# An Abstract Beurling’s Theorem for Several Complex Variables I

Charles W. Neville  1 Chatfield Drive, Apartment 136
West Hartford, CT 06110
USA  chip.neville at gmail.com

###### Abstract.

How to extend Beurling’s theorem on the shift invariant subspaces of Hardy
class $H^{2}$ of the unit disk to several complex variables has been an open
problem at least since 1964. In this paper, we prove a generalization of
Beurling’s theorem to valuation Hilbert modules over valuation algebras. We
shall apply our abstract Beurling’s theorem to obtain a complete description
of the closed invariant subspaces of $H^{2}$ of the polydisk. As we shall show
in a subsequent paper, further consequences of our abstract Beurling’s theorem
are complete extensions of Beurling’s theorem to many more Hilbert spaces of
analytic functions in several complex variables.

###### 1991 Mathematics Subject Classification:

42B30 (47A15 32A35)

Dedicated to the memory of the late Lee Rubel. Lee, this one’s for you.

##  1\. Introduction.

When confronted with an $R$ module $M$ over a ring $R$, one of the first
questions one can ask is, what are the $R$ submodules of $M$? Often, this
question is too difficult to answer, but if $M$ happens to be endowed with a
topology, one can ask, what are the _closed_ $R$ submodules of $M$. In a
celebrated 1949 paper, Arne Beurling did just this for the case where $R$ is
the algebra $P[\mathbb{C},z]$ of polynomials in the complex variable $z$, and
$M$ is Hardy class $H^{2}$ of the unit disk [4]. (See [11] for complete
details.) Beurling’s theorem completely characterizes the closed
$P[\mathbb{C},z]$ submodules of $H^{2}$ of the unit disk.

As for extensions of Beurling’s theorem to several complex variables, the
question was raised as far back as 1964 by Henry Helson ([10], p. 25 ff). In
1986, Agrawal, Clark, and Douglas observed, “Almost everyone who has thought
about this topic must have considered the corresponding problem for $H^{2}$ of
the polydisk” but that little progress had been made, except for counter-
examples and the important theorem of Ahern and Clark [1]. (The theorem of
Ahern and Clark characterizes the closed submodules of $H^{2}$ of the polydisk
of finite codimension as the closures of polynomial ideals of finite
codimension [2], cf. also the the exposition in [14].) Since Agrawal, Clark,
and Douglas wrote this, significant progress was made in related areas (cf.
the papers [1], [3], [13], [6], and [7]). In 2007, there was a
characterization of an important class of closed submodules of $H^{2}$ of the
polydisk with deep ties to operator theory [16]. But a complete
characterization of the closed submodules of $H^{2}$ of the polydisk had
eluded us.. In this paper, we shall prove an abstract generalization of
Beurling’s classical theorem, which specializes to a concrete generalization
of Beurling’s theorem to $H^{2}$ of the polydisk. In a second paper, we shall
apply our abstract generalization of Beurlings theorem to obtain.concrete
generalizations of Beurling’s theorem to a number of Hilbert modules of
analytic functions on domains in $\mathbb{C}^{n}$, and on multi-dimensional
complex analytic manifolds.

##  2\. Some Notation.

Throughout this paper, $\mathbb{C}$ will be the complex numbers and
$\mathbb{C}^{n}$ complex n-space. $\mathbb{R}$ will be the real numbers and
$\mathbb{R}_{+}$ the positive reals, $\\{r\in\mathbb{R}:r\geq 0\\}$.
$\mathbb{Z}$ will be the integers and $\mathbb{Z}_{+}$ the positive integers,
$\\{m\in\mathbb{Z}:m\geq 0\\}$. $E$ will be a complex Hilbert space.
$P[z_{1},\dots,z_{n}]=P[\mathbb{C},z_{1},\dots,z_{n}]$ will be the algebra of
polynomials with complex coefficients in the $n$ complex variables
$z_{1},\dots,z_{n}$, and $P[E,z_{1},\dots,z_{n}]$ will be the complex vector
space (and $P[\mathbb{C},z_{1},\dots,z_{n}]$ module) of polynomials with
$E$-valued coefficients in the $n$ complex variables $z_{1},\dots,z_{n}$. We
shall shamelessly identify polynomials with functions, so
$P[E,z_{1},\dots,z_{n}]$ can be thought of as the space of $E$-valued
polynomials in $z_{1},\dots,z_{n}$.

We shall denote the Hilbert space norm on $E$ by $\|\cdot\|_{E}$ and the inner
product by $\langle\cdot,\cdot\rangle_{E}$.

Throughout this paper, $R$ will be a (possibly non-commutative) complex
algebra (sometimes without a unit element) and $\mathbb{H}$ a left Hilbert
module over $R$. That is $\mathbb{H}$ will be both a Hilbert space and a left
$R$ module, and the multiplication maps $m_{r}\colon h\mapsto rh$ will be
continuous for each fixed $r\in R$. We shall denote the Hilbert space norm on
$\mathbb{H}$ by $\|\cdot\|$ and the inner product by
$\langle\cdot,\cdot\rangle$.

In the frequently occurring case where $\mathbb{H}$ is a Hilbert module of
$E$-valued analytic functions, the norm and inner product on $E$ will be
subscripted as indicated above, but they will be easy to distinguish from the
corresponding quantities on $\mathbb{H}$, which will be unsubscripted.

If $X$ is a subset of a larger topological space $Y$, we shall denote the
closure of $X$ by $\operatorname{cl}X=\operatorname{cl}_{Y}X$, the interior of
$X$ by $\operatorname{int}X=\operatorname{int}_{Y}X$, and the topological
boundary of $X$ by $\partial X=\partial_{Y}X$. Finally, $\text{card}(X)$ will
denote the cardinality of the set $X$.

##  3\. A Preview of Coming Attractions.

The special case where the Hilbert module $\mathbb{H}$ is Hardy class $H^{2}$
of the polydisk and the algebra $R$ is $P[\mathbb{C},z_{1},\dots,z_{n}]$
provides an excellent guide to our abstract Beurling’s theorem. It also
illustrates why our abstract Beurling’s theorem deserves to be considered a
generalization of Beurling’s theorem.

We need yet more standing notation. Throughout this paper, $D^{n}$ will be the
open $n$-dimensional unit polydisk,

$\\{z=(z_{1},\dots,z_{n})\in\mathbb{C}^{n}:|z_{j}|<1\text{ for
}j=1,\dots,n\\}$

and $H^{2}(E,D^{n})$ will be $E$-valued Hardy class $H^{2}$ of the polydisk.
Of course, $\mathbb{C}$ itself is a complex vector space (the simplest one),
so $H^{2}(\mathbb{C},D^{n})$ is the familiar scalar valued $H^{2}$ space of
the polydisk [14].

Throughout this paper, $T^{n}$ will be the $n$-dimensional unit torus,

$\\{z=(z_{1},\dots,z_{n})\in\mathbb{C}^{n}:|z_{j}|=1\text{ for
}j=1,\dots,n\\}.$

$T^{n}$ is the distinguished boundary of $D^{n}$ and a subset of the
topological boundary $\partial D^{n}$ of $D^{n}$. The standard parametrization
of $T^{n}$ is
$e^{\imath\theta}=(e^{\imath\theta_{1}},\dots,e^{\imath\theta_{n}})$,
$0\leq\theta_{j}\leq 2\pi$, for $j=1,\dots,n$. When speaking of $T^{n}$,
$d\sigma$ will be normalized Haar measure on $T^{n}$ with respect to the group
of coordinate rotations,

$z\mapsto
e^{\imath\theta}z=(e^{\imath\theta_{1}}z_{1},\dots,e^{\imath\theta_{n}}z_{n}),\>\theta\in\mathbb{R}^{n}$

The measure $d\sigma$ has the simple form, $d\sigma=d\theta_{1}\cdots
d\theta_{n}/(2\pi)^{n}$, so we shall often use the shorthand
$d\sigma=d\theta/(2\pi)^{n}.$

A function $h\colon D^{n}\mapsto E$ is $n$-harmonic if it is continuous, and
if at each point $a=(a_{1},\dots,a_{n})\in D^{n}$, each slice function

$(a_{1},\dots,a_{j}+z_{j},\dots,a_{n})\mapsto
h((a_{1},\dots,a_{j}+z_{j},\dots,a_{n}))$

is harmonic in $z_{j}$ near $a_{j}$ (c.f. [14]).

$H^{2}(E,D^{n})$ is the space of all $E$-valued functions $f$ analytic on
$D^{n}$, such that the $n$-subharmonic function $z\mapsto\|f(z)\|_{E}^{2}$ has
an $n$-harmonic majorant. The norm on $H^{2}(E,D^{n})$ is given by

$\|f\|^{2}=h_{f}(0)$

where $h_{f}$ is the least $n$-harmonic majorant of the function
$z\to\|f(z)\|_{E}^{2}$.

If the Hilbert space $E$ is of finite or countable dimension, there are no
measurability issues, and the well known boundary value theory for the scalar
valued case (cf [14]) carries over directly to the $E$-valued case. This
allows us to identify $H^{2}(E,D^{n})$ with the subspace of a.e. $d\sigma$
defined functions in $L^{2}(E,d\sigma)$ with Fourier coefficients supported in
the “first quadrant,” $\mathbb{Z}_{+}^{n}$, exactly as in the scalar valued
case.

However, if $E$ is of uncountable dimension, there are significant
measurability issues. Basically, these can be resolved by using the Dunford-
Pettis integral and weak measurability. ($f$ is weakly measurable if $\langle
f,e\rangle_{E}$ is measurable for each $e\in E$.)

We need to discuss how to extend an $H^{2}(E,D^{n})$ function $f$, only
defined a.e. on $T^{n}$, to the interior of $D^{n}$. The key is to use the
$n$-harmonic Poisson integral formuula

(3.1) $\begin{split}f(a)&=\int_{T^{n}}K(a,z)f(z)\,d\sigma(z)\\\
h_{f}(a)&=\int_{T^{n}}K(a,z)\|f(z)\|_{E}^{2}\,d\sigma(z)\end{split}$

for $a\in D^{n}$. Here, $K(a,z)$ is the $n$-harmonic Poisson kernel, equal to
the product $K_{1}(a_{1},z_{1})\cdots K_{1}(a_{n},z_{n})$ of ordinary Poisson
kernels for the unit disk $D^{1}$, and $h_{f}$ is, as before, the least
$n$-harmonic majorant of $\|f\|_{E}^{2}$. Note that $K(0,0)=1$, so the two
definitions of $H^{2}(E,D^{n})$ are equivalent even in norm.

We can also profitably use the Szegő integral formula for $f\in
H^{2}(E,D^{n})$:

(3.2) $f(a)=\int_{T^{n}}S(a,z)d\sigma(z)$

for $a\in D^{n}$. Here, $S(a,z)$ is the Szegő kernel for the polydisk, equal
to the product $S_{1}(a,z_{1})\cdots S_{1}(a,z_{n})$ of ordinary Szegő kernels
for the unit disk.

