y(z)^{*}$
for all $z\in S$.

######  Definition 1.7 ([38], p. 206).

Let $H,K$ be Hilbert spaces and let $T\colon H\to K$ be a compact operator.
Suppose that $s$ is a singular value of $T.$ A _Schmidt pair_ for $T$
corresponding to $s$ is a pair $(x,y)$ of non-zero vectors, with $x\in
H,\;y\in K,$ such that

$Tx=sy,\;T^{*}y=sx.$

The following lemma is elementary.

######  Lemma 1.8.

Let $T\in\mathcal{L}(H,K)$ be a compact operator and let $x\in H$, $y\in K$ be
such that $(x,y)$ is a Schmidt pair for $T$ corresponding to $s=\|T\|.$ Then
$x$ is a maximizing vector for $T,$ $y$ is a maximizing vector for $T^{*},$
and $\|x\|_{H}=\|y\|_{K}.$

######  Definition 1.9 ([17], p. 190).

(i). The matrix-valued bounded analytic function $\Theta\in
H^{\infty}(\mathbb{D},\mathbb{C}^{m\times n})$ is called _inner_ if
$\;\Theta(e^{it})$ is an isometry from $\mathbb{C}^{n}$ to $\mathbb{C}^{m}$
for almost every $e^{it}$ on $\mathbb{T}$.

(ii). An analytic $(m\times n)$-matrix-valued function $\Phi$ on $\mathbb{D}$
is said to be _outer_ if

$\Phi H^{2}(\mathbb{D},\mathbb{C}^{n})=\\{\Phi f:f\in
H^{2}(\mathbb{D},\mathbb{C}^{n})\\}$

is a norm-dense subspace of $H^{2}(\mathbb{D},\mathbb{C}^{m})$, and _co-outer_
if

$\Phi^{T}H^{2}(\mathbb{D},\mathbb{C}^{m})=\\{\Phi^{T}g:g\in
H^{2}(\mathbb{D},\mathbb{C}^{m})\\}$

is dense in $H^{2}(\mathbb{D},\mathbb{C}^{n})$.

The following is a brief summary of our algorithm. A full account of all the
steps, with definitions and justifications will be given in Section 4. Our
method makes use of exterior powers $\wedge^{p}E$ of a finite-dimensional
Hilbert space $E$ and of ‘pointwise wedge products’ of $E$-valued functions
$f,g$ on $\mathbb{D}$ or $\mathbb{T}$, defined by

$(f\dot{\wedge}g)(z)=f(z)\wedge g(z)\quad\mbox{ for all }z\in\mathbb{D}\mbox{
or for all }z\in\mathbb{T}.$

These notions are explained more fully in Subsection 3.2.

######  _Algorithm:_.

For a given $G\in H^{\infty}(\mathbb{D},\mathbb{C}^{m\times
n})+C(\mathbb{T},\mathbb{C}^{m\times n}),$ the superoptimal analytic
approximant $\mathcal{A}G\in H^{\infty}(\mathbb{D},\mathbb{C}^{m\times n})$
can be constructed as follows.

i)  Step 0. Let $T_{0}=H_{G}$ be the Hankel operator with symbol $G$. Let
$t_{0}=\|H_{G}\|.$ If $t_{0}=0,$ then $H_{G}=0,$ which implies $G\in
H^{\infty}(\mathbb{D},\mathbb{C}^{m\times n}).$ In this case, the algorithm
terminates, we define $r$ to be zero and the superoptimal approximant
$\mathcal{A}G$ is given by $\mathcal{A}G=G.$

Let $t_{0}\neq 0.$ The Hankel operator $H_{G}$ is a compact operator and so
there exists a Schmidt pair $(x_{0},y_{0})$ corresponding to the singular
value $t_{0}=\|H_{G}\|$ of $H_{G}.$ By the definition of a Schmidt pair
$(x_{0},y_{0})$,

$x_{0}\in H^{2}(\mathbb{D},\mathbb{C}^{n}),\quad y_{0}\in
H^{2}(\mathbb{D},\mathbb{C}^{m})^{\perp}$

are non-zero vector-valued functions such that

$H_{G}x_{0}=t_{0}y_{0},\quad H_{G}^{*}y_{0}=t_{0}x_{0}.$

The functions $x_{0}\in H^{2}(\mathbb{D},\mathbb{C}^{n})$ and
$\bar{z}\bar{y}_{0}\in H^{2}(\mathbb{D},\mathbb{C}^{m})$ admit the inner-outer
factorizations

$x_{0}=\xi_{0}h_{0},\quad\bar{z}\bar{y}_{0}=\eta_{0}h_{0}$ (1.1)

for some scalar outer factor $h_{0}\in H^{2}(\mathbb{D},\mathbb{C})$ and
column matrix inner functions $\xi_{0}\in
H^{\infty}(\mathbb{D},\mathbb{C}^{n})$, $\eta_{0}\in
H^{\infty}(\mathbb{D},\mathbb{C}^{m}).$ Then,

$\|x_{0}(z)\|_{\mathbb{C}^{n}}=|h_{0}(z)|=\|y_{0}(z)\|_{\mathbb{C}^{m}}\;\;\text{almost
everywhere on }\;\mathbb{T}.$ (1.2)

We write equations (1.1) as

$\xi_{0}=\frac{x_{0}}{h_{0}},\quad\eta_{0}=\frac{\bar{z}\bar{y}_{0}}{h_{0}}.$
(1.3)

Then

$\|\xi_{0}(z)\|_{\mathbb{C}^{n}}=1=\|\eta_{0}(z)\|_{\mathbb{C}^{m}}\;\text{almost
everywhere on}\;\mathbb{T}.$ (1.4)

There exists a function $Q_{1}\in H^{\infty}(\mathbb{D},\mathbb{C}^{m\times
n})$ which is at minimal distance from $G$; any such function satisfies

$(G-Q_{1})x_{0}=t_{0}y_{0},\quad y_{0}^{*}(G-Q_{1})=t_{0}x_{0}^{*}.$ (1.5)

Choose any function $Q_{1}\in H^{\infty}(\mathbb{D},\mathbb{C}^{m\times n})$
which satisfies the equations (1.5).

ii) Step 1. Let

$X_{1}=\xi_{0}\dot{\wedge}H^{2}(\mathbb{D},\mathbb{C}^{n})\subset
H^{2}(\mathbb{D},\wedge^{2}\mathbb{C}^{n}),\vspace{2ex}$ (1.6)

and let

$Y_{1}=\bar{\eta}_{0}\dot{\wedge}H^{2}(\mathbb{D},\mathbb{C}^{m})^{\perp}\subset
H^{2}(\mathbb{D},\wedge^{2}\mathbb{C}^{m})^{\perp}.$ (1.7)

$X_{1}$ is a closed linear subspace of
$H^{2}(\mathbb{D},\wedge^{2}\mathbb{C}^{n})$. $Y_{1}$ is a closed linear
subspace of $H^{2}(\mathbb{D},\wedge^{2}\mathbb{C}^{m})^{\perp}.$

Define the operator

$T_{1}:X_{1}\to Y_{1}$

by

$T_{1}(\xi_{0}\dot{\wedge}x)=P_{Y_{1}}(\bar{\eta}_{0}\dot{\wedge}(G-Q_{1})x)\;\text{for
all}\;x\in H^{2}(\mathbb{D},\mathbb{C}^{n}),$ (1.8)

where $P_{Y_{1}}$ is the projection from
$L^{2}(\mathbb{T},\wedge^{2}\mathbb{C}^{m})$ on $Y_{1}.$ We show that $T_{1}$
is well-defined.

If $T_{1}=0,$ then the algorithm terminates, we define $r$ to be $1$ and the
superoptimal approximant $\mathcal{A}G$ is given by the formula

$G-\mathcal{A}G=\displaystyle\sum\limits_{i=0}^{r-1}\frac{t_{i}y_{i}x_{i}^{*}}{|h_{i}|^{2}}=\frac{t_{0}y_{0}x_{0}^{*}}{|h_{0}|^{2}},$

and the solution is

$\mathcal{A}G=G-\frac{t_{0}y_{0}x_{0}^{*}}{|h_{0}|^{2}}.$

If $T_{1}\neq 0,$ let $t_{1}=\|T_{1}\|>0.$ $T_{1}$ is a compact operator and
so there exist $v_{1}\in H^{2}(\mathbb{D},\mathbb{C}^{n}),\;w_{1}\in
H^{2}(\mathbb{D},\mathbb{C}^{m})^{\perp}$ such that
$(\xi_{0}\dot{\wedge}v_{1},\bar{\eta}_{0}\dot{\wedge}w_{1})$ is a Schmidt pair
for $T_{1}$ corresponding to $t_{1}.$ Let $h_{1}$ be the scalar outer factor
of $\xi_{0}\dot{\wedge}v_{1}$ and let

$x_{1}=(I_{\mathbb{C}^{n}}-\xi_{0}\xi_{0}^{*})v_{1},\;\;y_{1}=(I_{\mathbb{C}^{m}}-\bar{\eta}_{0}\eta_{0}^{T})w_{1},$
(1.9)

where $\mathrm{I}_{\mathbb{C}^{n}}$ and $\mathrm{I}_{\mathbb{C}^{m}}$ are the
identity operators in $\mathbb{C}^{n}$ and $\mathbb{C}^{m}$ respectively. Then

$\|x_{1}(z)\|_{\mathbb{C}^{n}}=|h_{1}(z)|=\|y_{1}(z)\|_{\mathbb{C}^{m}}\;\text{almost
everywhere on}\;\mathbb{T}.$ (1.10)

There exists a function $Q_{2}\in H^{\infty}(\mathbb{D},\mathbb{C}^{m\times
n})$ such that both $s_{0}^{\infty}(G-Q_{2})$ and $s_{1}^{\infty}(G-Q_{2})$
are minimized and

$s_{1}^{\infty}(G-Q_{2})=t_{1}.$

Any such $Q_{2}$ satisfies

$\displaystyle(G-Q_{2})x_{0}=t_{0}y_{0},\quad
y_{0}^{*}(G-Q_{2})=t_{0}x_{0}^{*}$ (1.11)
$\displaystyle(G-Q_{2})x_{1}=t_{1}y_{1},\quad
y_{1}^{*}(G-Q_{2})=t_{1}x_{1}^{*}.$

Choose any function $Q_{2}\in H^{\infty}(\mathbb{D},\mathbb{C}^{m\times n})$
which satisfies the equations (1.11). Define

$\xi_{1}=\frac{x_{1}}{h_{1}},\quad\eta_{1}=\frac{\bar{z}\bar{y}_{1}}{h_{1}}.$
(1.12)

Then $\|\xi_{1}(z)\|_{\mathbb{C}^{n}}=1=\|\eta_{1}(z)\|_{\mathbb{C}^{n}}$
almost everywhere on $\mathbb{T}.$

######  Definition 1.10.

Let $E$ be a Hilbert space. We say that a collection $\\{\gamma_{j}\\}$ of
elements of $L^{2}(\mathbb{T},E)$ is _pointwise orthonormal on $\mathbb{T}$_
if, for almost all $z\in\mathbb{T}$ with respect to Lebesgue measure, the
collection of vectors $\\{\gamma_{j}(z)\\}$ is orthonormal in $E.$

iii) Inductive step. Suppose we have constructed

$\begin{array}[]{clllllllll}&t_{0}\geq t_{1}\geq\cdots\geq t_{j}>0\\\
&x_{0},x_{1},\cdots,x_{j}\in L^{2}(\mathbb{T},\mathbb{C}^{n})\\\
&y_{0},y_{1},\cdots,y_{j}\in L^{2}(\mathbb{T},\mathbb{C}^{m})\\\
&h_{0},h_{1},\cdots,h_{j}\in H^{2}(\mathbb{D},\mathbb{C})\;\text{outer}\\\
&\xi_{0},\xi_{1},\cdots,\xi_{j}\in
L^{\infty}(\mathbb{T},\mathbb{C}^{n})\;\text{pointwise orthonormal
on}\;\mathbb{T}\\\ &\eta_{0},\eta_{1},\cdots,\eta_{j}\in
L^{\infty}(\mathbb{T},\mathbb{C}^{m})\;\text{pointwise orthonormal
on}\;\mathbb{T}\\\
&X_{0}=H^{2}(\mathbb{D},\mathbb{C}^{n}),X_{1},\cdots,X_{j}\\\
&Y_{0}=H^{2}(\mathbb{D},\mathbb{C}^{m})^{\perp},Y_{1},\cdots,Y_{j}\\\
&T_{0},T_{1},\cdots,T_{j}\;\text{compact operators}.\par\end{array}$

There exists a function $Q_{j+1}\in H^{\infty}(\mathbb{D},\mathbb{C}^{m\times
n})$ such that

$\left(s_{0}^{\infty}(G-Q_{j+1}),s_{1}^{\infty}(G-Q_{j+1}),\cdots,s_{j}^{\infty}(G-Q_{j+1})\right)$

is lexicographically minimized. Any such function $Q_{j+1}$ satisfies

$(G-Q_{j+1})x_{i}=t_{i}y_{i},\quad y_{i}^{*}(G-Q_{j+1})=t_{i}x_{i}^{*},\quad
i=0,1,\cdots,j.$ (1.13)

Choose any function $Q_{j+1}\in H^{\infty}(\mathbb{D},\mathbb{C}^{m\times n})$
which satisfies the equations (1.13).

Define

$X_{j+1}=\xi_{0}\dot{\wedge}\xi_{1}\dot{\wedge}\cdots\dot{\wedge}\xi_{j}\dot{\wedge}H^{2}(\mathbb{D},\mathbb{C}^{n}),$
(1.14)

and let

$Y_{j+1}=\bar{\eta}_{0}\dot{\wedge}\bar{\eta}_{1}\dot{\wedge}\cdots\dot{\wedge}\bar{\eta}_{j}\dot{\wedge}H^{2}(\mathbb{D},\mathbb{C}^{m})^{\perp}.$
(1.15)

$X_{j+1}$ is a closed subset of
$H^{2}(\mathbb{D},\wedge^{j+2}\mathbb{C}^{n}),$ and $Y_{j+1}$ is a closed
subspace of $H^{2}(\mathbb{D},\wedge^{j+2}\mathbb{C}^{m})^{\perp}.$ Consider
the operator

$T_{j+1}:X_{j+1}\to Y_{j+1}$

given, for all $x\in H^{2}(\mathbb{D},\mathbb{C}^{n})$, by

$T_{j+1}(\xi_{0}\dot{\wedge}\xi_{1}\dot{\wedge}\cdots\dot{\wedge}\xi_{j}\dot{\wedge}x)=P_{Y_{j+1}}\left(\bar{\eta}_{0}\dot{\wedge}\bar{\eta}_{1}\dot{\wedge}\cdots\dot{\wedge}\bar{\eta}_{j}\dot{\wedge}(G-Q_{j+1})x\right).$
(1.16)

$T_{j+1}$ is well defined.

If $T_{j+1}=0,$ then the algorithm terminates, we define $r$ to be $j+1,$ and
the superoptimal approximant $\mathcal{A}G$ is given by the formula

$G-\mathcal{A}G=\sum\limits_{i=0}^{r-1}\frac{t_{i}y_{i}x_{i}^{*}}{|h_{i}|^{2}}=\sum\limits_{i=0}^{j}\frac{t_{i}y_{i}x_{i}^{*}}{|h_{i}|^{2}}.$

Otherwise, we define $t_{j+1}=\|T_{j+1}\|>0.$ Then $T_{j+1}$ is a compact
operator and hence there exist $v_{j+1}\in
H^{2}(\mathbb{D},\mathbb{C}^{n}),\;w_{j+1}\in
H^{2}(\mathbb{D},\mathbb{C}^{m})^{\perp}$ such that

$(\xi_{0}\dot{\wedge}\xi_{1}\dot{\wedge}\cdots\dot{\wedge}\xi_{j}\dot{\wedge}v_{j+1},\bar{\eta}_{0}\dot{\wedge}\bar{\eta}_{1}\dot{\wedge}\cdots\dot{\wedge}\bar{\eta}_{j}\dot{\wedge}w_{j+1})$
(1.17)

is a Schmidt pair for $T_{j+1}$ corresponding to the singular value $t_{j+1}.$

Let $h_{j+1}$ be the scalar outer factor of
$\xi_{0}\dot{\wedge}\xi_{1}\dot{\wedge}\cdots\dot{\wedge}\xi_{j}\dot{\wedge}v_{j+1},$
and let

$x_{j+1}=(I_{\mathbb{C}^{n}}-\xi_{0}\xi_{0}^{*}-\cdots-\xi_{j}\xi_{j}^{*})v_{j+1},\quad
y_{j+1}=(I_{\mathbb{C}^{m}}-\bar{\eta}_{0}\eta_{0}^{T}-\cdots-\bar{\eta}_{j}\eta_{j}^{T})w_{j+1},$
(1.18)

and define

$\xi_{j+1}=\frac{x_{j+1}}{h_{j+1}},\quad\eta_{j+1}=\frac{\bar{z}\bar{y}_{j+1}}{h_{j+1}}.$
(1.19)

One can show that $\|\xi_{j+1}(z)\|_{\mathbb{C}^{n}}=1$ and
$\|\eta_{j+1}(z)\|_{\mathbb{C}^{m}}=1$ almost everywhere on $\mathbb{T}.$

This completes the recursive step. The algorithm terminates after at most
$\min(m,n)$ steps, so that, $r\leq\min(m,n)$ and the superoptimal approximant
$\mathcal{A}G$ is given by the formula

$G-\mathcal{A}G=\sum\limits_{i=0}^{r-1}\frac{t_{i}y_{i}x_{i}^{*}}{|h_{i}|^{2}}.$

######  Remark 1.11.

Observe that, in step $j$ of the algorithm, we define an operator $T_{j}$ in
terms of any function $Q_{j}\in H^{\infty}(\mathbb{D},\mathbb{C}^{m\times n})$
that satisfies the equations

$(G-Q_{j})x_{i}=t_{i}y_{i},\quad y_{i}^{*}(G-Q_{j})=t_{i}x_{i}^{*},\quad
i=0,1,\cdots,j-1.$ (1.20)

This constitutes a system of linear equations for $Q_{j}$ in terms of the
computed quantities $x_{i},t_{i}$ and $y_{i}$ for $i=0,\dots,j-1$, and we
know, from Proposition 10.5, that the system has a solution for $Q_{j}$ in
$H^{\infty}(\mathbb{D},\mathbb{C}^{m\times n})$. By Proposition 8.1 $T_{j}$ is
independent of the choice of $Q_{j}$ that satisfies equations (1.20).

######  Remark 1.12.

At each step we need to find $\|T_{j}\|$ and a Schmidt pair

$(\xi_{0}\dot{\wedge}\xi_{1}\dot{\wedge}\cdots\dot{\wedge}\xi_{j-1}\dot{\wedge}v_{j},\bar{\eta}_{0}\dot{\wedge}\bar{\eta}_{1}\dot{\wedge}\cdots\dot{\wedge}\bar{\eta}_{j-1}\dot{\wedge}w_{j})$
(1.21)

for $T_{j}$ corresponding to the singular value $t_{j}.$ Then we compute the
scalar outer factor $h_{j}$ of
$\xi_{0}\dot{\wedge}\xi_{1}\dot{\wedge}\cdots\dot{\wedge}\xi_{j-1}\dot{\wedge}v_{j}\in
H^{2}(\mathbb{D},\wedge^{j+1}\mathbb{C}^{n})$. These are the only spectral
factorisations needed in the algorithm. Note that if $f\in
H^{2}(\mathbb{D},\mathbb{C}^{n})$ has the inner-outer factorisation $f=hg$,
with $h\in H^{2}(\mathbb{D},\mathbb{C})$ a scalar outer function and $g\in
H^{\infty}(\mathbb{D},\mathbb{C}^{n})$ inner, then $(f^{*}f)(z)=|h(z)|^{2}$
almost everywhere on $\mathbb{T}$, and so the calculation of $h$ requires us
to find a spectral factorisation of the positive scalar-valued function
$f^{*}f$ on the circle.

######  Remark 1.13.

In a numerical implementation of the algorithm one would need to find a way to
compute the norms and Schmidt vectors of the compact operators $T_{j}$. For
this purpose it would be natural to choose convenient orthonormal bases of the
cokernel $X_{j}\ominus\ker T_{j}$ and the range $\operatorname{ran}T_{j}$. It
is safe to assume that in most applications $G$ will be a rational function,
in which case the cokernel and range will be finite-dimensional. At step 0,
$T_{0}$ is a Hankel operator, and the calculation of the matrix of $T_{0}$
with respect to suitable orthonormal bases is a known task [36]; we believe
that similar methods will work for step $j$.

In Theorem 10.15 we arrive at the following conclusion about the superoptimal
approximant $\mathcal{A}G.$

######  Theorem 1.14.

Let $G\in H^{\infty}(\mathbb{D},\mathbb{C}^{m\times
n})+C(\mathbb{T},\mathbb{C}^{m\times n}).$ Let $T_{i},x_{i},y_{i},h_{i}$, for
$i\geq 0$, be defined by the algorithm above. Let $r$ be the least index
$j\geq 0$ such that $T_{j}=0$. Then $r\leq\min(m,n)$ and the superoptimal
approximant $\mathcal{A}G$ is given by the formula

$G-\mathcal{A}G=\displaystyle\sum\limits_{0}^{r-1}\frac{t_{i}y_{i}x_{i}^{*}}{|h_{i}|^{2}}.$

Wedge products, and in particular pointwise wedge products, along with their
properties are studied in detail in Section 3.

