# Matrix-valued truncated Toeplitz operators: unbounded symbols, kernels and
equivalence after extension

Ryan O’Loughlin  School of Mathematics
University of Leeds
Leeds
LS2 9JT
U.K<EMAIL_ADDRESS>
###### Abstract.

This paper studies matrix-valued truncated Toeplitz operators, which are a
vectorial generalisation of truncated Toeplitz operators. It is demonstrated
that, although there exist matrix-valued truncated Toeplitz operators without
a matrix symbol in $L^{p}$ for any $p\in(2,\infty]$, there is a wide class of
matrix-valued truncated Toeplitz operators which possess a matrix symbol in
$L^{p}$ for some $p\in(2,\infty]$. In the case when the matrix-valued
truncated Toeplitz operator has a symbol in $L^{p}$ for some $p\in(2,\infty]$,
an approach is developed which bypasses some of the technical difficulties
which arise when dealing with problems concerning matrix-valued truncated
Toeplitz operators with unbounded symbols. Using this new approach, two new
notable results are obtained. The kernel of the matrix-valued truncated
Toeplitz operator is expressed as an isometric image of an $S^{*}$-invariant
subspace. Also, a Toeplitz operator is constructed which is equivalent after
extension to the matrix-valued truncated Toeplitz operator. In a different yet
overlapping vein, it is also shown that multidimensional analogues of the
truncated Wiener-Hopf operators are unitarily equivalent to certain matrix-
valued truncated Toeplitz operators.

###### Key words and phrases:

Hardy space, Toeplitz operator, matrix-valued functions, model space.

###### 1991 Mathematics Subject Classification:

30H10, 47B35, 47A56, 46E20.

##  1\. Introduction

The purpose of this paper is to study the matrix-valued truncated Toeplitz
operator (abbreviated to MTTO). The MTTO is a vectorial generalisation of the
truncated Toeplitz operator. We make a powerful observation, that when
studying a given property of a MTTO it is often convenient to initially modify
the MTTO by changing its codomain (in a natural way), then one can deduce
results about the MTTO from the modified MTTO. This approach allows us to
tackle problems which were previously out of reach concerning MTTOs with
unbounded symbols. In particular for a MTTO which has a matrix symbol with
each entry lying in $L^{p}$ for $p\in(2,\infty]$, we describe the kernel of
the MTTO as an isometric image of a $S^{*}$-invariant subspace and we also
find a new form of Toeplitz operator which is equivalent after extension to
the MTTO. We emphasise that although the results in this paper are for MTTOs,
all the results are new even in the scalar case of the truncated Toeplitz
operator. As the study of MTTOs is a recent endeavour, we devote the final
section of this paper to discuss some applications of MTTOs to integral
equations.

The spaces $H^{p}$ and $L^{p}$ will be defined on the unit circle,
$\mathbb{T}$, where $1\leqslant p\leqslant\infty$. We write $(H^{p})^{n}$
(respectively $(L^{p})^{n})$ to mean the column vector of length $n$ with each
entry taking values in $H^{p}$ (respectively $L^{p}$). Background theory on
the classical Hardy space $H^{p}$ can be found in [14, 22]. For
$1<p\leqslant\infty$, we denote $L^{(p,n\times n)}$ to be the space of
$n$-by-$n$ matrices with each entry taking values in $L^{p}$. We make an
analogous definition for $H^{(p,n\times n)}$. For a matrix $M\in
L^{(\infty,n\times n)}$ the adjoint of $M\in L^{(\infty,n\times n)}$ is
denoted $M^{*}$. A $n$-by-$n$ matrix inner function $\Theta$ is an element of
$H^{(\infty,n\times n)}$ such that for almost every $z\in\mathbb{T}$, we have
$\Theta(z)$ is a unitary matrix. Throughout we use $\Theta$ to denote an
$n$-by-$n$ inner function.

For $f\in(H^{2})^{n}$, the backward shift, $S^{*}$, is defined by
$S^{*}(f)=\frac{f-f(0)}{z}$. We know from the Beurling-Lax Theorem that
$\Theta(H^{2})^{n}$ is a shift invariant subspace. Therefore using
orthogonality one can see that the (matricial) model space,
$K_{\Theta}:=\Theta\overline{(H^{2}_{0})^{n}}\cap(H^{2})^{n}$ is
$S^{*}$-invariant. If we define

$(\overline{H^{2}_{0}})^{n}:=\\{\overline{f}:f\in(H^{2})^{n}\text{ and
}f(0)\text{ is the zero vector}\\},$

then we have the orthogonal decompositions

$(H^{2})^{n}=K_{\Theta}\oplus\Theta(H^{2})^{n},$

and

$(L^{2})^{n}=(\overline{H^{2}_{0}})^{n}\oplus(H^{2})^{n}.$

This in turn gives us orthogonal projections
$P_{+}:(L^{2})^{n}\to(H^{2})^{n}$, $P_{\Theta}:(L^{2})^{n}\to K_{\Theta}$ and
$Q_{\Theta}:(L^{2})^{n}\to\Theta(H^{2})^{n}$. For $k>n$ we write
$P_{n}:(H^{2})^{k}\to(H^{2})^{n}$ to mean the projection onto the first $n$
coordinates.

Matrix-valued truncated Toeplitz operators were first defined in [21] as a
natural generalisation of truncated Toeplitz operators. They have further been
studied in [20, 19]. We define the MTTO as follows. Let $G\in L^{(2,n\times
n)}$, consider the map

$f\mapsto P_{\Theta}(Gf),$ (1.1)

defined on $K_{\Theta}\cap(H^{\infty})^{n}$. It is shown in Section 4 of [21]
that $K_{\Theta}\cap(H^{\infty})^{n}$ is dense in $K_{\Theta}$, so in the case
when (1.1) is bounded this uniquely defines an operator $K_{\Theta}\to
K_{\Theta}$, which we denote $A_{G}^{\Theta}$ and call a matrix-valued
truncated Toeplitz operator (MTTO). We note that with this definition, all
MTTOs are implicitly bounded. We call $G$ the symbol of the MTTO, and we note
that if we have the additional assumption that $G\in L^{(\infty,n\times n)}$
then (1.1) can always be extended to a bounded operator. In the case when
$n=1$, we recover the well known bounded truncated Toeplitz operator.

We say $\Theta$ is pure if $||\Theta(0)||<1$. Matrix valued truncated Toeplitz
operators with a pure inner function appear naturally in the Sz.-Nagy and
Foias model theory for Hilbert space contractions. In particular, every
bounded linear operator between two Hilbert spaces $T:H_{1}\to H_{2}$ with
defect indices $(n,n)$ and with the property that for all $h\in H_{1}$,
$T^{*n}(h)\to 0$ (S.O.T) is unitarily equivalent to $A_{z}^{\Theta}$ for some
$n$-by-$n$ inner function $\Theta$. See Section 2, page 33, of [18] for a more
detailed discussion. Although this is one of the main motivations for interest
in the truncated Toeplitz operator (which is relevant when the defect indices
are $(1,1)$), there has been very little research done in to the general case
of the MTTO.

MTTOs also appear naturally in [6], where a dual band Toeplitz operator
(defined in [6]) is unitarily equivalent to a MTTO with a diagonal inner
function $\Theta$. Dual band Toeplitz operators have applications in speech
processing and signal transmission. We refer the reader to [6] and further
references thereafter for a detailed discussion.

Let $\theta\in H^{2}$ be a scalar inner function and let $\phi\in H^{\infty}$.
We denote the Hankel operator with symbol $g\in L^{\infty}$, by
$H_{g}:H^{2}\to\overline{H^{2}_{0}}$. This is defined by
$H_{\psi}(p)=P_{-}(\psi p)$, where $P_{-}:L^{2}\to\overline{H^{2}_{0}}$ is the
orthogonal projection. It is well known that many questions about Hankel
operators can be phrased in terms of truncated Toeplitz operators with an
analytic symbol. In particular the relation

$A_{\phi}^{\theta}=\theta H_{\overline{\theta}\phi}|_{K_{\theta}}$

has long been exploited. Making natural generalisations so that $\Psi\in
H^{(\infty,n\times n)}$ and $H:(H^{2})^{n}\to(\overline{H^{2}_{0}})^{n}$ is a
Hankel operator on the vector-valued Hardy space, we can also write the
relation

$A_{\Psi}^{\Theta}=\Theta H_{{\Theta}^{*}{\Psi}}|_{K_{\Theta}}.$

So, just as is true in the scalar case, the matricial Hankel operator and MTTO
are fundamentally linked. This has applications in minimisation problems and
Nehari’s Theorem, see Section 2.2 of [25].

MTTOs also have a link to complex symmetric operators. Direct sums of
truncated Toeplitz operators seem to play a role in some sort of model theory
for complex symmetric operators (see Section 9 of [16]). However, direct sums
of truncated Toeplitz operators are in fact special cases of MTTOs with
diagonal symbols and diagonal inner functions. So instead of considering what
role direct sums of truncated Toeplitz operators play in the theory of complex
symmetric operators, it may be more natural to consider what role MTTOs play
in the theory of complex symmetric operators.

In Section 2 we make some key observations which allow us to define the
modified MTTO. The modified MTTO turns out to be a crucial tool in later
sections, particularly when we are trying to understand properties of MTTOs
which do not possess a bounded symbol.

The kernel of a Toeplitz operator is easily checked to be nearly
$S^{*}$-invariant, and this has long been exploited to study the kernels of
vector-valued and scalar-valued Toeplitz operators [17] [10]. The kernel of a
(scalar) truncated Toeplitz operator with a bounded symbol was shown to be
nearly $S^{*}$-invariant with defect 1 in [23]. Then consequently, this
property was used to give a decomposition theorem for the kernel of a
truncated Toeplitz operator with a bounded symbol. In Section 3 we use the
modified MTTO to expand on previous studies to include the case where the
symbol of the operator is not bounded. In particular, we decompose the kernel
of a MTTO which possesses a symbol in $L^{(p,n\times n)}$ for $p\in(2,\infty)$
in to an isometric image of an $S^{*}$-invariant subspace.

In Section 4 we use the modified MTTO as a transitional device, which allows
us to find an operator which is equivalent after extension to a MTTO which has
a symbol in $L^{(p,n\times n)}$ for $p\in(2,\infty)$. Specifically, we first
find a Toeplitz operator which is equivalent after extension to the modified
MTTO, and then we change the codomain of this Toeplitz operator (as we have
done when defining the modified MTTO) to produce an operator which is
equivalent after extension to a MTTO which has a symbol in $L^{(p,n\times n)}$
for $p\in(2,\infty)$. This is a generalisation of the results in Section 6 of
[8], where the authors construct a Toeplitz operator which is equivalent after
extension to a truncated Toeplitz operator with a symbol in $L^{\infty}$.

In Section 5 we show there is a unitary equivalence between certain MTTOs and
matricial truncated Wiener-Hopf operators (which may also be called matricial
convolution operators on finite intervals). We show that we can use the theory
developed around the modified MTTO to test the continuity of matricial
truncated Wiener-Hopf operators. Finally, we show how the matricial truncated
Wiener-Hopf operators are naturally encountered when finding the solution to
multi input, multi output linear systems.

##  2\. The modified matrix-valued truncated Toeplitz operator

The Riesz projections $P_{q+}:(L^{q})^{n}\to(H^{q})^{n}$ and
$P_{q-}:=I-P_{q+}:(L^{q})^{n}\to(\overline{H^{q}_{0}})^{n}$ are bounded when
$q\in(1,\infty)$. Furthermore for $n=1$, $P_{q+}$ can be expressed by

$P_{q+}(f)(z)=\int_{\mathbb{T}}\frac{f(\zeta)}{1-\overline{\zeta}z}dm(\zeta),$

which is independent of $q\in(1,\infty)$. Which means we can deduce the
following;

######  Lemma 2.1.

For $q\in(1,2)$ and $f\in(L^{2})^{n}$, we have $P_{q+}(f)=P_{2+}(f)$ and
$P_{q-}(f)=P_{2-}(f)$.