Finally, to make this absolutely simple, $H^{2}(E,D^{n})$ is the space of
square summable $E$-valued power series,

(3.3) $\\{f(z)=\sum a_{m}z^{m}:\sum\|a_{m}\|_{E}^{2}<\infty\\}$

Here, and throughout this paper, we freely use standard multi-index notation:

$a_{m}=a_{m_{1},\dots,m_{n}}\quad\text{and}\quad z^{m}=z_{1}^{m_{1}}\cdots
z_{n}^{m_{n}}$

for $m=(m_{1},\dots,m_{n})\in\mathbb{Z}_{+}^{n}$ and
$z=(z_{1},\dots,z_{n})\in\mathbb{C}^{n}$.

In later papers, we shall consider $H^{2}(E,D^{n}(r))$, where $r$ is a real
number with $0<r<1$, or $r=r_{1},\dots r_{n})$ is a poly-radius with
$0<r_{j}<1$ for $j=1,\dots n$. One simply has to divide $a_{j}$ and $z_{j}$ by
$r_{j}$ or $r$ in the above formulas to reduce the study of
$H^{2}(E,D^{n}(r))$ to that of $H^{2}(E,D^{n}))$ ([14] p. 3).

Whichever way, using formula (3.1), (3.2), or (3.3), we note several
additional properties of $\mathbb{H}=H^{2}(E,D^{n})$ and
$R=P[\mathbb{C},z_{1},\dots,z_{n}]$:

  1. (3.4.1)

$\mathbb{H}$ is a Hilbert module over the complex algebra $R$.

  2. (3.4.2)

Both $\mathbb{H}$ and $R$ come equipped with a
$\mathbb{Z}_{+}\mspace{0.5mu}\mathbin{\cup}\mspace{0.5mu}\\{\infty\\}$ valued
ord function which measures the order of the zero of $f\in\mathbb{H}$ or $p\in
R$ at $0$.

  3. (3.4.3)

For $p\in R$, $\operatorname{ord}(p)=0$ if $p$ is a unit of $R$.

  4. (3.4.4)

For $f\in\mathbb{H}$ or $f\in R$, $\operatorname{ord}(f)=\infty$ if and only
if $f$ is identically $0$.

  5. (3.4.5)

The $\operatorname{ord}$ function almost respects multiplication:
$\operatorname{ord}(pf)\geq\operatorname{ord}(p)$
$+\operatorname{ord}(f)$, and
$\operatorname{ord}(pq)\geq\operatorname{ord}(p)+\operatorname{ord}(q)$, for
$f\in\mathbb{H}$, $p\in R$, and $q\in R$. (Of course, in this case, inequality
may be replaced by equality, but we want to state this in as general a form as
possible.)

  6. (3.4.6)

The $\operatorname{ord}$ function respects scalar multiplication:
$\operatorname{ord}(\lambda f)=\operatorname{ord}(f)$ for
$\lambda\in\mathbb{C}$, $\lambda\neq 0$, and $f\in\mathbb{H}$ or $f\in R$.

  7. (3.4.7)

The $\operatorname{ord}$ function (almost) respects addition:
$\operatorname{ord}(f+g)\geq\\\
\min(\operatorname{ord}(f),\operatorname{ord}(g))$ for $f\text{ and
}g\in\mathbb{H}$ or $f\text{ and }g\in R$.

  8. (3.4.8)

The $\operatorname{ord}$ function is upper semi-continuous on $\mathbb{H}$.
Thus the subspaces $\mathbb{H}_{m}=\\{f\in\mathbb{H}:\operatorname{ord}(f)\geq
m\\}$ are closed for $m=1,2,\dots$ because $\mathbb{H}_{m}$ is the inverse
image of the set $R_{m}=\\{r\in R:\operatorname{ord}(r)\geq m\\}$.

There are, in fact, several possible $\operatorname{ord}$ functions. The one
we shall use is

######  Definition 3.5.

$\operatorname{ord}(f)=$ the degree of the lowest degree term in the power
series expansion of $f$ for $f\in\mathbb{H}$, and similarly for $R$.

We still need to define the degree of a monomial: The _degree_ of
$z^{m}=z_{1}^{m_{1}}\cdots z_{n}^{m_{n}}$ is $|m|$, where here and throughout
this paper, $|m|=|m_{1}|+\dots+|m_{n}|$. This, of course, is the standard
definition.

Another equivalent definition of the $\operatorname{ord}$ function is

######  Definition 3.6.

$\operatorname{ord}(f)=$ the lowest degree term in the expansion of $f$ in
homogeneous polynomials for $f\in\mathbb{H}$, and similarly for $p\in R$.

Of course, the degree of a homogeneous polynomial is the common degree of its
monomial terms. (See [15] for homogeneous expansions of analytic functions.)

In the next section, we shall call a left Hilbert module $\mathbb{H}$ over a
(possibly non-commutative) complex algebra $R$ a _valuation Hilbert module_ ,
and $R$ a _valuation algebra_ , if both are equipped with an
$\operatorname{ord}$ function satisfying properties (3.4.1) through (3.4.8).
Our abstract Beurling’s theorem will characterize the closed left $R$
submodules of of an arbitrary valuation Hilbert module. (Truth in advertising:
Actually, the closed $R_{1}$ submodules, where $R_{1}=\\{r\in
R:\operatorname{ord}(r)\geq 1\\}$. But if $R$ has a unit element $1_{R}$ and
$R=\mathbb{C}\cdot 1_{R}\,+\,R_{1}$, these are the same as the closed left $R$
submodules. See definition 4.10 and proposition 4.11.) As hinted above,
homogeneous subspaces, similar to subspaces of of homogeneous polynomials all
of the same degree or $\\{0\\}$, will play a prominent role.

To gain an inkling of what our actual abstract Beurling’s theorem will look
like in the polydisk case, recall the wandering subspace approach to the
classical $1$-variable Hilbert space valued Beurling’s theorem due to Halmos
[8]. (See [11] for a complete exposition.) This theorem characterizes the
closed $P[\mathbb{C},z]$ submodules (classically called closed shift invariant
subspaces) of $H^{2}(E,D^{1})$ as the subspaces of the form

(3.7) $V=I\cdot H^{2}(W,D^{1})$

where $W$ is a complex Hilbert space and $I$ is an operator valued inner
function. This means

(3.8) $I(z)\colon W\mapsto E$

is a (usually partial) isometry a.e. $d\theta$ on $T^{1}$, the unit circle.
Given a closed $P[\mathbb{C},z]$ submodule $V$, the operator valued inner
function is constructed from the _wandering subspace_

(3.9) $W=V\ominus_{\perp}zV$

Here and throughout this paper, $\ominus_{\perp}$ will denote the orthogonal
complement. Then an equivalent statement of the $1$-variable Beurling’s
theorem is

(3.10) $V=W\oplus_{\perp}zW\oplus_{\perp}z^{2}W\oplus_{\perp}\cdots$

where here and throughout this paper, $\oplus_{\perp}$ denotes the orthogonal
direct sum.

Our abstract Beurling’s theorem uses a similar idea. Specialized to the case
of $H^{2}(E,D^{n})$, our theorem may be briefly stated as follows:

Let $V$ be a closed subspace of $H^{2}(E,D^{n})$. In place of the single
wandering subspace $W$ in equations (3.9) and (3.10), we consider the
decreasing sequence of closed subspaces

(3.11) $\begin{split}V_{0}=V,\quad V_{1}&=\\{f\in V:\operatorname{ord}(f)\geq
1\\},\\\ V_{2}&=\\{f\in V:\operatorname{ord}(f)\geq 2\\},\>\dots\end{split}$

and the sequence of _near homogeneous_ subspaces,

(3.12) $W_{0}=V_{0}\ominus_{\perp}V_{1},\quad
W_{1}=V_{1}\ominus_{\perp}V_{2},\>\dots$

(A subspace is _near homogeneous_ if the $\operatorname{ord}$ function is
constant on all non-zero elements of the subspace.) Then (as we shall prove
later although it is fairly obvious)

(3.13) $V=W_{0}\oplus_{\perp}W_{1}\oplus_{\perp}W_{2}\oplus_{\perp}\cdots$

We call this orthogonal decomposition the _near homogeneous decomposition_ of
$V$.

Any closed subspace has a near homogeneous decomposition, but closed left
submodules also have a _near inner_ decomposition. In the polydisk case, a
closed subspace $W$ of $H^{2}(E,D^{n})$ is _near inner_ if $W\perp z^{m}W$ for
all multi-indices $m\in\mathbb{Z}_{+}^{n}$ with
$|m|=|m_{1}|+\cdots+|m_{n}|\geq 1$. We can define a _near inner decomposition_
of a closed subspace $V$ to be an orthogonal decomposition

(3.14) $V=W^{0}\oplus_{\perp}W^{1}\oplus_{\perp}W^{2}\oplus_{\perp}\cdots$

of near inner subspaces such that

(3.15) $z^{k}W^{l}\perp W^{m}$

whenever $|k|+l>m$. (The actual definition is slightly more complicated – see
definition 6.3.) We have superscripted the subspaces
$W^{0},W^{1},W^{2},\ldots$ to distinguish them from the near homogeneous
decomposition of $V$. However, there is a close connection between near
homogeneous decompositions and near inner decompositions, and this connection
constitutes the core of our abstract Beurling’s theorem:

######  Theorem 3.16.

Let $V$ be a closed subspace of $H^{2}(E,D^{n})$. Then $V$ is a closed
$P[\mathbb{C},z_{1},\dots,z_{n}]$ submodule of $V$ if and only if the near
homogeneous decomposition of $V$ is near inner and has the full projection
property.

The full projection property, which we shall not go into until section 7, is
designed to eliminate situations such as

(3.17)
$V=\operatorname{LH}\\{I\\}\oplus_{\perp}\operatorname{LH}\\{z^{2}I\\}\oplus_{\perp}\operatorname{LH}\\{z^{4}I\\}\oplus_{\perp}\cdots$

where $I$ is an inner function in $H^{2}(\mathbb{C},D^{1})$. (Here and
throughout this paper, $\operatorname{LH}$ denotes the linear hull of a set.)
In this case, the orthogonal decomposition (3.17) is the near homogeneous
decomposition of $V$, and it is near inner, but it does not have the full
projection property. This is a good thing because $V$ is not a
$P[\mathbb{C},z]$ submodule of $H^{2}(\mathbb{C},D^{1})$.

In the classical $1$-variable case, our theorem 3.16 says the the closed
$P[\mathbb{C},z]$ submodules of $H^{2}(\mathbb{C},D^{1})$ are exactly the
closed subspaces of the form (3.10), although this takes some proving. In
fact, the decomposition (3.10) turns out to be the near homogeneous
decomposition of $V$. Furthermore, it is near inner and has the full
projection property, as we shall show in a second paper. Thus our abstract
Beurling’s theorem really is a generalization of the classical $1$-variable
Beurling’s theorem in the case where the Hilbert space of values equals
$\mathbb{C}$.