##  2\. History and recent work

The Nehari problem of approximating an essentially bounded Lebesgue measurable
function on the unit circle $\mathbb{T}$ by a bounded analytic function on the
unit disk $\mathbb{D}$, has been attracting the interest of both pure
mathematicians and engineers since the middle of the 20th century. The problem
was first formulated and studied from the viewpoint of scalar-valued
functions, and, in the years that followed, from the operator-valued
perspective also, which motivated research into the superoptimal approximation
problem.

The Nehari problem in the scalar case first appeared in the paper of Nehari
[18]. Given an essentially bounded complex valued function $g$ on
$\mathbb{T},$ one seeks its distance from $H^{\infty}$ with respect to the
essential supremum norm, and wishes to determine for which elements of
$H^{\infty}$ this distance is attained. It is also of interest to know whether
the distance is attained at a uniquely determined function. Such problems have
been studied in detail by Nehari [18], Sarason [31] and Adamjan, Arov and
Krein in [1] and [2]. These authors proved that the distance of $g$ from
$H^{\infty}$ is equal to the norm of the Hankel operator $H_{g}$ with symbol
$g.$ Moreover, if $H_{g}$ has a maximizing vector in $H^{2},$ then the bounded
analytic complex-valued function $q$ that minimizes the essential supremum
norm $\|g-q\|_{L^{\infty}}$ is uniquely determined and can be explicitly
calculated (see, for example, [38, p. 196]). Furthermore, if the essential
norm $\|H_{g}\|_{e}$ is less than $\|H_{g}\|$, then $g$ has a unique best
approximant.

Pure mathematicians and engineers started seeking analogues of those results
for matrix- and operator-valued functions. These generalizations are not only
mathematically interesting, but are essential for applications in engineering,
and especially in control theory. There has accordingly been an explosion of
research in this field since 1980, on the part of both pure mathematicians and
engineers.

Page [19] and Treil [34] gave various extensions of the results of Adamjan,
Arov and Krein to operator-valued functions. Page proved that for operator-
valued mappings $T\in L^{\infty}(\mathbb{T},\mathcal{L}(E_{1},E_{2})),$
$\inf\\{\|T-\Phi\|:\Phi\in H^{\infty}(\mathbb{D},\mathcal{L}(E_{1},E_{2})\\}$
is equal to $\|H_{T}\|$. Here $E_{1},E_{2}$ are Hilbert spaces and
$\mathcal{L}(E_{1},E_{2})$ denotes the Banach space of bounded linear
operators from $E_{1}$ to $E_{2}.$ Treil extended the Adamjan, Arov and Krein
theorem in [2] to an operator-valued analogue.

However, in the matrix-valued setting there are typically infinitely many
functions that minimize the $L^{\infty}$ norm of the error function. This fact
is simply illustrated by the following example. Let
$G(z)=\mathrm{diag}\\{\bar{z},0\\}$, for $z\in\mathbb{T}.$ The norm of $H_{G}$
in this case is easily seen to be $1,$ and hence _all_ matrix-valued functions
$Q\in H^{\infty}(\mathbb{D},\mathbb{C}^{2\times 2})$ of the form
$Q(z)=\mathrm{diag}\\{0,q(z)\\},$ where $q\in H^{\infty}$ and
$\|q\|_{H^{\infty}}\leq 1,$ minimize the norm $\|G-Q\|_{\infty}$, yielding the
error $1$. However, if one goes on to minimize in turn the essential suprema
of both singular values of $G(z)-Q(z)$ over $Q\in
H^{\infty}(\mathbb{D},\mathbb{C}^{2\times 2})$, one finds that such a minimum
occurs uniquely when $q(z)$ is equal to $0.$ This type of example suggests
that the enhanced approximation criterion based on successive singular values
generates the “very best” amongst the best approximants to $G$ by an element
of $H^{\infty}(\mathbb{D},\mathbb{C}^{2\times 2})$.

Such reflections led to the formulation of a strengthened approximation
problem, the superoptimal approximation problem 1.2 as explained on above. In
[37] N.J. Young introduced this strengthened notion of optimal analytic
approximation, subsequently called superoptimal approximation. Given a $G$ as
above, find a $Q\in H^{\infty}(\mathbb{D},\mathbb{C}^{m\times n})$ such that
the sequence $s^{\infty}(G-Q)$ is lexicographically minimized. This criterion
obviously constitutes a considerable strengthening of the notion of
optimality, as one needs to determine a $Q\in
H^{\infty}(\mathbb{D},\mathbb{C}^{m\times n})$ that not only minimizes
$\|G-Q\|_{L^{\infty}},$ but minimizes the $L^{\infty}$ norm of all the
subsequent singular values $s_{j}(G(z)-Q(z))$ for $j\geq 0$.

A good starting point for the superoptimal approximation problem of matrix
functions is [26]. As we have said, the problem is to find, for a given

$G\in H^{\infty}(\mathbb{D},\mathbb{C}^{m\times
n})+C(\mathbb{T},\mathbb{C}^{m\times n}),$

a function $Q\in H^{\infty}(\mathbb{D},\mathbb{C}^{m\times n})$ such that the
sequence $s^{\infty}(G-Q)$ is lexicographically minimized. Peller and Young
proved some requisite preparatory results on “thematic factorizations”, on the
analyticity of the minors of unitary completions of inner matrix columns and
on the compactness of some Hankel-type operators with matrix symbols. These
results provided the foundation for their main theorem, namely that if $G$
belongs to $H^{\infty}(\mathbb{D},\mathbb{C}^{m\times
n})+C(\mathbb{T},\mathbb{C}^{m\times n}),$ then there exists a unique $Q\in
H^{\infty}(\mathbb{D},\mathbb{C}^{m\times n})$ such that the sequence
$s^{\infty}(G-Q)$ is lexicographically minimized as $Q$ varies over
$H^{\infty}(\mathbb{D},\mathbb{C}^{m\times n})$; moreover for this $Q$, the
singular values $s_{j}(G(z)-Q(z)$ are constant almost everywhere for
$z\in\mathbb{T}$, for $j=0,1,2,\dots\;.$

Later, in [27] Peller and Young presented a conceptual algorithm for the
computation of the superoptimal approximant. Their algorithm is based on the
theory developed in [26]. Also in [27], the algorithm was applied to a
concrete example of a rational $2\times 2$ matrix-valued function $G$ in
$H^{\infty}(\mathbb{D},\mathbb{C}^{2\times
2})+C(\mathbb{T},\mathbb{C}^{2\times 2})$ and the superoptimal approximant
$\mathcal{A}G$ was calculated by hand.

Additionally, Peller and Young in [28] studied superoptimal approximation by
meromorphic matrix-valued functions, that is, matrix-valued functions in
$H^{\infty}$ that have at most $k$ poles for some prescribed integer $k$. They
modified the results of [26] and established a uniqueness criterion in the
case that the given matrix-valued function $G$ is in $H^{\infty}+C$ and has at
most $k$ poles. In addition, they provided an algorithm for the calculation of
the superoptimal approximant.

One can extend the above results to operator-valued functions on the circle;
the operator-valued superoptimal approximation problem was studied by Peller
in [21]. He generalized the notions of [26] and proved that there exists a
unique superoptimal approximant in $H^{\infty}(\mathcal{B})$ for functions
that belong to $H^{\infty}(\mathcal{B})+C(\mathcal{C}),$ where $\mathcal{B}$
denotes the space of bounded linear operators and $C(\mathcal{C})$ denotes the
space of continuous functions on the circle taking values in the space of
compact operators.

Very badly approximable functions, that is, functions that have the zero
function as a superoptimal approximant, were studied in the years that
followed and a considerable amount of work was published. Peller and Young’s
paper [26] provided the motivation for the study of this problem, where they
were able to algebraically characterise the very badly approximable matrix
functions of class $H^{\infty}(\mathbb{D},\mathbb{C}^{m\times
n})+C(\mathbb{T},\mathbb{C}^{m\times n}).$ Their results were extended in [24]
to the case of matrix functions $G$ for which $\|H_{G}\|_{e}$ is less than the
smallest non-zero superoptimal singular value of $G.$ Very badly approximable
matrix functions with entries in $H^{\infty}+C$ were completely characterised
in [25].

Recent work in [4] by Baratchart, Nazarov and Peller explores the analytic
approximation of matrix-valued functions in $L^{p}$ of the unit circle by
matrix-valued functions from $H^{p}$ of the unit disk in the $L^{p}$ norm for
$p\leq 2.$ They proved that if a given matrix-valued function $\Psi\in
L^{p}(\mathbb{T},\mathbb{C}^{m\times n})$ is a ‘respectable’ matrix function,
then its distance from $H^{p}(\mathbb{D},\mathbb{C}^{m\times n})$ is equal to
$\|H_{\Psi}\|,$ and they obtained a characterisation of that distance also in
the case $\Psi$ is a ‘weird’ matrix-valued function. Furthermore, they
established the notion of $p$-superoptimal approximation and illustrated the
fact that every $n\times n$ rational matrix function has a unique
$p$-superoptimal approximant for $2\leq p<\infty.$ For the case $p=\infty$
they provided a counterexample.

In a more recent paper of Condori [6], the author considered the relation
between the sum of the superoptimal singular values of admissible functions in
$L^{\infty}(\mathbb{T},\mathbb{C}^{m\times n})$ and the superoptimal analytic
approximation problem in the space $L^{\infty}(\mathbb{T},S_{p}^{m,n}),$ where
$S_{p}^{m,n}$ denotes the space of $m\times n$ matrices endowed with the
Schatten-von Neumann norm $\|\cdot\|_{S_{p}^{m,n}}.$ He illustrated the fact
that if $\Phi\in L^{\infty}(\mathbb{T},\mathbb{C}^{n\times n})$ is an
admissible matrix function of order $k$, then $Q\in
H^{\infty}(\mathbb{D},\mathbb{C}^{n\times n})$ is a best approximant function
under the $L^{\infty}(\mathbb{T},S_{1}^{n,n})$-norm and the singular values
$s_{j}((\varphi-Q)(z))$ are constant almost everywhere on $\mathbb{T}$ for
$j=0,1,\dots,k-1$ if and only if $Q$ is a superoptimal approximant to $\Phi,$

$\mathrm{ess}\sup_{z\in\mathbb{T}}s_{j}((\Phi-Q)(z))=0$

for $j\geq k,$ and the sum of the superoptimal singular values of $\Phi$ is
equal to

$\sup\left|\int_{\mathbb{T}}\mathrm{trace}(\Phi(\zeta)\Psi(\zeta))\;dm(\zeta)\right|,$

where $m,n>1,$ $1\leq k\leq\min(m,n)$ and the supremum is taken over all
$\Psi\in H_{0}^{1}(\mathbb{D},\mathbb{C}^{n\times m})$ for which
$\|\Psi\|_{L^{1}(\mathbb{T},\mathbb{C}^{n\times m})}\leq 1$ and
$\mathrm{rank}\Psi(\zeta)\leq k$ almost everywhere on $\mathbb{T}.$

##  3\. Exterior powers of Hilbert spaces

In this section we recall the well-established notion of the wedge product of
Hilbert spaces. One can find definitions and properties of wedge products in
[8], [13], [20], [35] and [32, 33]. Here we present a concise version of this
theory which we need for the new superoptimal algorithm.

###  3.1. Exterior powers

In this subsection, we first present some results concerning the action of
permutation operators on tensors, then we recall the definition of
antisymmetric tensors and we define an inner product on the space of all
antisymmetric tensors. In the following $E$ denotes a Hilbert space. We shall
assume known the notion of the algebraic tensor product of vector spaces,
which is precisely explained in [14]. For the Hilbert space tensor product of
Hilbert spaces, see [9]. One can find proofs of many statements given below,
in our paper [5].

######  Definition 3.1.

$\otimes^{p}E$ is the $p$-fold algebraic tensor product of $E$ and is spanned
by tensors of the form $x_{1}\otimes x_{2}\otimes\dots\otimes x_{p},$ where
$x_{j}\in E$ for $j=1,\dots,p.$

######  Definition 3.2.

An inner product on $\otimes^{p}E$ is defined on elementary tensors by

$\langle x_{1}\otimes x_{2}\otimes\dots\otimes x_{p},y_{1}\otimes
y_{2}\otimes\dots\otimes y_{p}\rangle_{\otimes^{p}E}=p!\langle
x_{1},y_{1}\rangle_{E}\cdots\langle x_{p},y_{p}\rangle_{E},$

for any $x_{1},\dots,x_{p},y_{1},\dots,y_{p}\in E$, and is extended to
$\otimes^{p}E$ by sesqui-linearity.

######  Definition 3.3.

$\otimes_{H}^{p}E$ is the completion of $\otimes^{p}E$ with respect to the
norm $\|u\|=\langle u,u\rangle^{1/2}_{\otimes^{p}E},$ for $u\in\otimes^{p}E.$

######  Definition 3.4.

Let $\mathfrak{S}_{p}$ denote the symmetric group on $\\{1,\dots,p\\},$ with
the operation of composition. For $\sigma\in\mathfrak{S}_{p}$, we define

$S_{\sigma}:\otimes^{p}E\to\otimes^{p}E$

on elementary tensors by

$\displaystyle S_{\sigma}(x_{1}\otimes x_{2}\otimes\dots\otimes
x_{p})=x_{\sigma(1)}\otimes x_{\sigma(2)}\otimes\dots\otimes x_{\sigma(p)},$

and we extend $S_{\sigma}$ to $\otimes^{p}E$ by linearity, that is, for
$u=\sum_{i=1}^{n}\lambda_{i}x_{1}^{i}\otimes\cdots\otimes x_{p}^{i},$ we
define

$S_{\sigma}(u)=\sum\limits_{i=1}^{n}\lambda_{i}S_{\sigma}(x_{1}^{i}\otimes\cdots\otimes
x_{p}^{i}).$

for any $x_{j}^{i}\in E$ and $\lambda_{i}\in\mathbb{C}$.

######  Remark 3.5.

Clearly if $\sigma$ is a bijective self-map of $\\{1,\dots,p\\}$, then so is
its inverse map $\sigma^{-1}$, and $(\mathfrak{S}_{p},\circ)$ is a group under
composition. Moreover,

$\sigma\circ\sigma^{-1}={\mathrm{id}}=\sigma^{-1}\circ\sigma,$

where ${\mathrm{id}}\in\mathfrak{S}_{p}$ is the identity map on
$\\{1,\dots,p\\}.$ Then, if $\epsilon_{\sigma}$ denotes the signature of the
permutation $\sigma$,

$\epsilon_{\sigma\circ\sigma^{-1}}=\epsilon_{\sigma}\epsilon_{\sigma^{-1}}=1,$

hence $\epsilon_{\sigma}=\epsilon_{\sigma^{-1}}.$

######  Proposition 3.6.

Let $E$ be a Hilbert space, and let $p$ be any positive integer. Then, for any
$\sigma\in\mathfrak{S}_{p}$, $S_{\sigma}$ is a linear operator on the normed
space $(\otimes^{p}E,\|\cdot\|)$, which extends to an isometry
$\mathbf{S}_{\sigma}$ on $(\otimes^{p}_{H}E,\|\cdot\|)$. Furthermore,
$\mathbf{S}_{\sigma}$ is a unitary operator on $\otimes_{H}^{p}E$,
$\mathbf{S}_{\sigma}^{*}=\mathbf{S}_{\sigma^{-1}}$, and therefore

$\mathbf{S}_{\sigma}^{*}\mathbf{S}_{\sigma}=\mathbf{S}_{\sigma^{-1}}\mathbf{S}_{\sigma}=I$

is the identity operator on $\otimes^{p}_{H}E$.

Henceforth we shall denote the extended operator $\mathbf{S}_{\sigma}$ by
$S_{\sigma}$.

######  Definition 3.7.

A tensor $u\in\otimes_{H}^{p}E$ is said to be _symmetric_ if $S_{\sigma}(u)=u$
for all $\sigma\in\mathfrak{S}_{p}.$ A tensor $u\in\otimes_{H}^{p}E$ is said
to be _antisymmetric_ if $u=\epsilon_{\sigma}S_{\sigma}u$ for all
$\sigma\in\mathfrak{S}_{p}$ where $\epsilon_{\sigma}$ is the signature of
$\sigma.$

######  Definition 3.8.

The space of all antisymmetric tensors in $\otimes_{H}^{p}E$ will be denoted
by $\wedge^{p}E$.

######  Theorem 3.9.

Let $E$ be a Hilbert space. Then $\bigwedge^{p}E$ is a closed linear subspace
of the Hilbert space $\otimes_{H}^{p}E$ for any $p\geq 2.$

###### Proof.

For $\sigma\in\mathfrak{S}_{p}$ define the operator

$f_{\sigma}\stackrel{{\scriptstyle\emph{def}}}{{=}}S_{\sigma}-\epsilon_{\sigma}I\;\colon\otimes_{H}^{p}E\to\otimes_{H}^{p}E,$

where $I$ denotes the identity operator on $\otimes_{H}^{p}E$. Since
$S_{\sigma}$ is a continuous linear operator on $\otimes_{H}^{p}E$,
$f_{\sigma}$ is a continuous linear operator. The kernel of the operator
$f_{\sigma}$ is

$\begin{array}[]{cllllllllll}\ker
f_{\sigma}&=\\{u\in\otimes_{H}^{p}E\colon(S_{\sigma}-\epsilon_{\sigma}I)(u)=0\\}\\\
&=\\{u\in\otimes_{H}^{p}E\colon\epsilon_{\sigma}S_{\sigma}(u)=u\\}.\end{array}$

Since $f_{\sigma}$ is a continuous linear operator on $\otimes_{H}^{p}E,$
$\ker f_{\sigma}$ is a closed linear subspace of $\otimes_{H}^{p}E.$ Thus
$\wedge^{p}E$ is a closed linear subspace of $\otimes_{H}^{p}E,$ since

$\wedge^{p}E=\\{u\in\otimes_{H}^{p}E\;\colon\epsilon_{\sigma}S_{\sigma}(u)=u\mbox{
for all
}\sigma\in\mathfrak{S}_{p}\\}=\bigcap\limits_{\sigma\in\mathfrak{S}_{p}}\ker
f_{\sigma}.$

∎

Theorem 3.9 implies that the orthogonal projection from $\otimes^{p}_{H}E$
onto $\wedge^{p}E$ is well defined on $\otimes^{p}_{H}E$.

######  Definition 3.10.

Let $E$ be a Hilbert space. For $x_{1},\dots,x_{p}\in E,$ define $x_{1}\wedge
x_{2}\wedge\dots\wedge x_{p}$ to be the orthogonal projection of the
elementary tensor $x_{1}\otimes x_{2}\otimes\dots\otimes x_{p}$ onto
$\wedge^{p}E$, that is

$x_{1}\wedge x_{2}\wedge\dots\wedge
x_{p}=P_{\wedge^{p}E}(x_{1}\otimes\cdots\otimes x_{p}).$

One can find a proof of the following statement in [5].

######  Theorem 3.11.

Let $E$ be a Hilbert space. For all $u\in\otimes_{H}^{p}E,$

$P_{\wedge^{p}E}(u)=\displaystyle\frac{1}{p!}\sum\limits_{\sigma\in\mathfrak{S}_{p}}\epsilon_{\sigma}S_{\sigma}(u).$

######  Proposition 3.12.

Let $E$ be a Hilbert space. The inner product in $\wedge^{p}_{H}E$ is given by
:

$\langle x_{1}\wedge\dots\wedge x_{p},y_{1}\wedge\dots\wedge
y_{p}\rangle_{\wedge^{p}_{H}E}=\det\begin{pmatrix}\langle
x_{1},y_{1}\rangle_{E}&\dots&\langle x_{1},y_{p}\rangle_{E}\\\
\vdots&\ddots&\vdots\\\ \langle x_{p},y_{1}\rangle_{E}&\dots&\langle
x_{p},y_{p}\rangle_{E}\par\par\end{pmatrix},$

for all $x_{1},\dots,x_{p},y_{1},\dots,y_{p}\in E$.