Using the projection $P_{\Theta,q}:=P_{q+}\Theta P_{q-}\Theta^{*}$ we can
decompose, as we have done in the case of $q=2$, $(H^{q})^{n}$ as

$(H^{q})^{n}=K_{\Theta}^{q}\oplus\Theta(H^{q})^{n},$

where $K_{\Theta}^{q}=\Theta\overline{(H^{q}_{0})^{n}}\cap(H^{q})^{n}$ and

$(L^{q})^{n}=(\overline{H^{q}_{0}})^{n}\oplus
K_{\Theta}^{q}\oplus\Theta(H^{q})^{n}.$

As the orthogonal projection $P_{\Theta}=P_{\Theta,2}:L^{2}\to K_{\Theta}^{2}$
can also be written $P_{\Theta,2}=P_{2+}\Theta P_{2-}\Theta^{*}$ we can
conclude the following;

######  Lemma 2.2.

For $q\in(1,2)$ and $f\in(L^{2})^{n}$, we have

$P_{\Theta,2}(f)=P_{\Theta,q}(f).$

We can also deduce that for $Q_{\Theta,q}:=\Theta
P_{q+}\Theta^{*}=P_{q+}-P_{\Theta,q}:(L^{q})^{n}\to\Theta(H^{q})^{n}$ we have
the following;

######  Lemma 2.3.

For $q\in(1,2)$ and $f\in(L^{2})^{n}$, we have

$Q_{\Theta,2}(f)=Q_{\Theta,q}(f).$

We will use Lemmas 2.1, 2.2 and 2.3 freely throughout. When we are clearly
working in the context of $(L^{2})^{n}$, we will just write $P_{\Theta}$,
respectively $Q_{\Theta}$, as opposed to $P_{\Theta,2}$, respectively
$Q_{\Theta,2}$.

######  Definition 2.4.

Let $p\in(2,\infty]$, let $G\in L^{(p,n\times n)}$ and let
$\frac{1}{2}+\frac{1}{p}=\frac{1}{q}$. Then the bounded operator
$\tilde{A_{G}^{\Theta}}:K_{\Theta}^{2}\to K_{\Theta}^{q}$ is defined by
$\tilde{A_{G}^{\Theta}}(f)=P_{\Theta,q}(Gf)$. We call the operator
$\tilde{A_{G}^{\Theta}}$ the modified matrix-valued truncated Toeplitz
operator.

######  Remark 2.5.

Although $\tilde{A_{G}^{\Theta}}$ does have a specific $p$ dependence
depending on which space $G$ lies in, we will omit this from our notation.

The following proposition shows that when $A_{G}^{\Theta}:K_{\Theta}^{2}\to
K_{\Theta}^{2}$ is a MTTO, up to a change in codomain, $A_{G}^{\Theta}$ and
$\tilde{A_{G}^{\Theta}}$ are actually the same operator. In the next section
we will exploit this link to study the kernel of $A_{G}^{\Theta}$.

######  Proposition 2.6.

Let the assumptions of Definition 2.4 hold and let
$A_{G}^{\Theta}:K_{\Theta}^{2}\to K_{\Theta}^{2}$ be a MTTO. Then for each
$f\in K_{\Theta}^{2}$ we have $\tilde{A_{G}^{\Theta}}(f)=A_{G}^{\Theta}(f)$.

###### Proof.

For a given $f\in K_{\Theta}^{2}$, let $f_{n}\in
K_{\Theta}^{2}\cap(H^{\infty})^{n}$ be such that
$f_{n}\overset{(L^{2})^{n}}{\to}f$. It is easily checked that
$\tilde{A_{G}^{\Theta}}$ is bounded and so we have
$P_{\Theta,q}(Gf_{n})\overset{(L^{q})^{n}}{\to}P_{\Theta,q}(Gf)$. By Lemma 2.2
this means

$P_{\Theta,2}(Gf_{n})\overset{(L^{q})^{n}}{\to}P_{\Theta,q}(Gf)=\tilde{A_{G}^{\Theta}}(f).$
(2.1)

Because
$P_{\Theta,2}(Gf_{n})\overset{(L^{2})^{n}}{\to}P_{\Theta,2}(Gf)=A_{G}^{\Theta}(f)$
and convergence in $(L^{2})^{n}$ is stronger than $(L^{q})^{n}$ we must have

$P_{\Theta,2}(Gf_{n})\overset{(L^{q})^{n}}{\to}P_{\Theta,2}(Gf)=A_{G}^{\Theta}(f).$
(2.2)

Now comparing (2.1) and (2.2) uniqueness of limits implies that
$\tilde{A_{G}^{\Theta}}(f)=A_{G}^{\Theta}(f)$. ∎

######  Corollary 2.7.

Let the assumptions of Definition 2.4 hold and let
$A_{G}^{\Theta}:K_{\Theta}^{2}\to K_{\Theta}^{2}$ be a MTTO. Then
$\text{Im}\tilde{A_{G}^{\Theta}}\subseteq K_{\Theta}^{2}$.

In fact we have the following;

######  Proposition 2.8.

Let the assumptions of Definition 2.4 hold. Then
$\text{Im}\tilde{A_{G}^{\Theta}}\subseteq K_{\Theta}^{2}$ if and only if
$A_{G}^{\Theta}$ is a MTTO (i.e the map (1.1) is bounded).

###### Proof.

The above corollary shows that when $A_{G}^{\Theta}:K_{\Theta}^{2}\to
K_{\Theta}^{2}$ is a MTTO, then $\text{Im}\tilde{A_{G}^{\Theta}}\subseteq
K_{\Theta}^{2}$. To show the other implication, we first change the codomain
of $\tilde{A_{G}^{\Theta}}$, to view the map
$\tilde{A_{G}^{\Theta}}:K_{\Theta}^{2}\to K_{\Theta}^{2}$, which is well
defined by the assumption $\text{Im}\tilde{A_{G}^{\Theta}}\subseteq
K_{\Theta}^{2}$. We now use the closed graph theorem to show
$\tilde{A_{G}^{\Theta}}:K_{\Theta}^{2}\to K_{\Theta}^{2}$ is continuous. Let
$(f_{n})_{n\in\mathbb{N}}\in K_{\Theta}^{2}$ and let

$(f_{n},\tilde{A_{G}^{\Theta}}(f_{n}))\overset{K_{\Theta}^{2}\times
K_{\Theta}^{2}}{\to}(y_{1},y_{2}),$

then clearly $f_{n}\overset{K_{\Theta}^{2}}{\to}y_{1}$ and
$\tilde{A_{G}^{\Theta}}(f_{n})\overset{K_{\Theta}^{2}}{\to}y_{2}$. We also
know that
$\tilde{A_{G}^{\Theta}}(f_{n})\overset{K_{\Theta}^{q}}{\to}\tilde{A_{G}^{\Theta}}(y_{1})$,
and as $L^{2}$ convergence is stronger than $L^{q}$ convergence we can say
that $\tilde{A_{G}^{\Theta}}(f_{n})\overset{K_{\Theta}^{q}}{\to}y_{2}$.
Uniqueness of limits now shows
$(f_{n},\tilde{A_{G}^{\Theta}}(f_{n}))\overset{K_{\Theta}^{2}\times
K_{\Theta}^{2}}{\to}(y_{1},\tilde{A_{G}^{\Theta}}(y_{1}))$, and hence the
graph is closed. Now, again viewing $\tilde{A_{G}^{\Theta}}:K_{\Theta}^{2}\to
K_{\Theta}^{2}$, we have

$\tilde{A_{G}^{\Theta}}(f)=A_{G}^{\Theta}(f)$

for all $f\in K_{\Theta}^{2}\cap(H^{\infty})^{n}$. Thus boundedness of
$\tilde{A_{G}^{\Theta}}:K_{\Theta}^{2}\to K_{\Theta}^{2}$ ensures boundedness
of (1.1). ∎

In [2] the authors give an equivalent condition for a bounded truncated
Toeplitz operator to have a bounded symbol. They then go on to describe the
(scalar) inner functions, $\theta$, such that every bounded truncated Toeplitz
operator on $K_{\theta}^{2}$ has a bounded symbol. If we consider MTTOs with
symbols in $L^{(p,n\times n)}$ where $p\in(2,\infty]$, the above proposition
allows one to describe the set of all symbols of MTTOs (or when specialised to
the scalar case, symbols of all bounded truncated Toeplitz operators). This is
given by

$\\{G:P_{\Theta,q}(GK_{\Theta}^{2})\subseteq(L^{2})^{n}\\}.$

In a similar fashion to how we have changed the codomain of the MTTO to obtain
the modified MTTO, we can also change the codomain of the matricial Toeplitz
operator. Let $p\in(2,\infty]$ and let $G\in L^{(p,n\times n)}$. Define

$\mathcal{G}=\begin{pmatrix}{\Theta}^{*}&0\\\ G&\Theta\\\ \end{pmatrix},$
(2.3)

where $0$ denotes the $n$-by-$n$ matrix with each entry being 0. Throughout
all sections, given two Banach spaces $X_{1},X_{2}$ we will equip the space

$\begin{pmatrix}X_{1}\\\ X_{2}\end{pmatrix}=\left\\{\begin{pmatrix}f_{1}\\\
f_{2}\end{pmatrix}:f_{1}\in X_{1},f_{2}\in X_{2}\right\\}$

with the norm given by

$\left|\left|\begin{pmatrix}f_{1}\\\
f_{2}\end{pmatrix}\right|\right|=||f_{1}||_{X_{1}}+||f_{2}||_{X_{2}}.$

With this convention we can define
$T_{\mathcal{G}}:\begin{pmatrix}(H^{2})^{n}\\\
(H^{q})^{n}\end{pmatrix}\to\begin{pmatrix}(H^{2})^{n}\\\
(H^{q})^{n}\end{pmatrix}$, where if $f_{1}\in(H^{2})^{n}$ and
$f_{2}\in(H^{q})^{n}$ (where $\frac{1}{2}+\frac{1}{p}=\frac{1}{q}$),

$\begin{pmatrix}f_{1}\\\
f_{2}\end{pmatrix}\mapsto\begin{pmatrix}P_{2+}({\Theta}^{*}f_{1})\\\
P_{q+}(Gf_{1}+\Theta f_{2})\end{pmatrix}.$ (2.4)

An application of Hölder’s inequality shows $T_{\mathcal{G}}$ is bounded.

######  Proposition 2.9.

For the matrix $\mathcal{G}$ defined as (2.3) we have $P_{n}(\ker
T_{\mathcal{G}})=\ker\tilde{A_{G}^{\Theta}}$.

###### Proof.

Clearly, for $f_{1}\in(H^{2})^{n}$ and $f_{2}\in(H^{q})^{n}$, we have
$(f_{1},f_{2})\in\ker T_{\mathcal{G}}$ if and only if $f_{1}\in\ker
T_{\Theta^{*}}=K_{\Theta}$ and ${G}f_{1}+\Theta
f_{2}\in(\overline{H^{q}_{0}})^{n}.$ So $f_{1}\in\ker A^{\Theta}_{G}$, and
likewise given $f_{1}\in\ker A^{\Theta}_{G}$ there exist $f_{2}\in(H^{q})^{n}$
with $(f_{1},f_{2})\in\ker T_{\mathcal{G}}$. ∎

Although the above results are interesting in their own right, our main
motivation for introducing the modified MTTO is to study the properties of the
MTTOs which do not posses a bounded symbol.

