# Multiplication by finite Blaschke factors on a general class of Hardy spaces

Apoorva Singh  Department of Mathematics
School of Natural Sciences
Shiv Nadar University
Gautam Budh Nagar - 201314.
Uttar Pradesh, India<EMAIL_ADDRESS>Niteesh Sahni  Department of
MathematicsSchool of Natural SciencesShiv Nadar UniversityGautam Budh Nagar -
201314.Uttar Pradesh, India<EMAIL_ADDRESS>
###### Abstract.

A broader class of Hardy spaces and Lebesgue spaces have been introduced
recently on the unit circle by considering continuous $\|.\|_{1}$-dominating
normalized gauge norms instead of the classical norms on measurable functions
and a Beurling type result has been proved for the operator of multiplication
by the coordinate function. In this paper, we generalize the above Beurling
type result to the context of multiplication by a finite Blaschke factor
$B(z)$ and also derive the common invariant subspaces of $B^{2}(z)$ and
$B^{3}(z)$. These results lead to a factorization result for all functions in
the Hardy space equipped with a continuous rotationally symmetric norm.

###### Key words and phrases:

Normalized gauge norm, Blaschke factor, Invariant subspace, Rotationally
symmetric norm, Hardy space, Lebesgue space, Inner-outer factorization.

###### 1991 Mathematics Subject Classification:

Primary 47A15; Secondary 30H10, 47B38

##  1\. Introduction

The study of invariant subspaces of the shift operator has been an area of
extensive research over the past decades. After the landmark success of
obtaining the invariant subspaces of $H^{2}$ under the multiplication with the
monomial $z$ by Beurling [1], many new directions of extensions have emerged.
Some of the notable ones are the Helson-Lowdenslager theorem [13] which
characterizes the simply invariant subspaces of $L^{2}$, Weiner’s theorem [12]
which characterizes the doubly invariant subspaces of $L^{2}$, the theorems by
Lax [19] and Halmos [11] which study the invariance under the operator of
multiplication by $z^{n}$, and the characterization of contractively contained
sub Hilbert spaces of $H^{2}$ by de Branges [9]. Furthermore, in the recent
past, the validity of the above results have been proved in the context of
general Lebesgue $L^{p}$ and Hardy spaces $H^{p}$ $(0\leq p\leq\infty)$ by
[25], [26], [17], [20], and [23]. It is worth pointing out that in the works
cited above, the authors have studied invariance under the multiplier algebra
$H^{\infty}(z)$. For a finite Blaschke factor $B(z)$, invariance in $H^{p}$
and $L^{p}$ have been explored under the algebra $H^{\infty}(B)$ =
$\\{f(B(z)):f\in H^{\infty}\\}$ in [22], [16], and [18].

The common invariant subspaces have also been a focus of attention in the
recent past. This encapsulates the discovery of invariant subspaces of $H^{p}$
under the algebras $H_{1}^{\infty}(z)$ = $\\{f\in
H^{\infty}:f^{\prime}(0)=0\\}$, $H_{1}^{\infty}(B)$ = $\\{f(B(z)):f\in
H^{\infty}_{1}\\}$, and ${\mathbb{C}}+BH^{\infty}(z)$. Some notable works on
the same are [8], [22], and [21].

In [3], Chen defined the continuous $\|.\|_{1}$-dominating normalized gauge
norm ${\alpha}$ on the space of all measurable functions on the unit circle.
The closure of $L^{\infty}$ with respect to the norm ${\alpha}$ is called the
Lebesgue space and is denoted by $L^{\alpha}$, and the closure of $H^{\infty}$
under ${\alpha}$ is called the Hardy space $H^{\alpha}$. It has been proved in
[3] that both $L^{\alpha}$ and $H^{\alpha}$ are Banach spaces. For all $1\leq
p<\infty$, the classical Lebesgue spaces $L^{p}$ and the Hardy spaces $H^{p}$
on the unit circle are special cases of $L^{\alpha}$ and $H^{\alpha}$
respectively. It must be noted that $H^{\infty}$ does not belong to these
special classes. In fact $H^{\infty}\subsetneq H^{\alpha}$. The simply
invariant and doubly invariant subspaces of multiplication by the coordinate
function $e^{i\theta}$ on $L^{\alpha}$ have also been obtained [3]. Beurling’s
theorem, as usual, follows as a special case of the $L^{\alpha}$ result.