###### Proof.

By Theorem 3.11, we have

$\begin{array}[]{cllllllllllllllllll}&\langle x_{1}\wedge\dots\wedge
x_{p},y_{1}\wedge\dots\wedge y_{p}\rangle_{\wedge^{p}E}=\\\\[21.52771pt]
&=\left\langle\displaystyle\frac{1}{p!}\sum\limits_{\sigma\in\mathfrak{S}_{p}}\epsilon_{\sigma}S_{\sigma}(x_{1}\otimes
x_{2}\otimes\dots\otimes
x_{p}),\displaystyle\frac{1}{p!}\sum\limits_{\tau\in\mathfrak{S}_{p}}\epsilon_{\tau}S_{\tau}(y_{1}\otimes
y_{2}\otimes\dots\otimes y_{p})\right\rangle_{\otimes_{H}^{p}E}\vspace{2ex}\\\
&=\;\displaystyle\frac{1}{p!}\sum\limits_{\sigma^{\prime}\in\mathfrak{S}_{p}}\epsilon_{\sigma^{\prime}}\langle
x_{1}\otimes x_{2}\otimes\dots\otimes x_{p},S_{\sigma^{\prime}}(y_{1}\otimes
y_{2}\otimes\dots\otimes y_{p})\rangle_{\otimes_{H}^{p}E}\vspace{2ex}\\\
&=\;\displaystyle\sum\limits_{\sigma^{\prime}\in\mathfrak{S}_{p}}\epsilon_{\sigma^{\prime}}\prod\limits_{i=1}^{p}\langle
x_{i},y_{\sigma^{\prime}(i)}\rangle_{E}\vspace{2ex}\\\
&=\det\begin{pmatrix}\langle x_{1},y_{1}\rangle_{E}&\cdots&\langle
x_{1},y_{p}\rangle_{E}\\\ \vdots&\ddots&\vdots\\\ \langle
x_{p},y_{1}\rangle_{E}&\cdots&\langle
x_{p},y_{p}\rangle_{E}\par\end{pmatrix},\;\;\text{by Leibniz'
formula}.\par\end{array}$

∎

See [5, Proposition 2.14] for slightly more detail.

######  Corollary 3.13.

Let $E$ be a Hilbert space and let $x_{1},\dots,x_{p}\in E$. Then
$x_{1}\wedge\cdots\wedge x_{p}=0$ if and only if $x_{1},\dots,x_{p}$ are
linearly dependent.

###### Proof.

Note that $x_{1}\wedge\cdots\wedge x_{p}=0$ if and only if
$\|x_{1}\wedge\cdots\wedge x_{p}\|_{\wedge^{p}E}^{2}=0,$ which, by Proposition
3.12, holds if and only if

$\det[\langle x_{i},x_{j}\rangle]_{i,j=1}^{p}=0.$

Thus $x_{1}\wedge\cdots\wedge x_{p}=0$ if and only if there exist complex
numbers $\lambda_{1},\dots,\lambda_{p},$ which are not all zero, such that

$\begin{pmatrix}\langle x_{1},y_{1}\rangle_{E}&\dots&\langle
x_{1},y_{p}\rangle_{E}\\\ \vdots&\ddots&\vdots\\\ \langle
x_{p},y_{1}\rangle_{E}&\dots&\langle
x_{p},y_{p}\rangle_{E}\par\end{pmatrix}\begin{pmatrix}\bar{\lambda}_{1}\\\
\vdots\\\ \bar{\lambda}_{p}\end{pmatrix}=0.$

This holds if and only if there exist complex numbers
$\lambda_{1},\dots,\lambda_{p},$ which are not all zero, such that

$\langle
x_{i},\sum_{j=1}^{p}\lambda_{j}x_{j}\rangle_{E}=0\quad\text{for}\;i=1,\dots,p.$

The latter statement is equivalent to the assertion that there exist complex
numbers $\lambda_{1},\dots,\lambda_{p},$ which are not all zero, such that

$\langle\sum_{i=1}^{p}\lambda_{i}x_{i},\sum_{j=1}^{p}\lambda_{j}x_{j}\rangle_{E}=0,$

which in turn is equivalent to the condition that there exist complex numbers
$\lambda_{1},\dots,\lambda_{p},$ not all zero, such that

$\sum_{j=1}^{p}\lambda_{j}x_{j}=0.$

The latter statement is equivalent to the linear dependence of
$x_{1},\dots,x_{p}$ as required. ∎

######  Corollary 3.14.

Let $E$ be a Hilbert space. Suppose $x,y\in E,$ and $x$,$y$ are orthogonal in
$E$, that is, $\langle x,y\rangle_{E}=0.$ Then

$\|x\wedge y\|_{\wedge^{2}E}=\|x\|_{E}\|y\|_{E}.$

###### Proof.

By Proposition 3.12,

$\|x\wedge y\|_{\wedge^{2}E}^{2}=\langle x\wedge y,x\wedge
y\rangle_{\wedge^{2}E}=\det\begin{pmatrix}\langle x,x\rangle_{E}&\langle
x,y\rangle_{E}\\\ \langle y,x\rangle_{E}&\langle y,y\rangle_{E}\end{pmatrix}.$

If $x$ is orthogonal to $y$ in $E,$ the off-diagonal entries are zero and thus

$\|x\wedge y\|_{\wedge^{2}E}^{2}=\|x\|_{E}^{2}\|y\|_{E}^{2}.$

∎

######  Lemma 3.15.

Suppose $\\{u_{1},\cdots,u_{n}\\}$ is an orthonormal set in $E.$ Then, for
$j=1,\dots,n-1$ and for every $x\in E,$

$\|u_{1}\wedge\cdots\wedge u_{j}\wedge
x\|_{\wedge^{j+1}E}=\|x-\displaystyle\sum\limits_{i=1}^{j}\langle
x,u_{i}\rangle u_{i}\|_{E}.$

See [5, Lemma 2.15].

######  Definition 3.16.

Let $(E,\|\cdot\|_{E})$ be a Hilbert space. The _$p$ -fold Cartesian product
of $E$_ is defined to be the set

$\underbrace{E\times\dots\times E}_{p-times}=\\{(x_{1},\dots,x_{p}):x_{i}\in
E\\}.$

Moreover, we define a norm on $\underbrace{E\times\dots\times E}_{p-times}$ by

$\|(x_{1},\dots,x_{p})\|=\\{\sum_{i=1}^{p}\|x_{i}\|_{E}^{2}\\}^{\tfrac{1}{2}}.$

######  Definition 3.17.

Let $E$ be a Hilbert space. We define the multilinear operator

$\Lambda\colon\underbrace{E\times\dots\times E}_{p-times}\to\wedge^{p}E$

by

$\Lambda(x_{1},\dots,x_{p})=x_{1}\wedge\dots\wedge x_{p}\quad\text{for
all}\quad x_{1},\dots,x_{p}\in E.$

######  Proposition 3.18.

[Hadamard’s inequality, [15], p. 477] For any matrix

$A=(a_{ij})\in\mathbb{C}^{n\times n},$
$|\det(A)|\leq\prod\limits_{j=1}^{n}\left(\sum\limits_{i=1}^{n}|a_{ij}|^{2}\right)^{1/2}\quad\text{and}\quad|\det(A)|\leq\prod\limits_{i=1}^{n}\left(\sum\limits_{j=1}^{n}|a_{ij}|^{2}\right)^{1/2}.$

######  Proposition 3.19.

Let $E$ be a Hilbert space. Then the multilinear mapping

$\Lambda\colon\underbrace{E\times\dots\times E}_{p-times}\to\wedge^{p}E$

is bounded and

$\|\Lambda(x_{1},\dots,x_{p})\|_{\wedge^{p}E}^{2}\leq\prod\limits_{j=1}^{p}\|x_{j}\|_{E}\left(\sum\limits_{i=1}^{p}\|x_{i}\|_{E}^{2}\right)^{1/2}.$
(3.1)

See [5, Proposition 2.19] for more detail.

###  3.2. Pointwise wedge products and pointwise creation operators

For the purposes of this paper we need to consider the wedge product of
mappings defined on the unit circle or in the unit disk that take values in
Hilbert spaces. To this end we introduce a notion of pointwise wedge product
and we study its properties.

######  Definition 3.20.

Let $E$ be a Hilbert space and let $f,g\colon\mathbb{D}\to E$
$\mathrm{(}f,g\colon\mathbb{T}\to E\mathrm{)}$ be $E$-valued maps. We define
the _pointwise wedge product of $f$ and $g,$_

$f\dot{\wedge}g\colon\mathbb{D}\to\wedge^{2}E\quad\mathrm{(}f\dot{\wedge}g\colon\mathbb{T}\to\wedge^{2}E\mathrm{)}$

by

$(f\dot{\wedge}g)(z)=f(z)\wedge g(z)\quad\text{for
all}\;z\in\mathbb{D}\quad\mathrm{(}\text{for almost
all}\;z\in\mathbb{T}\mathrm{)}.$

######  Definition 3.21.

Let $E$ be a Hilbert space and let $\chi_{1},\dots,\chi_{n}\colon\mathbb{D}\to
E$
$\mathrm{(}\chi_{1},\dots,\chi_{n}\colon\mathbb{T}\to E\mathrm{)}$ be
$E$-valued maps. We call $\chi_{1},\dots\chi_{n}$ _pointwise linearly
dependent_ on $\mathbb{D}$ _(_ or on $\mathbb{T}$_)_ if for all
$z\in\mathbb{D}$ (for almost all $z\in\mathbb{T}$ respectively) the vectors
$\chi_{1}(z),\dots,\chi_{n}(z)$ are linearly dependent in $E$.

######  Remark 3.22.

If $x_{1},\dots,x_{n}$ are pointwise linearly dependent on $\mathbb{T}$, then

$(x_{1}\dot{\wedge}\dots\dot{\wedge}x_{n})(z)=0$

for almost all $z\in\mathbb{T}.$

######  Proposition 3.23.

Let $E$ be a separable Hilbert space and let
$\displaystyle\frac{1}{p}+\frac{1}{q}=1,$ where
$1\leq p,q\leq\infty$. Suppose that $x\in L^{p}(\mathbb{T},E),$ $y\in
L^{q}(\mathbb{T},E).$ Then

$x\dot{\wedge}y\in L^{1}(\mathbb{T},\wedge^{2}E).$

and

$\|x\dot{\wedge}y\|_{L^{1}(\mathbb{T},\wedge^{2}E)}\leq\|x\|_{L^{p}(\mathbb{T},E)}\|y\|_{L^{q}(\mathbb{T},E)}.$
(3.2)

###### Proof.

By Proposition 3.12, for all $z\in\mathbb{T},$

$\begin{array}[]{cllllllll}\|(x\dot{\wedge}y)(z)\|_{\wedge^{2}E}^{2}&=\langle
x(z)\wedge y(z),x(z)\wedge y(z)\rangle_{\wedge^{2}E}\\\ &=\langle
x(z),x(z)\rangle_{E}\cdot\langle y(z),y(z)\rangle_{E}-|\langle
x(z),y(z)\rangle_{E}|^{2}\\\
&\leq\|x(z)\|_{E}^{2}\|y(z)\|_{E}^{2}.\end{array}$

Thus, for all $z\in\mathbb{T},$

$\|(x\dot{\wedge}y)(z)\|_{\wedge^{2}E}\leq\|x(z)\|_{E}\|y(z)\|_{E}.$ (3.3)

By Definition 1.4,

$\|x\dot{\wedge}y\|_{L^{1}(\mathbb{T},\wedge^{2}E)}=\displaystyle\frac{1}{2\pi}\int\limits_{0}^{2\pi}\|(x\dot{\wedge}y)(e^{i\theta})\|_{\wedge^{2}E}\;d\theta\leq\frac{1}{2\pi}\int\limits_{0}^{2\pi}\|x(e^{i\theta})\|_{E}\|y(e^{i\theta})\|_{E}\;d\theta.$
(3.4)

Now by Hölder’s inequality,

$\frac{1}{2\pi}\int\limits_{0}^{2\pi}\|x(e^{i\theta})\|_{E}\|y(e^{i\theta})\|_{E}\;d\theta\leq\left(\frac{1}{2\pi}\int\limits_{0}^{2\pi}\|x(e^{i\theta})\|_{E}^{p}\;d\theta\right)^{1/p}\left(\frac{1}{2\pi}\int\limits_{0}^{2\pi}\|y(e^{i\theta})\|_{E}^{q}\;d\theta\right)^{1/q}.$
(3.5)

By inequalities (3.4) and (3.5), $x\dot{\wedge}y\in
L^{1}(\mathbb{T},\wedge^{2}E)$ and the inequality (3.2) holds. ∎

######  Proposition 3.24.

Let $E$ be a Hilbert space and $x_{1},x_{2},\dots,x_{n}\colon\mathbb{D}\to E$
be analytic $E$-valued maps on $\mathbb{D}.$ Then,

$x_{1}\dot{\wedge}x_{2}\dot{\wedge}\dots\dot{\wedge}x_{n}\colon\mathbb{D}\to\wedge^{n}E$

is also analytic on $\mathbb{D}$ and

$(x_{1}\dot{\wedge}x_{2}\dot{\wedge}\dots\dot{\wedge}x_{n})^{\prime}(z)=x_{1}^{\prime}(z)\wedge
x_{2}(z)\wedge\dots\wedge x_{n}(z)+\dots+x_{1}(z)\wedge
x_{2}(z)\wedge\dots\wedge x_{n}^{\prime}(z)$

for all $z\in\mathbb{D}.$

The proof is straightforward. It follows from Proposition 3.12, continuity of
$\Lambda$ (Proposition 3.19) and Hadamard’s inequalities 3.1.

######  Definition 3.25.

Let $E$ be a separable Hilbert space. If $x\in L^{2}(\mathbb{T},E)$ and $y\in
L^{\infty}(\mathbb{T},E)$, then $y^{*}x\in L^{2}(\mathbb{T},E)$ is given by
$(y^{*}x)(z)=\langle x(z),y(z)\rangle_{E}$ almost everywhere on $\mathbb{T}$.

######  Proposition 3.26.

Let $E$ be a separable Hilbert space, let $x\in H^{2}(\mathbb{D},E)$ and let
$y\in H^{\infty}(\mathbb{D},E).$ Then

$x\dot{\wedge}y\in H^{2}(\mathbb{D},\wedge^{2}E).$

See [5, Proposition 3.8].

######  Definition 3.27.

Let $E$ be a Hilbert space. We say that a family
$\\{f_{\lambda}\\}_{\lambda\in\Lambda}$ of maps from $\mathbb{T}$ to $E$ is
pointwise orthonormal on $\mathbb{T}$, if for all $z$ in a set of full measure
in $\mathbb{T}$, the family of vectors
$\\{f_{\lambda}(z)\\}_{\lambda\in\Lambda}$ is orthonormal in $E$.

######  Proposition 3.28.

Let $E$ be a separable Hilbert space, and let
$\;\xi_{0},\xi_{1},\cdots,\xi_{j}\in L^{\infty}(\mathbb{T},E)$ be a pointwise
orthonormal set on $\mathbb{T}$, and let $x\in L^{2}(\mathbb{T},E)$. Then

$\xi_{0}\dot{\wedge}\xi_{1}\dot{\wedge}\cdots\dot{\wedge}\xi_{j}\dot{\wedge}x\in
L^{2}(\mathbb{T},\wedge^{j+2}E),$

and

$\|\xi_{0}\dot{\wedge}\xi_{1}\dot{\wedge}\cdots\dot{\wedge}\xi_{j}\dot{\wedge}x\|_{L^{2}(\mathbb{T},\wedge^{j+2}E)}\leq\|x\|_{L^{2}(\mathbb{T},E)}.$

It follows from Lemma 3.15. See [5, Proposition 3.11] for a proof of this
proposition.

######  Definition 3.29.

Let $E$ be a separable Hilbert space. Let $\xi\in H^{\infty}(\mathbb{D},E)$.
We define the _pointwise creation operator_

$C_{\xi}:H^{2}(\mathbb{D},E)\to H^{2}(\mathbb{D},\wedge^{2}E)$

by

$C_{\xi}f=\xi\dot{\wedge}f\;\text{for}\;f\in H^{2}(\mathbb{D},E).$

######  Remark 3.30.

Let $E$ be a separable Hilbert space. Let $\xi\in H^{\infty}(\mathbb{D},E)$
and let $f\in H^{2}(\mathbb{D},E)$. By the generalized Fatou’s Theorem [17,
Chapter V], the radial limits

$\lim_{r\to
1}\xi(re^{i\theta})\underset{\|\cdot\|_{E}}{=}\tilde{\xi}(e^{i\theta}),\quad\lim_{r\to
1}f(re^{i\theta})\underset{\|\cdot\|_{E}}{=}\tilde{f}(e^{i\theta})\;\;(0<r<1)$

exist almost everywhere on $\mathbb{T}$ and define functions $\tilde{\xi}\in
L^{\infty}(\mathbb{T},E)$ and $\tilde{f}\in L^{2}(\mathbb{T},E)$ respectively,
which satisfy the relations

$\lim_{r\to
1}\|\xi(re^{i\theta})-\tilde{\xi}(e^{i\theta})\|_{E}=0,\quad\lim_{r\to
1}\|f(re^{i\theta})-\tilde{f}(e^{i\theta})\|_{E}=0\;\;(0<r<1)$

for almost all $e^{i\theta}\in\mathbb{T}$.

######  Lemma 3.31.

Let $E$ be a separable Hilbert space. Let $\xi\in H^{\infty}(\mathbb{D},E)$
and let $f\in H^{2}(\mathbb{D},E).$ Then the radial limits $\lim_{r\to
1}(\xi(re^{i\theta})\wedge f(re^{i\theta}))$ exist for almost all
$e^{i\theta}\in\mathbb{T}$ and define functions in
$L^{2}(\mathbb{T},\wedge^{2}E).$

One can find a proof of this statement in [5, Lemma 4.3].

######  Remark 3.32.

Let $E$ be a separable Hilbert space. By [17, Chapter 5, Section 1], for any
separable Hilbert space $E$, the map $f\mapsto\tilde{f}$ is an isometric
embedding of $H^{2}(\mathbb{D},E)$ in $L^{2}(\mathbb{T},E)$, where
$\tilde{f}(e^{i\theta})=\lim_{r\to 1}f(re^{i\theta}).$ Since
$H^{2}(\mathbb{D},E)$ is complete and the embedding is isometric, the image of
the embedding is complete, and therefore is closed in $L^{2}(\mathbb{T},E).$
Therefore, the space $H^{2}(\mathbb{D},E)$ is identified isometrically with a
closed linear subspace of $L^{2}(\mathbb{T},E).$ In future we shall use the
same notation for $f$ and $\tilde{f}.$

######  Definition 3.33.

Let $E$ be a separable Hilbert space. Let $F$ be a subspace of
$L^{2}(\mathbb{T},E)$ and let $X$ be a subset of $L^{2}(\mathbb{T},E).$

We define the _pointwise orthogonal complement_ of $X$ in $F$ to be the set

$\operatorname{POC}(X,F)=\\{f\in F:f(z)\perp\\{x(z):x\in X\\}\;\text{for
almost all}\;z\in\mathbb{T}\\}.$

######  Proposition 3.34.

Let $E$ be a separable Hilbert space. Let $\eta\in L^{2}(\mathbb{D},E)$. Then

  1. (i)

The space $V=\\{f\in H^{2}(\mathbb{D},E):\langle
f(z),\eta(z)\rangle_{E}=0\;\text{for almost all}\;z\in\mathbb{T}\\}$ is a
closed subspace of $H^{2}(\mathbb{D},E).$

  2. (ii)

The space $V=\\{f\in L^{2}(\mathbb{T},E):\langle
f(z),\eta(z)\rangle_{E}=0\;\text{for almost all}\;z\in\mathbb{T}\\}$ is a
closed subspace of $L^{2}(\mathbb{T},E).$

One can find a proof of this statement in [5, Proposition 4.6].

##  4\. Superoptimal analytic approximation

In this section we present our main result, which is an algorithm for the
superoptimal analytic approximation of a matrix-valued function on the circle.
In Subsection 4.1 we recall certain known results and Peller and Young’s
algorithm (Theorem 4.19). In Subsection 4.2 we present an alternative
algorithm for the superoptimal approximant, based on exterior powers of
Hilbert spaces. The proof of the validity of the new algorithm relies on the
cited work given in Subsection 4.1.

###  4.1. Known results

######  Theorem 4.1 (Hartman’s Theorem, [22], p. 74).

Let $E,F$ be separable Hilbert spaces and let $\Phi\in
L^{\infty}(\mathbb{T},\mathcal{L}(E,F)).$ The following statements are
equivalent:

  1. i)

The Hankel operator $H_{\Phi}$ is compact on $H^{2}(\mathbb{D},E)$;

  2. ii)

$\Phi\in
H^{\infty}(\mathbb{D},\mathcal{L}(E,F))+C(\mathbb{T},\mathcal{K}(E,F))$;

  3. iii)

there exists a function $\Psi\in C(\mathbb{T},\mathcal{K}(E,F))$ such that
$\hat{\Phi}(n)=\hat{\Psi}(n)$ for $n<0.$

######  Theorem 4.2 ([19]).

For any matrix-valued $\varphi\in L^{\infty}(\mathbb{T},\mathbb{C}^{m\times
n}),$

$\inf\limits_{Q\in H^{\infty}(\mathbb{D},\mathbb{C}^{m\times
n})}\|\varphi-Q\|_{\infty}=\|H_{\varphi}\|$

and the infimum is attained.

######  Definition 4.3 ([26], p. 306).

The class of _quasi-continuous_ functions is defined by

$QC=(H^{\infty}(\mathbb{D},\mathbb{C}^{m\times
n})+C(\mathbb{T},\mathbb{C}^{m\times
n}))\cap\overline{(H^{\infty}(\mathbb{D},\mathbb{C}^{m\times
n})+C(\mathbb{T},\mathbb{C}^{m\times n}))}.$

In other words this class consists of functions on the circle which belong to
$H^{\infty}+C$ and have the property that their complex conjugates belong to
$H^{\infty}+C$ as well.

We shall also need the class of functions of vanishing mean oscillation, as
described, for example, in [22, Appendix 2, Section 5].