The condition that we no longer require a bounded symbol to study
$A_{G}^{\Theta}$ is a significant extension to previous studies. This is
because there are MTTOs which do not have a bounded symbol but do have a
symbol in $L^{(p,n\times n)}$, where $p\in(2,\infty)$. This can be shown in
the case where $n=1$ by using Theorem 5.3 in [3], which is the following;

######  Theorem 2.10.

Suppose $\theta$ is a (scalar) inner function which has an angular derivative
(or ADC for short) at $\zeta\in\mathbb{T}$. Let $p\in(2,\infty)$. Then the
following are equivalent:

  1. (1)

the bounded truncated Toeplitz operator $k_{\zeta}^{\theta}\otimes
k_{\zeta}^{\theta}$ has a symbol $\phi\in L^{p}$ ;

  2. (2)

$k_{\zeta}^{\theta}\in L^{p}$.

In the above Theorem
$k_{\zeta}^{\theta}=\frac{1-\overline{\theta(\zeta)}\theta(z)}{1-\overline{\zeta}z}\in
K_{\theta}$ is the reproducing kernel at $\zeta$. In particular, the above
theorem shows that if $2<p_{1}<p_{2}<\infty$ and $k_{\zeta}^{\theta}\in
L^{p_{1}}$ but $k_{\zeta}^{\theta}\notin L^{p_{2}}$, then
$k_{\zeta}^{\theta}\otimes k_{\zeta}^{\theta}$ does not have a bounded symbol
but does have a symbol in $L^{p_{1}}$.

The precise conditions for $k_{\zeta}^{\theta}$ to lie in $L^{p}$ for
$p\in(1,\infty)$ are given in [1] and [12]. In particular, for a Blaschke
product with zeros $(a_{k})$ we have $k_{\zeta}^{\theta}\in L^{p}$ if and only
if

$\sum_{k}\frac{1-|a_{k}|^{2}}{|\zeta-a_{k}|^{p}}<\infty.$ (2.5)

To obtain a bounded truncated Toeplitz operator which does not have a bounded
symbol but does have a symbol in $L^{p_{1}}$, for some $p_{1}\in(2,\infty)$,
it is sufficient to have a point $\zeta\in\mathbb{T}$, and a Blaschke product
which has an ADC at $\zeta$ such that (2.5) is true for some
$p=p_{1}\in(2,\infty)$ but not for some strictly larger value of $p$. An
explicit example of this is a Blaschke product with zeros $(a_{k})$
accumulating to the point 1 such that

$\sum_{k}\frac{1-|a_{k}|^{2}}{|1-a_{k}|^{p_{1}}}<\infty\hskip 11.38092pt\text{
for some $2<p_{1}<\infty$},$

but

$\sum_{k}\frac{1-|a_{k}|^{2}}{|1-a_{k}|^{p_{2}}}=\infty\hskip 11.38092pt\text{
for some $p_{1}<p_{2}<\infty$}.$

Theorem 5.1(b) in [26] states that if $\theta$ has an ADC at
$\zeta\in\mathbb{T}$, then $k_{\zeta}^{\theta}\otimes k_{\zeta}^{\theta}$ is a
bounded truncated Toeplitz operator. Therefore by Theorem 5.1(b) in [26] and
the above theorem, we can construct an example of a bounded truncated Toeplitz
operator which has a symbol in $L^{2}$, but does not have a symbol in $L^{p}$
for any $p\in(2,\infty)$. Similar to our previous example, in order to do this
it is sufficient to have a point $\zeta\in\mathbb{T}$ and a Blaschke product
with an ADC at $\zeta$ such that (2.5) is true for $p=2$ but not for any
$p\in(2,\infty)$. A numerical example of such a point $\zeta\in\mathbb{T}$ and
Blaschke product is the Blaschke product with zeros (accumulating to 1) given
by $a_{k}=(1-\epsilon_{k})e^{i\delta_{k}}$ where
$\epsilon_{k}=\frac{1}{k^{2}}$ and $\delta_{k}=\frac{log(k)}{k^{1/2}}$. This
observation shows that not every bounded truncated Toeplitz operator has a
symbol in $L^{p}$ for some $p\in(2,\infty)$.

##  3\. The kernel

A closed subspace $M\subseteq(H^{2})^{n}$ is said to be nearly
$S^{*}$-invariant with defect $m$ if and only if there exists a
$m$-dimensional subspace $D$ (which may be taken to be orthogonal to $M$) such
that if $f\in M$ and $f(0)$ is the zero vector then $S^{*}f\in M\oplus D$. We
call $D$ the defect space. If $M$ is nearly $S^{*}$-invariant with defect 0
then it is said to be nearly $S^{*}$-invariant. Similarly, we say a closed
subspace $N\subseteq\begin{pmatrix}(H^{2})^{n}\\\ (H^{q})^{n}\end{pmatrix}$ is
nearly $S^{*}$-invariant if and only if all functions $f\in N$ with the
property $f(0)$ is the zero vector satisfy $S^{*}(f)=\frac{f}{z}\in N$.

In this section we decompose the kernel of a MTTO into an isometric image of
an $S^{*}$-invariant subspace.

Define $W:=\ker T_{\mathcal{G}}(0)=\\{F(0):F\in\ker
T_{\mathcal{G}}\\}\subseteq\mathbb{C}^{2n}$. Let $\dim W=r$, and pick
$W_{1},\ldots,W_{r}\in\ker T_{\mathcal{G}}$ such that
$W_{1}(0),\ldots,W_{r}(0)$ are a basis for W.

######  Proposition 3.1.

The space $P_{n}(\ker T_{\mathcal{G}})$ is nearly $S^{*}$-invariant with a
defect space

$\left(\frac{\text{span}\\{P_{n}(W_{1}),\ldots
P_{n}(W_{r})\\}}{z}\cap(H^{2})^{n}\right).$ (3.1)

######  Remark 3.2.

This may be viewed as a generalisation of Corollary 3.2 in [23], but the
delicate issue here is that we are no longer working with a Hilbert space and
so we can not use orthogonality.

###### Proof.

Let $f_{1}\in P_{n}(\ker T_{\mathcal{G}})$ with $f_{1}(0)$ equal to the zero
vector. Pick $f_{2}\in(H^{q})^{n}$ such that $\begin{pmatrix}f_{1}\\\
f_{2}\end{pmatrix}\in\ker T_{\mathcal{G}}$ and pick constants
$\lambda_{1}\ldots\lambda_{r}$ such that $\begin{pmatrix}f_{1}\\\
f_{2}\end{pmatrix}-\lambda_{1}W_{1}-\ldots\lambda_{r}W_{r}$ evaluated at $0$
is the zero vector, then

$\begin{pmatrix}f_{1}\\\
f_{2}\end{pmatrix}-\lambda_{1}W_{1}-\ldots\lambda_{r}W_{r}\in\ker
T_{\mathcal{G}}\cap z\begin{pmatrix}(H^{2})^{n}\\\ (H^{q})^{n}\end{pmatrix}.$

Near invariance of $\ker T_{\mathcal{G}}$ now ensures

$\frac{f_{1}}{z}-\frac{\lambda_{1}P_{n}(W_{1})-\ldots\lambda_{r}P_{n}(W_{r})}{z}\in
P_{n}(\ker T_{\mathcal{G}}),$

and therefore

$\frac{f_{1}}{z}\in P_{n}(\ker
T_{\mathcal{G}})+\left(\frac{\text{span}\\{P_{n}(W_{1}),\ldots
P_{n}(W_{r})\\}}{z}\cap(H^{2})^{n}\right).$

∎

Previous results on the kernel of the truncated Toeplitz operator (see [23],
[9] and [8]) have been under the assumption that the symbol for the operator
is bounded. Now using the operator $\tilde{A_{G}^{\Theta}}$ as an intermediate
tool, this allows us to obtain a Hitt-style characterisation for the kernel of
a MTTO and, unlike previous results, we do not require that the symbol of the
MTTO is bounded for this characterisation to hold.

######  Theorem 3.3.

Let $p\in(2,\infty]$, and let $G\in L^{(p,n\times n)}$ be such that
$A_{G}^{\Theta}$ is a MTTO. Then $\ker A_{G}^{\Theta}$ is nearly
$S^{*}$-invariant with defect $m$, where $m\leqslant n$.

###### Proof.

From Proposition 2.6 it is clear that $\ker
A_{G}^{\Theta}=\ker\tilde{A_{G}^{\Theta}}$, and Proposition 2.9 shows that
$\ker\tilde{A_{G}^{\Theta}}=P_{n}(\ker T_{\mathcal{G}})$, so from Proposition
3.1 we can deduce that $\ker A_{G}^{\Theta}$ is a nearly invariant subspace
with a defect space given by (3.1). If $r\leqslant n$ it is clear that the
dimension of (3.1) is less than or equal to $n$, so it remains to prove that
if $r=n+i$ for $i>0$ then the dimension of (3.1) is at most $n$. Suppose
$r=n+i$ for $i>0$. We form a matrix

$[W_{1}(0),\ldots,W_{n+i}(0)],$

then for $\begin{pmatrix}s_{1}\\\ \vdots\\\
s_{n+i}\end{pmatrix}\in\mathbb{C}^{n+i}$ we have that
$s_{1}P_{n}(W_{1})+\ldots s_{n+i}P_{n}(W_{n+i})\in z(H^{2})^{n}$ if and only
if

$P_{n}\left([W_{1}(0),\ldots,W_{n+i}(0)]\begin{pmatrix}s_{1}\\\ \vdots\\\
s_{n+i}\end{pmatrix}\right)$

is the zero vector. Hence the dimension of (3.1) is given by the dimension of

$S=\left\\{\begin{pmatrix}s_{1}\\\ \vdots\\\
s_{n+i}\end{pmatrix}\in\mathbb{C}^{n+i}:P_{n}\left([W_{1}(0),\ldots,W_{n+i}(0)]\begin{pmatrix}s_{1}\\\
\vdots\\\ s_{n+i}\end{pmatrix}\right)=\begin{pmatrix}0\\\ \vdots\\\ 0\\\
\end{pmatrix}\right\\}.$

As $W_{1}(0),...W_{n+i}(0)\in\mathbb{C}^{2n}$ are linearly independent, we may
pick vectors

$X_{1},\ldots X_{n-i}\in\mathbb{C}^{2n}$

such that the vectors $W_{1}(0),\ldots$ $W_{n+i}(0),X_{1},\ldots,X_{n-i}$ are
linearly independent. We then define $S^{{}^{\prime}}$ as

$\left\\{\begin{pmatrix}s_{1}\\\ \vdots\\\ s_{n+i}\\\ 0\\\ \vdots\\\
0\end{pmatrix}\in\mathbb{C}^{2n}:P_{n}\left([W_{1}(0),\ldots,W_{n+i}(0),X_{1},\ldots,X_{n-i}]\begin{pmatrix}s_{1}\\\
\vdots\\\ s_{n+i}\\\ 0\\\ \vdots\\\ 0\end{pmatrix}\right)=\begin{pmatrix}0\\\
\vdots\\\ \vdots\\\ \vdots\\\ 0\end{pmatrix}\right\\}.$

It is clear $\dim S=\dim S^{{}^{\prime}}$, and moreover $S^{{}^{\prime}}$ is
contained in

$\left\\{[W_{1}(0),\ldots,W_{n+i}(0),X_{1},\ldots,X_{n-i}]^{-1}V:V\in\mathbb{C}^{2n}\text{and
}P_{n}(V)=0\right\\},$

which has dimension $n$. Thus we can conclude that the dimension of (3.1) is
equal to $\dim S=\dim S^{{}^{\prime}}\leqslant n$. ∎

Theorem 3.4 in [23] (which was also independently proved in [11]) gives a
decomposition for vector-valued nearly $S^{*}$-invariant subspaces with a
defect. So combining the above theorem and Theorem 3.4 in [23] we obtain the
following decomposition for the kernels of MTTOs in terms of $S^{*}$-invariant
subspaces.