The statement of the general Beurling’s theorem in [3] is:

######  Theorem 1.1.

Suppose ${\alpha}$ is a continuous $\|.\|_{1}$-dominating normalized gauge
norm and $\mathcal{M}$ is a non-trivial closed subspace of $H^{\alpha}$
invariant under the multiplication by the coordination function $e^{i\theta}$.
Then, there is an inner function $\phi$ such that $\mathcal{M}=\phi
H^{\alpha}$. (A function $\phi\in H^{\infty}$ is called inner if
$\lvert\phi\rvert=1$ a.e.)

The key components of the proof for Theorem 1.1 includes the description of
the dual space of $L^{\alpha}$ and a factorization theorem on $L^{\infty}$.

In this paper, we study the invariance in a general class of Hardy spaces
equipped with continuous $\|.\|_{1}$-dominating normalized gauge norms
introduced by Chen [3] in the context of the algebras $H^{\infty}(B)$ and
$H_{1}^{\infty}(B)$. Results are also obtained for a special class of
continuous rotationally symmetric norms [2], [5]. Furthermore, we present a
general inner-outer factorization of $H^{\alpha}$ functions in terms of
$n$-inner and $n$-outer functions when ${\alpha}$ is a continuous rotationally
symmetric norm. This result is a generalization of the inner-outer
factorization of functions in $H^{2}$ obtained in [24].

The proofs presented in the current paper do not rely on any factorization
results concerning $L^{\infty}$ functions. In that sense, our arguments are
more elementary than those in [3]. We only make use of the well known inner-
outer factorization of $H^{2}$ functions. This allows the techniques to work
in the context of a more general algebra $H^{\infty}(B)$. We further describe
the common invariant subspaces of $H^{\alpha}$. We also obtain sharp
descriptions of the invariant and common invariant subspaces in the
rotationally symmetric case. This generalizes the main results of [3], [5] in
addition to the main results in [17] and [22].

The rest of the paper has been organized as follows. Section 2 deals with the
basic definitions and preliminary results of classical Hardy spaces and
general $H^{\alpha}$ spaces. In Section 3, we characterize the invariant and
common invariant subspaces of $H^{\alpha}$ under multiplication by finite
Blaschke factors. This is equivalent to characterizing invariance under the
algebras $H^{\infty}(B)$ and $H_{1}^{\infty}(B)$, respectively. In Section 4,
we demonstrate that the descriptions of the invariant subspaces can be made
sharper if $\alpha$ is taken to be a continuous rotationally symmetric norm.
Here, the underlying Blaschke factor is the monomial $B(z)=z^{n}$. The proofs
of the main results in this section rely on the decomposition of $H^{\alpha}$
in terms of the space $M_{\alpha}(z^{n})$, defined as the closure of
$H^{\infty}(z^{n})$ in the ${\alpha}$ norm. Lastly, in Section 5, we define
the $n$-outer functions in Hardy spaces equipped with a continuous
rotationally symmetric norm and obtain a factorization of $H^{\alpha}$
functions in terms of $n$-inner and $n$-outer functions.

##  2\. Notation and Preliminary Results

Throughout the paper, ${\mathbb{D}}$ stands for the open unit disk in the
complex plane and it’s boundary, the unit circle, is denoted by
${\mathbb{T}}$. The classical Lebesgue space on the unit circle $L^{p}$,
$1\leq p<\infty$, is a Banach space under the norm

$\|f\|_{p}=\left(\int_{\mathbb{T}}\lvert f\rvert^{p}dm\right)^{\frac{1}{p}},$

where $dm$ is the normalized Lebesgue measure on ${\mathbb{T}}$. The space
$L^{\infty}$ is a Banach space under the essential supremum norm

$\|f\|_{\infty}=inf\\{\;M\;:\;m\\{e^{i\theta}:\lvert
f(e^{i\theta})\rvert>M\\}=0\\}.$