######  Definition 4.4.

For any function $f\in L^{1}(\mathbb{T})$ and any arc $I$ in $\mathbb{T}$ let

$f_{I}\stackrel{{\scriptstyle\rm def}}{{=}}\frac{1}{m(I)}\int_{I}fdm$

where $m$ is Lebesgue measure on $\mathbb{T}$. Thus, $f_{I}$ is the mean of
$f$ over $I$. The function $f$ is said to have vanishing mean oscillation if

$\lim_{m(I)\to 0}\frac{1}{m(I)}\int_{I}|f-f_{I}|\,dm=0.$

The space of functions of vanishing mean oscillation on $\mathbb{T}$ is
denoted by $\mathrm{VMO}$.

$\mathrm{VMO}$ is also related to the compactness of Hankel operators. The
following is [22, Theorem 5.8].

######  Theorem 4.5.

Let $\varphi\in L^{2}$. Then $H_{\varphi}$ is compact if and only if
$P_{-}\varphi\in\mathrm{VMO}$.

It is therefore not surprising that the spaces $QC$ and $\mathrm{VMO}$ are
closely related. In fact

######  Theorem 4.6.

[22, Page 729]

$QC=\mathrm{VMO}\cap L^{\infty}.$

It follows from another characterization of $\mathrm{VMO}$, to wit

$\mathrm{VMO}\ =\ \\{f+\tilde{g}:f,g\in C(\mathbb{T})\\},$

where $\tilde{g}$ denotes the harmonic conjugate of $g$ [22, Theorem A2.8].

######  Remark 4.7.

For $G\in H^{\infty}(\mathbb{D},\mathbb{C}^{m\times
n})+C(\mathbb{T},\mathbb{C}^{m\times n})$ we will say that a function $Q\in
H^{\infty}(\mathbb{D},\mathbb{C}^{m\times n})$ which minimizes the norm
$\|G-Q\|_{L^{\infty}}$ is a function at minimal distance from $G.$ By Nehari’s
Theorem, all such functions $Q$ satisfy $\|G-Q\|_{L^{\infty}}=\|H_{G}\|.$

######  Definition 4.8 ([17], p. 190).

For a separable Hilbert space $E,$ a function $\xi\in
H^{\infty}(\mathbb{D},E)$ will be called _inner_ if for almost every
$z\in\mathbb{T},$

$\|\xi(z)\|_{E}=1.$

######  Theorem 4.9 ([26], Theorem 1.1).

Let $\varphi$ be an $n\times 1$ inner matrix function. There exists a co-outer
function $\varphi_{c}\in H^{\infty}(\mathbb{D},\mathbb{C}^{n\times(n-1)})$
such that

$\Phi=\begin{pmatrix}\varphi&\bar{\varphi}_{c}\end{pmatrix}$

is unitary-valued on $\mathbb{T}$ and all minors of $\Phi$ on the first column
are in $H^{\infty}.$

For a function $G:\mathbb{T}\to\mathbb{C}^{m\times n}$ and a space $X$ of
scalar functions on $\mathbb{T}$, we write $G\in X$ to mean that each entry of
$G$ belongs to $X$.

Next we describe some properties that a space $X$ of equivalence classes of
scalar functions on the circle may possess [26, Page 330]. Define the non-
linear operator $\mathcal{A}=\mathcal{A}^{(m,n)}$ on the space of $m\times n$
functions $G\in H^{\infty}(\mathbb{D},\mathbb{C}^{m\times
n})+C(\mathbb{T},\mathbb{C}^{m\times n})$ by saying that
$\mathcal{A}^{(m,n)}G$ is the unique superoptimal approximation in
$H^{\infty}(\mathbb{D},\mathbb{C}^{m\times n})$ to $G$.

We say that $X$ is hereditary for $\mathcal{A}$ if, for every scalar function
$g\in X$, the best analytic approximation $\mathcal{A}g$ of $g$ belongs to
$X$.

Consider the following conditions on $X$ from [23]:

  1. ($\alpha$1)

$X$ contains all polynomial functions and $X\subset\mathrm{VMO}$;

  2. ($\alpha$2)

$X$ is hereditary for $\mathcal{A}$;

  3. ($\alpha$3)

if $f\in X$ then $\bar{z}\bar{f}\in X$ and $P_{+}f\in X$;

  4. ($\alpha$4)

if $f,g\in X\cap L^{\infty}$ then $fg\in X\cap L^{\infty}$;

  5. ($\alpha$5)

if $f\in X\cap H^{2}$ and $h\in H^{\infty}$ then $T_{\bar{h}}f\in X\cap
H^{2}$.

The relevance of these properties is contained in the following statement,
which is [26, Lemma 5.3]. Recall that a function $f\in L^{\infty}$ is said to
be badly approximable if the best analytic approximant to $f$ is the zero
function. In view of Nehari’s Theorem, $f$ is badly approximable if and only
if $\|f\|_{\infty}=\|H_{f}\|$.

######  Lemma 4.10.

Let $X$ satisfy $(\alpha 1)$ to $(\alpha 5)$ and let $\varphi\in X$ be an
$n\times 1$ inner function. Let $\varphi_{c}$ be an $n\times(n-1)$ function in
$H^{\infty}$ such that $[\varphi\ \bar{\varphi}_{c}]$ is unitary-valued a.e.
on $\mathbb{T}$ and has all its minors on the first column in $H^{\infty}$.
Then each entry of $\varphi_{c}$ belongs to $X$.

Below we shall use a modified version of [26, Theorem 0.2].

######  Theorem 4.11.

_[26, Theorem 0.2]_ Let $\varphi\in L^{\infty}(\mathbb{T},\mathbb{C}^{m\times
n})$ be such that $H_{\varphi}$ has a Schmidt pair $(v,w)$ corresponding to
the singular value $t=\|H_{\varphi}\|$. Let $Q$ be a function in
$H^{\infty}(\mathbb{D},\mathbb{C}^{m\times n})$ at minimal
$L^{\infty}$-distance from $\varphi$. Then

$(\varphi-Q)v=tw$

and

$(\varphi-Q)^{*}w=tv.$

Moreover

$\|w(z)\|_{\mathbb{Cc}^{m}}=\|v(z)\|_{\mathbb{Cc}^{n}}~{}\text{almost
everywhere on}~{}\mathbb{T}$ (4.1)

and

$\|\varphi(z)-Q(z)\|=t~{}\text{almost everywhere on}~{}\mathbb{T}.$

###### Proof.

By Nehari’s Theorem, $\|\varphi-Q\|_{L^{\infty}}=t$ and, by hypothesis,

$H_{\varphi}v=tw,\qquad H_{\varphi}^{*}w=tv.$

If $t=0$ then $\varphi\in H^{\infty}(\mathbb{D},\mathbb{C}^{m\times n})$, so
that $\varphi=Q$ and the statement of the theorem is trivially true. We may
therefore assume $t>0$. Thus $H_{\varphi}^{*}H_{\varphi}v=t^{2}v$, and so $v$
is a maximising vector for $H_{\varphi}$. We can assume that $v$ is a unit
vector in $H^{2}(\mathbb{D},\mathbb{C}^{n})$, and then $w$ is a unit vector in
$H^{2}(\mathbb{D},\mathbb{C}^{m})^{\perp}$ and is a maximising vector for
$H_{\varphi}^{*}$. We have

$t=\|H_{\varphi}v\|=\|H_{\varphi-Q}v\|=\|P_{-}(\varphi-Q)v\|\leq\|(\varphi-Q)v\|\leq\|\varphi-Q\|_{L^{\infty}}=t.$

The inequalities must hold with equality throughout, and therefore

$\|P_{-}(\varphi-Q)v\|=\|(\varphi-Q)v\|$, which implies that
$(\varphi-Q)v\perp H^{2}$ and so

$H_{\varphi}v=P_{-}(\varphi-Q)v=(\varphi-Q)v.$

Furthermore $\|(\varphi-Q)v\|$ = $\|(\varphi-Q)\|_{L^{\infty}}\|v\|$ and since
$v(z)$ is therefore a maximizing vector for $\varphi(z)-Q(z)$ for almost all
$z$, we have $\|\varphi(z)-Q(z)\|=\|H_{\varphi}\|.$

Likewise,

$\displaystyle t=\|H_{\varphi}^{*}\|=\|H_{\varphi-Q}^{*}\|$
$\displaystyle=\|H_{\varphi-Q}^{*}w\|=\|P_{+}(\varphi-Q)^{*}w\|_{L^{2}}\leq\|(\varphi-Q)^{*}w\|_{L^{2}}$
$\displaystyle\leq\|(\varphi-Q)^{*}\|_{L^{\infty}}\|w\|_{L^{2}}=\|(\varphi-Q)^{*}\|_{L^{\infty}}=t.$

Again, the inequalities hold with equality throughout, and in particular

$\|P_{+}(\varphi-Q)^{*}w\|_{L^{2}}=\|(\varphi-Q)^{*}w\|_{L^{2}},$

so that $(\varphi-Q)^{*}w\in H^{2}$ and

$(\varphi-Q)^{*}w=H_{\varphi}^{*}w=tv.$

∎

######  Lemma 4.12.

[26, p. 315-316] Let $m,n>1,$ let $G\in
H^{\infty}(\mathbb{D},\mathbb{C}^{m\times n})+C(\mathbb{T},\mathbb{C}^{m\times
n})$ and $t_{0}=\|H_{G}\|\neq 0.$ Suppose that $v$ is a maximizing vector of
$H_{G}$ and let

$H_{G}v=t_{0}w.$ (4.2)

Then $v,\bar{z}\bar{w}\in H^{2}(\mathbb{D},\mathbb{C}^{n})$ have the
factorizations

$v=v_{0}h,\;\;\bar{z}\bar{w}=\varphi w_{0}h$ (4.3)

for some scalar outer function $h,$ some scalar inner $\varphi,$ and column-
matrix inner functions $v_{0},w_{0}.$ Moreover there exist unitary-valued
functions $V,W$ of types $n\times n,\;m\times m$ respectively, of the form

$V=\begin{pmatrix}v_{0}&\bar{\alpha}\end{pmatrix},\;W^{T}=\begin{pmatrix}w_{0}&\bar{\beta}\end{pmatrix},$
(4.4)

where $\alpha,\beta$ are inner, co-outer functions, quasi-continuous functions
of types $n\times(n-1),\;m\times(m-1)$ respectively, and all minors on the
first columns of $V,W^{T}$ are in $H^{\infty}.$

Furthermore every $Q\in H^{\infty}(\mathbb{D},\mathbb{C}^{m\times n})$ which
is at minimal distance from $G$ satisfies

$W(G-Q)V=\begin{pmatrix}t_{0}u_{0}&~{}&0\\\ 0&~{}&F\end{pmatrix}$ (4.5)

for some $F\in
H^{\infty}(\mathbb{D},\mathbb{C}^{(m-1)\times(n-1)})+C(\mathbb{T},\mathbb{C}^{(m-1)\times(n-1)})$
and some quasi-continuous function $u_{0}$ given by

$u_{0}=\frac{\bar{z}\bar{\varphi}\bar{h}}{h}$ (4.6)

with $|u_{0}(z)|=1$ almost everywhere on $\mathbb{T}.$

In the statement of the lemma, in saying that an $m\times n$ matrix-valued
function $\alpha$ is co-outer we mean that each column of $\alpha$ is in
$H^{\infty}_{m}$ and $\alpha^{T}H^{2}_{m}$ is dense in $H^{2}_{n}$. (In [17,
Page 190], such a function $\alpha$ is said to be *-outer).

###### Proof.

First we construct $V$ and $W$ with the properties (4.2) to (4.5). By equation
(4.1), $\|v(z)\|=\|w(z)\|$ almost everywhere, and so the column-vector
functions $v,\bar{z}\bar{w}$ in $H^{2}$ have the same (scalar) outer factor
$h.$ This property yields the inner-outer factorizations (4.3) for some column
inner functions $v_{0},w_{0}$. By Theorem 4.9, there exists an inner co-outer
function $\alpha$ of type $n\times(n-1)$ such that
$V\stackrel{{\scriptstyle\rm def}}{{=}}[v_{0}\ \bar{\alpha}]$ is unitary-
valued almost everywhere on $\mathbb{T}$ and all minors on the first column of
$V$ are in $H^{\infty}$. Likewise there exists an inner co-outer function
$\beta$ of type $m\times(m-1)$ such that $W\stackrel{{\scriptstyle\rm
def}}{{=}}[w_{0}\ \bar{\beta}]^{T}$ is unitary-valued almost everywhere on
$\mathbb{T}$ and all minors on the first column of $W^{T}$ are in
$H^{\infty}$.

Next we show that $u_{0}$ given by equation (4.6) is quasicontinuous. Let
$Q\in H^{\infty}(\mathbb{D},\mathbb{C}^{m\times n})$ be at minimal distance
from $G.$ Then

$\|G-Q\|_{\infty}=\|H_{G}\|=t_{0}.$

By Theorem 4.11,

$(G-Q)v=t_{0}w$

and by the factorizations (4.3) we have

$(G-Q)v_{0}h=t_{0}\bar{z}\bar{\varphi}\bar{h}\bar{w}_{0}$

and by equations (4.4) and (4.6)

$(G-Q)V\begin{pmatrix}1&0&\cdots&0\end{pmatrix}^{T}=W^{*}\begin{pmatrix}t_{0}u_{0}&0&\cdots&0\end{pmatrix}^{T}.$

Thus

$W(G-Q)V=\begin{pmatrix}t_{0}u_{0}&f\\\ 0&F\end{pmatrix}$

for some $f\in L^{\infty}(\mathbb{T},\mathbb{C}^{1\times(n-1)}),\;F\in
L^{\infty}(\mathbb{T},\mathbb{C}^{(m-1)\times(n-1)}).$

Because $t_{0}=\|H_{G}\|,$ it follows that $|u_{0}|=1$ almost everywhere, and
from Nehari’s Theorem

$\|W(G-Q)V\|_{\infty}=\|G-Q\|=\|H_{G}\|=t_{0},$

and we have $f=0.$ So, $W(G-Q)V$ has the form (4.5). Now,
$\|H_{u_{0}}\|\leq\|u_{0}\|_{\infty}=1$ and
$\|H_{u_{0}}h\|=\|\bar{z}\bar{\varphi}\bar{h}\|=\|h\|.$ Hence

$\|H_{u_{0}}\|=1=\|u_{0}\|_{\infty},$

which implies that $u_{0}$ is badly approximable. The $(1,1)$ entries of
equation (4.5) are

$w_{0}^{T}(G-Q)v_{0}=t_{0}u_{0}.$

Since $v_{0}\in H^{\infty}(\mathbb{D},\mathbb{C}^{n}),\;w_{0}\in
H^{\infty}(\mathbb{D},\mathbb{C}^{m})$ and
$H^{\infty}(\mathbb{D},\mathbb{C})+C(\mathbb{T},\mathbb{C})$ is an algebra,
$u_{0}\in H^{\infty}+C.$ By a result in [23, Section 3.1], if $u_{0}\in
H^{\infty}+C$ and $u_{0}$ is badly approximable then $\bar{u}_{0}\in
H^{\infty}+C.$ Thus $u_{0}$ is quasi-continuous.

Now we show that $v_{0},w_{0}\in QC$. It follows from Nehari’s Theorem that

$(G-Q)^{*}w=t_{0}v,$

much as in the proof of [26, Theorem 0.2]. Indeed, since $H_{G}^{*}w=t_{0}v$
and $H_{G}^{*}=P_{+}M_{(G-Q)^{*}}|{H^{2}}^{\perp}$, we have (assuming, as we
may, that $v$ and $w$ are unit vectors),

$\displaystyle t_{0}=\|H_{G}^{*}w\|$ $\displaystyle=\|P_{+}(G-Q)^{*}w\|$
$\displaystyle\leq\|(G-Q)^{*}w\|\leq\|G-Q\|_{\infty}\|w\|=t_{0}.$

It follows that the inequalities hold with equality, and so

$\|P_{+}(G-Q)^{*}w\|=\|(G-Q)^{*}w\|,$

whence

$P_{+}(G-Q)^{*}w=(G-Q)^{*}w,$

and so

$(G-Q)^{*}w=H_{G}^{*}w=t_{0}v,$ (4.7)

as claimed.

Taking complex conjugates in the last equation we have

$(G-Q)^{T}\bar{w}=t_{0}\bar{v}.$

Thus, by equation (4.3),

$(G-Q)^{T}z\varphi w_{0}h=t_{0}\bar{h}\bar{v}_{0}$

for some outer function $h$ and scalar inner $\varphi$. Therefore

$\bar{v}_{0}=\frac{(G-Q)^{T}z\varphi w_{0}h}{t_{0}\bar{h}}.$

Recall that $u_{0}=\bar{z}\bar{\varphi}\bar{h}/h$, and so

$\bar{v}_{0}=\frac{1}{t_{0}}(G-Q)^{T}\bar{u}_{0}w_{0}.$

Since $u_{0}\in QC,\ G-Q\in H^{\infty}+C$ and $w_{0}\in H^{\infty}$, it
follows that $\bar{v}_{0}\in H^{\infty}+C$. Since also $v_{0}\in H^{\infty}$,
we have $v_{0}\in QC$.

To complete the proof of Lemma 4.12, all that remains is to show that
$\alpha,\beta$ are quasicontinuous and $F\in H^{\infty}+C$. This will follow
from Lemma 4.10 above.

The space $\mathrm{VMO}$ satisfies conditions $(\alpha 1)$ to $(\alpha 5)$, as
stated on [26, Page 335], and we have $v_{0}\in QC\subset\mathrm{VMO}$. Hence
we may apply Lemma 4.10 with $\varphi=v_{0}$ to deduce that
$\alpha\in\mathrm{VMO}$. Since also $\alpha\in L^{\infty}$, it follows from
Theorem 4.6 that $\alpha\in QC$. Likewise, $\beta\in QC$.

To show that $F\in H^{\infty}+C$, for $1<i\leq m,\;1<j\leq n$ consider the
$2\times 2$ minor of equation (4.5) with indices $1i,1j:$

$\sum\limits_{r<s,\;k<l}W_{1i,rs}(G-Q)_{rs,kl}V_{kl,1j}=t_{0}u_{0}F_{i-1,j-1}.$
(4.8)

By the analytic minors property of $W,V,$

$V_{kl,1j},W_{1i,rs}\in H^{\infty}.$

Since $G-Q\in H^{\infty}(\mathbb{D},\mathbb{C}^{m\times
n})+C(\mathbb{T},\mathbb{C}^{m\times n}),$ the left hand side of equation
(4.8) is in $H^{\infty}(\mathbb{D},\mathbb{C}^{2\times
2})+C(\mathbb{T},\mathbb{C}^{2\times 2})$ and hence $u_{0}F\in
H^{\infty}(\mathbb{D},\mathbb{C}^{(m-1)\times(n-1)}+C(\mathbb{T},\mathbb{C}^{(m-1)\times(n-1)}).$
Thus

$F=\bar{u}_{0}(u_{0}F)\in
H^{\infty}(\mathbb{D},\mathbb{C}^{(m-1)\times(n-1)})+C(\mathbb{T},\mathbb{C}^{(m-1)\times(n-1)}).$

∎

######  Definition 4.13.

We say that a unitary-matrix-valued function $V$ is a _thematic completion_ of
a column-matrix inner function $v_{0}\in
H^{\infty}(\mathbb{D},\mathbb{C}^{n})$ if
$V=\begin{pmatrix}v_{0}&\bar{\alpha}\end{pmatrix}$, for some co-outer function
$\alpha\in H^{\infty}(\mathbb{D},\mathbb{C}^{n\times(n-1)})$ such that $V(z)$
is a unitary matrix for almost all $z\in\mathbb{T}$ and such that all minors
on the first column of $V$ are analytic.

######  Remark 4.14.

By Theorem $1.1$ of [26], every column-matrix inner function has a thematic
completion. Thematic completions are not unique, for if
$V=\begin{pmatrix}v_{0}&\bar{\alpha}\end{pmatrix}$ is a thematic completion of
$v_{0}$, then so is $\begin{pmatrix}v_{0}&\bar{\alpha}U\end{pmatrix}$ for any
constant $(n-1)$-square unitary matrix $U.$ However, by Corollary $1.6$ of
[26], the thematic completion of $v_{0}$ _is_ unique up to multiplication on
the right by a constant unitary matrix of the form $\mathrm{diag}\\{1,U\\}$
for some constant $(n-1)$\- square matrix $U,$ and so it is permissible to
speak of “the thematic completion of $v_{0}$”.

Furthermore, by Theorem $1.2$ of [26], thematic completions have constant
determinants almost everywhere on $\mathbb{T},$ and hence $\alpha,\beta$ are
inner matrix functions. Observe that, as we showed above, if the column
$v_{0}$ belongs to $\mathrm{VMO},$ then the thematic completion of $v_{0}$ is
quasi-continuous. Similarly, if the column $w_{0}$ belongs to $\mathrm{VMO}$,
then the thematic completion of $w_{0}$ is quasi-continuous. Thus
$\alpha,\beta$ are inner, co-outer, quasi-continuous functions of types
$n\times(n-1)$ and $m\times(m-1)$ respectively.