######  Theorem 3.4.

Let $p\in(2,\infty]$, and let $G\in L^{(p,n\times n)}$ be such that
$A_{G}^{\Theta}$ is a MTTO. Let $\\{e_{1},\ldots e_{m}\\}$ be an orthonormal
basis for the $m$-dimensional defect space (where $m\leqslant n$) for $\ker
A_{G}^{\Theta}$ given by (3.1) and set $r=\dim\ker A_{G}^{\Theta}\ominus(\ker
A_{G}^{\Theta}\cap z(H^{2})^{n})$. Then

  1. (1)

in the case where there are functions in $\ker A_{G}^{\Theta}$ that do not
vanish at $0$,

$\ker
A_{G}^{\Theta}=\\{F:F(z)=F_{0}(z)k_{0}(z)+z\sum_{j=1}^{m}k_{j}(z)e_{j}(z):(k_{0},\ldots,k_{m})\in
K\\},$

where $F_{0}$ is the matrix with each column being an element of an
orthonormal basis for $\ker A_{G}^{\Theta}\ominus(\ker A_{G}^{\Theta}\cap
z(H^{2})^{n})$, $k_{0}\in(H^{2})^{r}$, $k_{1},\ldots k_{m}\in H^{2}$, and
$K\subseteq(H^{2})^{(r+m)}$ is a closed $S^{*}$-invariant subspace.
Furthermore $||F||^{2}=\sum_{j=0}^{m}||k_{j}||^{2}$.

  2. (2)

In the case where all functions in $\ker A_{G}^{\Theta}$ vanish at $0$,

$\ker
A_{G}^{\Theta}=\\{F:F(z)=z\sum_{j=1}^{m}k_{j}(z)e_{j}(z):(k_{1},\ldots,k_{m})\in
K\\},$

with the same notation as in 1, except that $K$ is now a closed
$S^{*}$-invariant subspace of $(H^{2})^{m}$, and
$||F||^{2}=\sum_{j=1}^{m}||k_{j}||^{2}$.

######  Remark 3.5.

We remark that the above theorem is a generalisation of the results of Section
3 of [23] in two ways. We are now considering the MTTO instead of the scalar
truncated Toeplitz operator. We are also now allowing for the MTTO to have a
unbounded symbol whereas [23] only considers bounded symbols.

We now give an example to show that under the conditions of Theorem 3.3, $n$
is the smallest dimension of defect space for $\ker A_{G}^{\Theta}$, i.e. it
is not true that for all inner functions $\Theta$ and symbols $G\in
L^{(p,n\times n)}$, that $\ker A_{G}^{\Theta}$ has a $j$-dimensional defect
where $j<n$.

######  Example.

Let $\Theta=\begin{pmatrix}z^{2}&0\\\ 0&z^{2}\\\ \end{pmatrix}$, and
$G=\begin{pmatrix}z&0\\\ 0&z\\\ \end{pmatrix}$ then

$\ker A_{G}^{\Theta}=\left\\{\begin{pmatrix}\lambda z\\\ \mu z\\\
\end{pmatrix}:\lambda,\mu\in\mathbb{C}\right\\},$

which is clearly nearly $S^{*}$-invariant with defect 2.

##  4\. Equivalence after extension

For Banach spaces $X,\tilde{X},Y,\tilde{Y}$ the operators $T:X\to\tilde{X}$
and $S:Y\to\tilde{Y}$ are said to be (algebraically and topologically)
equivalent if and only if $T=ESF$, where $E$ and $F$ are invertible operators.
More generally $T$ and $S$ are equivalent after extension (abbreviated to EAE)
if and only if there exist (possibly trivial) Banach spaces $X_{0},Y_{0}$,
called extension spaces and invertible linear operators $E:\tilde{Y}\oplus
Y_{0}\to\tilde{X}\oplus X_{0}$ and $F:X\oplus X_{0}\to Y\oplus Y_{0}$, such
that

$\begin{pmatrix}T&0\\\ 0&I_{X_{0}}\\\ \end{pmatrix}=E\begin{pmatrix}S&0\\\
0&I_{Y_{0}}\\\ \end{pmatrix}F.$

In this case we write that $T\overset{*}{\backsim}S$.

The relation $\overset{*}{\backsim}$ is an equivalence relation. Operators
that are equivalent after extension have many features in common. In
particular, using the notation $X\simeq Y$ to say that two Banach spaces $X$
and $Y$ are isomorphic, i.e., that there exists an invertible operator from
$X$ onto $Y$ , and the notation $\text{Im}A$ to denote the range of an
operator $A$, we have the following.

######  Theorem 4.1 ([4]).

Let $T:X\to\tilde{X},S:Y\to\tilde{Y}$ be operators and assume that
$T\overset{*}{\backsim}S$. Then

  1. (1)

$\ker T\simeq\ker S$;

  2. (2)

$\text{Im }T$ is closed if and only if $\text{Im }S$ is closed and, in that
case, $\tilde{X}/\text{Im }T\simeq\tilde{Y}/\text{Im }S$;

  3. (3)

if one of the operators $T,S$ is generalised (left, right) invertible, then
the other is generalised (left, right) invertible too;

  4. (4)

$T$ is Fredholm if and only if $S$ is Fredholm and in that case $\dim\ker
T=\dim\ker S$ and $\text{codim }\text{Im }T=\text{codim }\text{Im }S$.

The above theorem highlights that when one wants to consider invertibility,
Fredholmness and spectral properties, EAE extension results are very useful.
Section 6 of [8] shows that a truncated Toeplitz operator with a bounded
symbol is EAE to a matricial Toeplitz operator, and then consequently the
spectral properties of the truncated Toeplitz operator were studied in [7].
For $\theta$ a scalar inner function and $g\in L^{\infty}$, the dual truncated
Toeplitz operator $D_{g}^{\theta}:(K_{\theta})^{\perp}\to(K_{\theta})^{\perp}$
is defined by $f\mapsto(Q_{\theta}+P_{-})(gf)$, where $P_{-}=I-P_{+}$. Section
5 of [5] shows the dual truncated Toeplitz operator is EAE to a paired
operator on $(L^{2})^{2}$.

Throughout this section, unless otherwise stated, we assume that $G\in
L^{(p,n\times n)}$ where $p\in(2,\infty]$. We let $q\in(1,2]$ be such that
$\frac{1}{2}+\frac{1}{p}=\frac{1}{q}$. In this context, we write
$T_{G}:(H^{2})^{n}\to(H^{q})^{n}$ to mean the map $f\mapsto P_{q+}(Gf)$.

In the first part of this section we initially adapt the results in Section 6
of [8] to show that $T_{\mathcal{G}}$ is EAE to $\tilde{A_{G}^{\Theta}}$. We
then build on this result to construct a Toeplitz operator which is EAE to
$A_{G}^{\Theta}$. Unlike the works of [8] we consider MTTOs which only have
unbounded symbols, and in order to overcome the problem of $G$ not being
bounded (and then necessarily the domain and codomain of
$\tilde{A_{G}^{\Theta}}$ being different spaces) one must define a new normed
space which mixes $H^{p}$ and $H^{q}$ spaces.

Consider the operator

$P_{\Theta,q}GP_{\Theta,2}+Q_{\Theta,2}:(H^{2})^{n}\to
K_{\Theta}^{q}+\Theta(H^{2})^{n},$

where here the norm of $k+\Theta f\in K_{\Theta}^{q}+\Theta(H^{2})^{n}$ is
given by $||k||_{(L^{q})^{n}}+||\Theta f||_{(L^{2})^{n}}$. We first show that

$\tilde{A_{G}^{\Theta}}\overset{*}{\backsim}P_{\Theta,q}GP_{\Theta,2}+Q_{\Theta,2}.$
(4.1)

We have

$\begin{pmatrix}\tilde{A_{G}^{\Theta}}&0\\\ 0&I_{\Theta(H^{2})^{n}}\\\
\end{pmatrix}=E_{1}\begin{pmatrix}P_{\Theta,q}GP_{\Theta,2}+Q_{\Theta,2}&0\\\
0&I_{0}\\\ \end{pmatrix}F_{1},$

where

$F_{1}:K_{\Theta}^{2}\oplus\Theta(H^{2})^{n}\to(H^{2})^{n}\oplus\\{0\\}$

is such that

$\begin{pmatrix}k\\\ \Theta f\\\ \end{pmatrix}\mapsto\begin{pmatrix}k+\Theta
f\\\ 0\\\ \end{pmatrix},$

and

$E_{1}:K_{\Theta}^{q}+\Theta(H^{2})^{n}\oplus\\{0\\}\to
K_{\Theta}^{q}\oplus\Theta(H^{2})^{n}$

is such that

$\begin{pmatrix}k+\Theta f\\\ 0\\\ \end{pmatrix}\mapsto\begin{pmatrix}k\\\
\Theta f\\\ \end{pmatrix}.$

On the other hand it is clear that

$P_{\Theta,q}GP_{\Theta,2}+Q_{\Theta,2}\overset{*}{\backsim}\begin{pmatrix}P_{\Theta,q}GP_{\Theta,2}+Q_{\Theta,2}&0\\\
0&P_{q+}\\\ \end{pmatrix}.$ (4.2)

If we denote $I$ to be the identity operator on
$K_{\Theta}^{q}+\Theta(H^{2})^{n}$, we also have

$P_{\Theta,q}GP_{\Theta,2}+Q_{\Theta,2}=(I-P_{\Theta,q}T_{G}Q_{\Theta,q})(P_{\Theta,q}T_{G}+Q_{\Theta,2}).$

Furthermore adapting Lemma 6.3 in [8] we can deduce:

######  Lemma 4.2.

The operator
$I-P_{\Theta,q}T_{G}Q_{\Theta,q}:K_{\Theta}^{q}+\Theta(H^{2})^{n}\to
K_{\Theta}^{q}+\Theta(H^{2})^{n}$ is invertible with inverse
$I+P_{\Theta,q}T_{G}Q_{\Theta,q}$.

We now mimic the factorisations given in Section 6 of [8], however as we are
working with a mixed $H^{p}$-$H^{q}$ space we must also manage our
factorisation in such a way that the domain and codomain in our consecutive
factors match up. As the results of [8] purely deal with operators on $H^{2}$,
this did not have to be considered.

In the following argument for ease of notation we write the domain and co-
domain above the operator. For example, if the operator $A:X\to Y$, we will
label this as $\overbrace{A}^{X\to Y}$. In the case when $A:X\to X$ we will
denote this by $\overbrace{A}^{X}$. With this notation we will omit the
specific $q$ or $2$ notation from the projections in the following matrices.