We shall close with a brief outline of things to come. In section 4, we shall
carefully develop the theory of valuation Hilbert modules over valuation
algebras. In section 5, we shall discuss near homogeneous subspaces and
decompositions. In section 6, we shall discuss near inner subspaces and
decompositions, and give the full, correct definition of a near inner
decomposition. In section 7, we shall discuss the full projection property. In
section 8, we shall fully state and prove our abstract Beurling’s theorem
characterizing the closed submodules of valuation Hilbert modules over
valuation algebras. And in section 9, we shall apply our abstract Beurling’s
theorem to completely characterize the closed invariant subspaces of
$H^{2}(E,D^{n})$.

In our second paper mentioned above, we shall discuss analytic algebras and
analytic Hilbert modules, which provide a general setting for examples where
our abstract Beurling’s theorem applies. Our second paper paper will then
present concrete examples of the application of our abstract Beurling’s
theorem to several complex variables, including to $H^{2}$ spaces and Bergman
$A^{2}$ spaces. And at the end of our second paper, we shall list of a number
of open problems.

##  4\. Valuation Algebras and Hilbert Modules.

The notion of a valuation on a field will probably be familiar to most
readers. An extension of one form of this notion to algebras and Hilbert
modules will provide the setting for our abstract Beurling’s theorem. As you
may have already guessed from section 3, the motivating example for a
_valuation algebra_ and _valuation Hilbert module_ is the algebra
$P[\mathbb{C},z_{1},\dots,z_{n}]$ and Hilbert module $H^{2}(E,D^{n})$, with
the $\operatorname{ord}$ function introduced there as the valuation.

There are several notions of a valuation in abstract algebra. The one we shall
borrow (and amend slightly) is often called an _exponential valuation_ [12].
Perhaps we should call our notion of a valuation an‘ _inverse exponential
valuation_ , but the phrase is way too cumbersome, so we shall simply use the
term _analytic valuation_ to avoid confusion and distinguish it from the usual
notion of a valuation in abstract algebra. For concision, we shall usually
drop the word analytic and smply call our notion a _valuation_.

We state here and now that ALL algebras and Hilbert spaces will be _complex_.

######  Definition 4.1.

Let $R$ be a (possibly non-commutative) algebra. An _analytic valuation_ on
$R$ is a function
$\operatorname{ord}\colon\\!R\mapsto\mathbb{Z}_{+}\mspace{0.5mu}\mathbin{\cup}\mspace{0.5mu}\\{\infty\\}$
such that for all $r$ and $s\in R$,

  1. (4.1.1)

$\operatorname{ord}(r)=0$ if $r$ is a left or right unit of $R$.

  2. (4.1.2)

$\operatorname{ord}(r)=\infty$ if and only if $r=0$.

  3. (4.1.3)

$\operatorname{ord}(rs)\geq\operatorname{ord}(r)+\operatorname{ord}(s)$.

  4. (4.1.4)

$\operatorname{ord}(\lambda r)=\operatorname{ord}(r)$ for
$\lambda\in\mathbb{C},\lambda\neq 0$

  5. (4.1.5)

$\operatorname{ord}(r+s)\geq\min(\operatorname{ord}(r),\operatorname{ord}(s))$

Of course, condition (4.1.1) is satisfied vacuously if $R$ does not have a two
sided identity element.

Here and throughout, we follow the usual conventions with respect to $\infty$:
$m<\infty$ for all $m\in\mathbb{Z}$, and for such $m$,
$m\cdot\infty=\infty\cdot m=\infty$ if $m\neq 0$. Further, $\infty\cdot
0=0\cdot\infty=\infty$ and $\infty\cdot\infty=\infty$.

######  Definition 4.2.

A _valuation algebra_ is an ordered pair $(R,\operatorname{ord})$, where $R$
is a (possibly non-commutative) algebra and $\operatorname{ord}$ is an
analytic valuation on $R$.

######  Definition 4.3.

Let $(R,\operatorname{ord}_{R})$ be a valuation algebra, and let $\mathbb{H}$
be a Hilbert space which is a left $R$ Hilbert module. A _Hilbert module
valuation_ on $\mathbb{H}$ (with respect to $(R,\operatorname{ord}_{R})$) is a
function
$\operatorname{ord}\colon\mathbb{H}\mapsto\mathbb{Z}_{+}\mspace{0.5mu}\mathbin{\cup}\mspace{0.5mu}\\{\infty\\}$
such that for all $h$, $h_{1}$ and $h_{2}\in\mathbb{H}$, and for all $r\in R$,

  1. (4.3.1)

$\operatorname{ord}(h)=\infty$ if and only if $h=0$.

  2. (4.3.2)

$\operatorname{ord}(rh)\geq\operatorname{ord}_{R}(r)+\operatorname{ord}(h)$

  3. (4.3.3)

$\operatorname{ord}(\lambda h)=\operatorname{ord}(h)$ for
$\lambda\in\mathbb{C},\lambda\neq 0$.

  4. (4.3.4)

$\operatorname{ord}(h_{1}+h_{2})\geq\min(\operatorname{ord}(h_{1}),\operatorname{ord}(h_{2}))$.

  5. (4.3.5)

The $\operatorname{ord}$ function is upper semi-continuous on $\mathbb{H}$.

Note that each set $\mathbb{H}_{m}$ is a subspace by properties (4.3.3) and
(4.3.4). The subspaces
$\mathbb{H}_{m}=\\{h\in\mathbb{H}:\operatorname{ord}(h)\geq m\\}$ are closed
for $m=0,1,2,\,\dots$ because $\operatorname{ord}$ is upper semicontinuous.
The subspace $\mathbb{H}_{0}$ is automatically a closed subspace because
$\mathbb{H}_{0}=\mathbb{H}$. Fiinally, as part of the definition of an $R$
Hilbert module, the multiplication map $m_{r}\colon h\mapsto rh$ is continuous
for each $r\in R$.

We need to review some of the properties of upper semicontinuous functions
([5], [9]): Here is the easiest way to see that the prototype of the
$\operatorname{ord}$ function is upper semicontinuous at $h$, but that we
cannot define $\operatorname{ord}$ to be a continuous function: Recall that
the one of the conditions that $\operatorname{ord}$ is upper semicontinuous at
$h\in\mathbb{H}$ is

$\limsup\operatorname{ord}(h_{k})\leq\operatorname{ord}(h)\text{ if
}h_{k}\text{ is a sequence in }\mathbb{H}\text{ converging to }h.$

Now suppose that $\mathbb{H}=H^{2}$, and recall that for $h\in H^{2}$,
$\operatorname{ord}(h)$ is the order of the zero of $h$ at $0$. The upper
semicontinuity condition is clearly satisfied by $\operatorname{ord}$ in
$H^{2}$.

Now if $h(0)\neq 0$, $\operatorname{ord}(h_{k})$ converges to
$\operatorname{ord}(h)=0$, so $\operatorname{ord}$ is continuous at $h$. But
if $h$ has a zero at $0$, then $\operatorname{ord}(h)>0$. Choose $h_{k}=h+1/k$
and observe that $h_{k}$ converges to $h$, but $\operatorname{ord}(h_{k})=0$
does not converge to $\operatorname{ord}(h)$. Thus $\operatorname{ord}$ is not
continuous at $h$.

######  Definition 4.4.

Let $(R,\operatorname{ord}_{R})$ be a valuation algebra. A _valuation Hilbert
module_ over $(R,\operatorname{ord}_{R})$ is an ordered pair
$(\mathbb{H},\operatorname{ord})$, where $\mathbb{H}$ is a left Hilbert module
over $R$ and $\operatorname{ord}$ is a Hilbert module valuation on
$\mathbb{H}$ with respect ot $(R,\operatorname{ord}_{R})$.

Where it will cause no confusion, we shall denote both the algebra valuation
on $R$ and the Hilbert module valuation on $\mathbb{H}$ by the same symbol,
$\operatorname{ord}$. We shall also make the gloss, whenever convenient, of
denoting the valuation algebra $(R,\operatorname{ord})$ by $R$ alone, and the
valuation Hilbert module $(\mathbb{H},\operatorname{ord})$ by $\mathbb{H}$
alone. Finally, we note that the concepts of a valuation algebra and a
valuation Hilbert module apply equally well to _real_ algebras and _real_
Hilbert modules, but we shall not consider these here. Our focus will be
exclusively on _complex_ valuation algebras and Hilbert modules.

The inequalities in properties (4.1.3) and (4.3.2) will doubtless raise the
eyebrows of some readers, as will the fact that the algebra $R$ might be non-
commutative, might have divisors of zero, and might not have an identity
element. Normally in abstract algebra, a valuation algebra must be an integral
domain (ie. be commutative and have no divisors of zero) with an identity
element, and there must be equality rather than inequality in property
(4.1.3). There are examples of valuation Hilbert modules over valuation
algebras of bounded operator valued functions which are non-commutative and
have divisors of zero. In these examples, there will sometimes be inequality
in properties (4.1.3) and (4.3.2). However, we shall not consider these in
this or our subsequent paper.

In the most common examples of valuation algebras and valuation Hilbert
modules, equality does hold. We need a name for this situation:

######  Definition 4.5.

An algebra valuation $\operatorname{ord}$ on an algebra $R$ is _strict_ if
equality holds in property (4.1.3), that is if

(4.5.1) $\operatorname{ord}(rs)=\operatorname{ord}(r)+\operatorname{ord}(s)$

for all $r$ and $s\in R$. A _strict_ valuation algebra is a valuation algebra
for which the valuation $\operatorname{ord}_{R}$ is strict.

A Hilbert module valuation $\operatorname{ord}$ on a left Hilbert module
$\mathbb{H}$ is _strict_ if equality holds in property (4.3.2), that is if

(4.5.2) $\operatorname{ord}(rh)=\operatorname{ord}(r)+\operatorname{ord}(h)$

for all $r\in R$ and $h\in\mathbb{H}$. A _strict_ valuation Hilbert module is
a valuation Hilbert module for which the valuation $\operatorname{ord}$ is
strict.

If an algebra $R$ or a left Hilbert module over $R$ admits a strict valuation,
then the algebra must be quite well behaved.

######  Proposition 4.6.

Let $R$ be a strict valuation algebra. Then $R$ has no left divisors of zero
and no right divisors of zero. If $R$ is commutative, then it is an integral
domain.

###### Proof.

Suppose $R$ had divisors of zero. Let $r$ be a left divisor of zero and $s$ be
a right divisor of zero, such that $r\neq 0,s\neq 0$, but $rs=0$. Then
$\operatorname{ord}(rs)=\infty>\operatorname{ord}(r)+\operatorname{ord}(s)$,
which contradicts the strictness of $R$. ∎

Note that the second assertion of the proposition, that $R$ is an integral
domain, follows immediately from the definition of an integral domain as a
commutative algebra with no divisors of zero.

######  Proposition 4.7.

Let $\mathbb{H}$ be a strict valuation Hilbert module over a valuation algebra
$R$. Then $R$ is a strict valuation algebra and has no left divisors of zero,
and no right divisors of zero. Further, if $R$ is commutative, then $R$ is an
integral domain.