######  Lemma 4.15 ([26], p. 316).

Let $m,n>1,$ let $G\in H^{\infty}(\mathbb{D},\mathbb{C}^{m\times
n})+C(\mathbb{T},\mathbb{C}^{m\times n}),$ let $\|H_{G}\|=t_{0}$ and let
$Q_{1}\in H^{\infty}(\mathbb{D},\mathbb{C}^{m\times n})$ be at minimal
distance from $G,$ so that in the notation of Lemma 4.12,

$W(G-Q_{1})V=\begin{pmatrix}t_{0}u_{0}&0\\\ 0&F\end{pmatrix}$ (4.9)

for some $F\in H^{\infty}(\mathbb{D},\mathbb{C}^{m-1\times
n-1})+C(\mathbb{T},\mathbb{C}^{m-1\times n-1}).$ Let

$\mathcal{E}=\\{G-Q:Q\in H^{\infty}(\mathbb{D},\mathbb{C}^{m\times
n}),\|G-Q\|_{\infty}=t_{0}\\}.$

Then

$W\mathcal{E}V=\begin{pmatrix}t_{0}u_{0}&0\\\
0&F+H^{\infty}(\mathbb{D},\mathbb{C}^{m-1\times n-1})\end{pmatrix}\cap
B(t_{0}),$ (4.10)

where $B(t_{0})$ is the closed ball of radius $t_{0}$ in
$L^{\infty}(\mathbb{T},\mathbb{C}^{m\times n}).$

###### Proof.

Let $E_{1}=G-Q_{1}\in\mathcal{E}$ and for $Q\in
H^{\infty}(\mathbb{D},\mathbb{C}^{m\times n})$ consider

$E=E_{1}-Q.$

By Lemma 4.12, there exists a function $g\in
L^{\infty}(\mathbb{D},\mathbb{C}^{m-1\times n-1})$ such that

$WEV=\begin{pmatrix}t_{0}u_{0}&0\\\ 0&g\end{pmatrix}.$

The latter equation combined with equation (4.9), yields

$\begin{array}[]{clllll}WQV&=W(G-Q_{1})V-WEV\vspace{2ex}\\\
&=\begin{pmatrix}0&0\\\ 0&F-g\end{pmatrix}\in\begin{pmatrix}0&0\\\
0&L^{\infty}(\mathbb{T},\mathbb{C}^{m-1\times n-1})\end{pmatrix}\cap
WH^{\infty}(\mathbb{D},\mathbb{C}^{m\times n})V.\end{array}$

By ([26], Lemma 1.5), $WQV\in H^{\infty}(\mathbb{D},\mathbb{C}^{m\times n}),$
say $F-g=q\in H^{\infty}(\mathbb{D},\mathbb{C}^{m\times n}).$ Then

$WEV=\begin{pmatrix}t_{0}u_{0}&0\\\
0&g\end{pmatrix}=\begin{pmatrix}t_{0}u_{0}&0\\\ 0&F-q\end{pmatrix},$

which proves the inclusion ($\subseteq$) in equation (4.10).

Conversely, suppose $q\in H^{\infty}(\mathbb{D},\mathbb{C}^{m-1\times n-1})$
and

$\|F-q\|_{\infty}\leq t_{0}.$

By ([26], Lemma 1.5), there exists a function $Q\in
H^{\infty}(\mathbb{D},\mathbb{C}^{m\times n})$ such that

$WQV=\begin{pmatrix}0&0\\\ 0&q\end{pmatrix}$

and thus

$W(E_{1}-Q)V=\begin{pmatrix}t_{0}u_{0}&0\\\ 0&F-q\end{pmatrix}.$

Then

$E_{1}-Q=G-(Q_{1}-Q)\in\mathcal{E},$

and so

$\begin{pmatrix}t_{0}u_{0}&0\\\ 0&F-q\end{pmatrix}\in W\mathcal{E}V.$

Hence equality holds in equation (4.10). ∎

######  Lemma 4.16 ([27], p. 16).

Let $G\in H^{\infty}(\mathbb{D},\mathbb{C}^{m\times
n})+C(\mathbb{T},\mathbb{C}^{m\times n})$ and let $(x_{0},y_{0})$ be a Schmidt
pair for the Hankel operator $H_{G}$ corresponding to the singular value
$t_{0}=\|H_{G}\|.$ Let $x_{0}=\xi_{0}h_{0}$ be the inner-outer factorization
of $x_{0},$ where $\xi_{0}\in H^{\infty}(\mathbb{D},\mathbb{C}^{n})$ is the
inner and $h_{0}\in H^{2}(\mathbb{D},\mathbb{C})$ is the scalar outer factor
of $x_{0}\in H^{2}(\mathbb{D},\mathbb{C}^{n}),$ and let

$V_{0}=\begin{pmatrix}\xi_{0}&\overline{\alpha_{0}}\end{pmatrix}$

be a unitary-valued function on $\mathbb{T}$, where $\alpha_{0}\in
H^{\infty}(\mathbb{D},\mathbb{C}^{n\times(n-1)})$ is co-outer. Then

$V_{0}\begin{pmatrix}0&H^{2}(\mathbb{D},\mathbb{C}^{n-1})\end{pmatrix}^{T}$

is the orthogonal projection of $H^{2}(\mathbb{D},\mathbb{C}^{n})$ onto the
pointwise orthogonal complement of $x_{0}$ in
$L^{2}(\mathbb{T},\mathbb{C}^{n}).$ Similarly

$V_{0}^{*}\begin{pmatrix}0&H^{2}(\mathbb{D},\mathbb{C}^{n-1})^{\perp}\end{pmatrix}$

is the orthogonal projection of $H^{2}(\mathbb{D},\mathbb{C}^{n})^{\perp}$
onto the pointwise orthogonal complement of $x_{0}$ in
$L^{2}(\mathbb{T},\mathbb{C}^{n}).$

######  Lemma 4.17 ([27], p. 16).

Let $G,x_{0},y_{0}$ be defined as in Lemma 4.16 and let
$\mathcal{K},\mathcal{L}$ be the projections of
$H^{2}(\mathbb{D},\mathbb{C}^{n}),H^{2}(\mathbb{D},\mathbb{C}^{m})^{\perp}$
onto the pointwise orthogonal complements of $x_{0},y_{0}$ in
$L^{2}(\mathbb{T},\mathbb{C}^{n}),L^{2}(\mathbb{T},\mathbb{C}^{m})$
respectively. Let $Q_{0}\in H^{\infty}(\mathbb{D},\mathbb{C}^{m\times n})$ be
at minimal distance from $G,$ let $F$ be the $(2,2)$ block of
$W_{0}(G-Q_{0})V_{0},$ as in Lemma 4.12, that is,

$V_{0}=\begin{pmatrix}\xi_{0}&\overline{\alpha_{0}}\end{pmatrix},\quad
W_{0}=\begin{pmatrix}\eta_{0}&\overline{\beta_{0}}\end{pmatrix}^{T}$ (4.11)

are unitary-valued functions on $\mathbb{T},$ $\alpha_{0},\beta_{0}$ are co-
outer functions of size $n\times(n-1),m\times(m-1)$ respectively and all
minors on the first columns of $V_{0},W_{0}^{T}$ are in $H^{\infty}.$ Let
$Q\in H^{\infty}(\mathbb{D},\mathbb{C}^{m\times n})$ satisfy

$(G-Q)x_{0}=\|H_{G}\|y_{0},\quad y_{0}^{*}(G-Q)=\|H_{G}\|x_{0}^{*}.$

Then $H_{F}$ is a unitary multiple of the operator

$\Gamma:=P_{\mathcal{L}}M_{G-Q}|\mathcal{K},$ (4.12)

where $M_{G-Q}:L^{2}(\mathbb{T},\mathbb{C}^{n})\to
L^{2}(\mathbb{T},\mathbb{C}^{m})$ is the operator of multiplication by $G-Q.$
More explicitly, if $U_{1}:H^{2}(\mathbb{D},\mathbb{C}^{n-1})\to\mathcal{K},$
$U_{2}:H^{2}(\mathbb{D},\mathbb{C}^{m-1})^{\perp}\to\mathcal{L}$ are defined
by

$U_{1}\chi=V_{0}\begin{pmatrix}0\\\ \chi\end{pmatrix},\quad
U_{2}\psi=W_{0}^{*}\begin{pmatrix}0\\\ \psi\end{pmatrix}\quad\text{for
all}\quad\chi\in H^{2}(\mathbb{D},\mathbb{C}^{n-1}),\psi\in
H^{2}(\mathbb{D},\mathbb{C}^{m-1}),$ (4.13)

then $U_{1},U_{2}$ are unitaries and

$H_{F}=U_{2}^{*}\Gamma U_{1}.$

######  Lemma 4.18 ([26], p. 337).

Let $\alpha\in\mathrm{QC}$ of type $m\times n,$ where $m\geq n,$ be inner and
co-outer. There exists $A\in H^{\infty}(\mathbb{D},\mathbb{C}^{n\times m})$
such that $A\alpha=I_{n}.$ Here $I_{n}$ denotes the $n\times n$ identity
matrix.

Theorem 4.19 gives the algorithm for the superoptimal analytic approximant
constructed in [27].

######  Theorem 4.19 ([27], p. 17).

Let $G\in H^{\infty}(\mathbb{D},\mathbb{C}^{m\times
n})+C(\mathbb{T},\mathbb{C}^{m\times n}).$ The superoptimal approximant
$\mathcal{A}G$ to $G$ is given by the following formula.

If $H_{G}=0,$ then $\mathcal{A}G=G.$ Otherwise define spaces $K_{j}\subset
L^{2}(\mathbb{T},\mathbb{C}^{n}),N_{j}\subset
L^{2}(\mathbb{T},\mathbb{C}^{m}),$ vectors $\chi_{j}\in K_{j},$ $\psi_{j}\in
N_{j},$ $H^{\infty}$ functions $Q_{j},$ operators $\Gamma_{j}$ and positive
$\lambda_{j}$ as follows.

Let

$K_{0}=H^{2}(\mathbb{D},\mathbb{C}^{n}),\quad
N_{0}=H^{2}(\mathbb{D},\mathbb{C}^{m})^{\perp},\quad Q_{0}=0.$

Let

$\Gamma_{j}=P_{N_{j}}M_{G-Q_{j}}|K_{j}:K_{j}\to
N_{j},\quad\lambda_{j}=\|\Gamma_{j}\|,$

where $P_{N_{j}}$ is the orthogonal projection onto $N_{j}$. If
$\lambda_{j}=0$ set $r=j$ and terminate the construction. Otherwise let
$\chi_{j},\psi_{j}$ be a Schmidt pair for $\Gamma_{j}$ corresponding to the
singular value $\lambda_{j}.$ Let $K_{j+1}$ be the range of the orthogonal
projection of $K_{j}$ onto the pointwise orthogonal complement of
$\chi_{0},\cdots,\chi_{j}$ in $L^{2}(\mathbb{T},\mathbb{C}^{n}).$ Let
$N_{j+1}$ be the projection of $N_{j}$ onto the pointwise orthogonal
complement of $\psi_{0},\cdots,\psi_{j}$ in
$L^{2}(\mathbb{T},\mathbb{C}^{m}).$ Let $Q_{j+1}\in
H^{\infty}(\mathbb{D},\mathbb{C}^{m\times n})$ be chosen to satisfy, for
$0\leq k\leq j,$

$Q_{j+1}\chi_{k}=G\chi_{k}-t_{k}\psi_{k},\quad\psi_{k}^{*}Q_{j+1}=\psi_{k}^{*}G-t_{k}\chi_{k}^{*}.$
(4.14)

Then each $\Gamma_{j}$ is a compact operator, $Q_{j}$ with the above
properties does exist, the construction terminates with $r\leq\min(m,n)$ and

$G-\mathcal{A}G=\displaystyle\sum\limits_{j=0}^{r-1}\frac{\lambda_{j}\psi_{j}\chi_{j}^{*}}{|h_{j}|^{2}}.$
(4.15)

We shall derive a similar formula for the superoptimal analytic approximant
$\mathcal{A}G,$ by making use of exterior products of Hilbert spaces.

###  4.2. Algorithm for superoptimal analytic approximation

In this section we consider the superoptimal analytic approximation problem
for a function $G\in H^{\infty}(\mathbb{D},\mathbb{C}^{m\times
n})+C(\mathbb{T},\mathbb{C}^{m\times n}).$ We first state the algorithm for
the solution of Problem 1.2; later we shall prove the claims that are made in
this description of the algorithm. We will assume here the result of Peller
and Young [26] that Problem 1.2 has a unique solution (see Theorem 1.3). For
convenience, we give citations of the steps in this paper where the
corresponding claims are proved.

######  _Algorithm:_.

Let $G\in H^{\infty}(\mathbb{D},\mathbb{C}^{m\times
n})+C(\mathbb{T},\mathbb{C}^{m\times n}).$ In this subsection we shall give a
fuller and more precise statement of the algorithm for $\mathcal{A}G$ outlined
in the Introduction, Section 1, in preparation for a subsequent formal proof
of Theorem 10.15, which asserts that if entities $r,t_{i},x_{i},y_{i},h_{i}$
for $i=0,\dots,r-1,$ are generated by the algorithm, then the superoptimal
approximant is given by equation

$\mathcal{A}G=\displaystyle
G-\sum\limits_{i=0}^{r-1}\frac{t_{i}y_{i}x_{i}^{*}}{|h_{i}|^{2}}.$

The proof will be by induction on $r,$ which is the least index $j\geq 0$ such
that $T_{j}=0,$ where $T_{0}=H_{G},T_{1},T_{2},\dots$ is a sequence of
operators recursively generated by the algorithm.

Step 0. Let $t_{0}=\|H_{G}\|.$ If $t_{0}=0,$ then $H_{G}=0,$ which implies
$G\in H^{\infty}(\mathbb{D},\mathbb{C}^{m\times n}).$ In this case, the
algorithm terminates, we define $r$ to be zero and the superoptimal
approximant $\mathcal{A}G$ is given by $\mathcal{A}G=G$, in agreement with the
formula

$G-\mathcal{A}G=\sum\limits_{j=0}^{r-1}\frac{t_{j}y_{j}x_{j}^{*}}{x_{j}^{*}x_{j}}$
(4.16)

contained in the statement of Theorem 10.15 (since the sum on the right hand
side of equation (4.16) is empty, and therefore by convention is interpreted
as being zero).

Otherwise, $t_{0}>0$. By Theorem 4.1 and Lemma 4.12, $H_{G}$ is a compact
operator and so there exists a Schmidt pair $(x_{0},y_{0})$ corresponding to
the singular value $t_{0}$ of $H_{G}.$ By the definition of the Schmidt pair
$(x_{0},y_{0})$ corresponding to $t_{0}$ for the Hankel operator
$H_{G}:H^{2}(\mathbb{D},\mathbb{C}^{n})\to
H^{2}(\mathbb{D},\mathbb{C}^{m})^{\perp},$

$x_{0}\in H^{2}(\mathbb{D},\mathbb{C}^{n}),\quad y_{0}\in
H^{2}(\mathbb{D},\mathbb{C}^{m})^{\perp}$

are non-zero vector-valued functions such that

$H_{G}x_{0}=t_{0}y_{0}\quad\mbox{ and }\quad H_{G}^{*}y_{0}=t_{0}x_{0}.$

By Lemma 4.12, $x_{0}\in H^{2}(\mathbb{D},\mathbb{C}^{n})$ and
$\bar{z}\bar{y}_{0}\in H^{2}(\mathbb{D},\mathbb{C}^{m})$ admit inner-outer
factorizations

$x_{0}=\xi_{0}h_{0},\quad\bar{z}\bar{y}_{0}=\eta_{0}h_{0}$ (4.17)

for some scalar outer factor $h_{0}\in H^{2}(\mathbb{D},\mathbb{C})$ and
column matrix inner functions $\xi_{0}\in
H^{\infty}(\mathbb{D},\mathbb{C}^{n})$, $\eta_{0}\in
H^{\infty}(\mathbb{D},\mathbb{C}^{m}).$ Then

$\|x_{0}(z)\|_{\mathbb{C}^{n}}=|h_{0}(z)|=\|y_{0}(z)\|_{\mathbb{C}^{m}}\;\;\text{almost
everywhere on }\;\mathbb{T}.$ (4.18)

We write equations (4.17) as

$\xi_{0}=\frac{x_{0}}{h_{0}}\;,\;\eta_{0}=\frac{\bar{z}\bar{y}_{0}}{h_{0}}.$
(4.19)

By equations (4.18) and (4.19),

$\|\xi_{0}(z)\|_{\mathbb{C}^{n}}=1=\|\eta_{0}(z)\|_{\mathbb{C}^{m}}\;\text{almost
everywhere on}\;\mathbb{T}.$ (4.20)

Let $t_{0}\neq 0.$ By Lemma 4.12, every function $Q_{1}\in
H^{\infty}(\mathbb{D},\mathbb{C}^{m\times n})$ which is at minimal distance
from $G$ satisfies

$(G-Q_{1})x_{0}=t_{0}y_{0},\quad y_{0}^{*}(G-Q_{1})=t_{0}x_{0}^{*}.$ (4.21)

Step 1. Let

$X_{1}\stackrel{{\scriptstyle\text{def}}}{{=}}\xi_{0}\dot{\wedge}H^{2}(\mathbb{D},\mathbb{C}^{n})\vspace{2ex}.$
(4.22)

By Proposition 6.1, $X_{1}$ is a closed subspace of
$H^{2}(\mathbb{D},\wedge^{2}\mathbb{C}^{n})$. Similarly,

$\eta_{0}\dot{\wedge}zH^{2}(\mathbb{D},\mathbb{C}^{m})\subset
zH^{2}(\mathbb{D},\wedge^{2}\mathbb{C}^{m})$

and therefore

$\bar{\eta}_{0}\dot{\wedge}\overline{zH^{2}(\mathbb{D},\mathbb{C}^{m})}\subset\bar{z}\overline{H^{2}(\mathbb{D},\wedge^{2}\mathbb{C}^{m})},$

that is, if

$Y_{1}\stackrel{{\scriptstyle\text{def}}}{{=}}\bar{\eta}_{0}\dot{\wedge}H^{2}(\mathbb{D},\mathbb{C}^{m})^{\perp},$
(4.23)

then

$Y_{1}\subset H^{2}(\mathbb{D},\wedge^{2}\mathbb{C}^{m})^{\perp}.$

By Proposition 7.1, $Y_{1}$ is a closed subspace of
$H^{2}(\mathbb{D},\wedge^{2}\mathbb{C}^{m})^{\perp}.$

Choose any function $Q_{1}\in H^{\infty}(\mathbb{D},\mathbb{C}^{m\times n})$
which satisfies the equations (4.21). Consider the operator $T_{1}:X_{1}\to
Y_{1}$ defined by

$T_{1}(\xi_{0}\dot{\wedge}x)=P_{Y_{1}}(\bar{\eta}_{0}\dot{\wedge}(G-Q_{1})x)\;\text{for
all}\;x\in H^{2}(\mathbb{D},\mathbb{C}^{n}),$ (4.24)

where $P_{Y_{1}}$ is the projection from
$L^{2}(\mathbb{T},\wedge^{2}\mathbb{C}^{m})$ on $Y_{1}.$

By Corollary 7.2 and Proposition 8.1, $T_{1}$ is well-defined.

If $T_{1}=0,$ then the algorithm terminates, we define $r$ to be $1$ and, in
agreement with Theorem 10.15, the superoptimal approximant $\mathcal{A}G$ is
given by the formula

$G-\mathcal{A}G=\displaystyle\sum\limits_{i=0}^{r-1}\frac{t_{i}y_{i}x_{i}^{*}}{|h_{i}|^{2}}=\frac{t_{0}y_{0}x_{0}^{*}}{|h_{0}|^{2}},$

and the solution is

$\mathcal{A}G=G-\frac{t_{0}y_{0}x_{0}^{*}}{|h_{0}|^{2}}.$

If $T_{1}\neq 0,$ let $t_{1}=\|T_{1}\|>0.$ By Theorem 9.1, $T_{1}$ is a
compact operator and so there exist $v_{1}\in
H^{2}(\mathbb{D},\mathbb{C}^{n}),\;w_{1}\in
H^{2}(\mathbb{D},\mathbb{C}^{m})^{\perp}$ such that
$(\xi_{0}\dot{\wedge}v_{1},\bar{\eta}_{0}\dot{\wedge}w_{1})$ is a Schmidt pair
for $T_{1}$ corresponding to $t_{1}.$ Let $h_{1}$ be the scalar outer factor
of $\xi_{0}\dot{\wedge}v_{1}$ and let

$x_{1}=(I_{\mathbb{C}^{m}}-\xi_{0}\xi_{0}^{*})v_{1},\;\;y_{1}=(I_{\mathbb{C}^{m}}-\bar{\eta}_{0}\eta_{0}^{T})w_{1},$
(4.25)

where $\mathrm{I}_{\mathbb{C}^{n}}$ and $\mathrm{I}_{\mathbb{C}^{m}}$ are the
identity operators in $\mathbb{C}^{n}$ and $\mathbb{C}^{m}$ respectively.
Then, by Proposition 5.1,

$\|x_{1}(z)\|_{\mathbb{C}^{n}}=|h_{1}(z)|=\|y_{1}(z)\|_{\mathbb{C}^{m}}\;\text{almost
everywhere on}\;\mathbb{T}.$ (4.26)

By Theorem 1.3, there exists a function $Q_{2}\in
H^{\infty}(\mathbb{D},\mathbb{C}^{m\times n})$ such that both
$s_{0}^{\infty}(G-Q_{2})$ and $s_{1}^{\infty}(G-Q_{2})$ are minimized and

$s_{1}^{\infty}(G-Q_{2})=t_{1}.$

By Proposition 9.16, any such $Q_{2}$ satisfies

$\displaystyle(G-Q_{2})x_{0}=t_{0}y_{0},\quad
y_{0}^{*}(G-Q_{2})=t_{0}x_{0}^{*}$ (4.27)
$\displaystyle(G-Q_{2})x_{1}=t_{1}y_{1},\quad
y_{1}^{*}(G-Q_{2})=t_{1}x_{1}^{*}.$

Choose any function $Q_{2}\in H^{\infty}(\mathbb{D},\mathbb{C}^{m\times n})$
which satisfies the equations (4.27).