Thus with

$T=\begin{pmatrix}\overbrace{I-P_{\Theta}T_{G}Q_{\Theta}}^{K_{\Theta}^{q}+\Theta(H^{2})^{n}}&\overbrace{0}^{(H^{q})^{n}\to\\{0\\}}\\\
\overbrace{0}^{K_{\Theta}^{q}+\Theta(H^{2})^{n}\to\\{0\\}}&\overbrace{P_{+}}^{(H^{q})^{n}}\\\
\end{pmatrix},$ (4.3)

we can write

$\begin{pmatrix}\overbrace{P_{\Theta}GP_{\Theta}+Q_{\Theta}}^{(H^{2})^{n}\to
K_{\Theta}^{q}+\Theta(H^{2})^{n}}&\overbrace{0}^{(H^{q})^{n}\to
K_{\Theta}^{q}+\Theta(H^{2})^{n}}\\\
\overbrace{0}^{(H^{2})^{n}\to(H^{q})^{n}}&\overbrace{P_{+}}^{(H^{q})^{n}}\\\
\end{pmatrix}=$
$T\begin{pmatrix}\overbrace{P_{\Theta}T_{G}+Q_{\Theta}}^{(H^{2})^{n}\to
K_{\Theta}^{q}+\Theta(H^{2})^{n}}&\overbrace{0}^{(H^{q})^{n}\to
K_{\Theta}^{q}+\Theta(H^{2})^{n}}\\\
\overbrace{0}^{(H^{2})^{n}\to(H^{q})^{n}}&\overbrace{P_{+}}^{(H^{q})^{n}}\\\
\end{pmatrix}=$
$T\begin{pmatrix}\overbrace{T_{\Theta}}^{(H^{2})^{n}\to\Theta(H^{2})^{n}}&\overbrace{P_{\Theta}}^{(H^{q})^{n}\to
K_{\Theta}^{q}}\\\
\overbrace{-P_{+}}^{(H^{2})^{n}}&\overbrace{T_{{\Theta}^{*}}}^{(H^{q})^{n}}\\\
\end{pmatrix}\begin{pmatrix}\overbrace{T_{{\Theta}^{*}}}^{(H^{2})^{n}}&\overbrace{0}^{(H^{q})^{n}\to(H^{2})^{n}}\\\
\overbrace{{T_{G}-Q_{\Theta}T_{G}+Q_{\Theta}P_{+}}}^{(H^{2})^{n}\to(H^{q})^{n}}&\overbrace{T_{\Theta}}^{(H^{q})^{n}}\\\
\end{pmatrix},$

where the last line follows by using the identity
$P_{+}-Q_{\Theta}=P_{\Theta}$ and $T_{{\Theta}^{*}}P_{\Theta}=0$. This can be
factorised further to equal

$T\begin{pmatrix}\overbrace{T_{\Theta}}^{(H^{2})^{n}\to\Theta(H^{2})^{n}}&\overbrace{P_{\Theta}}^{(H^{q})^{n}\to
K_{\Theta}^{q}}\\\
\overbrace{-P_{+}}^{(H^{2})^{n}}&\overbrace{T_{{\Theta}^{*}}}^{(H^{q})^{n}}\\\
\end{pmatrix}T_{\mathcal{G}}\begin{pmatrix}\overbrace{P_{+}}^{(H^{2})^{n}}&\overbrace{0}^{(H^{q})^{n}\to(H^{2})^{n}}\\\
\overbrace{-T_{{\Theta}^{*}}T_{G}+T_{{\Theta}^{*}}P_{+}}^{(H^{2})^{n}\to(H^{q})^{n}}&\overbrace{P_{+}}^{(H^{q})^{n}}\\\
\end{pmatrix},$ (4.4)

where $T_{\mathcal{G}}$ is defined as in (2.4). In the above, we label the
second factor as $T_{1}$ and the final factor as $T_{2}$.

  1. (1)

The first factor, $T$, is invertible with inverse given by

$\begin{pmatrix}\overbrace{I+P_{\Theta}T_{G}Q_{\Theta}}^{K_{\Theta}^{q}+\Theta(H^{2})^{n}}&\overbrace{0}^{(H^{q})^{n}\to\\{0\\}}\\\
\overbrace{0}^{K_{\Theta}^{q}+\Theta(H^{2})^{n}\to\\{0\\}}&\overbrace{P_{+}}^{(H^{q})^{n}}\\\
\end{pmatrix}.$

This is verified by Lemma 4.2.

  2. (2)

Adapting Lemma 6.4 from [8] one can show the second factor, $T_{1}$, is
invertible as a map $\begin{pmatrix}(H^{2})^{n}\\\
(H^{q})^{n}\end{pmatrix}\to\begin{pmatrix}K_{\Theta}^{q}+\Theta(H^{2})^{n}\\\
(H^{q})^{n}\end{pmatrix}$.

  3. (3)

Adapting Lemma 6.5 from [8] one can show the last factor, $T_{2}$, is
invertible in $\begin{pmatrix}(H^{2})^{n}\\\ (H^{q})^{n}\end{pmatrix}$.

We can now conclude the following;

######  Theorem 4.3.

$T_{\mathcal{G}}$ is equivalent after extension to $\tilde{A_{G}^{\Theta}}$.

###### Proof.

Using (4.1), (4.2) and the fact that $\overset{*}{\backsim}$ is an equivalence
relation, we see that

$\tilde{A_{G}^{\Theta}}\overset{*}{\backsim}\begin{pmatrix}P_{\Theta,q}GP_{\Theta,2}+Q_{\Theta,2}&0\\\
0&P_{q+}\\\ \end{pmatrix}.$

Now (4.4) and the reasoning immediately following (4.4) shows

$\begin{pmatrix}P_{\Theta,q}GP_{\Theta,2}+Q_{\Theta,2}&0\\\ 0&P_{q+}\\\
\end{pmatrix}\overset{*}{\backsim}T_{\mathcal{G}}$

and so transitivity of $\overset{*}{\backsim}$ gives us

$\tilde{A_{G}^{\Theta}}\overset{*}{\backsim}T_{\mathcal{G}}.$

∎

######  Remark 4.4.

In the case when $n=1$ and $p=\infty$, Theorem 4.3 specialises to become (the
symmetric case of) Theorem 6.6 in [8].

When $G$ is bounded we have $\tilde{A_{G}^{\Theta}}=A_{G}^{\Theta}$, so we may
specialise Theorem 4.3 to find an operator which is EAE to $A_{G}^{\Theta}$
when $G$ is bounded.

######  Theorem 4.5.

Let $G\in L^{(\infty,n\times n)}$. Then
$T_{\mathcal{G}}:(H^{2})^{2n}\to(H^{2})^{2n}$ is equivalent after extension to
${A_{G}^{\Theta}}$.

As operators which are EAE have isomorphic kernels and cokernels, Theorem 4.3
and Proposition 2.6 suggest that restricting the codomain of $T_{\mathcal{G}}$
may provide an operator which is EAE to $A_{G}^{\Theta}$, where $G\in
L^{(p,n\times n)}$, for $p\in(2,\infty)$. We now pursue this idea.

Throughout the remainder of this section we now continue to assume that $G\in
L^{(p,n\times n)}$ where $p\in(2,\infty]$, but we now we also make the extra
assumption that ${A_{G}^{\Theta}}$ is a MTTO (and hence bounded).

The image of $T_{\mathcal{G}}$ is computed to be

$\begin{pmatrix}0\\\ \Theta(H^{q})^{n}\\\ \end{pmatrix}+\begin{pmatrix}0\\\
P_{q+}(GK_{\Theta}^{2})\\\ \end{pmatrix}+\left\\{\begin{pmatrix}f\\\
P_{q+}(G\Theta f)\\\ \end{pmatrix}:f\in(H^{2})^{n}\right\\},$

where for $A\subseteq(L^{q})^{n}$, $\begin{pmatrix}0\\\ A\\\ \end{pmatrix}$ is
the set of all vectors of length $2n$ with the last $n$ coordinates taking a
value $a\in A$. We now define the Banach space

$\text{Co-d}:=\begin{pmatrix}0\\\ \Theta(H^{q})^{n}\\\
\end{pmatrix}+\begin{pmatrix}0\\\ K_{\Theta}^{2}\\\
\end{pmatrix}+\left\\{\begin{pmatrix}f\\\ P_{q+}(G\Theta f)\\\
\end{pmatrix}:f\in(H^{2})^{n}\right\\},$ (4.5)

where for $p_{1}\in(H^{q})^{n},p_{2}\in K_{\Theta}^{2},p_{3}\in(H^{2})^{n}$ we
have the well defined norm

$\left|\left|\begin{pmatrix}0\\\ \Theta p_{1}\\\
\end{pmatrix}+\begin{pmatrix}0\\\ p_{2}\\\
\end{pmatrix}+\begin{pmatrix}p_{3}\\\ P_{q+}(G\Theta p_{3})\\\
\end{pmatrix}\right|\right|_{\text{Co-d}}:=||\Theta
p_{1}||_{(H^{q})^{n}}+||p_{2}||_{K_{\Theta}^{2}}+||p_{3}||_{(H^{2})^{n}}.$

We note that completeness of each of the spaces $(H^{q})^{n},K_{\Theta}^{2}$
and $(H^{2})^{n}$ ensures completeness of Co-d. Corollary 2.7 ensures that
$P_{q+}(GK_{\Theta}^{2})\subseteq K_{\Theta}^{2}+\Theta(H^{q})^{n}$ so this
gives us a well defined bounded map

$T_{\mathcal{G}}^{r}:\begin{pmatrix}(H^{2})^{n}\\\
(H^{q})^{n}\end{pmatrix}\to\text{Co-d},$

where for $f_{1}\in(H^{2})^{n}$ and $f_{2}\in(H^{q})^{n}$

$\begin{pmatrix}f_{1}\\\
f_{2}\end{pmatrix}\mapsto\begin{pmatrix}P_{2+}({\Theta}^{*}f_{1})\\\
P_{q+}(Gf_{1}+\Theta
f_{2})\end{pmatrix}=T_{\mathcal{G}}\begin{pmatrix}f_{1}\\\
f_{2}\end{pmatrix}.$

######  Remark 4.6.

In the case when $p=\infty$ and so $q=2$, as sets we have
$\text{Co-d}=\begin{pmatrix}(H^{2})^{n}\\\ (H^{2})^{n}\end{pmatrix}$ and
furthermore the Co-d norm is equivalent to the $\begin{pmatrix}(H^{2})^{n}\\\
(H^{2})^{n}\end{pmatrix}$ norm.

Similar to the proof of Theorem 4.3, we can show that

${A_{G}^{\Theta}}\overset{*}{\backsim}\begin{pmatrix}\overbrace{P_{\Theta,q}GP_{\Theta,2}+Q_{\Theta,2}}^{(H^{2})^{n}\to
K_{\Theta}^{2}+\Theta(H^{2})^{n}}&\overbrace{0}^{(H^{q})^{n}\to
K_{\Theta}^{q}+\Theta(H^{2})^{n}}\\\
\overbrace{0}^{(H^{2})^{n}\to(H^{q})^{n}}&\overbrace{P_{q+}}^{(H^{q})^{n}}\\\
\end{pmatrix},$

where we know by Corollary 2.7 that $P_{\Theta,q}(GK_{\Theta}^{2})\subseteq
K_{\Theta}^{2}$. It is also clear that for $\begin{pmatrix}f_{1}\\\
f_{2}\end{pmatrix}\in\begin{pmatrix}(H^{2})^{n}\\\ (H^{q})^{n}\end{pmatrix}$
using (4.4) we still have

$TT_{1}T_{\mathcal{G}}^{r}T_{2}\begin{pmatrix}f_{1}\\\
f_{2}\end{pmatrix}=\begin{pmatrix}\overbrace{P_{\Theta,q}GP_{\Theta,2}+Q_{\Theta,2}}^{(H^{2})^{n}\to
K_{\Theta}^{2}+\Theta(H^{2})^{n}}&\overbrace{0}^{(H^{q})^{n}\to
K_{\Theta}^{q}+\Theta(H^{2})^{n}}\\\
\overbrace{0}^{(H^{2})^{n}\to(H^{q})^{n}}&\overbrace{P_{q+}}^{(H^{q})^{n}}\\\
\end{pmatrix}\begin{pmatrix}f_{1}\\\ f_{2}\end{pmatrix}.$

One can also check that the operator
$TT_{1}:\text{Co-d}\to\begin{pmatrix}(H^{2})^{n}\\\ (H^{q})^{n}\end{pmatrix}$
is well defined, bounded and invertible. We know from an adaptation of Lemma
6.5 in [8] that $T_{2}:\begin{pmatrix}(H^{2})^{n}\\\
(H^{q})^{n}\end{pmatrix}\to\begin{pmatrix}(H^{2})^{n}\\\
(H^{q})^{n}\end{pmatrix}$ is invertible. So we can conclude;

######  Theorem 4.7.

${A_{G}^{\Theta}}\overset{*}{\backsim}T_{\mathcal{G}}^{r}$.

##  5\. Application to integral equations

In this section we concretely relate the theory of MTTOs to integral
equations. We show the matricial truncated Wiener-Hopf operator is unitarily
equivalent to a MTTO and give a neat application of the theory we have
developed around the modified MTTO which allows us to test continuity of the
matricial truncated Wiener-Hopf operator by considering only the image of a
modified MTTO.