###### Proof.

That $R$ has no left divisors of zero and no right divisors of zero follows
immediately from proposition 4.6 once we prove that $R$ is strict. To that
end, let $r$ and $s$ be non-zero elements of $R$, and let $h$ be a non-zero
element of $\mathbb{H}$. Then

(4.8) $\begin{split}\operatorname{ord}((rs)h)&=ord(rs)+ord(h)\\\
\operatorname{ord}((rs)h)&=\operatorname{ord}(r(sh))=ord(r)+ord(sh)\\\
&=\operatorname{ord}(r)+\operatorname{ord}(s)+\operatorname{ord}(h)\end{split}$

Thus $\operatorname{ord}((rs)h)\neq\infty$, and $\operatorname{ord}(h)\text{\
certainly}\neq\infty$, so we can set

$ord(rs)+ord(h)=\operatorname{ord}(r)+\operatorname{ord}(s)+\operatorname{ord}(h)$

and cancel out $\operatorname{ord}(h)$ to obtain
$\operatorname{ord}(rs)=\operatorname{ord}(r)+\operatorname{ord}(s)$. Hence
$R$ is strict. ∎

The subalgebra $R_{1}=\\{r\in R:\operatorname{ord}(r)\geq 1\\}$ will be
particularly important in this paper. In fact, the natural form of our
abstract Beurling’s theorem will concern $R_{1}$ invariant subspaces.

######  Definition 4.9.

Let $R$ be a valuation algebra and let $\mathbb{H}$ be a valuation Hilbert
module over $R$. Then a closed subspace $V$ of $\mathbb{H}$ is $R_{1}$
_invariant_ if $R_{1}V\subseteq V$.

A natural question is, when is an $R_{1}$ invariant subspace $V$ of
$\mathbb{H}$ a left submodule, that is when is $RV\subseteq V$? A useful
condition which guarantees this is given by the following definition and
proposition:

######  Definition 4.10.

Let $R$ be a valuation algebra with unit element (ie. identity element)
$1_{R}$. Then $R$ is a _unital_ valuation algebra if

(4.10.1) $R=\\{\lambda 1_{R}:\lambda\in\mathbb{C}\\}+R_{1}$

Note that the $+$ in equation 4.10.1 represents simple addition. It does not
represent a direct sum.

######  Proposition 4.11.

Let $R$ be a valuation algebra and let $\mathbb{H}$ be a left Hilbert module
over $R$. Then every $R_{1}$ invariant subspace of $\mathbb{H}$ is a left $R$
submodule of $\mathbb{H}$ if either of the following conditions hold:

  1. (4.11.1)

$R=R_{1}$, or

  2. (4.11.2)

$R$ is unital.

###### Proof.

Clear. ∎

A simple example of a valuation algebra satisfying condition 4.11.1 is

$P_{1}[\mathbb{C},z_{1},\dots,z_{n}]=\\{f\in
P[\mathbb{C},z_{1},\dots,z_{n}]:f(0)=0\\}$

Two iconic examples of valuation algebras satisfying condition 4.11.2 are the
algebra of polynomials $P[\mathbb{C},z_{1},\dots,z_{n}]$ and the algebra
$H^{\infty}(\mathbb{C},(M,p))$ of bounded complex valued analytic functions on
a complex manifold $M$ with basepoint $p$. In the first case,
$\operatorname{ord}(f)=$ the order of the zero of $f$ at $0$, and in the
second $\operatorname{ord}(f)=$ the order of the zero of $f$ at $p$.

##  5\. Near Homogeneous Decompositions.

The additional structure provided by the valuation on a valuation Hilbert
module allows us to decompose its elements into parts analogous to the
decomposition of a power series into a sum of homogeneous polynomials. We
shall call these decompositions _near homogeneous decompositions_.

It will be convenient to introduce some more standing notation. Unless
otherwise stated, $R$ will be a valuation algebra, $\mathbb{H}$ will be a
valuation Hilbert module over $R$, and $V$ will be a closed subspace of
$\mathbb{H}$. To avoid trivial issues, unless otherwise stated,
$V\neq\\{0\\}$. Remember from our standing notation in section 2 and
definition 4.4 that if $R$ is non-commutative, then $\mathbb{H}$ is a _left_
(as opposed to a _right_) Hilbert module over $R$.

######  Definition 5.1.

The valuation subspace series for $V$ is the decreasing sequence of closed
subspaces

(5.1.1) $V=V_{0}\supseteq V_{1}\supseteq V_{2}\supseteq\cdots$

where

(5.1.2) $V_{k}=\\{h\in V:\operatorname{ord}(h)\geq k\\}$

The valuation ideal series for $R$ is the decreasing sequence of two sided
ideals

(5.1.3) $R=R_{0}\supseteq R_{1}\supseteq R_{2}\supseteq\cdots$

where

(5.1.4) $R_{k}=\\{r\in R:\operatorname{ord}(r)\geq k\\}$

We have already met the valuation subspace series for $\mathbb{H}$; it is the
decereasing sequence of closed subspaces
$\mathbb{H}=\mathbb{H}_{0}\supseteq\mathbb{H}_{1}\supseteq\mathbb{H}_{2}\supseteq\cdots$
following property (4.3.5). Note that each of the sets $V_{k}$ is a subspace
by properties (4.3.3) and (4.3.4). Because
$V_{k}=\mathbb{H}_{k}\mathbin{\cap}V$, each of the sets $V_{k}$ in the
valuation subspace series (5.1.1) for $V$ is a closed subspace of $V$.

That each of the sets $R_{k}$ in the valuation ideal series (5.1.3) is in fact
a two sided ideal follows from properties (4.1.3), (4.1.4), and (4.1.5). That

(5.2) $\bigcap R_{k}=\\{0\\},\quad\bigcap
V_{k}=\\{0\\},\quad\bigcap\mathbb{H}_{k}=\\{0\\}$

all follow from properties (4.1.2) and (4.3.1) To see this, suppose, that
$h\in\bigcap V_{k}$. Then for each $k$, $h\in V_{k}$, so
$\operatorname{ord}(h)\geq k$ for each $k<\infty$. Thus
$\operatorname{ord}(h)=\infty$, so $h=0$ by property (4.3.1).

The valuation ideal series for $R$ is entirely analogous to the valuation
subspace series for $V$, except that we can use the Hilbert space structure of
$V$ to form orthogonal complements:

######  Definition 5.3.

The _near homogeneous decomposition_ of $V$ is the orthogonal direct sum

(5.3.1) $V=W_{0}\oplus_{\perp}W_{1}\oplus_{\perp}W_{2}\oplus_{\perp}\,\cdots$

where

(5.3.2)
$W_{0}=V_{0}\ominus_{\perp}V_{1},\;W_{1}=V_{1}\ominus_{\perp}V_{2},\;W_{2}=V_{2}\ominus_{\perp}V_{3},\,\dots$

and $V=V_{0}\supseteq V_{1}\supseteq V_{2}\supseteq\,\cdots$ is the valuation
subspace series for $V$. The orthogonal subspaces $W_{0}$, $W_{1}$,
$W_{2},\,\dots$ are called the _near homogeneous components_ of $V$.

We need an all too obvious comment here. For algebraists, a direct sum like
(5.3.1) is the subspace of all _finite_ sums of elements from the subspaces
$W_{k}$, $k\in\mathbb{Z}_{+}$. But for analysts like us, the orthogonal direct
sum (5.3.1) is the closed subspace of all norm convergent sums, whether
_finite_ or _infinite_ , of elements from the subspaces $W_{k}$,
$k\in\mathbb{Z}_{+}$.

Note that the near homogeneous components $W_{0},W_{1},W_{2},\,\dots$ are
closed because they are orthogonal complements of sets in closed subspaces.
Further, they are orthogonal by construction. However, that the orthogonal
direct sum (5.3.1) equals all of $V$ requires proof:

######  Proposition 5.4.

The orthogonal direct sum

$W=W_{0}\oplus_{\perp}W_{1}\oplus_{\perp}W_{2}\oplus_{\perp}\,\cdots$

equals all of $V$.

###### Proof.

By construction, $W\subseteq V$. We shall show that if $h\in V$ and $h\perp
W$, then $h=0$. This will prove $W=V$.

So let $h\in V$ and $h\perp W$. Then

$h\perp W_{0}\oplus_{\perp}W_{1}\oplus_{\perp}\cdots\oplus_{\perp}W_{m}$

for $m=0,1,2,\cdots$. By construction,

$V=W_{0}\oplus_{\perp}W_{1}\oplus_{\perp}\cdots\,\oplus_{\perp}W_{m}\oplus_{\perp}V_{m+1}$

so $h\in V_{m+1}$ for all $m\in\mathbb{Z}_{+}$. By the equations (5.2), $h=0$.
∎

It will be convenient at this point to introduce some additional standing
notation: Unless otherwise stated,
$\mathbb{H}=\mathbb{H}_{0}\supseteq\mathbb{H}_{1}\supseteq\mathbb{H}_{2}\supseteq\cdots$
will denote the valuation subspace series for $\mathbb{H}$, and
$\mathbb{H}=H_{0}\oplus_{\perp}H_{1}\oplus_{\perp}H_{2}\oplus_{\perp}\cdots$
will denote the near homogeneous decomposition of $\mathbb{H}$.
$V=V_{0}\supseteq V_{1}\supseteq V_{2}\supseteq\cdots$ will denote the
valuation subspace series for $V$, and
$V=W_{0}\oplus_{\perp}W_{1}\oplus_{\perp}W_{2}\oplus_{\perp}\cdots$ will
denote the near homogeneous decomposition of $V$. $R=R_{0}\supseteq
R_{1}\supseteq R_{2}\supseteq\cdots$ will denote the valuation ideal series
for $R$. Thus the meaning of $\mathbb{H}_{k}$, $H_{k}$, $V_{k}$, $W_{k}$, and
$R_{k}$ will be clear without further explanation.

To continue, unless otherwise stated,

(5.5) $\begin{split}P_{k}^{\mathbb{H}}\colon\mathbb{H}&\mapsto H_{k},\quad
Q_{k+1}^{\mathbb{H}}\colon\mathbb{H}\mapsto\mathbb{H}_{k+1}\\\ P_{k}^{V}\colon
V&\mapsto W_{k},\quad Q_{k+1}^{V}\colon V\mapsto V_{k+1}\end{split}$

will denote the orthogonal projections onto $H_{k}$, $\mathbb{H}_{k+1}$,
$W_{k}$, and $V_{k+1}$ respectively. Note that

(5.6) $\begin{split}P_{k}^{\mathbb{H}}+Q_{k+1}^{\mathbb{H}}&=I\text{ on
}\mathbb{H}_{k}\\\ P_{k}^{V}+Q_{k+1}^{V}&=I\text{ on }V_{k}\end{split}$

where $I$ is the identity map, and

(5.7)
$\begin{split}\sum_{j=0}^{k}P_{j}^{\mathbb{H}}+Q_{k+1}^{\mathbb{H}}&=I\text{
on }\mathbb{H}\\\ \sum_{j=0}^{k}P_{j}^{V}+Q_{k+1}^{V}&=I\text{ on
}V\end{split}$

Now we are equipped to talk about the series of near homogeneous components of
individual elements of $\mathbb{H}$ and $V$.