Define

$\xi_{1}=\frac{x_{1}}{h_{1}},\quad\eta_{1}=\frac{\bar{z}\bar{y}_{1}}{h_{1}}.$
(4.28)

By equations (4.26) and (4.28),
$\|\xi_{1}(z)\|_{\mathbb{C}^{n}}=1=\|\eta_{1}(z)\|_{\mathbb{C}^{n}}$ almost
everywhere on $\mathbb{T}.$

Step 2. Define

$\begin{array}[]{cllll}X_{2}&\stackrel{{\scriptstyle\text{def}}}{{=}}\xi_{0}\dot{\wedge}\xi_{1}\dot{\wedge}H^{2}(\mathbb{D},\mathbb{C}^{n})\vspace{2ex}\\\
Y_{2}&\stackrel{{\scriptstyle\text{def}}}{{=}}\bar{\eta}_{0}\dot{\wedge}\bar{\eta}_{1}\dot{\wedge}H^{2}(\mathbb{D},\mathbb{C}^{m})^{\perp}.\end{array}$

Note that, by Proposition 6.2, $X_{2}$ is a closed linear subspace of
$H^{2}(\mathbb{D},\wedge^{3}\mathbb{C}^{n}),$ and, by Proposition 7.3, $Y_{2}$
is a closed linear subspace of
$H^{2}(\mathbb{D},\wedge^{3}\mathbb{C}^{m})^{\perp}.$

Now consider the operator $T_{2}:X_{2}\to Y_{2}$ given by

$T_{2}(\xi_{0}\dot{\wedge}\xi_{1}\dot{\wedge}x)=P_{Y_{2}}(\bar{\eta}_{0}\dot{\wedge}\bar{\eta}_{1}\dot{\wedge}(G-Q_{2})x),$
(4.29)

where $P_{Y_{2}}$ is the projection from $L^{2}(\mathbb{T},\mathbb{C}^{m})$ on
$Y_{2}.$

By Corollary 7.4 and Proposition 8.1, $T_{2}$ is well defined, that is, it
does not depend on the choice of $Q_{2}\in
H^{\infty}(\mathbb{D},\mathbb{C}^{m\times n})$ satisfying equations (4.27). If
$T_{2}=0,$ then the algorithm terminates, we define $r$ to be $2$ and,
according to Theorem 10.15, the superoptimal approximant $\mathcal{A}G$ is
given by the formula

$G-\mathcal{A}G=\sum\limits_{i=0}^{r-1}\frac{t_{i}y_{i}x_{i}^{*}}{|h_{i}|^{2}}=\frac{t_{0}y_{0}x_{0}^{*}}{|h_{0}|^{2}}+\frac{t_{1}y_{1}x_{1}^{*}}{|h_{1}|^{2}}.$

If $T_{2}\neq 0$, then let $t_{2}=\|T_{2}\|.$ By Theorem 9.26, $T_{2}$ is a
compact operator and hence there exist $v_{2}\in
H^{2}(\mathbb{D},\mathbb{C}^{n}),\;w_{2}\in
H^{2}(\mathbb{D},\mathbb{C}^{m})^{\perp}$ such that

$(\xi_{0}\dot{\wedge}\xi_{1}\dot{\wedge}v_{2},\;\bar{\eta}_{0}\dot{\wedge}\bar{\eta}_{1}\dot{\wedge}w_{2})$

is a Schmidt pair for $T_{2}$ corresponding to $\|T_{2}\|=t_{2}.$

Let $h_{2}$ be the scalar outer factor of
$\xi_{0}\dot{\wedge}\xi_{1}\dot{\wedge}v_{2}$. Note that, by Proposition 6.2,
$\xi_{0}\dot{\wedge}\xi_{1}\dot{\wedge}v_{2}\in
H^{2}(\mathbb{D},\wedge^{3}\mathbb{C}^{n}).$ Let

$x_{2}=(I_{\mathbb{C}^{m}}-\xi_{0}\xi_{0}^{*}-\xi_{1}\xi_{1}^{*})v_{2},\;\;y_{2}=(I_{\mathbb{C}^{m}}-\bar{\eta}_{0}\eta_{0}^{T}-\bar{\eta}_{1}\eta_{1}^{T})w_{2}.$
(4.30)

Then, by Proposition 5.1,

$\|x_{2}(z)\|_{\mathbb{C}^{n}}=|h_{2}(z)|=\|y_{2}(z)\|_{\mathbb{C}^{m}}\;\text{almost
everywhere on}\;\mathbb{T}.$ (4.31)

Define

$\xi_{2}=\frac{x_{2}}{h_{2}},\;\eta_{2}=\frac{\bar{z}\bar{y}_{2}}{h_{2}}.$
(4.32)

Clearly $\|\xi_{2}(z)\|_{\mathbb{C}^{n}}=1$ and
$\|\eta_{2}(z)\|_{\mathbb{C}^{m}}=1$ almost everywhere on $\mathbb{T}.$

Recursive step. Suppose that, for $j\leq\min(m,n)-2$, we have constructed

$\displaystyle t_{0}\geq t_{1}\geq\cdots\geq t_{j}>0$ (4.33) $\displaystyle
x_{0},x_{1},\cdots,x_{j}\in L^{2}(\mathbb{T},\mathbb{C}^{n})$ $\displaystyle
y_{0},y_{1},\cdots,y_{j}\in L^{2}(\mathbb{T},\mathbb{C}^{m})$ $\displaystyle
h_{0},h_{1},\cdots,h_{j}\in H^{2}(\mathbb{D},\mathbb{C})\;\text{outer}$
$\displaystyle\xi_{0},\xi_{1},\cdots,\xi_{j}\in
L^{\infty}(\mathbb{T},\mathbb{C}^{n})\;\text{pointwise orthonormal
on}\;\mathbb{T}$ $\displaystyle\eta_{0},\eta_{1},\cdots,\eta_{j}\in
L^{\infty}(\mathbb{T},\mathbb{C}^{m})\;\text{pointwise orthonormal
on}\;\mathbb{T}$ $\displaystyle
X_{0}=H^{2}(\mathbb{D},\mathbb{C}^{n}),X_{1},\cdots,X_{j}$ $\displaystyle
Y_{0}=H^{2}(\mathbb{D},\mathbb{C}^{m})^{\perp},Y_{1},\cdots,Y_{j}$
$\displaystyle T_{0},T_{1},\cdots,T_{j}\;\text{compact operators}.$

By Theorem 1.3, there exists a function $Q_{j+1}\in
H^{\infty}(\mathbb{D},\mathbb{C}^{m\times n})$ such that

$\left(s_{0}^{\infty}(G-Q_{j+1}),s_{1}^{\infty}(G-Q_{j+1}),\cdots,s_{j+1}^{\infty}(G-Q_{j+1})\right)$

is lexicographically minimized. By Proposition 10.5, any such function
$Q_{j+1}$ satisfies

$(G-Q_{j+1})x_{i}=t_{i}y_{i},\quad y_{i}^{*}(G-Q_{j+1})=t_{i}x_{i}^{*},\quad
i=0,1,\cdots,j.$ (4.34)

Choose any function $Q_{j+1}\in H^{\infty}(\mathbb{D},\mathbb{C}^{m\times n})$
which satisfies the equations (4.34). Define

$X_{j+1}\stackrel{{\scriptstyle\text{def}}}{{=}}\xi_{0}\dot{\wedge}\xi_{1}\dot{\wedge}\cdots\dot{\wedge}\xi_{j}\dot{\wedge}H^{2}(\mathbb{D},\mathbb{C}^{n})$
(4.35)
$Y_{j+1}\stackrel{{\scriptstyle\text{def}}}{{=}}\bar{\eta}_{0}\dot{\wedge}\bar{\eta}_{1}\dot{\wedge}\cdots\dot{\wedge}\bar{\eta}_{j}\dot{\wedge}H^{2}(\mathbb{D},\mathbb{C}^{m})^{\perp}.$
(4.36)

Note that, by Proposition 6.1, $X_{j+1}$ is a subset of
$H^{2}(\mathbb{D},\wedge^{j+2}\mathbb{C}^{n}),$ and, by Proposition 10.8,
$Y_{j+1}$ is a closed subspace of
$H^{2}(\mathbb{D},\wedge^{j+2}\mathbb{C}^{m})^{\perp}.$ Consider the operator

$T_{j+1}:X_{j+1}\to Y_{j+1}$

given by

$T_{j+1}(\xi_{0}\dot{\wedge}\xi_{1}\dot{\wedge}\cdots\dot{\wedge}\xi_{j}\dot{\wedge}x)=P_{Y_{j+1}}\left(\bar{\eta}_{0}\dot{\wedge}\bar{\eta}_{1}\dot{\wedge}\cdots\dot{\wedge}\bar{\eta}_{j}\dot{\wedge}(G-Q_{j+1})x\right).$
(4.37)

By Corollary 7.4 and Proposition 8.1, $T_{j+1}$ is well-defined and does not
depend on the choice of $Q_{j+1}$ subject to equations (4.34).

If $T_{j+1}=0,$ then the algorithm terminates, we define $r$ to be $j+1,$ and,
according to Theorem 10.15, the superoptimal approximant $\mathcal{A}G$ is
given by the formula

$G-\mathcal{A}G=\sum\limits_{i=0}^{r-1}\frac{t_{i}y_{i}x_{i}^{*}}{|h_{i}|^{2}}.$

Otherwise, we define $t_{j+1}=\|T_{j+1}\|>0.$ By Theorem 9.1, $T_{j+1}$ is a
compact operator and hence there exist $v_{j+1}\in
H^{2}(\mathbb{D},\mathbb{C}^{n}),\;w_{j+1}\in
H^{2}(\mathbb{D},\mathbb{C}^{m})^{\perp}$ such that

$(\xi_{0}\dot{\wedge}\xi_{1}\dot{\wedge}\cdots\dot{\wedge}\xi_{j}\dot{\wedge}v_{j+1},\bar{\eta}_{0}\dot{\wedge}\bar{\eta}_{1}\dot{\wedge}\cdots\dot{\wedge}\bar{\eta}_{j}\dot{\wedge}w_{j+1})$
(4.38)

is a Schmidt pair for $T_{j+1}$ corresponding to the singular value $t_{j+1}.$

Let $h_{j+1}$ be the scalar outer factor of
$\xi_{0}\dot{\wedge}\xi_{1}\dot{\wedge}\cdots\dot{\wedge}\xi_{j}\dot{\wedge}v_{j+1},$
and let

$x_{j+1}=(I_{\mathbb{C}^{n}}-\xi_{0}\xi_{0}^{*}-\cdots-\xi_{j}\xi_{j}^{*})v_{j+1},\quad
y_{j+1}=(I_{\mathbb{C}^{m}}-\bar{\eta}_{0}\eta_{0}^{T}-\cdots-\bar{\eta}_{j}\eta_{j}^{T})w_{j+1}.$
(4.39)

Then, by Proposition 5.1,

$\|x_{j+1}(z)\|_{\mathbb{C}^{n}}=|h_{j+1}(z)|=\|y_{j+1}(z)\|_{\mathbb{C}^{m}}\;\text{almost
everywhere on}\;\mathbb{T}.$ (4.40)

We define

$\xi_{j+1}=\frac{x_{j+1}}{h_{j+1}},\quad\eta_{j+1}=\frac{\bar{z}\overline{y_{j+1}}}{h_{j+1}}.$
(4.41)

Clearly $\|\xi_{j+1}(z)\|_{\mathbb{C}^{n}}=1$ and
$\|\eta_{j+1}(z)\|_{\mathbb{C}^{m}}=1$ almost everywhere on $\mathbb{T}.$

This completes the recursive step. The algorithm terminates after at most
$\min(m,n)$ steps, so that $r\leq\min(m,n)$ and, in accordance with Theorem
10.15 the superoptimal approximant $\mathcal{A}G$ is given by the formula

$G-\mathcal{A}G=\sum\limits_{i=0}^{r-1}\frac{t_{i}y_{i}x_{i}^{*}}{|h_{i}|^{2}}.$

∎

##  5\. Pointwise orthonormality of $\\{\xi_{i}\\}_{i=1}^{j}$ and
$\\{\bar{\eta}_{i}\\}_{i=1}^{j}$ almost everywhere on $\mathbb{T}$

These orthonormality properties will be needed for the justification of the
main algorithm.

######  Proposition 5.1.

Let $m,n$ be positive integers with $\min(m,n)\geq 2,$ let
$G\in H^{\infty}(\mathbb{D},\mathbb{C}^{m\times
n})+C(\mathbb{T},\mathbb{C}^{m\times n})$ and let $0\leq j\leq\min(m,n)-2.$
Suppose we have applied steps $0,\dots,j$ of the superoptimal analytic
approximation algorithm from Subsection 4.2 to $G$ and we have obtained
$x_{i},y_{i}$ as in equations (4.39), and $\xi_{i},\eta_{i}$ as in equations
(4.41) for $i=0,\cdots,j.$ Then

  1. _(i)_

$\xi_{0}\dot{\wedge}v_{1}=\xi_{0}\dot{\wedge}x_{1},\quad\xi_{0}\dot{\wedge}\cdots\dot{\wedge}\xi_{j-1}\dot{\wedge}v_{j}=\xi_{0}\dot{\wedge}\cdots\dot{\wedge}\xi_{j-1}\dot{\wedge}x_{j},$
$\bar{\eta}_{0}\dot{\wedge}w_{1}=\bar{\eta}_{0}\dot{\wedge}y_{1},$
and
$\bar{\eta}_{0}\dot{\wedge}\cdots\dot{\wedge}\bar{\eta}_{j-1}\dot{\wedge}w_{j}=\bar{\eta}_{0}\dot{\wedge}\cdots\dot{\wedge}\bar{\eta}_{j-1}\dot{\wedge}y_{j};$

  2. _(ii)_

$\|x_{j}(z)\|_{\mathbb{C}^{n}}=\|y_{j}(z)\|_{\mathbb{C}^{m}}=|h_{j}(z)|$
almost everywhere on $\mathbb{T};$

  3. _(iii)_

The sets $\\{\xi_{i}(z)\\}_{i=0}^{j}$ and $\\{\bar{\eta}_{i}(z)\\}_{i=0}^{j}$
are orthonormal in $\mathbb{C}^{n}$ and $\mathbb{C}^{m}$ respectively for
almost every $z\in\mathbb{T}.$

###### Proof.

We will prove statement (ii) in Propositions 9.13 and 9.29. Statement (i) is
proven below in equations (5.5), (5.9), (5). Let us prove assertion (iii).

Since the function $G$ belongs to $H^{\infty}(\mathbb{D},\mathbb{C}^{m\times
n})+C(\mathbb{T},\mathbb{C}^{m\times n}),$ by Hartman’s theorem, the Hankel
operator with symbol $G$, denoted by $H_{G},$ is a compact operator, and so
there exist functions

$x_{0}\in H^{2}(\mathbb{D},\mathbb{C}^{n}),\quad y_{0}\in
H^{2}(\mathbb{D},\mathbb{C}^{m})^{\perp}$

such that $(x_{0},y_{0})$ is a Schmidt pair corresponding to the singular
value $t_{0}=\|H_{G}\|\neq 0.$ By Lemma 4.12, $x_{0},\bar{z}\bar{y}_{0}$ admit
the inner-outer factorizations

$x_{0}=\xi_{0}h_{0},\quad\bar{z}\bar{y}_{0}=\eta_{0}h_{0}$

for column matrix inner functions $\xi_{0}\in
H^{\infty}(\mathbb{D},\mathbb{C}^{n})$, $\eta_{0}\in
H^{\infty}(\mathbb{D},\mathbb{C}^{m})$ and some scalar outer factor $h_{0}\in
H^{2}(\mathbb{D},\mathbb{C}).$ By Theorem 4.11,

$\|x_{0}(z)\|_{\mathbb{C}^{n}}=|h_{0}(z)|=\|y_{0}(z)\|_{\mathbb{C}^{m}}\;\;\text{almost
everywhere on }\;\mathbb{T}.$ (5.1)

Thus

$\|\xi_{0}(z)\|_{\mathbb{C}^{n}}=1\;\text{almost everywhere on}\;\mathbb{T}.$
(5.2)

Hence (iii) of Proposition 5.1 holds for $\\{\xi_{i}(z)\\}_{i=0}^{j}$ in the
case that $j=0.$

Let $T_{1}$ be given by equation (4.24). By the hypothesis (4.33), $T_{1}$ is
a compact operator, and if $T_{1}\neq 0,$ then there exist $v_{1}\in
H^{2}(\mathbb{D},\mathbb{C}^{n})$ and $w_{1}\in
H^{2}(\mathbb{D},\mathbb{C}^{m})^{\perp}$ such that
$(\xi_{0}\dot{\wedge}v_{1},\bar{\eta}_{0}\dot{\wedge}w_{1})$ is a Schmidt pair
corresponding to $\|T_{1}\|=t_{1}.$ By Proposition 3.26,
$\xi_{0}\dot{\wedge}v_{1}\in H^{2}(\mathbb{D},\wedge^{2}\mathbb{C}^{n}).$ Let
$h_{1}$ be the scalar outer factor of $\xi_{0}\dot{\wedge}v_{1}.$ We define

$x_{1}=(I_{n}-\xi_{0}\xi_{0}^{*})v_{1}$ (5.3)

and

$\xi_{1}=\frac{x_{1}}{h_{1}}.$ (5.4)

Then, for $z\in\mathbb{D},$

$\xi_{1}(z)=\displaystyle\frac{1}{h_{1}(z)}v_{1}(z)-\frac{1}{h_{1}(z)}\xi_{0}(z)\xi_{0}(z)^{*}v_{1}(z).$

Note that by equation (5.2),

$\xi_{0}^{*}(z)\xi_{0}(z)=\langle\xi_{0}(z),\xi_{0}(z)\rangle_{\mathbb{C}^{n}}=1\quad\text{almost
everywhere on}\;\mathbb{T},$

hence

$\langle\xi_{1}(z),\xi_{0}(z)\rangle_{\mathbb{C}^{n}}=\xi_{0}^{*}(z)\xi_{1}(z)=\displaystyle\frac{1}{h_{1}(z)}\xi_{0}(z)^{*}v_{1}(z)-\displaystyle\frac{1}{h_{1}(z)}\xi_{0}(z)^{*}\xi_{0}(z)\xi_{0}(z)^{*}v_{1}(z)=0$

almost everywhere on $\mathbb{T}.$ Note that, by equation (5.3), for almost
every $z\in\mathbb{T},$

$\displaystyle\xi_{0}(z)\wedge v_{1}(z)$
$\displaystyle=\xi_{0}(z)\wedge(x_{1}(z)+\xi_{0}(z)\xi_{0}(z)^{*}v_{1}(z))\vspace{2ex}$
$\displaystyle=\xi_{0}(z)\wedge
x_{1}(z)+\xi_{0}(z)\wedge\xi_{0}(z)\xi_{0}(z)^{*}v_{1}(z))\vspace{2ex}$
$\displaystyle=\xi_{0}(z)\wedge x_{1}(z),$ (5.5)

the last equality following from the pointwise linear dependence of the
vectors $\xi_{0}$ and $z\mapsto\xi_{0}(z)\langle
v_{1}(z),\xi_{0}(z)\rangle_{\mathbb{C}^{n}}$ almost everywhere on
$\mathbb{T}.$

Moreover, since $h_{1}$ is the scalar outer factor of
$\xi_{0}\dot{\wedge}v_{1},$ for almost every $z\in\mathbb{T},$ we have

$\begin{array}[]{clll}|h_{1}(z)|&=\|\xi_{0}(z)\wedge
v_{1}(z)\|_{\wedge^{2}{\mathbb{C}^{n}}}=\|\xi_{0}(z)\wedge
x_{1}(z)\|_{\wedge^{2}\mathbb{C}^{n}},\end{array}$