We consider $H^{2}(\mathbb{C}_{+})$, the Hardy space of the upper half plane.
Section 3 of [8] shows there is an isometric isomorphism between the Hardy
space on the disc and the Hardy space on the half plane, which we denote
$V:H^{2}\to H^{2}(\mathbb{C}_{+})$. It further shows that via the unitary map
$V$, truncated Toeplitz operators in the two different settings are unitarily
equivalent. The functions $\theta_{1}(w)=e^{iaw}$ and $\theta_{2}(w)=e^{ibw}$
are inner functions in $H^{2}(\mathbb{C}_{+})$ and it is well known that the
inverse Fourier transform of $L^{2}(0,a)$, $L^{2}(0,b)$ is
$K_{\theta_{1}}^{2}$, $K_{\theta_{2}}^{2}$ respectively. We denote
$\mathcal{F}:H^{2}(\mathbb{C}_{+})\to L^{2}(0,\infty)$ to be the Fourier
transform. The Fourier transform of a matrix is understood by taking the
Fourier transform of each entry of the matrix.

MTTOs on the model space $K_{\Theta}^{2}$ where
$\Theta=\begin{pmatrix}\theta_{1}&0\\\ 0&\theta_{2}\\\ \end{pmatrix}$, are
closely connected with matricial truncated Wiener-Hopf operators. For
$c\in\mathbb{R}_{+}$ and $f=\begin{pmatrix}f_{1}\\\ f_{2}\end{pmatrix}$ where
$f_{1},f_{2}\in L^{1}(\mathbb{R})$, we denote
$\text{int}_{c}^{1}(f)=\int_{0}^{c}f_{1}(t)dt$ and
$\text{int}_{c}^{2}(f)=\int_{0}^{c}f_{2}(t)dt$. Let
$G=\begin{pmatrix}g_{11}&g_{12}\\\ g_{21}&g_{22}\end{pmatrix}\in
L^{1}(\mathbb{R})^{2\times 2}$, let $k=\begin{pmatrix}k_{1}\\\
k_{2}\end{pmatrix}$ where $k_{1}\in L^{2}(0,a)$ and $k_{2}\in L^{2}(0,b)$ for
$a,b\in\mathbb{R}_{+}$. The matricial truncated Wiener-Hopf operator is an
operator on $\begin{pmatrix}L^{2}(0,a)\\\ L^{2}(0,b)\end{pmatrix}$ (equipped
with the product norm) densely defined by

$\left(W_{G}\begin{pmatrix}k_{1}\\\
k_{2}\end{pmatrix}\right)(x)=\begin{pmatrix}\text{int}_{a}^{1}(G(x-t)k(t))\\\
\text{int}_{b}^{2}(G(x-t)k(t))\end{pmatrix},$ (5.1)

for $k_{1}\in L^{2}(0,a)\cap L^{\infty}(0,a)$, $k_{2}\in L^{2}(0,b)\cap
L^{\infty}(0,b)$. We mostly only consider the case when $a=b$. In this case

$\left(W_{G}\begin{pmatrix}k_{1}\\\
k_{2}\end{pmatrix}\right)(x)=\int_{0}^{a}G(x-t)k(t)dt.$

If $W$ extends to a bounded operator and if $G=\hat{H}$, where
$H\mathcal{F}^{-1}\begin{pmatrix}L^{\infty}(0,a)\\\
L^{\infty}(0,b)\end{pmatrix}\subseteq(L^{2}(\mathbb{R}))^{2}$ (this condition
on $H$ is necessary to densely define a MTTO with symbol $H$ on the domain
$\mathcal{F}^{-1}\begin{pmatrix}L^{\infty}(0,a)\\\
L^{\infty}(0,b)\end{pmatrix})$ we have

$W_{G}(k)=\mathcal{F}P_{\Theta}(H\check{k}),$

for $k_{1}\in L^{2}(0,a)\cap L^{\infty}(0,a)$, $k_{2}\in L^{2}(0,b)\cap
L^{\infty}(0,b)$. Thus, in this case $W_{G}$ is unitarily equivalent to the
MTTO on the half plane $A_{H}^{\Theta}$, where we adopt the convention that
$A_{H}^{\Theta}$ is initially densely defined on
$\mathcal{F}^{-1}\begin{pmatrix}L^{\infty}(0,a)\\\
L^{\infty}(0,b)\end{pmatrix}$.

A computation shows that $V^{-1}\mathcal{F}^{-1}L^{\infty}(0,a)$ is bounded
and lies in the model space $K_{u}^{2}$, where $u=e^{ia\frac{1-z}{1+z}}$ is an
inner function on the disc. As $V$ and $\mathcal{F}$ are isomorphic and
$L^{\infty}(0,a)$ is dense in $L^{2}(0,a)$, we must have
$V^{-1}\mathcal{F}^{-1}L^{\infty}(0,a)$ is contained in $K_{u}^{\infty}$ and
is dense subspace of $K_{u}^{2}$. A similar reasoning holds for
$L^{\infty}(0,b)$. If we again assume that $G=\hat{H}$, where
$H\mathcal{F}^{-1}\begin{pmatrix}L^{\infty}(0,a)\\\
L^{\infty}(0,b)\end{pmatrix}\subseteq(L^{2}(\mathbb{R}))^{2}$ then we can
deduce that $W_{G}$ is bounded if and only if $A_{H}^{\Theta}$ densely defined
on $\mathcal{F}^{-1}\begin{pmatrix}L^{\infty}(0,a)\\\
L^{\infty}(0,b)\end{pmatrix}$ is bounded if and only if the corresponding map
on the disc, given by $A_{H\circ m}^{\Theta\circ m}$ where
$m:\mathbb{D}\to\mathbb{C}_{+}$ is defined by
$m(z)=i\left(\frac{1-z}{1+z}\right)$ is bounded, i.e. if $A_{H\circ
m}^{\Theta\circ m}$ is an MTTO. Now with the above reasoning and Proposition
2.8, whenever $G=\hat{H}$ with
$H\mathcal{F}^{-1}\begin{pmatrix}L^{\infty}(0,a)\\\
L^{\infty}(0,b)\end{pmatrix}\subseteq(L^{2}(\mathbb{R}))^{2}$ and $H\circ m\in
L^{(p,n\times n)}$ for $p\in(2,\infty]$, we can test the continuity of (5.1)
by considering only the image of a corresponding modified MTTO.

In the case when $H\mathcal{F}^{-1}\begin{pmatrix}L^{\infty}(0,a)\\\
L^{\infty}(0,b)\end{pmatrix}\subseteq(L^{2}(\mathbb{R}))^{2}$ and $W_{G}$ is
bounded we have that $W_{G}$ is unitarily equivalent to $A_{H\circ
m}^{\Theta\circ m}$ via the unitary mapping $\mathcal{F}\circ V$.

Although the above demonstration is in the case of the 2-by-2 matrix $G$, it
is easily generalised to the $n$-by-$n$ case. Matricial truncated Wiener-Hopf
operators (which may also be called matricial convolution operators on finite
intervals) are studied in detail in Chapter 7 of [15]. They are then shown to
play an important role in Chapter 8 of [15], where a continuous analogue of
Krein’s Theorem for matrix polynomials is given.

Matricial truncated Wiener-Hopf operators are also encountered naturally when
finding the solution to MIMO (multi input, multi output) linear systems.
Consider a matrix indicator function $\mathbbm{1}_{\mathbb{R}_{-}}$, where
$\mathbbm{1}_{\mathbb{R}_{-}}(x)=I_{2\times 2}$ if $x\leqslant 0$ and
$\mathbbm{1}_{\mathbb{R}_{-}}(x)=0_{2\times 2}$ if $x>0$, and set
$G^{^{\prime}}(x)=\exp(Ax)\mathbbm{1}_{\mathbb{R}_{-}}(x)B$, where $A$ and $B$
are constant $2$-by-$2$ matrices. If we set $a=b$, then $x\mapsto
W_{G^{^{\prime}}}(u)(x)+\exp(Ax)v_{0}$ is the solution found for $v$ when
solving the system of MIMO state space equations given by

$\displaystyle\dot{v}$ $\displaystyle=Av+Bu,$ $\displaystyle v(0)=v_{0},$
(5.2) $\displaystyle y$ $\displaystyle=Cv+Du,$ (5.3)

where $u(x),v(x),y(x)\in\mathbb{C}^{2}$ are defined for all $x\in(0,a)$,
additionally $v$ is defined at 0 with the condition $v(0)=v_{0}$, $C$ and $D$
are constant $2$-by-$2$ matrices and $u\in(L^{2}(0,a))^{2}$. See Chapter 8 of
[24], and in particular the solution of equation 8.1 for a more detailed
discussion. The above working is easily generalised to the $n$-by-$n$ case.
MIMO systems are encountered in control theory, dynamical systems, electrical
engineering (see Chapter 5 of [13]) and find further applications in wireless
communication and multi-channel digital transmission.

### Acknowledgments

The author is grateful to the EPSRC for financial support.
The author is grateful to Professor Partington for his valuable comments.

### Declarations

This research was supported by the EPSRC (grant number 1972662).
Conflicts of interest/Competing interests- none.
Availability of data and material- not applicable.
Code availability- not applicable.
Data sharing not applicable to this article as no datasets were generated or
analysed during the current study.

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# Extended Møller-Plesset perturbation theory for dynamical and static
correlations

Takashi Tsuchimochi<EMAIL_ADDRESS>Troy Van Voorhis  Department of
Chemistry, Massachusetts Institute of Technology, 77 Massachusetts Ave.,
Cambridge MA 02139

###### Abstract

We present a novel method that appropriately handles both dynamical and static
electron correlation in a balanced manner, using a perturbation theory on a
spin-extended Hartree-Fock (EHF) wave function reference. While EHF is a
suitable candidate for degenerate systems where static correlation is
ubiquitous, it is known that most of dynamical correlation is neglected in
EHF. In this work, we derive a perturbative correction to a fully spin-
projected self-consistent wave function based on second-order Møller-Plesset
perturbation theory (MP2). The proposed method efficiently captures the
ability of EHF to describe static correlation in degeneracy, combined with
MP2’s ability to treat dynamical correlation effects. We demonstrate drastic
improvements on molecular ground state and excited state potential energy
curves and singlet-triplet splitting energies over both EHF and MP2 with
similar computational effort to the latter.

Introduction. An efficient and accurate treatment of both dynamical and static
electron correlation effects has been elusive in electronic structure theory.
Single reference methods such as second-order Møller-Plesset perturbation
theory (MP2) and coupled-cluster singles and doubles (CCSD) achieve high
accuracy in computed observables for non-degenerate systems,Bartlett07  but it
is well known that they cannot describe static correlation in degenerate
systems. This failure is undoubtedly attributed to the reference wave
function: Hartree-Fock (HF). A HF reference is qualitatively inadequate for
(nearly-) degenerate systems where the true wave function is multi-
determinantal in nature. Complete active space self-consistent field (CASSCF)
resolves this problem by treating all the configurations in an active space,
yielding a multi-reference state, and usually represents a good starting point
when an appropriate active space is chosen. When the residual dynamical
correlation is included through a perturbative correctionHirao92 ; Andersson94
or configuration interaction (CI), CASSCF can achieve very accurate results
both for the ground state and excited states. However, none of these are
black-box, and their computational cost is very expensive.