######  Definition 5.8.

Let $h\in\mathbb{H}$. The _near homogeneous decomposition_ of $h$ is the
convergent series of orthogonal terms

(5.8.1)
$h=P_{0}^{\mathbb{H}}(h)\oplus_{\perp}P_{1}^{\mathbb{H}}(h)\oplus_{\perp}P_{2}^{\mathbb{H}}(h)\oplus_{\perp}\,\cdots$

The individual terms $P_{k}^{\mathbb{H}}$ are called the _near homogeneous
components_ of $h$.

If $h\in V$, the _near homogeneous decomposition_ of $h$ _with respect_ to $V$
is the convergent series of orthogonal terms

(5.8.2)
$h=P_{0}^{V}(h)\oplus_{\perp}P_{1}^{V}(h)\oplus_{\perp}P_{2}^{V}(h)\oplus_{\perp}\,\cdots$

The individual terms $P_{k}^{V}$ are called the _near homogeneous components_
of $h$ _with respect_ to $V$

We shall postpone the proof that the series (5.8.1) and (5.8.2) are in fact
convergent and converge to $h$ until after we establish their connection to
the $\operatorname{ord}$ function:

######  Proposition 5.9.

For $h\in V$, $\operatorname{ord}(h)$ is the index (subscript) of the first
non-vanishing term in the near homogeneous decomposition of $h$ with respect
to $V$ if $h\neq 0$, or $\operatorname{ord}(h)=\infty$ if all the terms
vanish, that is if $h=0$.

###### Proof.

Suppose $h\in\mathbb{H}$ and $h\neq 0$. Then $\operatorname{ord}(h)=m<\infty$,
so $h\in V_{m}=W_{m}\oplus_{\perp}V_{m+1}$, and $h\notin V_{m+1}$. If
$P_{m}^{V}(h)=0$, then $h$ would belong to $V_{m+1}$ which is not possible.
Thus $P_{m}^{V}(h)\neq 0$.

On the other hand,

$V=W_{0}\oplus_{\perp}W_{1}\oplus_{\perp}\cdots\oplus_{\perp}W_{m-1}\oplus_{\perp}V_{m}$

so if one or more of the terms $P_{j}^{V}(h)\neq 0$ for $j=0,\dots,m-1$, then
$h$ would not belong to $V_{m}$, which is also not possible. Thus
$P_{j}^{V}=0$, for $j=0,\dots,m-1$, so the index of the first non-vanishing
term in the near homogeneous decomposition of $h$ with respect to $V$ is
$\operatorname{ord}(h)$.

Finally, all the terms $P_{j}^{V}(h)$ vanish if and only if $h\in\bigcap
V_{j}$, which is true if and only if $h=0$ by the equations (5.2). Thus all
the terms vanish if and only if $\operatorname{ord}(h)=\infty$. ∎

######  Corollary 5.10.

Let $h\in\mathbb{H}$. Then $\operatorname{ord}(h)$ is the index (subscript) of
the first non-vanishing term in the near homogeneous decomposition of $h$ if
$h\neq 0$, or $\operatorname{ord}(h)=\infty$ if all the terms vanish.

###### Proof.

Consider the special case where $\mathbb{H}=V$. ∎

######  Corollary 5.11.

If $h$ and $g$ belong to $V$, and $P_{k}^{V}(h)=P_{k}^{V}(g)$ for all
$k\in\mathbb{Z}_{+}$, then $h=g$.

###### Proof.

$P_{k}^{V}(h-g)=0$ for all $k\in\mathbb{Z}_{+}$. Apply proposition 5.9 ∎

######  Corollary 5.12.

Let $h$ and $g$ belong to $\mathbb{H}$. Suppose
$P_{k}^{\mathbb{H}}(h)=P_{k}^{\mathbb{H}}(g)$ for all $k\in\mathbb{Z}_{+}$.
Then $h=g$.

###### Proof.

Consider the special case where $\mathbb{H}=V$. ∎

We used the phrase “ _index_ of the first non-vanishing term” in the statement
of the proposition and its first corollary, even though we could have used the
phrase “ _order_ of the first non-vanishing term.” The reason is that that if
$P_{m}^{V}(h)\neq 0$, then $m=\operatorname{ord}(P_{m}^{V}(h))$. From now on,
we shall use the two phrases interchangeably.

One way of interpreting proposition 5.9 and corollary 5.10 is that all order
functions look a lot like the familiar order function on $H^{2}(E,D^{n})$
introduced in section 3, where $\operatorname{ord}(f)$ was the degree of the
first non-vanishing homogeneous polynomial in the expansion (3.6) of $f$ as a
sum of ordinary homogeneous polynomials. In fact, in this case, the expansion
mentioned in definition 3.6 is the near homogeneous decomposition of $f$, and
this motivates the use of the phrase “near homogeneous” in the general case of
an arbitrary valuation Hilbert module $\mathbb{H}$.

Until now, we have been treating the near homogeneous decomposition of $h$ as
a formal series. We are now in a position to prove that it converges to $h$.
At the considerable risk of boring the experts, we do so now:

######  Proposition 5.13.

Let $h\in V$. Then the near homogeneous decomposition (5.8.2) of $h$ with
respect to $V$ converges in norm to $h$.

###### Proof.

By equation (5.7),

$h=\sum_{j=0}^{k}P_{j}^{V}(h)\oplus_{\perp}Q_{k+1}^{V}(h)$

By orthogonality, the partial sums

$\|\sum_{j=0}^{k}P_{j}^{k}(h)\|^{2}=\sum_{j=0}^{k}\|P_{j}^{V}(h)\|^{2}\leq\|h\|^{2}$

so the partial sums form a Cauchy sequence in $V$. Because $V$ is closed, the
partial sums converge to some element $g\in V$. Now because $P_{k}^{V}$ is a
projection and a bounded linear operator on $\mathbb{H}$, and because
$P_{k}^{V}(P_{j}^{V})=0$ for $j\neq k$,

$P_{k}^{V}(g)=P_{k}^{V}(\lim_{l\to\infty}\sum_{j=0}^{l}P_{j}^{V}(h))=(P_{k}^{V})^{2}(h)=P_{k}^{V}(h)$

by orthogonality, so $g=h$ by corollary 5.11. ∎

######  Corollary 5.14.

Let $h\in\mathbb{H}$. Then the near homogeneous decomposition (5.8.1) of $h$
converges in norm to $h$.

###### Proof.

Consider the special case where $\mathbb{H}=V$. ∎

We turn now to the study of the near homogeneous components $W_{0}$, $W_{1}$,
$W_{2}$, $\\!\dots\\!$ of $V$. These have a number of striking properties.
First, each is a near homogeneous subspace. (Think: A linear combination of
ordinary homogeneous polynomials of the same degree is a homogeneous
polynomial.) Second, and most importantly for us, the near homogeneous part of
an element $h\in W_{k}$ determines the element $h$ itself. We shall now make
this precise:

######  Definition 5.15.

A closed linear subspace $W$ of $\mathbb{H}$ is _near homogeneous_ if the
$\operatorname{ord}$ function is constant on $W\mathbin{\sim}\\{0\\}$.

######  Proposition 5.16.

The near homogeneous components of $V$ are all near homogeneous subspaces. In
fact, $\operatorname{ord}(h)=m$ for all non-zero $h\in W_{m}$.

###### Proof.

$W_{m}$ is a closed subspace because it is an orthogonal complement. That
$\operatorname{ord}(h)=m$ follows immediately from proposition 5.9, but here
is another short proof: Let $h\in W_{m}$ and suppose that
$\operatorname{ord}(h)\neq m$. We shall show that $h$ must equal $\\{0\\}$.
Since $W_{m}=V_{m}\ominus_{\perp}V_{m+1}$, $h\in V_{m}$, and so
$\operatorname{ord}(h)\geq m$. Since $\operatorname{ord}(h)\neq m$,
$\operatorname{ord}(h)$ must be $>m$, and so $h\in V_{m+1}$. But $W_{m}\perp
V_{m+1}$, so $h\perp h$. Thus $h=0$. ∎

It can happen that one or more of the near homogeneous components of $V$ may
be $\\{0\\}$. For example, simply delete a non-zero homogeneous component of a
larger subspace to obtain $V$.

We now turn to the crucial property that projection of an element $h\in W_{k}$
onto $H_{k}$ determines the element $h$ itself. In fact, this is true for any
near homogeneous subspace $W$, not just for the near homogeneous components of
$V$:

######  Proposition 5.17.

Let $W$ be a near homogeneous subspace of $\mathbb{H}$ not equal to $\\{0\\}$,
and let $m$ be the common value of the $\operatorname{ord}$ function on
$W\mathbin{\sim}\\{0\\}$. Let $L_{W}$ be $P_{m}^{\mathbb{H}}$ restricted to
$W$, so that

(5.17.1) $L_{W}\colon W\mapsto P_{m}^{\mathbb{H}}(W)\subseteq H_{m}$

Then $L_{W}$ is an invertible bounded linear transformation.

###### Proof.

Linearity and boundedness are trivial. In fact,
$L_{W}={P_{m}^{\mathbb{H}}|}_{W}$ and $\|L_{W}\|\leq\|P_{m}^{\mathbb{H}}\|=1$.
To show that $L_{W}$ is invertible, let $h\in W$ and $h\neq 0$. Then
$m=\operatorname{ord}(h)$, and by corollary 5.10, $P_{m}^{\mathbb{H}}(h)$ is
the first non-vanishing term in the near homogeneous decomposition (5.8.1) of
$h$. Thus $L_{W}(h)=P_{m}^{\mathbb{H}}(h)\neq 0$, so $\ker(L_{W})=\\{0\\}$,
that is $L_{W}$ is invertible. ∎

In the exceptional case where $W=\\{0\\}$, the conclusion of the proposition
still holds because $h=0$ is the _only_ element of $W$.

An interesting question is, when does $L_{W}$ have a _bounded_ inverse? By the
closed graph theorem, the question is equivalent to, when is
$P_{m}^{\mathbb{H}}(W)$ closed?

The near homogeneous components of $V$ also have an interesting maximality
property, which we mention for completeness:

######  Proposition 5.18.

The non-zero near homogeneous components of $V$ are all maximal. In fact, if
$W_{m}\neq\\{0\\}$ and $W^{\prime}$ is a near homogeneous subspace of $V$ with
$W_{m}\subseteq W^{\prime}$, then $W_{m}=W^{\prime}$.

Our first proof of our abstract Beurling’s theorem used this proposition, but,
as it turns out, we can base a simpler proof on the projection lemmas in the
next section. Hence we shall not need proposition 5.18 in this paper, and so
we leave its proof to the reader.