By Lemma 3.15,

$\|\xi_{0}(z)\wedge x_{1}(z)\|_{\wedge^{2}\mathbb{C}^{n}}=\|x_{1}(z)-\langle
x_{1}(z),\xi_{0}(z)\rangle_{\mathbb{C}^{n}}\xi_{0}(z)\|_{\mathbb{C}^{n}}=\|x_{1}(z)\|_{\mathbb{C}^{n}}$

almost everywhere on $\mathbb{T}.$ Hence, for almost every $z\in\mathbb{T},$

$|h_{1}(z)|=\|x_{1}(z)\|_{\mathbb{C}^{n}}$ (5.6)

and thus

$\|\xi_{1}(z)\|_{\mathbb{C}^{n}}=\displaystyle\frac{\|x_{1}(z)\|_{\mathbb{C}^{n}}}{|h_{1}(z)|}=1\quad\text{almost
everywhere on}\;\mathbb{T}.$

Consequently, $\\{\xi_{0}(z),\xi_{1}(z)\\}$ is an orthonormal set in
$\mathbb{C}^{n}$ for almost every $z\in\mathbb{T}.$ Hence (iii) of Proposition
5.1 holds for $\\{\xi_{i}(z)\\}_{i=0}^{j}$ in the case that $j=1.$

Recursive step: Suppose the entities in equations (4.33) have been constructed
and have the stated properties. Since by the inductive hypothesis $T_{j}$ is a
compact operator, there exist

$v_{j}\in H^{2}(\mathbb{D},\mathbb{C}^{n}),\quad w_{j}\in
H^{2}(\mathbb{D},\mathbb{C}^{m})^{\perp}$

such that

$(\xi_{0}\dot{\wedge}\xi_{1}\dot{\wedge}\cdots\dot{\wedge}\xi_{j-1}\dot{\wedge}v_{j},\overline{\eta}_{0}\dot{\wedge}\overline{\eta}_{1}\dot{\wedge}\cdots\dot{\wedge}\overline{\eta}_{j-1}\dot{\wedge}w_{j})$

is a Schmidt pair for $T_{j}$ corresponding to $\|T_{j}\|=t_{j}.$ Let us first
prove that
$\xi_{0}\dot{\wedge}\xi_{1}\dot{\wedge}\cdots\dot{\wedge}\xi_{j-1}\dot{\wedge}v_{j}$
is an element of $H^{2}(\mathbb{D},\wedge^{j+1}\mathbb{C}^{n}).$ By
hypothesis,

$x_{i}=(I_{n}-\xi_{0}\xi_{0}^{*}-\dots-\xi_{i-1}\xi_{i-1}^{*})v_{i}\quad\text{and}\quad\xi_{i}=\frac{x_{i}}{h_{i}}$

for $i=0,\dots,j-1.$ Then, for all $z\in\mathbb{D},$

$\displaystyle\left(\xi_{0}\dot{\wedge}\xi_{1}\dot{\wedge}\cdots\dot{\wedge}\xi_{j-1}\dot{\wedge}v_{j}\right)(z)$
$\displaystyle=\left(\xi_{0}\dot{\wedge}\frac{x_{1}}{h_{1}}\dot{\wedge}\cdots\dot{\wedge}\frac{x_{j-1}}{h_{j-1}}\dot{\wedge}v_{j}\right)(z)$
$\displaystyle=\left(\frac{1}{h_{1}}\cdots\frac{1}{h_{j-1}}\xi_{0}\dot{\wedge}x_{1}\dot{\wedge}\cdots\dot{\wedge}x_{j-1}\dot{\wedge}v_{j}\right)(z).$

We obtain

$(\xi_{0}\dot{\wedge}\xi_{1}\dot{\wedge}\cdots\dot{\wedge}\xi_{j-1}\dot{\wedge}v_{j})(z)=\left(\frac{1}{h_{1}}\cdots\frac{1}{h_{j-1}}\xi_{0}\dot{\wedge}v_{1}\dot{\wedge}\cdots\dot{\wedge}v_{j-1}\dot{\wedge}v_{j}\right)(z),\;\text{for
all}\;z\in\mathbb{D},$

due to pointwise linear dependence of $\xi_{k}$ and $z\mapsto\xi_{k}(z)\langle
v_{j}(z),\xi_{k}(z)\rangle_{\mathbb{C}^{n}}$ on $\mathbb{D},$ for all
$k=0,\dots,j-1.$ By Proposition 3.24,

$\frac{1}{h_{1}}\cdots\frac{1}{h_{j-1}}\xi_{0}\dot{\wedge}v_{1}\dot{\wedge}\cdots\dot{\wedge}v_{j-1}\dot{\wedge}v_{j}$

is analytic on $\mathbb{D}.$ Moreover, by Proposition 3.28, since
$\xi_{0},\xi_{1},\dots,\xi_{j-1}$ are pointwise orthogonal on $\mathbb{T},$

$\|\xi_{0}\dot{\wedge}\xi_{1}\dot{\wedge}\cdots\dot{\wedge}\xi_{j-1}\dot{\wedge}v_{j}\|_{L^{2}(\mathbb{T},\wedge^{j+1}\mathbb{C}^{n})}<\infty.$

Therefore

$\xi_{0}\dot{\wedge}\xi_{1}\dot{\wedge}\cdots\dot{\wedge}\xi_{j-1}\dot{\wedge}v_{j}\in
H^{2}(\mathbb{D},\wedge^{j+1}\mathbb{C}^{n}).$

Let $h_{j}$ be the scalar outer factor of
$\xi_{1}\dot{\wedge}\xi_{2}\dot{\wedge}\cdots\dot{\wedge}\xi_{j-1}\dot{\wedge}v_{j}.$
We define

$x_{j}=(I_{n}-\xi_{0}\xi_{0}^{*}-\cdots-\xi_{j-1}\xi_{j-1}^{*})v_{j}$ (5.7)

and

$\xi_{j}=\frac{x_{j}}{h_{j}}.$ (5.8)

Let us show that $\\{\xi_{0}(z),\cdots,\xi_{j-1}(z),\xi_{j}(z)\\}$ is an
orthonormal set in $\mathbb{C}^{n}$ almost everywhere on $\mathbb{T}.$ We have

$\xi_{j}=\displaystyle\frac{1}{h_{j}}v_{j}-\displaystyle\frac{1}{h_{j}}\xi_{0}\xi_{0}^{*}v_{j}-\cdots-\displaystyle\frac{1}{h_{j}}\xi_{j-1}\xi_{j-1}^{*}v_{j},$

and so, for $i=0,\dots,j-1,$

$\begin{array}[]{cll}&\langle\xi_{j}(z),\xi_{i}(z)\rangle_{\mathbb{C}^{n}}=\displaystyle\frac{1}{h_{j}(z)}\xi_{i}^{*}(z)v_{j}(z)-\displaystyle\frac{1}{h_{j}(z)}\xi_{i}^{*}(z)\xi_{0}(z)\xi_{0}^{*}(z)v_{1}(z)-\cdots\vspace{2ex}\\\
&\hskip
159.30505pt-\displaystyle\frac{1}{h_{j}(z)}\xi_{i}^{*}(z)\xi_{j-1}(z)\xi_{j-1}^{*}(z)v_{1}(z)\end{array}$

almost everywhere on $\mathbb{T}.$ Note that by the inductive hypothesis, for
$i,k=0,1,\cdots,j-1$ and for almost all $z\in\mathbb{T}$,

$\xi_{i}^{*}(z)\xi_{k}(z)=\left\\{\begin{array}[]{ll}0,\quad\text{for}\;i\neq
k\\\ 1,\quad\text{for}\;i=k\end{array}.\right.$

Thus, for $i=0,\dots,j-1,$

$\langle\xi_{j}(z),\xi_{i}(z)\rangle_{\mathbb{C}^{n}}=\displaystyle\frac{1}{h_{j}(z)}\xi_{i}^{*}(z)v_{j}(z)-\displaystyle\frac{1}{h_{j}(z)}\xi_{i}^{*}(z)\xi_{i}(z)\xi_{i}^{*}v_{j}(z)=0$

almost everywhere on $\mathbb{T},$ and hence, by induction on $j$ and for all
integers $j=0,\dots,r-1,$ $\\{\xi_{0}(z),\cdots,\xi_{j-1}(z),\xi_{j}(z)\\}$ is
an orthogonal set in $\mathbb{C}^{n}$ for almost all $z\in\mathbb{T}.$

Let us show that

$\xi_{0}(z)\wedge\cdots\wedge\xi_{j-1}(z)\wedge
v_{j}(z)=\xi_{0}(z)\wedge\cdots\wedge\xi_{j-1}(z)\wedge x_{j}(z)$

almost everywhere on $\mathbb{T}.$ Equation (5.7) yields

$\begin{array}[]{cllll}&\xi_{0}(z)\wedge\cdots\wedge\xi_{j-1}(z)\wedge
v_{j}(z)\vspace{2ex}\\\
&=\xi_{0}(z)\wedge\cdots\wedge\xi_{j-1}(z)\wedge(x_{j}(z)+\xi_{0}(z)\xi_{0}^{*}(z)v_{j}(z)\\\
&\hskip 137.77734pt+\cdots+\xi_{j-1}(z)\xi_{j-1}^{*}(z)v_{j}(z))\\\
&=\xi_{0}(z)\wedge\cdots\wedge\xi_{j-1}(z)\wedge(x_{j}(z)+\xi_{0}(z)\langle
v_{j}(z),\xi_{0}(z)\rangle_{\mathbb{C}^{n}}+\cdots\\\ &\hskip
137.77734pt+\cdots+\xi_{j-1}(z)\langle
v_{j}(z),\xi_{j-1}(z)\rangle_{\mathbb{C}^{n}})\end{array}$

almost everywhere on $\mathbb{T}.$ Notice that, for $i=0,\cdots,j-1,$ the
vectors $\xi_{i}$ and $z\mapsto\xi_{i}(z)\langle
v_{j}(z),\xi_{i}(z)\rangle_{\mathbb{C}^{n}}$ are pointwise linearly dependent
almost everywhere on $\mathbb{T}.$ Thus for all $i=0,\cdots,j-1,$

$\xi_{0}(z)\wedge\cdots\wedge\xi_{j-1}(z)\wedge\xi_{i}(z)\langle
v_{i+1}(z),\xi_{i}(z)\rangle_{\mathbb{C}^{n}}=0$

almost everywhere on $\mathbb{T}.$

Hence

$\xi_{0}(z)\wedge\cdots\wedge\xi_{j-1}(z)\wedge
v_{j}(z)=\xi_{0}(z)\wedge\cdots\wedge\xi_{j-1}(z)\wedge
x_{j}(z)\quad\text{almost everywhere on}\;\mathbb{T}.$ (5.9)

Next, we shall show that $\|\xi_{j}(z)\|_{\mathbb{C}^{n}}=1$ for almost all
$z\in\mathbb{T}.$ Recall that $h_{j}$ is the scalar outer factor of
$\xi_{1}\dot{\wedge}\xi_{2}\dot{\wedge}\cdots\dot{\wedge}\xi_{j-1}\dot{\wedge}v_{j},$
and therefore

$\begin{array}[]{cllll}|h_{j}(z)|&=\|\xi_{0}(z)\wedge\cdots\wedge\xi_{j-1}(z)\wedge
v_{j}(z)\|_{\wedge^{j+1}\mathbb{C}^{n}}=\|\xi_{0}(z)\wedge\cdots\wedge\xi_{j-1}(z)\wedge
x_{j}(z)\|_{\wedge^{j+1}\mathbb{C}^{n}}\\\ \end{array}$

almost everywhere on $\mathbb{T}.$

By the inductive hypothesis, $\\{\xi_{0}(z),\cdots,\xi_{j-1}(z)\\}$ is an
orthonormal set in $\mathbb{C}^{n}$ for almost all $z\in\mathbb{T},$ hence, by
Lemma 3.15,

$\displaystyle|h_{j}(z)|$
$\displaystyle=\|\xi_{0}(z)\wedge\cdots\wedge\xi_{j-1}(z)\wedge
x_{j}(z)\|_{\wedge^{j+1}\mathbb{C}^{n}}\vspace{2ex}$
$\displaystyle=\|x_{j}(z)-\sum\limits_{i=0}^{j-1}\langle
x_{j}(z),\xi_{i}(z)\rangle\xi_{i}(z)\|_{\mathbb{C}^{n}}\vspace{2ex}$
$\displaystyle=\|x_{j}(z)\|_{\mathbb{C}^{n}}\;\text{almost everywhere
on}\;\mathbb{T}.$ (5.10)

Thus

$\|\xi_{j}(z)\|_{\mathbb{C}^{n}}=\frac{\|x_{j}(z)\|_{\mathbb{C}^{n}}}{|h_{j}(z)|}=1$

almost everywhere on $\mathbb{T},$ and hence, by induction on $j,$
$\\{\xi_{0}(z),\cdots,\xi_{j-1}(z),\xi_{j}(z)\\}$ is an orthonormal set in
$\mathbb{C}^{n}$ for almost all $z\in\mathbb{T},$ and for all integers
$j=0,\dots,r-1.$

Next, we will prove inductively that the set
$\\{\bar{\eta}_{i}(z)\\}_{i=0}^{j},$ defined in equations (4.41), is
orthonormal. For $i=0,$ by equation (5.1), we have

$\|\bar{\eta}_{0}(z)\|_{\mathbb{C}^{m}}=1\;\text{almost everywhere
on}\;\mathbb{T}.$ (5.11)

Let $T_{1}$ be given by equation (4.24). $T_{1}$ is assumed to be a compact
operator, and if $T_{1}\neq 0,$ there exist $v_{1}\in
H^{2}(\mathbb{D},\mathbb{C}^{n})$ and $w_{1}\in
H^{2}(\mathbb{D},\mathbb{C}^{m})^{\perp}$ such that
$(\xi_{0}\dot{\wedge}v_{1},\bar{\eta}_{0}\dot{\wedge}w_{1})$ is a Schmidt pair
corresponding to $\|T_{1}\|=t_{1}.$ Suppose $h_{1}$ is the scalar outer factor
of $\xi_{0}\dot{\wedge}v_{1}.$ Let

$y_{1}=(I_{m}-\bar{\eta}_{0}\eta_{0}^{T})w_{1}=w_{1}-\bar{\eta}_{0}\eta_{0}^{T}w_{1}$
(5.12)

and let

$\eta_{1}(z)=\frac{\bar{z}\bar{y}_{1}(z)}{h_{1}(z)}\quad\text{almost
everywhere on}\;\mathbb{T}.$

Then,

$\bar{\eta}_{1}(z)=\frac{zy_{0}(z)}{\bar{h}_{1}(z)}=\frac{zw_{1}(z)}{\bar{h}_{1}(z)}-\frac{z\bar{\eta}_{0}(z)\eta_{0}^{T}(z)w_{1}(z)}{\bar{h}_{1}(z)}\quad\text{almost
everywhere on}\;\mathbb{T}.$

By equation (5.11), $\left\|\bar{\eta}_{0}(z)\right\|_{\mathbb{C}^{m}}=1$
almost everywhere on $\mathbb{T}.$ Hence

$\begin{array}[]{cllllll}\langle\bar{\eta}_{1}(z),\bar{\eta}_{0}(z)\rangle_{\mathbb{C}^{m}}&=\eta_{0}^{T}(z)\bar{\eta}_{1}(z)\vspace{2ex}\\\
&=\displaystyle\frac{z\eta_{0}^{T}(z)w_{1}(z)}{\bar{h}_{1}(z)}-\frac{z\eta_{0}^{T}(z)\bar{\eta}_{0}(z)\eta_{0}^{T}(z)w_{1}(z)}{\bar{h}_{1}(z)}\vspace{2ex}\\\
&=\displaystyle\frac{z\eta_{0}^{T}(z)w_{1}(z)}{\bar{h}_{1}(z)}-\frac{z\langle\bar{\eta}_{0}(z),\bar{\eta}_{0}(z)\rangle_{\mathbb{C}^{m}}\eta_{0}^{T}(z)w_{1}(z)}{\bar{h}_{1}(z)}\vspace{2ex}\\\
&=\displaystyle\frac{z\eta_{0}^{T}(z)w_{1}(z)}{\bar{h}_{1}(z)}-\frac{z\eta_{0}^{T}(z)w_{1}(z)}{\bar{h}_{1}(z)}\vspace{2ex}\\\
&=0\quad\text{almost everywhere on}\;\mathbb{T}.\par\end{array}$

Recall that $h_{1}$ is the scalar outer factor of $\xi_{0}\dot{\wedge}v_{1}.$
By equation (5.6) and Proposition 9.13,

$\|x_{1}(z)\|_{\mathbb{C}^{n}}=\|y_{1}(z)\|_{\mathbb{C}^{m}}=|h_{1}(z)|$

almost everywhere on $\mathbb{T},$ thus

$\|\bar{\eta}_{1}(z)\|_{\mathbb{C}^{m}}=\frac{\|zy_{1}(z)\|_{\mathbb{C}^{m}}}{|\bar{h}_{1}(z)|}=1\quad\text{almost
everywhere on}\;\mathbb{T}.$

Consequently, $\\{\bar{\eta}_{0}(z),\bar{\eta}_{1}(z)\\}$ is an orthonormal
set in $\mathbb{C}^{m}$ for almost every $z\in\mathbb{T}.$ Hence (iii) of
Proposition 5.1 holds for $\\{\bar{\eta}_{i}\\}_{i=0}^{j}$ in the case $j=1.$

Recursive step: Suppose the entities in equations (4.33) have been constructed
and have the stated properties. Since by the inductive hypothesis $T_{j}$ is a
compact operator, there exist

$v_{j}\in H^{2}(\mathbb{D},\mathbb{C}^{n}),\quad w_{j}\in
H^{2}(\mathbb{D},\mathbb{C}^{m})^{\perp}$

such that

$(\xi_{1}\dot{\wedge}\xi_{2}\dot{\wedge}\cdots\dot{\wedge}\xi_{j-1}\dot{\wedge}v_{j},\bar{\eta}_{0}\dot{\wedge}\bar{\eta}_{1}\dot{\wedge}\cdots\dot{\wedge}\bar{\eta}_{j-1}\dot{\wedge}w_{j})$

is a Schmidt pair for $T_{j}$ corresponding to $\|T_{j}\|=t_{j}.$ We have
proved above that

$\xi_{0}\dot{\wedge}\cdots\dot{\wedge}\xi_{j-1}\dot{\wedge}v_{j}\in
H^{2}(\mathbb{D},\wedge^{j+1}\mathbb{C}^{n}).$

Let $h_{j}$ be the scalar outer factor of
$\xi_{0}\dot{\wedge}\xi_{1}\dot{\wedge}\cdots\dot{\wedge}\xi_{j-1}\dot{\wedge}v_{j}.$
We define

$y_{j}=(I_{m}-\bar{\eta}_{0}\eta_{0}^{T}-\cdots-\bar{\eta}_{j-1}\eta_{j-1}^{T})w_{j}$

and

$\bar{\eta}_{j}=\frac{zy_{j}}{\bar{h}_{j}}$ (5.13)

Let us show that $\\{\bar{\eta}_{0}(z),\dots,\bar{\eta}_{j}(z)\\}$ is an
orthonormal set in $\mathbb{C}^{m}$ almost everywhere on $\mathbb{T}.$ We have

$\bar{\eta}_{j}=\frac{zw_{j}}{\bar{h}_{j}}-\cdots-\frac{{z}\bar{\eta}_{j-1}\eta_{j-1}^{T}w_{j}}{\bar{h}_{j}}$

and so, for $i=0,\dots,j-1,$

$\begin{array}[]{cllllll}\langle\bar{\eta}_{j}(z),\bar{\eta}_{i}(z)\rangle_{\mathbb{C}^{m}}&=\eta_{i}^{T}(z)\bar{\eta}_{j}(z)\vspace{2ex}\\\
&=\displaystyle\frac{z\eta_{i}^{T}(z)w_{j}(z)}{\bar{h}_{j}(z)}-\cdots-\frac{z\eta_{i}^{T}(z)\bar{\eta}_{j}(z)\eta_{j}^{T}(z)w_{j}(z)}{\bar{h}_{j}(z)}\end{array}$

almost everywhere on $\mathbb{T}.$

Notice that, by the inductive hypothesis, for $i,k=0,\dots,j-1$ and for almost
all $z\in\mathbb{T},$

$\eta_{i}^{T}(z)\bar{\eta}_{k}(z)=\left\\{\begin{array}[]{ll}0,\quad\text{for}\;i\neq
k\\\ 1,\quad\text{for}\;i=k\end{array}.\right.$

Hence, for $i=0,\dots,j-1,$

$\langle\bar{\eta}_{j}(z),\bar{\eta}_{i}(z)\rangle_{\mathbb{C}^{m}}=\displaystyle\frac{z\eta_{i}^{T}(z)w_{j}(z)}{\bar{h}_{j}(z)}-\displaystyle\frac{z\eta_{i}^{T}(z)w_{j}(z)}{\bar{h}_{j}(z)}=0$

almost everywhere on $\mathbb{T}.$ Thus by induction on $j$, for all integers
$j=0,\dots,r-1,$ $\\{\bar{\eta}_{0}(z),\dots,\bar{\eta}_{j}(z)\\}$ is an
orthogonal set in $\mathbb{C}^{m}$ almost everywhere on $\mathbb{T}.$

To complete the proof, we have to prove that
$\left\|\bar{\eta}_{j}(z)\right\|_{\mathbb{C}^{m}}=1$ for almost all
$z\in\mathbb{T}.$ Recall that $h_{j}$ is the scalar outer factor of
$\xi_{0}\dot{\wedge}\xi_{1}\dot{\wedge}\cdots\dot{\wedge}\xi_{j-1}\dot{\wedge}v_{j}.$
By Proposition 10.13,