Yet another approach to tackling static correlation may be spin-extended HF
(EHF),Lowdin55 ; Mayer80  which is also called spin-projected HF. The idea
behind it is to optimize orbitals of a broken symmetry Slater determinant
$|\Phi_{0}\rangle$, called a deformed state, projected by a spin-projection
operator $\hat{P}$ so that the total energy of the projected state,

$\displaystyle E_{\rm
EHF}=\frac{\langle\Phi_{0}|\hat{P}^{{\dagger}}\hat{H}\hat{P}|\Phi_{0}\rangle}{\langle\Phi_{0}|\hat{P}^{{\dagger}}\hat{P}|\Phi_{0}\rangle}=\frac{\langle\Phi_{0}|\hat{H}\hat{P}|\Phi_{0}\rangle}{\langle\Phi_{0}|\hat{P}|\Phi_{0}\rangle},$
(1)

is variationally minimized. This approach in particular is called variation-
after-projection (VAP), not to be confused with projection-after-variation
(PAV), which has been widely used in quantum chemistry.
$\hat{P}|\Phi_{0}\rangle$ spans a large part of the Hilbert space, and thus is
expected to capture most of static correlation in a black-box manner, i.e., no
active space is required. At the same time, for this reason, it has long been
thought in quantum chemistry that the full spin-projection is computationally
demanding and horribly complicated, even for PAV. Recently,
Jim$\rm\acute{e}$nez-Hoyos et al.,Jimenez12  however, have shown a feasible
and clear way to accomplish VAP by using the spin-projection operator of the
general integral form,

$\displaystyle{\hat{P}}^{s}_{mk}=|s;m\rangle\langle
s;k|=\frac{2s+1}{8\pi^{2}}\int D_{mk}^{s*}(\Omega)\hat{R}(\Omega)d\Omega,$ (2)

instead of the famous L${\rm\ddot{o}}$wdin projector.Lowdin55  Here $s$ is the
total spin, $m$ and $k$ are the spin angular momentum,
$D_{mk}^{s}(\Omega)=\langle s;m|{\hat{R}}(\Omega)|s;k\rangle$ is the Wigner
matrix, and
$\hat{R}(\alpha,\beta,\gamma)=e^{i\alpha\hat{S}_{z}}e^{i\beta\hat{S}_{y}}e^{i\gamma\hat{S}_{z}}$
is a unitary rotation operator. With this formalism, the computational effort
for EHF is known to be similar to that of mean-field methods.

Although EHF efficiently describes static correlation, it neglects a vast
amount of dynamical correlation, which is necessary for chemical accuracy. In
order to remedy this, there have been recently extensive work attempting to
incorporate the residual dynamical correlation into EHF, in the context of
density functional correlationGarza13  as well as non-orthogonal
CI,Rodriguez13  with promising results. In this work, we propose a
perturbative approach based on MP2, which we shall hereby term extended MP2
(EMP2). Since MP2 correlation is almost exclusively of dynamical character,
EMP2 should provide a seamless description of both static and dynamical
correlation effects. A similar idea was pursued for PAV in the late
1980s,Schlegel86 ; Knowles88A  but was immediately abandoned due to its
enormous computational cost even for approximate projection and many undesired
features such as pronounced derivative discontinuities in potential energy
surfaces. Below we show that, with the present scheme, all of these obstacles
can be thoroughly resolved with full spin-projection.

Theory. Throughout this Communication, we restrict ourselves to the cases
where $|\Phi_{0}\rangle$ is an eigenstate of ${\hat{S}}_{z}$ but not of
${\hat{S}}^{2}$, i.e. an unrestricted HF type determinant, and thus
$\hat{P}=\hat{P}_{mm}^{s}$. We will also adopt the conventional notations of
orbital indices: $i,j,k,l$ for occupied, $a,b,c,d$ for virtual, and $p,q,r,s$
for all orbitals.

Perturbation approaches for projected wave functions have been proposed by
many others.Amos70 ; Peierls73 ; Knowles88A  Here we will derive our own
scheme. We start by partitioning the Hamiltonian into
$\hat{H}=\hat{H}_{0}+\lambda\hat{V}$, such that

$\displaystyle\hat{H}_{0}|\Phi_{0}\rangle=E_{0}|\Phi_{0}\rangle.$ (3)

We remind the reader that $|\Phi_{0}\rangle$ is the broken symmetry deformed
state. Given the Sch${\rm\ddot{o}}$dinger equation,

$\displaystyle\hat{H}|\Psi\rangle={\cal E}|\Psi\rangle,$ (4)

$\cal E$ and $|\Psi\rangle$ are expanded around $E_{0}$ and $|\Phi_{0}\rangle$
to find $n$-th order energies and wave functions, $E_{n}$ and
$|\Phi_{n}\rangle$. The MP2 energy expression then becomes

$\displaystyle E^{(0)}_{\rm
MP2}=\langle\Phi_{0}|\hat{H}|\Phi_{0}+\Phi_{1}\rangle=E_{\rm
HF}^{(0)}+E_{2}^{(0)}$ (5)

where $E_{\rm HF}^{(0)}$ and $E_{2}^{(0)}$ are the HF energy and the second
order perturbation correlation energy of the deformed state, and
$|\Phi_{1}\rangle$ is the first order wave function, which we will define
later for our case. Note that we have not yet defined $\hat{H}_{0}$.
Nevertheless, it is an independent particle symmetry broken Hamiltonian, and
$|\Phi_{1}\rangle$ consists of up to doubly excited determinants from
$|\Phi_{0}\rangle$.

Here our goal is to derive a perturbation theory that begins with $|\Psi_{\rm
EHF}\rangle\equiv\hat{P}|\Phi_{0}\rangle$ and accomplishes the exact energy at
the infinite order limit. Because $|\Psi_{\rm EHF}\rangle$ has no well-defined
independent particle Hamiltonian, however, one faces the difficulty of
defining an appropriate zeroth order Hamiltonian. It should be clear that
$\hat{H}_{0}$ defined in Eq.(3) is not suitable, as $|\Psi_{\rm EHF}\rangle$
is not its eigenstate. Hence, we consider the expansion of $\cal E$ and
$|\Psi\rangle$ in the projected space around $|\Psi_{\rm EHF}\rangle$. In the
present scheme, our expansion for the wave function is based on the MP
partitioning, given by

$\displaystyle|\Psi\rangle$
$\displaystyle=\hat{P}|\Phi_{0}\rangle+\lambda\hat{P}|\Phi_{1}\rangle+\cdots.$
(6)

This is possible because the exact wave function can always be chosen as an
eigenstate of $\hat{P}$, i.e., $\hat{P}|\Psi\rangle=\Lambda|\Psi\rangle$. Note
that the spaces spanned by $\hat{P}|\Phi_{n}\rangle$ are not orthogonal one
another, and are necessarily overcomplete.Amos70 ; Peierls73  Eq. (6) allows
us to write the exact energy in the intermediate normalization,

$\displaystyle{\cal E}(\lambda)$ $\displaystyle=\frac{\langle\Psi_{\rm
EHF}|\hat{H}\hat{P}|\Phi_{0}+\lambda\Phi_{1}+\cdots\rangle}{\langle\Psi_{\rm
EHF}|\hat{P}|\Phi_{0}+\lambda\Phi_{1}+\cdots\rangle}$ (7)
$\displaystyle=E_{\rm EHF}+\lambda{\cal E}_{2}+\lambda^{2}{\cal E}_{3}\cdots,$
(8)

which achieves our goal, i.e., ${\cal E}(0)=E_{\rm EHF}$ and ${\cal
E}(1)={\cal E}$. All the perturbative information is then carried in Eq.(6)
and one is free from defining a zeroth order Hamiltonian. ${\cal E}_{2}$
parallels second order Rayleigh-Schr${\rm\ddot{o}}$dinger perturbation theory.
Thus, we refer to it as the second order energy. Expanding ${\cal E}(\lambda)$
around $\lambda_{0}=0$ and setting $\lambda=1$, we find

$\displaystyle E_{\rm EMP2}=E_{\rm EHF}+{\cal E}_{2},$ (9) $\displaystyle{\cal
E}_{2}=\frac{\langle\Phi_{0}|(\hat{H}-E_{\rm
EHF})\hat{P}|\Phi_{1}\rangle}{\langle\Phi_{0}|\hat{P}|\Phi_{0}\rangle}.$ (10)

This formalism has various desired features. First, each term is rigorously
defined by the magnitude of order parameter $\lambda$. Second, there is no
singles contribution from $|\Phi_{1}\rangle$ due to the generalized Brillouin
theorem,

$\displaystyle\langle\Phi_{0}|(\hat{H}-E_{\rm
EHF})\hat{P}a_{a}^{\dagger}a_{i}|\Phi_{0}\rangle=0,$ (11)

when the EHF state is stationary, similar to the property in the conventional
MP$n$ theory.111In fact, Eq.(11) is the EHF Fock matrix element defined as
$\partial E_{\rm EHF}/\partial P^{(0)}_{ia}$ with normalization
$\langle\Phi_{0}|\hat{P}|\Phi_{0}\rangle$. Last, and perhaps most importantly,
the perturbation series are spin-projected at all orders, including
$|\Phi_{1}\rangle$. In fact, it can be shown that Eq.(10) may be seen as the
fully spin-projected MP2 if $|\Phi_{0}\rangle$ is the stationary unrestricted
HF state. That is, by defining a projector onto the complementary space
orthogonal to $|\Psi_{\rm EHF}\rangle$,

$\displaystyle\hat{O}=1-\frac{\hat{P}|\Phi_{0}\rangle\langle\Phi_{0}|\hat{P}}{\langle\Phi_{0}|\hat{P}|\Phi_{0}\rangle}=1-\frac{|\Psi_{\rm
EHF}\rangle\langle\Psi_{\rm EHF}|}{\langle\Psi_{\rm EHF}|\Psi_{\rm
EHF}\rangle},$ (12)

Eqs.(9-10) are elegantly rewritten as

$\displaystyle E_{\rm EMP2}$
$\displaystyle=\frac{\langle\Phi_{0}|\hat{H}\hat{P}|\Phi_{0}+\hat{O}\Phi_{1}\rangle}{\langle\Phi_{0}|\hat{P}|\Phi_{0}\rangle}.$
(13)

This clearly indicates that the last term in the numerator of Eq.(13) lives in
the space orthogonal to $|\Psi_{\rm EHF}\rangle$ to eliminate double-counting
of correlation effects, and is subject to spin-projection. We also note that
the occurrence of double-counting is a natural consequence because, again, the
basis Eq.(6) is overcomplete.

Now we shall move our attention to the definition of ${\hat{H}}_{0}$ and thus
$|\Phi_{1}\rangle$ for EMP2. The performance of a perturbation theory
critically depends on ${\hat{H}}_{0}$. Since we rely on the MP expansion of
the deformed state, i.e. Eq.(6), a physical choice for $\hat{H}_{0}$ is given
by HF-like orbital energies evaluated from $|\Phi_{0}\rangle$. They are indeed
an appropriate candidate in view of Eq.(13): EMP2 may be regarded as the full
spin-projection of broken-symmetry MP2 for some special case. Also, this
scheme is guaranteed to reduce to the regular MP2 when $|\Phi_{0}\rangle$ is
already a spin-eigenstate or $\hat{P}=1$, providing a seamless connection.

At the stationary state, $|\Phi_{0}\rangle$ is not an eigenfunction of a sum
of Fock operators, which is $\hat{H}_{0}$ of the conventional MP2. However,
$|\Phi_{0}\rangle$ is still a Slater determinant, and $|\Psi_{\rm EHF}\rangle$
is invariant with respect to a unitary rotation among $|\Phi_{0}\rangle$.
Hence, we diagonalize the occupied-occupied ($oo$) block and virtual-virtual
($vv$) block of the deformed Fock matrix, defined as

$\displaystyle F^{(0)}_{pq}=h^{(0)}_{pq}+\sum_{rs}P^{(0)}_{rs}\langle
pr||qs\rangle,$ (14)

with the deformed density matrix
$P^{(0)}_{rs}=\langle\Phi_{0}|a_{s}^{\dagger}a_{r}|\Phi_{0}\rangle$, and
choose $\hat{H}_{0}=\sum_{p}\varepsilon_{p}a_{p}^{\dagger}a_{p}$ with
$\varepsilon_{p}=F^{(0)}_{pp}$. The orbital basis of this particular choice
has been referred to as semi-canonical orbitals in literature. Consequently,
we will have not only doubles but also, potentially, singles contributions in
$|\Phi_{1}\rangle$,

$\displaystyle|\Phi_{1}\rangle$
$\displaystyle=\sum_{ia}|\Phi_{i}^{a}\rangle\frac{F^{(0)}_{ia}}{\varepsilon_{i}-\varepsilon_{a}}+\frac{1}{4}\sum_{ijab}|\Phi_{ij}^{ab}\rangle\frac{\langle
ij||ab\rangle}{\varepsilon_{i}+\varepsilon_{j}-\varepsilon_{a}-\varepsilon_{b}}$
$\displaystyle=\sum_{ia}t_{i}^{a}|\Phi_{i}^{a}\rangle+\frac{1}{4}\sum_{ijab}t_{ij}^{ab}|\Phi_{ij}^{ab}\rangle,$
(15)

because $F_{ia}^{(0)}$ are nonzero in general. As mentioned above, however,
all the singles contribution strictly vanish through $\hat{P}$ due to the
generalized Brillouin theorem, Eq.(11). Thus we only require the second term.