##  6\. Near Inner Decompositions and the Projection Lemmas.

In the one variable case of Beurling’s theorem for the closed shift invariant
subspaces of $H^{2}(E,D^{1})$, inner functions play a prominent role, and
these arise from closed subspaces $W$ with the property that $W\perp z^{m}W$
for all $m>0$. We might call such subspaces _inner_ subspaces.

We can easily extend this idea to closed subspaces of a valuation Hilbert
module $\mathbb{H}$ over a valuation algebra $R$:

######  Definition 6.1.

A closed subspace $W$ of $\mathbb{H}$ is _near inner_ if

(6.1.1) $W\perp R_{1}W$

where $R=R_{0}\supseteq R_{1}\supseteq R_{2}\supseteq\cdots$ is the valuation
ideal series of $R$. An element $h$ of $\mathbb{H}$ is _near inner_ if the
$1$-dimensional subspace generated by it is near inner.

In the $1$-dimensional scalar valued case, a near inner function in
$H^{2}(\mathbb{C},D^{1})$ is simply a scalar multiple of an inner function,
hence the name “near inner” in the abstract case.

Our abstract version of Beurling’s theorem will involve a sequence of near
inner subspaces. The simple motivating example is

######  Example 6.2.

Let $V$ be the closed $P[\mathbb{C},z_{1},z_{2}]$ submodule of
$H^{2}(\mathbb{C},D^{2})$ generated by the coordinate functions $z_{1}$ and
$z_{2}$:

(6.2.1) $V=W_{0}\oplus_{\perp}W_{1}\oplus_{\perp}W_{2}\oplus_{\perp}\cdots$

where, as usual, the orthogonal direct sum is the near homogeneous
decomposition of $V$: $W_{0}=\\{0\\}$,
$W_{1}=\operatorname{LH}\\{z_{1},z_{2}\\}$,
$W_{2}=\operatorname{LH}\\{z_{1}^{2},z_{1}z_{2},z_{2}^{2}\\}$, etc. Here,
again as usual, $\operatorname{LH}$ denotes the linear hull.

Thus we are led to

######  Definition 6.3.

Let $V$ be a closed subspace of $\mathbb{H}$. A _near inner_ decomposition of
$V$ is an orthogonal direct sum of subspaces

(6.3.1) $V=W^{0}\oplus_{\perp}W^{1}\oplus_{\perp}W^{2}\oplus_{\perp}\cdots$

such that if $k$ and $m\in\mathbb{Z}_{+}$, $h\in W^{k}$, $g\in V$, $r\in
R_{1}$, and $\operatorname{ord}(rh+g)>m$, then

(6.3.2) $rh+g\perp W^{m}$

A few comments are in order. First, we have denoted the subspaces $W^{0}$,
$W^{1}$, $W^{2},$ $\dots$ using superscripts rather than subscripts because
they are not necessarily equal to the near homogeneous components
$W_{0},W_{1},W_{2},\,\dots$ of $V$. Second, each of the subspaces $W^{m}$ in a
near inner decomposition of $V$ is closed because the decomposition is an
orthogonal direct sum, and orthogonal complements in closed subspaces are
closed. (We could equally easily have dropped the condition that $V$ be
closed, but instead insisted that each subspace $W^{m}$ be closed. In other
words, a subspace $V$ given by an orthogonal direct sum of closed subspaces is
closed.) Third, if $\operatorname{ord}(h)\geq m$, we may choose $g=0$ to make
$\operatorname{ord}(rh+g)>m$. Thus $rh\perp W^{m}$ by equation (6.3.2). In
particular, $R_{1}W^{m}\perp W^{m}$, so each of the closed subspaces $W^{m}$
is itself near inner. For future reference, we shall state this last
observation as a proposition:

######  Proposition 6.4.

Let $W^{0}\oplus_{\perp}W^{1}\oplus_{\perp}W^{2}\oplus_{\perp}\cdot$ be a near
inner decomposition of $V$. Then each of the subspaces $W^{m}$ is closed and
near inner, and if $h\in W^{k}$, $m\in\mathbb{Z}_{+}$, $r\in R_{1}$, and
$\operatorname{ord}(rh)>m$, then $rh\perp W^{m}$.

###### Proof.

See the remarks above. ∎

It can be hard to verify in practice that a given orthogonal decomposition of
$V$ is near inner, but it is often easier to verify that the decomposition
satisfies the conclusion of proposition 6.4. Thus decompositions satisfying
this conclusion deserve a name:

######  Definition 6.5.

Let $V$ be a closed subspace of $\mathbb{H}$. A _weakly near inner_
decomposition of $V$ is an orthogonal direct sum of subspaces

(6.5.1) $V=W^{0}\oplus_{\perp}W^{1}\oplus_{\perp}W^{2}\oplus_{\perp}\cdots$

such that if $k$ and $m\in\mathbb{Z}_{+}$, $h\in W^{k}$, $r\in R_{1}$, and
$\operatorname{ord}(rh)>m$, then

(6.5.2) $rh\perp W^{m}$

Note that a weakly near inner decomposition is simply one which satisfies the
$g=0$ case of the definition of a near inner decomposition. Note also that the
subspaces in a weakly near inner decomposition are closed (by the remarks
above) and are themselves near inner because the definition of a near inner
subspace does not involve the auxiliary function $g$.

An interesting question is, when is a weakly near inner decomposition near
inner? In particular, is this always true for near homogeneous decompositions
which are weakly near inner?

Near inner decompositions are closely related to $R_{1}$ invariant subspaces.
Recall from definition 4.9 that $V$ is $R_{1}$ invariant if $R_{1}V\subseteq
V$. Recall also, from the standing notation introduced in section 5, that $V$
is always a _closed_ subspace of $\mathbb{H}$, that
$V=W_{0}\oplus_{\perp}W_{1}\oplus_{\perp}W_{2}\oplus_{\perp}\cdots$ is the
near homogeneous decomposition of $V$, and $V=V_{0}\supseteq V_{1}\supseteq
V_{2}\supseteq\cdots$ is the valuation subspace series for $V$.

The reader can quickly verify that the near homogeneous decomposition in our
simple motivating example 6.2 is, in fact, a near inner decomposition. This
immediately leads us to the following general proposition:

######  Proposition 6.6.

If $V$ is $R_{1}$ invariant, then the near homogeneous decomposition of $V$ is
near inner.

###### Proof.

Suppose $V$ is $R_{1}$ invariant. Let $h\in W_{k}$, $g\in V$, and $r\in
R_{1}$. Then $rh\in V$, so $rh+g\in V$. If $\operatorname{ord}(rh+g)>m$, then
$rh+g\in V_{m+1}$, so $rh+g\perp W_{m}$. ∎

The reader might ask, where did we use the hypothesis that $V$ was closed in
the above? The answer is that only closed subspaces are guaranteed to have
near homogeneous decompositions.

It would be strikingly convenient if the near inner property were also a
sufficient condition for $V$ to be $R_{1}$ invariant, but the simplest
examples show us this is not so:

######  Example 6.7.

Let $R=P[\mathbb{C},z_{1},z_{2}]$, and let
$\mathbb{H}=H^{2}(\mathbb{C},D^{2})$, scalar valued $H^{2}$ of the polydisk
$D^{2}$. Let the $\operatorname{ord}$ function on $R$ and $\mathbb{H}$ be the
usual one, the degree of the first non-vanishing term in the power series
expansion for $f$. Let

(6.7.1) $V=W_{0}\oplus_{\perp}W_{1}\oplus_{\perp}W_{2}\oplus_{\perp}\cdots$

where

$\begin{split}W_{0}=\\{0\\},\;W_{1}&=\operatorname{LH}\\{z_{1},z_{2}\\},\;W_{2}=\operatorname{LH}\\{z_{1}^{2},z_{2}^{2}\\},\\\
W_{3}&=\operatorname{LH}\\{z_{1}^{3},z_{1}^{2}z_{2},z_{1}z_{2}^{2},z_{2}^{3}\\},\;\dots\end{split}$

and, as usual, $\operatorname{LH}$ denotes the linear hull. Then $V$ is
closed, the decomposition (6.7.1) is the near homogeneous decomposition of
$V$, and it is near inner. However, $V$ is _not_ $R_{1}$ invariant because
$W_{2}$ lacks the needed element $z_{1}z_{2}$. And, in fact, the closed
$R_{1}$ invariant subspace generated by $V$ is the closed
$P[\mathbb{C},z_{1},z_{2}]$ submodule generated by $z_{1}$ and $z_{2}$.

We need to add an additional property, which we call the _full projection
property_ , to obtain necessary and sufficient conditions for a closed
subspace $V$ to be $R_{1}$ invariant. We shall discuss the full projection
property in the next section.

To return to the topic of near inner decompositions, the next two _projection
lemmas_ will provide most of the remaining part of the proof of our abstract
Beurling’s theorem in section 8. To set the stage, let

(6.8) $V=W^{0}\oplus_{\perp}W^{1}\oplus_{\perp}W^{2}\oplus_{\perp}\cdots$

be a near inner decomposition of $V$. As before, because the decomposition
(6.8) is an arbitrary near inner decomposition, and does not necessarily
coincide with the near homogeneous decomposition of $V$, we use superscripts
rather than subscripts throughout.

The _first projection lemma_ is:

######  Lemma 6.9.

Let $r\in R_{1}$ and $h\in W^{k}$. Suppose $f^{m}=g^{0}+g^{1}+\cdots g^{m}$,
where $g^{0}\in W^{0}$, $g^{1}\in W^{1}$, $\dots$ $g^{m}\in W^{m}$. Suppose
that $\operatorname{ord}(rh-f^{m})>m$. Then the projection of $rh$ on the
subspace $W=W^{0}\oplus_{\perp}W^{1}\oplus_{\perp}\cdots\oplus_{\perp}W^{m}$
is $f^{m}$. Furthermore,

(6.9.1)
$\|f^{m}\|^{2}=\|g^{0}\|^{2}+\|g^{1}\|^{2}+\cdots+\|g^{m}\|^{2}\leq\|rh\|^{2}$

###### Proof.

For any element $w\in W$, $w=w^{0}+w^{1}+\cdots+w^{m}$, where $w^{0}\in
W^{0}$, $w^{1}\in W^{1}$, $\dots$, $w^{m}\in W^{m}$. By the definition of a
near inner decomposition, that is definition 6.3, $rh-f^{m}\perp W^{j}$ for
$j=0,1,\dots,m$. Thus

$\langle rh-f^{m},w\rangle=\sum_{j=0}^{m}\langle rh-f^{m},w^{j}\rangle=0$

so $rh-f^{m}\perp W$. Hence $f^{m}$ is the projection of $rh$ on $W$.