It is well known that any function $f\in L^{p}$ can be represented as a
Fourier series $f(z)=$ $\sum\limits_{n=-\infty}^{\infty}a_{n}z^{n}$. The
Fourier coefficients are given by $a_{n}=\int_{{\mathbb{T}}}fz^{-n}dm$. The
Hardy space $H^{p}$ is defined as the norm-closure (weak*-closure if
$p=\infty$) of the analytic polynomials in $L^{p}$. $H^{p}$ turns out to be a
closed subspace of $L^{p}$ which can be expressed as:

$H^{p}=\\{f\in L^{p}\;:\;a_{n}=\int_{{\mathbb{T}}}fz^{-n}dm=0,\;\text{for
all}\;n<0\\}.$

The space $H^{\infty}=H^{1}\cap L^{\infty}$ is a Banach space under the
essential supremum norm and it multiplies all $H^{p}$ spaces back into
$H^{p}$.

A norm ${\alpha}$ on $L^{\infty}$ is called a $\|.\|_{1}$-dominating
normalized gauge norm if:
(1) ${\alpha}(1)=1$,
(2) ${\alpha}(\lvert f\rvert)={\alpha}(f)$ for every $f\in L^{\infty}$,
(3) ${\alpha}(f)\geq\|f\|_{1}$ for every $f\in L^{\infty}$.

Further, we say that a norm ${\alpha}$ is continuous if

$\lim\limits_{m(E)\rightarrow 0^{+}}{\alpha}(\chi_{E})=0,$

where $\chi_{E}$ stands for a characteristic function of
$E\subset{\mathbb{T}}$.
Furthermore, we can also define ${\alpha}$ norm over all the measurable
functions $f$ on ${\mathbb{T}}$ as

${\alpha}(f)=\sup\\{{\alpha}(s):s~{}\text{is a simple function},~{}0\leq
s\leq\lvert f\rvert\\}.$

The space $\mathcal{L}^{\alpha}=\\{f:{\mathbb{T}}\rightarrow{\mathbb{C}}$
measurable such that ${\alpha}(f)<\infty$} is a Banach space under the
continuous $\|.\|_{1}$-dominating normalized gauge norm ${\alpha}$. The
${\alpha}$-closure of $L^{\infty}$ is the general Lebesgue space $L^{\alpha}$.
Some important facts about $L^{\alpha}$ and $\mathcal{L}^{\alpha}$ spaces
established in [3] are $L^{\infty}\subsetneq\mathcal{L}^{\alpha}$, and
$L^{\infty}\subsetneq L^{\alpha}\subset\mathcal{L}^{\alpha}\subset L^{1}$. The
space $L^{\infty}$ multiplies $L^{\alpha}$ back into $L^{\alpha}$ and the
following inequality is satisfied: ${\alpha}(fg)\leq\|f\|_{\infty}{\alpha}(g)$
for all $f\in L^{\infty}$ and $g\in L^{\alpha}$. In fact this inequality holds
for all $g\in\mathcal{L}^{{\alpha}}$.

The ${\alpha}$-closure of $H^{\infty}$ is denoted by $H^{\alpha}$ and it is a
closed subspace of $L^{\alpha}$. The general Hardy space $H^{\alpha}$ is a
Banach space under the norm ${\alpha}$. We also have a simpler description of
$H^{\alpha}$ proved in [3], which is $H^{\alpha}=H^{1}\cap L^{\alpha}$.
Further, we have an important relationship that $H^{\infty}\subsetneq
H^{\alpha}\subset H^{1}$.

A special class of norms on $L^{\infty}$, called rotationally symmetric norms,
are also $\|.\|_{1}$-dominating normalized gauge norms. We say that a norm
${\alpha}$ on $L^{\infty}$ is rotationally symmetric if

(1) ${\alpha}(1)=1$,
(2) ${\alpha}(\lvert f\rvert)={\alpha}(f)$ for every $f\in L^{\infty}$,
(3) ${\alpha}(f_{w})={\alpha}(f)$. Here $f_{w}(z)=f(\overline{w}z)$ for each
$w\in{\mathbb{T}}$ and $f\in L^{\infty}$.