$|h_{j}(z)|=\|x_{j}(z)\|_{\mathbb{C}^{n}}=\|y_{j}(z)\|_{\mathbb{C}^{m}}$

almost everywhere on $\mathbb{T},$ thus

$\left\|\bar{\eta}_{j}(z)\right\|_{\mathbb{C}^{m}}=\displaystyle\|\frac{zy_{j}(z)}{\bar{h}_{j}(z)}\|_{\mathbb{C}^{m}}=1$

almost everywhere on $\mathbb{T},$ and hence,
$\\{\bar{\eta}_{0}(z),\dots,\bar{\eta}_{j}(z)\\}$ is an orthonormal set in
$\mathbb{C}^{m}$ almost everywhere on $\mathbb{T}.$

Note that, for $j=1,\dots,r-1$,

$\displaystyle\bar{\eta}_{0}\dot{\wedge}\cdots\dot{\wedge}\bar{\eta}_{j-1}\dot{\wedge}y_{j}$
$\displaystyle=\bar{\eta}_{0}\dot{\wedge}\cdots\dot{\wedge}\bar{\eta}_{j-1}\dot{\wedge}(I_{m}-\bar{\eta}_{0}\eta_{0}^{T}-\dots-\bar{\eta}_{j-1}\eta_{j-1}^{T})w_{j}\vspace{2ex}$
$\displaystyle=\bar{\eta}_{0}\dot{\wedge}\cdots\dot{\wedge}\bar{\eta}_{j-1}\dot{\wedge}w_{j}-\sum\limits_{k=0}^{j-1}\bar{\eta}_{0}\dot{\wedge}\cdots\dot{\wedge}\bar{\eta}_{j-1}\dot{\wedge}\bar{\eta}_{k}\eta_{k}^{T}w_{j}\vspace{2ex}$
$\displaystyle=\bar{\eta}_{0}\dot{\wedge}\cdots\dot{\wedge}\bar{\eta}_{j-1}\dot{\wedge}w_{j}$
(5.14)

on account of the pointwise linear dependence of $\bar{\eta}_{k}$ and
$z\mapsto\bar{\eta}_{k}(z)\langle
w_{j}(z),\bar{\eta}_{k}(z)\rangle_{\mathbb{C}^{m}}$ almost everywhere on
$\mathbb{T}$. ∎

##  6\. The closed subspace $X_{j+1}$ of
$H^{2}(\mathbb{D},\wedge^{j+2}\mathbb{C}^{n})$

Notice that, although $x_{0}\in H^{2}(\mathbb{D},\mathbb{C}^{n})$ and
$\xi_{0}$ is inner, $x_{i}$ and $\xi_{i}$ might not be in
$H^{2}(\mathbb{D},\mathbb{C}^{n})$ in general for $i=1,\cdots,\min(m,n)-1.$
However, for every $x\in H^{2}(\mathbb{D},\mathbb{C}^{n}),$ the pointwise
wedge product

$\xi_{0}\dot{\wedge}\cdots\dot{\wedge}\xi_{j}\dot{\wedge}x$

_is_ an element of $H^{2}(\mathbb{D},\wedge^{j+2}\mathbb{C}^{n})$ as the
following proposition asserts.

######  Proposition 6.1.

Let $G\in H^{\infty}(\mathbb{D},\mathbb{C}^{m\times
n})+C(\mathbb{T},\mathbb{C}^{m\times n}),$ and let $j\leq n-1.$ Let the
vector-valued functions $\xi_{0},\xi_{1},\cdots,\xi_{j}$ be constructed after
applying steps $0,\dots,j$ of the algorithm above and be given by equations
(4.41). Then

$\xi_{0}\dot{\wedge}\dots\dot{\wedge}\xi_{j}\dot{\wedge}H^{2}(\mathbb{D},\mathbb{C}^{n})$

is a subset of $H^{2}(\mathbb{D},\wedge^{j+2}\mathbb{C}^{n}).$

###### Proof.

For $j=0,$ since $G\in H^{\infty}(\mathbb{D},\mathbb{C}^{m\times
n})+C(\mathbb{T},\mathbb{C}^{m\times n}),$ the Hankel operator $H_{G}$ is
compact. There exist $x_{0}\in H^{2}(\mathbb{D},\mathbb{C}^{n}),y_{0}\in
H^{2}(\mathbb{D},\mathbb{C}^{m})^{\perp}$ such that $(x_{0},y_{0})$ is a
Schmidt pair for the Hankel operator $H_{G}$ corresponding to the singular
value $\|H_{G}\|.$ By Lemma $\ref{2.2},$ $x_{0},y_{0}$ admit the inner-outer
factorizations

$x_{0}=\xi_{0}h_{0},\quad\bar{z}\bar{y}_{0}=\eta_{0}h_{0}$

for some inner $\xi_{0}\in H^{\infty}(\mathbb{D},\mathbb{C}^{n}),\eta_{0}\in
H^{\infty}(\mathbb{D},\mathbb{C}^{m})$ and some scalar outer $h_{0}\in
H^{2}(\mathbb{D},\mathbb{C}).$

Then, by Proposition 3.26,
$\xi_{0}\dot{\wedge}H^{2}(\mathbb{D},\mathbb{C}^{n})\subset
H^{2}(\mathbb{D},\wedge^{2}\mathbb{C}^{n}).$

Let us now consider the case where $j=1.$ By definition,

$X_{1}=\xi_{0}\dot{\wedge}H^{2}(\mathbb{D},\mathbb{C}^{n}),\quad
Y_{1}=\bar{\eta}_{0}\dot{\wedge}H^{2}(\mathbb{D},\mathbb{C}^{m})^{\perp}$

and, by the inductive hypothesis, $T_{1}\colon X_{1}\to Y_{1}$ given by
equation (4.24) is a compact operator. Suppose $\|T_{1}\|\neq 0$ and let
$(\xi_{0}\dot{\wedge}v_{1},\bar{\eta}_{0}\dot{\wedge}w_{1})$ be a Schmidt pair
corresponding to $\|T_{1}\|,$ where $v_{1}\in
H^{2}(\mathbb{D},\mathbb{C}^{n})$ and $w_{1}\in
H^{2}(\mathbb{D},\mathbb{C}^{m})^{\perp}.$ We define

$x_{1}=(I_{n}-\xi_{0}\xi_{0}^{*})v_{1}.$

Note that, by Proposition 3.26, $\xi_{0}\dot{\wedge}v_{1}\in
H^{2}(\mathbb{D},\wedge^{2}\mathbb{C}^{n}).$ Let $h_{1}\in
H^{2}(\mathbb{D},\mathbb{C})$ be the scalar outer factor of
$\xi_{0}\dot{\wedge}v_{1}\in H^{2}(\mathbb{D},\wedge^{2}\mathbb{C}^{n}).$ Then
we define

$\xi_{1}=\frac{x_{1}}{h_{1}}.$

Note that $\xi_{0}$ and $z\mapsto\xi_{0}(z)\langle
v_{1}(z),\xi_{0}(z)\rangle_{\mathbb{C}^{n}}$ are pointwise linearly dependent
on $\mathbb{D},$ since $\xi_{0}^{*}v_{1}$ is a mapping from $\mathbb{D}$ to
$\mathbb{C}.$ Thus, for all $x\in H^{2}(\mathbb{D},\mathbb{C}^{n})$ and
$z\in\mathbb{D},$ we have

$(\xi_{0}\dot{\wedge}\xi_{1}\dot{\wedge}x)(z)=\displaystyle\xi_{0}(z)\wedge\xi_{1}(z)\wedge
x(z)\vspace{2ex}\\\
=\displaystyle\xi_{0}(z)\wedge\frac{x_{1}(z)}{h_{1}(z)}\wedge x(z),$

and by substituting the value of $x_{1}$, we find

$\begin{array}[]{cllll}&\displaystyle\xi_{0}(z)\wedge\frac{x_{1}(z)}{h_{1}(z)}\wedge
x(z)\vspace{2ex}\\\
&=\displaystyle\frac{1}{h_{1}(z)}\xi_{0}(z)\wedge(v_{1}(z)-\xi_{0}(z)\xi_{0}(z)^{*}v_{1}(z))\wedge
x(z)\vspace{2ex}\\\ &=\displaystyle\frac{1}{h_{1}(z)}\xi_{0}(z)\wedge
v_{1}(z)\wedge
x(z)-\displaystyle\frac{1}{h_{1}(z)}\xi_{0}(z)\wedge\xi_{0}(z)\xi_{0}(z)^{*}v_{1}(z)\wedge
x(z)\vspace{2ex}\\\
&=\left(\displaystyle\frac{1}{h_{1}}\xi_{0}\dot{\wedge}v_{1}\dot{\wedge}x\right)(z).\end{array}$

Note that $v_{1}\in H^{2}(\mathbb{D},\mathbb{C}^{n}),$ $\xi_{0}\in
H^{\infty}(\mathbb{D},\mathbb{C}^{n})$ and $h_{1}\in
H^{2}(\mathbb{D},\mathbb{C})$ is the scalar outer factor of
$\xi_{0}\dot{\wedge}v_{1}$. By Proposition 3.24, for every $x\in
H^{2}(\mathbb{D},\mathbb{C}^{n}),$

$\frac{1}{h_{1}}\xi_{0}\dot{\wedge}v_{1}\dot{\wedge}x$

is analytic on $\mathbb{D}.$ By Proposition 3.28, since $\xi_{0}$ and
$\xi_{1}$ are pointwise orthogonal almost everywhere on $\mathbb{T},$

$\|\xi_{0}\dot{\wedge}\xi_{1}\dot{\wedge}x\|_{L^{2}(\mathbb{T},\wedge^{3}\mathbb{C}^{n})}<\infty.$

Hence,

$\xi_{0}\dot{\wedge}\xi_{1}\dot{\wedge}x\in\frac{1}{h_{1}}\xi_{0}\dot{\wedge}v_{1}\dot{\wedge}H^{2}(\mathbb{D},\mathbb{C}^{n})\subset
H^{2}(\mathbb{D},\wedge^{3}\mathbb{C}^{n}).$

Recursive step: suppose we have constructed vector-valued functions
$\xi_{0},\dots,\xi_{j-1},$ $\eta_{0},\dots,\eta_{j-1},$ spaces $X_{j},Y_{j}$
and a compact operator $T_{j}\colon X_{j}\to Y_{j}$ after applying steps
$0,\dots,j$ of the algorithm from Subsection 4.2 satisfying

$\xi_{0}\dot{\wedge}\cdots\dot{\wedge}\xi_{j-1}\dot{\wedge}H^{2}(\mathbb{D},\mathbb{C}^{n})\subset
H^{2}(\mathbb{D},\wedge^{j+1}\mathbb{C}^{n}).$ (6.1)

Since $T_{j}$ is a compact operator, there exist vector-valued functions
$v_{j}\in H^{2}(\mathbb{D},\mathbb{C}^{n}),w_{j}\in
H^{2}(\mathbb{D},\mathbb{C}^{m})^{\perp}$ such that

$(\xi_{0}\dot{\wedge}\cdots\dot{\wedge}\xi_{j-1}\dot{\wedge}v_{j},\overline{\eta}_{0}\dot{\wedge}\dots\dot{\wedge}\overline{\eta}_{j-1}\dot{\wedge}w_{j})$

is a Schmidt pair for $T_{j}$ corresponding to $\|T_{j}\|.$ Define

$x_{j}=(I_{n}-\xi_{0}\xi_{0}^{*}-\dots-\xi_{j-1}\xi_{j-1}^{*})v_{j}.$ (6.2)

By assumption,
$\xi_{0}\dot{\wedge}\cdots\dot{\wedge}\xi_{j-1}\dot{\wedge}v_{j}$ lies in
$H^{2}(\mathbb{D},\wedge^{j+1}\mathbb{C}^{n}).$ Let $h_{j}\in
H^{2}(\mathbb{D},\mathbb{C})$ be the scalar outer factor of
$\xi_{0}\dot{\wedge}\cdots\dot{\wedge}\xi_{j-1}\dot{\wedge}v_{j}.$ Define
$\xi_{j}=\frac{x_{j}}{h_{j}}.$ Note that $\xi_{i}(z)$ and
$z\mapsto\xi_{i}(z)\langle v_{j}(z),\xi_{i}(z)\rangle_{\mathbb{C}^{n}}$ are
pointwise linearly dependent on $\mathbb{D}$ for $i=0,\dots,j-1.$ Thus, for
all $x\in H^{2}(\mathbb{D},\mathbb{C}^{n})$ and all $z\in\mathbb{D},$

$\displaystyle(\xi_{0}\dot{\wedge}\cdots\dot{\wedge}\xi_{j-1}\dot{\wedge}\xi_{j}\dot{\wedge}x)(z)$
$\displaystyle=(\xi_{0}\dot{\wedge}\cdots\dot{\wedge}\xi_{j-1}\dot{\wedge}\frac{x_{j}}{h_{j}}\dot{\wedge}x)(z)\vspace{2ex}$
$\displaystyle=\xi_{0}(z)\wedge\cdots\wedge\xi_{j-1}(z)\wedge\frac{1}{h_{j}(z)}\bigg{(}v_{j}(z)-\xi_{0}(z)\xi_{0}^{*}(z)v_{j}(z)-\cdots\bigg{.}$
$\displaystyle\hskip
129.16626pt-\bigg{.}\xi_{j-1}(z)\xi_{j-1}^{*}(z)v_{j}(z)\bigg{)}\wedge
x(z)\vspace{2ex}$
$\displaystyle=\xi_{0}(z)\wedge\cdots\wedge\xi_{j-1}(z)\wedge\frac{1}{h_{j}(z)}v_{j}(z)\wedge
x(z)$
$\displaystyle-\sum\limits_{i=0}^{j-1}\xi_{0}(z)\wedge\dots\wedge\xi_{j-1}(z)\wedge\xi_{i}(z)\xi_{i}^{*}(z)v_{j}(z)\wedge
x(z)\vspace{2ex}$
$\displaystyle=\left(\frac{1}{h_{j}}\xi_{0}\dot{\wedge}\cdots\dot{\wedge}\xi_{j-1}\dot{\wedge}v_{j}\dot{\wedge}x\right)(z).$
(6.3)

Recall that, for $i=0,\dots,j-1,$ by the algorithm from Subsection $3.2.1,$

$x_{i}=(I_{n}-\xi_{0}\xi_{0}^{*}-\dots-\xi_{i-1}\xi_{i-1}^{*})v_{i}$

and

$\xi_{i}=\frac{x_{i}}{h_{i}}.$

By equation (6.3), for all $z\in\mathbb{D},$

$(\xi_{0}\dot{\wedge}\cdots\dot{\wedge}\xi_{j-1}\dot{\wedge}\xi_{j}\dot{\wedge}x)(z)=\left(\frac{1}{h_{j}}\xi_{0}\dot{\wedge}\cdots\dot{\wedge}\xi_{j-1}\dot{\wedge}v_{j}\dot{\wedge}x\right)(z).$

Substituting $\frac{x_{i}}{h_{i}}$ for $\xi_{i}$ in the latter equation, where
$x_{i}$ are given by equation (6.2) for $i=1,\dots,j-1,$ we obtain

$(\xi_{0}\dot{\wedge}\cdots\dot{\wedge}\xi_{j-1}\dot{\wedge}\xi_{j}\dot{\wedge}x)(z)=\left(\frac{1}{h_{1}}\frac{1}{h_{2}}\cdots\frac{1}{h_{j}}\xi_{0}\dot{\wedge}v_{1}\dot{\wedge}\cdots\dot{\wedge}v_{j-1}\dot{\wedge}v_{j}\dot{\wedge}x\right)(z),\;z\in\mathbb{D}$

on account of the pointwise linear dependence of $\xi_{k}$ and
$z\mapsto\langle v_{k}(z),\xi_{i}(z)\rangle_{\mathbb{C}^{n}}\xi_{k}(z)$ on
$\mathbb{D},$ for $k=0,\dots,j.$ By Proposition 3.24, for every $x\in
H^{2}(\mathbb{D},\mathbb{C}^{n}),$

$\frac{1}{h_{1}}\frac{1}{h_{2}}\cdots\frac{1}{h_{j}}\xi_{0}\dot{\wedge}v_{1}\dot{\wedge}\cdots\dot{\wedge}v_{j}\dot{\wedge}x$

is analytic on $\mathbb{D}.$ By Proposition 3.28, since
$\xi_{0},\xi_{1},\dots,\xi_{j}$ are pointwise orthogonal on $\mathbb{T},$

$\|\xi_{0}\dot{\wedge}\xi_{1}\dot{\wedge}\cdots\xi_{j}\dot{\wedge}x\|_{L^{2}(\mathbb{T},\wedge^{j+2}\mathbb{C}^{n})}<\infty.$

Thus, for every $x\in H^{2}(\mathbb{D},\mathbb{C}^{n}),$

$\xi_{0}\dot{\wedge}\xi_{1}\dot{\wedge}\cdots\dot{\wedge}\xi_{j}\dot{\wedge}x\in
H^{2}(\mathbb{D},\wedge^{j+2}\mathbb{C}^{n})$

and the claim has been proved.

∎

######  Proposition 6.2.

In the notation of Proposition 6.1,

$\xi_{0}\dot{\wedge}\dots\dot{\wedge}\xi_{j}\dot{\wedge}H^{2}(\mathbb{D},\mathbb{C}^{n})$

is a closed subspace of $H^{2}(\mathbb{D},\wedge^{j+2}\mathbb{C}^{n}).$

###### Proof.

Let us first show that $\xi_{0}\dot{\wedge}H^{2}(\mathbb{D},\mathbb{C}^{n})$
is a closed subspace of $H^{2}(\mathbb{D},\wedge^{2}\mathbb{C}^{n}).$ Observe
that, by Proposition 3.26,
$\xi_{0}\dot{\wedge}H^{2}(\mathbb{D},\mathbb{C}^{n})\subset
H^{2}(\mathbb{D},\wedge^{2}\mathbb{C}^{n}).$ Let

$\Xi_{0}=\\{f\in H^{2}(\mathbb{D},\mathbb{C}^{n}):\langle
f(z),\xi_{0}(z)\rangle_{\mathbb{C}^{n}}=0\quad\text{almost everywhere
on}\;\mathbb{T}\\}.$

Consider a vector-valued function $w\in H^{2}(\mathbb{D},\mathbb{C}^{n}).$ For
all $z\in\mathbb{D},$ we may write $w$ as

$w(z)=w(z)-\langle w(z),\xi_{0}(z)\rangle_{\mathbb{C}^{n}}\xi_{0}(z)+\langle
w(z),\xi_{0}(z)\rangle_{\mathbb{C}^{n}}\xi_{0}(z).$

Then, for all $w\in H^{2}(\mathbb{D},\mathbb{C}^{n})$ and for all
$z\in\mathbb{D},$

$\begin{array}[]{clllll}(\xi_{0}\dot{\wedge}w)(z)&=\xi_{0}(z)\wedge\big{(}w(z)-\langle
w(z),\xi_{0}(z)\rangle_{\mathbb{C}^{n}}\xi_{0}(z)+\langle
w(z),\xi_{0}(z)\rangle_{\mathbb{C}^{n}}\xi_{0}(z)\big{)}\vspace{2ex}\\\
&=\xi_{0}(z)\wedge\big{(}w(z)-\langle
w(z),\xi_{0}(z)\rangle_{\mathbb{C}^{n}}\xi_{0}(z)\big{)}\end{array}$

on account of the the pointwise linear dependence of $\xi_{0}$ and
$z\mapsto\langle w(z),\xi_{0}(z)\rangle_{\mathbb{C}^{n}}\xi_{0}(z)$ on
$\mathbb{D}.$

Note that

$w(z)-\langle w(z),\xi_{0}(z)\rangle_{\mathbb{C}^{n}}\xi_{0}(z)\in\Xi_{0},$

thus

$\xi_{0}\dot{\wedge}H^{2}(\mathbb{D},\mathbb{C}^{n})\subset\xi_{0}\dot{\wedge}\Xi_{0}.$

By Corollary 3.34, $\Xi_{0}$ is a closed subspace of
$H^{2}(\mathbb{D},\mathbb{C}^{n}),$ hence

$\xi_{0}\dot{\wedge}H^{2}(\mathbb{D},\mathbb{C}^{n})\supset\xi_{0}\dot{\wedge}\Xi_{0},$

and so,

$\xi_{0}\dot{\wedge}H^{2}(\mathbb{D},\mathbb{C}^{n})=\xi_{0}\dot{\wedge}\Xi_{0}.$

Consider the mapping

$C_{\xi_{0}}\colon\Xi_{0}\to\xi_{0}\dot{\wedge}\Xi_{0}$

given by

$C_{\xi_{0}}w=\xi_{0}\dot{\wedge}w$

for all $w\in\Xi_{0}.$ Notice that, by Proposition 5.1,
$\|\xi_{0}(e^{i\theta})\|_{\mathbb{C}^{n}}^{2}=1$ for almost every
$e^{i\theta}\in\mathbb{T}.$ Therefore, for any $w\in\Xi_{0},$ we have