As for the norm of $f^{m}$,

$\|f^{m}\|^{2}=\|g^{0}\|^{2}+\|g^{1}\|^{2}+\cdots+\|g^{m}\|^{2}$

because the near inner decomposition (6.8) is orthogonal. Furthermore,

$\|rh\|^{2}=\|rh-f^{m}\|^{2}+\|f^{m}\|^{2}$

because $rh-f^{m}\perp f^{m}$. Thus inequality (6.9.1) holds. ∎

The _second projection lemma_ is:

######  Lemma 6.10.

Let $r\in R_{1}$, and $h\in W_{k}$. Consider the formal series

(6.10.1) $g^{0}+g^{1}+g^{2}+\cdots$

where $g^{0}\in W^{0}$, $g^{1}\in W^{1}$, $g^{2}\in W^{2}$, $\cdots$, and let
$f^{m}=g^{0}+g^{1}+\cdots+g^{m}$. Suppose that
$\operatorname{ord}(rh-f^{m})>m$ for $m=0,1,2,\dots$. Then the formal series
(6.10.1) converges in norm to an element $f$ in $V$, and $rh=f$.

###### Proof.

We shall consider the formal series of positive terms

(6.11) $\|g^{0}\|^{2}+\|g^{1}\|^{2}+\|g^{2}\|^{2}+\cdots$

By the first projection lemma, the partial sums are

(6.12)
$\|f^{m}\|^{2}=\|g^{0}\|^{2}+\|g^{1}\|^{2}+\cdots+\|g^{m}\|^{2}\leq\|rh\|^{2}$

Thus the sequence of partial sums $\|f^{m}\|^{2}$, $m=0,1,2,\cdots$ is Cauchy.
Furthermore, if $l<m$,
$\|f^{m}-f^{l}\|^{2}=\|g^{l+1}\|^{2}+\cdots+\|g^{m}\|^{2}$, so the sequence of
partial sums of the formal series (6.10.1) is Cauchy in norm. Because each
$f_{m}\in V$ and $V$ is complete, the formal series (6.10.1) converges to an
element $f\in V$.

Now let $l\in\mathbb{Z}_{+}$. For all $m>l$, $\operatorname{ord}(rh-f^{m})>l$,
so $rh-f^{m}\in\mathbb{H}_{l}$. By the comments following definition (4.3),
$\mathbb{H}_{l}$ is closed, so $rh-f\in\mathbb{H}_{l}$ Thus

$rh-f\in\bigcap_{l=0}^{\infty}\mathbb{H}_{l}=\\{0\\}$

where the right hand equality follows from the equations (5.2). Thus $f=rh$. ∎

Note that since $r\in R_{1}$, $\operatorname{ord}(rh)>0$. We have included
$g^{0}$ in the above even though $g^{0}$ must equal $0$ in order for
$\operatorname{ord}(rh-f^{m})$ to be $>m$. In fact, $g^{0}$ through $g^{k}$
must all be equal to $0$ for the same reason.

##  7\. The Full Projection Property.

In example 6.7, we saw that for a closed subspace $V$ to be $R_{1}$ invariant,
it was not sufficient for the near homogeneous decomposition of $V$ to be near
inner because some of the components $W_{k}$ might not have enough elements.
The _full projection property_ , which we shall define below, guarantees that
this can’t happen.

Recall from our standing notation introduced in section 5 that
$\mathbb{H}=\mathbb{H}_{0}\supseteq\mathbb{H}_{1}\supseteq\mathbb{H}_{2}\supseteq\cdots$
is the valuation subspace series for $\mathbb{H}$,
$\mathbb{H}=H_{0}\oplus_{\perp}H_{1}\oplus_{\perp}H_{2}\oplus_{\perp}\cdots$
is the near homogeneous decomposition of $\mathbb{H}$, and
$P_{k}^{\mathbb{H}}\colon\mathbb{H}\mapsto H_{k}$ is orthogonal projection
onto $H_{k}$. Also, because we shall be speaking of near homogeneous
decompositions, we shall return to using subscripts rather than superscripts.

######  Definition 7.1.

The near homogeneous decomposition of $V$ has the full projection property if
for each $k$ and $m\in\mathbb{Z}_{+}$,

(7.1.1) $P_{m}^{\mathbb{H}}(rh+g)\in P_{m}^{\mathbb{H}}(W_{m}),$

whenever $r\in R_{1}$, $h\in W_{k}$, $g\in V$, and
$\operatorname{ord}(rh+g)\geq m$.

######  Proposition 7.2.

If $V$ is $R_{1}$ invariant, then the near homogeneous decomposition of $V$
has the full projection property.

###### Proof.

Suppose $V$ is $R_{1}$ invariant. Let $k$ and $m\in\mathbb{Z}_{+}$, $r\in
R_{1}$, $h\in W_{k}$, and $g\in V$. For simplicity, denote $rh+g$ by $f$, and
suppose $\operatorname{ord}(f)\geq m$.

Because $V$ is $R_{1}$ invariant, $rh\in V$ so $f=rh+g\in V$. Now there are
two cases: If $ord(f)>m$, then $P_{m}^{\mathbb{H}}(f)=0$ because
$f\in\mathbb{H}_{m+1}$ and $H_{m}\perp\mathbb{H}_{m+1}$. Thus
$P_{m}^{\mathbb{H}}\in P_{m}^{\mathbb{H}}(W_{m})$. On the other hand, if
$\operatorname{ord}(f)=m$, then we can write $f=f_{m}+F_{m+1}$, where
$f_{m}\in W_{m}$ and $F_{m+1}\in V_{m+1}\subseteq\mathbb{H}_{m+1}$. Since
$F_{m+1}\in\mathbb{H}_{m+1}$ and $H_{m}\perp\mathbb{H}_{m+1}$,
$P_{m}^{\mathbb{H}}(F_{m+1})=0$. Thus
$P_{m}^{\mathbb{H}}(f)=P_{m}^{\mathbb{H}}(f_{m})\in P_{m}^{\mathbb{H}}(W_{m})$
as required. ∎

When checking whether the near homogenous decomposition of $V$ has the full
projection property, certain simplifications can be made. First, we never used
the fact that $f=rh+g\in V$ in the $\operatorname{ord}(f)>m$ case. Thus
$P_{m}^{\mathbb{H}}(rf+g)\in P_{m}^{\mathbb{H}}(W_{m})$ always, whenever
$\operatorname{ord}(rh+g)>m$, even if $V$ is not $R_{1}$ invariant. Second,
$P_{0}^{\mathbb{H}}(rh+g)\in P_{0}^{\mathbb{H}}(W_{0})$ always, even if $V$ is
not $R_{1}$ invariant, because $r\in R_{1}$, and so
$\operatorname{ord}(rh)>0$. Thus
$P_{0}^{\mathbb{H}}(rh+g)=P_{0}^{\mathbb{H}}(g)=P_{0}^{\mathbb{H}}(g_{0})\in
P_{0}^{\mathbb{H}}(W_{0})$. Here we have written $g$ as $g_{0}+G_{1}$, where
$g_{0}\in W_{0}$ and $G_{1}\in V_{1}\subseteq\mathbb{H}_{1}$. Thus
$P_{0}^{\mathbb{H}}(g)=P_{0}^{\mathbb{H}}(g_{0})+P_{0}^{\mathbb{H}}(G_{1})=P_{0}^{\mathbb{H}}(g_{0})+0=P_{0}^{\mathbb{H}}(g_{0})$.

We can summarize these remarks in a corollary:

######  Corollary 7.3.

Let $V$ be an arbitrary closed subspace of $\mathbb{H}$, possibly not $R_{1}$
invariant. Let $k$, $m$, $h$, $r$, and $g$ be as in the proof of the
proposition. Then $P_{0}^{\mathbb{H}}(rh+g)$ always $\in
P_{0}^{\mathbb{H}}(W_{0})$, and $P_{m}^{\mathbb{H}}(rh+g)\in
P_{m}^{\mathbb{H}}(W_{m})$ whenever $\operatorname{ord}(rh+g)>m$.

###### Proof.

See the remarks above. ∎

As in the case of the near inner property, it would be strikingly convenient
if the full projection property were also a sufficient condition for $V$ to be
$R_{1}$ invariant. However, the following simple one-variable example shows
that this is not so:

######  Example 7.4.

Let $R=P[\mathbb{C},z]$, and let $\mathbb{H}=H^{2}(\mathbb{C},D^{1})$, scalar
valued $H^{2}$ of the unit disk $D^{1}$. Let $a\in D^{1}$ with $a\neq 0$, and
consider the simplest possible non-constant inner function not vanishing at
$0$, the simple Blaschke factor

(7.4.1) $B(z)=\frac{a-z}{1-\bar{a}z}$

Let

(7.4.2)
$V=\\{a_{0}+a_{1}z+a_{2}z^{2}B(z)+a_{3}z^{3}B(z)+\cdots\>:\>\sum|a_{j}|^{2}<\infty\\}$

Then $V$ is a closed subspace of $H^{2}(\mathbb{C},D^{1})$, the one-
dimensional subspaces

(7.4.3)
$\begin{split}W_{0}=\operatorname{LH}\\{1\\},W_{1}&=\operatorname{LH}\\{z\\},\\\
W_{2}&=\operatorname{LH}\\{z^{2}B(z)\\},W_{3}=\operatorname{LH}\\{z^{2}B(z)\\},\dots\end{split}$

are closed, near homogeneous, and mutually orthogonal (because $B$ is inner),
and

(7.4.4)
$V=W_{0}\oplus_{\perp}W_{1}\oplus_{\perp}W_{2}\oplus_{\perp}W_{3}\oplus_{\perp}\cdots$

is the near homogeneous decomposition of $V$. Clearly, $V$ has the full
projection property. However, $V$ is not $R_{1}$ invariant (ie. shift
invariant). In fact, the closed shift invariant subspace generated by $V$ is
all of $H^{2}(\mathbb{C},D^{1})$.

Variations on this example, except for the part about (7.4.4) being the near
homogeneous decomposition of $V$, are standard in the one-variable theory of
$H^{2}$ spaces.

So the full projection property is not sufficient for $V$ to be $R_{1}$
invariant. As our abstract Beurling’s theorem, theorem 8.1 below, shows, we
must add the property that the near homogeneous decomposition of $V$ is near
inner to obtain necessary and sufficient conditions for $V$ to be $R_{1}$
invariant.

However, we do have a partial converse to proposition 7.2:

######  Proposition 7.5.

Suppose the near homogeneous decomposition of $V$ has the full projection
property. Suppose $k\in\mathbb{Z}_{+}$, $r\in R_{1}$, $h\in W_{k}$, and
$V^{\prime}=V+\operatorname{LH}\\{rh\\}$. As in definition 5.3, let

(7.5.1)
$V^{\prime}=W_{0}^{\prime}\oplus_{\perp}W_{1}^{\prime}\oplus_{\perp}W_{2}^{\prime}\oplus_{\perp}\cdots$

be the near homogeneous decomposition of $V^{\prime}$. Suppose
$m\in\mathbb{Z}_{+}$ and $W_{m}\subseteq W_{m}^{\prime}$. Then
$W_{m}=W_{m}^{\prime}$.

We can summarize the conclusion of proposition 7.5 as “containment implies
equality.”