It has been proved in [5] that a rotationally symmetric norm ${\alpha}$ must
satisfy $\|.\|_{1}\leq{\alpha}(.)\leq\|.\|_{\infty}$. Therefore, these norms
are also $\|.\|_{1}$-dominating normalized gauge norms and are a special case
of the same. Besides the $p$-norms $(1\leq p<\infty)$, other important
examples of $\|.\|_{1}$-dominating normalized gauge norms are presented in
[4].

A Blaschke factor with $n$ zeros
${\alpha}_{1},{\alpha}_{2},\ldots,{\alpha}_{n}\in{\mathbb{D}}$ is defined as

$B(z)=\prod\limits_{i=1}^{n}\dfrac{z-{\alpha}_{i}}{1-\overline{{\alpha}_{i}}z}.$

Without loss of generality, we assume that ${\alpha}_{1}=0$ (due to the
conformal invariance of $H^{2}$). The operator of multiplication by $B(z)$
denoted by $T_{B}$ is an isometry on $H^{\alpha}$. This can be easily seen,
since $\lvert B(z)\rvert=1$ a.e on ${\mathbb{T}}$ and ${\alpha}(Bf)$ =
${\alpha}(\lvert Bf\rvert)$ = ${\alpha}(\lvert B\rvert\lvert f\rvert)$ =
${\alpha}(\lvert f\rvert)$ = ${\alpha}(f)$. A closed subspace $\mathcal{M}$ of
$H^{\alpha}$ is invariant under the operator $T_{B}$ if
$B\mathcal{M}\subset\mathcal{M}$. Moreover, $\mathcal{M}$ is invariant under
any subalgebra $A$ of $H^{\infty}$, if $af\in\mathcal{M}$ whenever $a\in A$
and $f\in\mathcal{M}$. The set $H^{\infty}(B)$ stands for the set $\\{foB:f\in
H^{\infty}\\}$ which turns out to be a subalgebra of $H^{\infty}$ generated by
$B$. We shall show in Lemma 2.5 that a closed subspace $\mathcal{M}$ of
$H^{\alpha}$ is invariant under the operator $T_{B}$ if and only if
$\mathcal{M}$ is invariant under the subalgebra $H^{\infty}(B)$. The
weak*-closure of the linear span of $\\{1,B^{2},B^{3},\ldots\\}$ is a
subalgebra of $H^{\infty}$ and is denoted by $H^{\infty}_{1}(B)$. It can be
shown that $\mathcal{M}$ is invariant under $H^{\infty}_{1}(B)$ if and only if
$\mathcal{M}$ is invariant under both $T_{B^{2}}$ and $T_{B^{3}}$.

Put $B_{j}(z)$ =
$\prod\limits_{i=1}^{j}\dfrac{z-\alpha_{i}}{1-\overline{\alpha_{i}}z}$. In
[24], it has been proved that the collection
$\left\\{e_{jm}=\dfrac{\sqrt{1-{\lvert{\alpha}_{j+1}\rvert}^{2}}}{1-\overline{{\alpha}_{j+1}}z}B_{j}B^{m}:0\leq
j\leq n-1,m=0,1,2,\ldots\right\\}$ is an orthonormal basis for $H^{2}$, and
that $H^{2}$ can be decomposed as an orthogonal direct sum

$H^{2}=e_{00}H^{2}(B)\oplus e_{10}H^{2}(B)\oplus\cdots\oplus
e_{n-1,0}H^{2}(B)$ (2.1)

where $H^{2}(B)$ is the $\|.\|_{2}$-closure of all the analytic polynomials in
$B$. A similar decomposition has also been proved for $H^{p}$ spaces [22].
Formally, for $1\leq p\leq\infty$, $H^{p}$ is the algebraic direct sum
$e_{00}H^{p}(B)\oplus e_{10}H^{p}(B)\oplus\cdots\oplus e_{n-1,0}H^{p}(B)$.
Here $H^{p}(B)$ is the $\|.\|_{p}$-closure of all the analytic polynomials in
$B$.