Finally, we discuss how one evaluates the projected coupling terms
$\langle\Phi_{0}|\hat{H}\hat{P}|\Phi_{ij}^{ab}\rangle$ and
$\langle\Phi_{0}|\hat{P}|\Phi_{ij}^{ab}\rangle$ that appear in Eq.(10). In
practice, each term can be decomposed to a discretized grid integration as

$\displaystyle\langle\Phi_{0}|\hat{H}\hat{P}|\Phi_{ij}^{ab}\rangle$
$\displaystyle=\sum_{g}^{N_{\rm
grid}}w_{g}\langle\Phi_{0}|\hat{H}\hat{R}_{g}|\Phi_{ij}^{ab}\rangle,$ (16)

where $w_{g}$ are the grid weights and $\hat{R}_{g}$ is the rotation operator
defined earlier but for each grid point $g$.Jimenez12  The brute-force
calculation of this term with the generalized Wick theoremRingSchuck  would
require ${\cal O}(N^{4})$ for each matrix element and therefore it gives rise
to a total complexity of ${\cal O}(N^{8}N_{\rm grid})$ for all the double
substitutions, which is intractable. To ameliorate the computational effort,
we will take a couple of steps.

Inserting the identity operator,
$1=|\Phi_{0}\rangle+\sum_{kc}|\Phi_{k}^{c}\rangle+\cdots$, between ${\hat{H}}$
and $\hat{R}_{g}$ in Eq. (16), we arrive at

$\displaystyle\langle\Phi_{0}|\hat{H}\hat{R}_{g}|\Phi_{ij}^{ab}\rangle=$
$\displaystyle E_{\rm
HF}^{(0)}\langle\Phi_{0}|\hat{R}_{g}|\Phi_{ij}^{ab}\rangle+\sum_{kc}F^{(0)}_{kc}\langle\Phi_{k}^{c}|\hat{R}_{g}|\Phi_{ij}^{ab}\rangle$
$\displaystyle+\frac{1}{4}\sum_{klcd}\langle
kl||cd\rangle\langle\Phi_{kl}^{cd}|\hat{R}_{g}|\Phi_{ij}^{ab}\rangle.$ (17)

In this way, only the rotation couplings
$\langle\Phi_{kl}^{cd}|\hat{R}_{g}|\Phi_{ij}^{ab}\rangle$, etc, are to be
evaluated.

Here, the important realization is that $\hat{R}_{g}$ will not mix orbitals
with one another, but instead independently rotate each spin orbital to give
general spin orbitals (i.e., $\alpha$ and $\beta$ spins are mixed). Therefore,
the ${\hat{R}}_{g}$ rotation on an excited determinant
$|\Phi_{ij}^{ab}\rangle=a_{a}^{\dagger}a_{b}^{\dagger}a_{j}a_{i}|\Phi_{0}\rangle$
is identical to the corresponding excitation of the rotated determinant
$|{{}^{g}\Phi}\rangle\equiv\hat{R}_{g}|\Phi_{0}\rangle$,

$\displaystyle\hat{R}_{g}|\Phi_{ij}^{ab}\rangle=c_{a}^{\dagger}c_{b}^{\dagger}c_{j}c_{i}|{{}^{g}\Phi}\rangle=|^{g}\Phi_{ij}^{ab}\rangle,$
(18)

where $c_{p}^{\dagger}$ and $c_{p}$ are the rotated creation and annihilation
operators, $c_{p}^{\dagger}=\hat{R}_{g}a^{\dagger}_{p}\hat{R}_{g}^{\dagger}$,
etc, and ${}^{g}|\Phi\rangle=\prod_{i}c_{i}^{\dagger}|-\rangle$ with
$|-\rangle$ being the bare vacuum. Then, the rotation couplings in Eq.(17) are
realized as just the overlaps between excited non-orthogonal general HF (GHF)
determinants.

This fact is particularly useful for our purpose, because all the simplicities
in HF determinants are still available for $|^{g}\Phi\rangle$. Among the most
important ones is the corresponding pair theorem.Amos61  One can
biorthogonalize the orbitals of $|\Phi_{0}\rangle$ and $|{{}^{g}\Phi}\rangle$,
$|p\rangle$ and $|{{}^{g}q}\rangle$, by performing a singular value
decomposition of the $oo$ and $vv$ blocks of the overlap matrix
${{}^{g}S}_{pq}=\langle p|{{}^{g}q}\rangle$. By the aforementioned theorem,
which of course holds for GHF determinants, the resulting ${{}^{g}\bf S}$
matrix in the corresponding orbital basis is banded: not only the $oo$ and
$vv$ blocks but also the $ov$ and $vo$ blocks can be chosen to be diagonal.
This greatly simplifies the overlap evaluationYost13  and makes it possible to
retain only ${\cal O}(N^{4})$ cost for the contraction of Eq.(17), using the
significant sparsity of $\langle\Phi_{kl}^{cd}|{{}^{g}\Phi_{ij}^{ab}}\rangle$
with a very simple algorithm. The limiting step of EMP2 is thus the
computation and transformation of $t$ amplitudes as well as two electron
integrals as in the regular MP2, which scales as ${\cal O}(N^{5})$. Note that
the final energy is invariant with respect to these orbital rotations.

Figure 1: $Top.$ Potential energy curves of the H2 molecule. $Bottom.$
Deviations from the FCI energy for FH.

Table 1: Non-parallelity error against FCI in kcal/mol. | EHF | EMP2 | MP2 | CCSD
---|---|---|---|---
H2222cc-pV5Z. | 4.5 | 0.7 | 13.3 | 0.0
FH | 2.8 | 0.8 | 10.2 | 2.4
H2O | 10.0 | 2.2 | 36.8 | 7.3
N23331$s$ orbitals are frozen in FCI. | 24.6 | 6.3 | 439.7 | 23.3
Table 2: Singlet-triplet splitting energies for diatomic molecules in kcal/mol ($\Delta E_{\rm ST}=E(^{1}\Delta)-E(^{3}\Sigma)$). | EHF | EMP2 | MP2 | CCSD | FCI
---|---|---|---|---|---
NH | 49.60 | 45.47 | 58.06 | 50.85 | 45.51
OH- | 62.58 | 58.06 | 74.83 | 64.46 | 58.34
O21111$s$ orbitals are frozen in FCI. | 35.99 | 28.80 | 30.75 | 32.71 | 25.54
NFa | 47.87 | 40.21 | 50.69 | 48.48 | 40.87
MAE | 6.44 | 1.06 | 11.02 | 6.56 |

Results. We have implemented Eq.(10) in our in-house quantum chemistry program
with the proposed contraction scheme. All the calculations were done with a
6-31G basis to enable the direct comparison with the exact full CI (FCI)
results, except the hydrogen molecule. In the top panel of Figure 1, we depict
the potential energy curve of H2 with cc-$p$V5Z. As is well known, the MP2
energy is accurate in the short range where a tremendous amount of dynamical
correlation is required, but it completely fails when a bond is stretched, due
to its inability to describe static correlation. It is evident that the almost
opposite event is observed in EHF. It dissociates H2 exactly, being less
accurate in the vicinity of the equilibrium bond length, Re. As one would
expect, EMP2 eliminates these disadvantages. It gives even slightly better
energies than MP2 near Re where static correlation is considered negligible,
while it starts to gain static correlation seamlessly toward the dissociation
limit. Overall, the potential curve of EMP2 is in excellent agreement with
FCI; the mean absolute error (MAE) is only 1.1 kcal/mol. These behaviors of
correlation effects can be seen generically. The bottom panel of Figure 1
shows the deviation of the total energy from FCI in the hydrogen fluoride
molecule dissociation. Again, the conventional MP2 becomes notoriously worse
after R${}_{\rm F-H}=1.5$ Å due to the degeneracy appearing. While the EHF
error is mostly flat and goes to the correct dissociation limit (although not
size-consistent Jimenez12 ), it vastly underestimates the dynamical
correlation. CCSD, which almost superposes on FCI near Re, loses its accuracy
significantly and is usually difficult to converge as the bond is stretched.
EMP2 yields the most accurate results over the entire region. The error
observed throughout the dissociation coordinate is almost constant for this
case. In Table 1, we list the non-parallelity errors (NPE), defined as the
error deviation from its MAE, i.e., NPE $={\rm avg}(|\Delta E-{\rm MAE}|)$, a
measure of how parallel the potential energy curve is to FCI. We also
performed the same analysis for the H2O (symmetric dissociation) and N2
molecules, all listed in Table 1, showing the good performance of EMP2.

Figure 2: Errors in potential energy curves of the $B{{}^{1}\Sigma}_{u}^{+}$
state in H2.

We also report singlet-triplet splitting energies, $\Delta E_{\rm ST}$, of
small diatomic molecules. The experimental geometries are used,CRC  and 1$s$
orbitals are frozen in the FCI calculations for O2 and NF. For triplet states,
we have used unrestricted methods for MP2 and CCSD. $\Delta E_{\rm ST}$ is
only accurate if a method offers a balanced description of dynamical and
static correlations. As tabulated in Table 2, we found EHF and CCSD share a
similar quality with MAEs of 6.44 and 6.56 kcal/mol, respectively. Although
CCSD includes the required (double) excitations, singlet states are not
treated as accurately as are triplet states because the reference closed-shell
HF orbitals are inadequate. This causes the consistent overestimation of
$\Delta E_{\rm ST}$. On the other hand, EMP2 outperforms other methods,
achieving an impressive improvement over EHF with a MAE of 1.06 kcal/mol.

Finally, we investigate the excited state of H2 by $\Delta$SCF where an
excited configuration is achieved by occupying electrons in virtual
orbitals.Gilbert08  This state-specific non-aufbau approach is also applicable
to EHF and allows us to compute low-lying excited states as a spin-pure state.
Therefore, as opposed to the conventional $\Delta$SCF using HF, which suffers
from significant spin-contamination, the spin-purification procedure is not
needed in EHF. Furthermore, since such EHF state is stationary, one can
directly perform EMP2. Figure 2 presents the error in potential energy curve
of the first excited $B{{}^{1}\Sigma_{u}^{+}}$ state of H2 against the FCI
result, using 6-31G**. While $\Delta$SCF (i.e., HF) gives a qualitatively
reasonable potential when purified, MP2 correction to $\Delta$SCF (denoted as
$\Delta$MP2) miserably diverges. This is due to the degeneracy appearing in
the dissociation limit with the second excited state. EMP2, however, has no
such issue. It improves the EHF energy by adding dynamical correlation on top
of it, and yields almost the exact potential curve. This encouraging result
demonstrates the applicability of EMP2 to excited states.

We close our discussions by stressing once again that the method presented
here achieves a black-box treatment of accurate dynamical and static
correlation with a moderate computational effort similar to the conventional
MP2.

The authors are grateful to Gustavo E. Scuseria for fruitful discussions. This
work was supported by NSF (CHE-1058219).

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