Our first proof of our abstract Beurling’s theorem used this proposition, but
we shall not need it here because we now base our proof on the projection
lemmas of section 6. Thus we shall leave the proof of proposition 7.5 to the
reader.

##  8\. The Abstract Beurling’s Theorem.

Now we can at last state and prove our abstract Beurling’s theorem. It fully
characterizes the closed $R_{1}$ invariant subspaces of valuation Hilbert
modules. For clarity, we shall repeat some of our standing notation in the
statement of the theorem and its proof:

######  Theorem 8.1.

Let $V$ be a closed subspace of a valuation Hilbert module over a valuation
algebra $R$. Then $V$ is $R_{1}$ invariant if and only if the near homogeneous
decomposition of $V$ is near inner and has the full projection property.

###### Proof.

Necessity follows from proposition 6.6, which states that the near homogeneous
decomposition of a closed $R_{1}$ invariant subspace is near inner, and
proposition 7.2, which states that a closed $R_{1}$ invariant subspace has the
full projection property.

As for sufficiency, suppose the near homogeneous decomposition of $V$ is near
inner and has the full projection property. Let $r\in R_{1}$ and $h\in W_{k}$,
where

(8.2) $V=W_{0}\oplus_{\perp}W_{1}\oplus_{\perp}W_{2}\oplus_{\perp}\cdots$

is the near homogeneous decomposition of $V$. We shall show that $rh\in V$.
First, we shall use the full projection property (definition 7.1) to show that
there are elements

(8.3) $g_{0}\in W_{0},\>g_{1}\in W_{1},\>g_{2}\in W_{2},\>\dots$

such that for $m=0,1,2,\dots$,

$\operatorname{ord}(rh-f_{m})>m$

Here, we have denoted the sum $g_{0}+\cdots g_{m}$ by $f_{m}$. Then we shall
use the second projection lemma (6.10) to conclude that the formal sum

(8.4) $g_{0}+g_{1}+g_{2}+\cdots$

converges to an element $f\in V$, and that $rh=f$. This will prove that $rh\in
V$.

First, let us construct the sequence (8.3) by induction. Because $r\in R_{1}$,
$\operatorname{ord}(rh)>0$. Thus $P_{0}^{\mathbb{H}}(rh)=0$, so we can (and
must) choose $g_{0}=0$. Now suppose $g_{0},\dots,g_{m-1}$ have been
constructed inductively, so that $g_{j}\in W_{j}$ and $\operatorname{ord}(rh-
f_{j})>j$ for $j=0,\dots,m-1$. We shall use the full projection property of
the near homogeneous decomposition (8.2) of $V$ to construct $g_{m}$. Note
that $\operatorname{ord}(rh-f_{m-1})\geq m$. By the full projection property,
$P_{m}^{\mathbb{H}}(rh-f_{m-1})\in P_{m}^{\mathbb{H}}(W_{m})$. If
$W_{m}\neq\\{0\\}$, we may use proposition 5.17 to conclude that there is a
unique element $g_{m}\in W_{m}$ such that
$P_{m}^{\mathbb{H}}(g_{m})=P_{m}^{\mathbb{H}}(rh-f_{m-1})$. In the exceptional
case where $W_{m}=\\{0\\}$, we can (and must) choose $g_{m}=0$. (See the
remark following proposition 5.17.) Let $f_{m}=f_{m-1}+g_{m}$. Then
$f_{m}=g_{1}+\cdots+g_{m}$, and $g_{m}\in W_{m}$ by construction.

But we still must prove that $\operatorname{ord}(rh-f_{m})>m$. To that end,
recall from corollary 5.10 that $\operatorname{ord}(rh-f_{m})$ is the index of
the first non-vanishing term in the near homogeneous decomposition of $rh-
f_{m}$, or $\infty$ if $rh-f_{m}=0$. Since $\operatorname{ord}(rh-
f_{m-1})>m-1$, all the terms in the near homogeneous decomposition of $rh-
f_{m-1}$ vanish up to and including the index $m-1$, that is
$P_{j}^{\mathbb{H}}(rh-f_{m-1})=0$ for $j=0,\dots,m-1$. Also, because
$g_{m}\in W_{m}$, $\operatorname{ord}(g_{m})=m$ (or $\infty$ if $g_{m}=0$).
Thus $P_{j}^{\mathbb{H}}(g_{m})=0$ as well for $j=0,\dots m-1$. Furthermore,
$P_{m}^{\mathbb{H}}(rh-f_{m-1})=P_{m}^{\mathbb{H}}(g_{m})$ by construction.
Thus

$P_{j}^{\mathbb{H}}(rh-f_{m})=P_{j}^{\mathbb{H}}(rh-
f_{m-1})-P_{j}^{\mathbb{H}}(g_{m})=0$

for $j=0,\dots,m$, and so $\operatorname{ord}(rh-f_{m})>m$.

This concludes the first part of the proof that $rh\in V$. The second part is
immediate. Because the near homogeneous decomposition (8.2) of $V$ is near
inner, we may apply the second projection lemma. By the second projection
lemma, the formal series 8.4 converges in norm to an element $f\in V$, and
$rh=f$. Thus $rh\in V$.

Now let $h\in V$, not necessarily an element of $W_{k}$. Let $r\in R_{1}$ as
before, and let

(8.5) $h=h_{0}\oplus_{\perp}h_{1}\oplus_{\perp}h_{2}\oplus_{\perp}\cdots$

be the near homogeneous decomposition of $h$ with respect to $V$. By
definition 5.8, each $h_{k}\in W_{k}$, and by proposition 5.13, the series
(8.5) converges in norm to $h$. By the definition of a Hilbert module, the
linear operators $m_{r}\colon f\to rf$, $f\in\mathbb{H}$ are bounded in
operator norm, so the series

$rh=rh_{0}+rh_{1}+rh_{2}+\cdots$

converges in norm to $rh$. Further, each term $rh_{k}\in V$, so, since $V$ is
closed, $rh\in V$ as required. Thus $V$ is $R_{1}$ invariant. ∎

The last part of this proof is useful enough to isolate as a separate
corollary:

######  Corollary 8.6.

A closed subspace $V$ is $R_{1}$ invariant if and only if, for each
$k\in\mathbb{Z}_{+}$ and $r\in R_{1}$, $rW_{k}\subseteq V$.

###### Proof.

See the above. ∎

A proper generalization of Beurling’s theorem should characterize, not just
closed $R_{1}$ invariant subspaces of $\mathbb{H}$, but closed $R$ submodules
as well. This we can do in two important cases. Recall from definition 4.10
that a valuation algebra $R$ is _unital_ if $R$ has a unit element $1_{R}$ and

(8.7) $R=\\{\lambda 1_{R}:\lambda\in\mathbb{C}\\}+R_{1}$

From proposition 4.11 we immediately have

######  Corollary 8.8.

Let $R$ be a valuation algebra satisfying either of the following conditions:

  1. (8.8.1)

$R=R_{1}$, or

  2. (8.8.2)

$R$ is unital

Then a closed subspace $V$ of a valuation Hilbert module $\mathbb{H}$ over $R$
is an $R$ submodule of $\mathbb{H}$ if and only if the near homogeneous
decomposition of $V$ is near inner and has the full projection property.

###### Proof.

Apply proposition 4.11 and theorem 8.1. ∎

We remind the reader that in the iconic case where
$R=P[\mathbb{C},z_{1},\dots,z_{n}]$, $R$ is unital, so corollary 8.8 fully
characterizes the closed $P[\mathbb{C},z_{1},\dots,z_{n}]$ submodules of
$\mathbb{H}$.

##  9\. Example – $H^{2}$ Spaces.

The closed invariant subspaces of many familiar Hilbert spaces of analytic
functions are completely described by our Abstract Beurling’s Theorem,
Theorem8.1 and its corollaries. In this section, we shall describe the closed
invariant subspaces of $\mathbb{H}=H^{2}(E,D^{n})$, where, as usual, $E$ is a
complex Hilbert space and $D^{n}$ is the $n$-dimensional polydisk. Again, as
usual, $R$ will be the familiar algebra of complex valued polynomials in the
variables $z_{1},\ldots,z_{n}$. And both $R$ and $\mathbb{H}$ will have the
usual ord function, $\operatorname{ord}(f)=$ the multiplicity of the zero of f
at $0$.

######  Example 9.1.

Theorem 8.1 and its corollaries completely characterize the closed invariant
subspaces of $\mathbb{H}=H^{2}(E,D^{n}$).

###### Proof.

All we have to show is that the $\operatorname{ord}$ function on $\mathbb{H}$
is upper semi-continuous. So let $f_{j}$, $j=1,2,3,\dots$ be a norm convergent
sequence in $\mathbb{H}$ converging to $f$. For what follows, we need to
briefly review one more thing about multi-index notation: Recall that
$|m|=|m_{1}|+\cdots|m_{n}|$.

Now by definition 3.5, the lowest degree terms in the power series expansion
of $f$ have degree $\operatorname{ord}(f)$. In multi-index notation, let
$a_{m}z^{m}$ be one of these terms (so $\operatorname{ord}(f)=|m|$), and let
$a_{j,m}$be the corresponding term in the power series expansion of $f_{j}$.
Then

$a_{m}=\int_{0}^{2\pi}f(e^{\imath\theta})e^{-\imath
m\theta}d\theta/(2\pi^{n})$

in multi-index notation, with a corresponding formula for $a_{j,m}$ with $f$
replaced by $f_{j}$. Here, the integral is repeated $n$ times,
$\theta=\theta_{1}\cdots\theta_{n}$, and $d\theta=d\theta_{1}\cdots
d\theta_{n}$.

The reader will note that $a_{m}$ is simply the $m^{\text{th}}$ Fourier
coefficient of $f$. Since the sequence $f_{j}$ converges to $f$ in $H^{2}$
norm (and $e^{-\imath m\theta}$ is bounded) $a_{j,m}$ converges to $a_{m}$ as
$j\rightarrow\infty$. Thus, for $j\geq\text{ some }J$,
$|a_{j,m}|\geq|a_{m}|/2\neq 0$. Hence, for each $j\geq J$, $f_{j}$ has terms
in its power series expansion of degree $|m|=\operatorname{ord}(f)$, so for
$j\geq J$, $\operatorname{ord}(f_{j})\leq\operatorname{ord}(f)$. Consequently,
$\limsup\operatorname{ord}(f_{j})\leq\operatorname{ord}(f)$, so the
$\operatorname{ord}$ function is upper semi-continuous on $\mathbb{H}$. ∎

In a second paper, we shall use theorem 8.1 to fully characterize the closed
invariant subspaces of a number of Hilbert spaces of analytic functions in
several complex variables. These will include $H^{2}$ of the unit ball in
$\mathbb{C}^{n}$, and of general bounded symmetric domains. Also, these will
include weighted Bergman spaces of analytic functions on domains in
$\mathbb{C}^{n}$, and weighted Bergman spaces of analytic functions and
analytic differentials on complex analytic manifolds.

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