The invariance under multiplication by $B(z)$ is characterized in terms of
$B$-inner functions which are more general than inner functions. Every inner
function is a $B$-inner function for $B(z)=z$. An $H^{\infty}$ function $\phi$
is called $B$-inner if $\\{B^{m}\varphi:m=0,1,2,\ldots\\}$ is an orthonormal
set in $H^{2}$. In general, for an $r$ tuple
$(\varphi_{1},\ldots,\varphi_{r})$ of $H^{\infty}$ functions, we can speak of
a $B$-inner matrix $A$ which is constructed as follows:
For each $1\leq j\leq r$, we can write $\varphi_{j}$ as
$\varphi_{j}=\sum\limits_{i=0}^{n-1}e_{i0}\;\varphi_{ij}$ where
$\varphi_{ij}\in H^{2}(B)$. The $n\times r$ matrix given by
$A=(\varphi_{ij})_{n\times r}$ is a $B$-inner matrix if $A^{*}A=I$. In [24],
the condition $A^{*}A=I$ has been shown to be equivalent to the fact that the
set $\\{B^{m}\varphi_{i}:1\leq i\leq r,m=0,1,2,\ldots\\}$ is orthonormal in
$H^{2}$. This is in line with the characterization of classical inner
functions and usually provides a convenient way of testing the $B$-inner
condition. When $B(z)=z^{n}$, we shall call $B$-inner functions simply as
$n$-inner functions.

Coming back to the basics of $L^{\alpha}$, the dual norm ${\alpha}^{\prime}$
of ${\alpha}$ on $L^{\infty}$ is defined as:

${\alpha}^{\prime}(h)=\sup\Big{\\{}\Big{\lvert}\int\limits_{{\mathbb{T}}}fhdm\Big{\rvert}$
: $f\in L^{\infty}$ and ${\alpha}(f)\leq 1\Big{\\}}$

$=\sup\Big{\\{}\int\limits_{{\mathbb{T}}}\lvert fh\rvert dm$ : $f\in
L^{\infty}$ and ${\alpha}(f)\leq 1\Big{\\}}$.

It has been shown that if $\alpha$ is a $\|.\|_{1}$-dominating normalized
gauge norm, or a rotationally symmetric norm, then ${{\alpha}}^{\prime}$ is
also a $\|.\|_{1}$-dominating normalized gauge norm[3], or a rotationally
symmetric norm [5], respectively. We can now define the space
$\mathcal{L}^{{\alpha}^{\prime}}$ along the same lines as
$\mathcal{L}^{{\alpha}}$. The dual space of $L^{\alpha}$ can be identified
with $\mathcal{L}^{{\alpha}^{\prime}}$. The precise formulation of an
isometric isomorphism is given in the proposition below:

######  Proposition 2.1 (Chen [3]).

Let ${\alpha}$ be a continuous $\|.\|_{1}$-dominating normalized gauge norm
(or, a continuous rotationally symmetric norm) and let ${\alpha}^{\prime}$ be
the dual norm of ${\alpha}$. Then $(L^{{\alpha}})^{\\#}$ =
$\mathcal{L}^{{\alpha}^{\prime}}$,i.e., for every $\phi\in(L^{\alpha})^{\\#}$,
there is an $h\in\mathcal{L}^{{\alpha}^{\prime}}$ such that
$\|\phi\|={\alpha}^{\prime}(h)$ and

$\phi(f)=\int_{{\mathbb{T}}}fhdm$

for all $f\in L^{\alpha}$.

######  Proposition 2.2 (Chen [3]).

Let ${\alpha}$ be a continuous $\|.\|_{1}$-dominating normalized gauge norm on
$\mathcal{L}^{\alpha}$ and if $0\leq f_{1}\leq f_{2}\leq\ldots$ and
$f_{n}\rightarrow f\;a.e.$. Then, ${\alpha}(f_{n})\rightarrow{\alpha}(f)$.

######  Remark 2.3.

In particular taking ${\alpha}=\|.\|_{1}$, we get the classical monotone
convergence theorem.

The lemma below states a version of Holder’s inequality valid on $L^{\alpha}$.
Its proof relies on the generalized monotone convergence theorem stated above
(Proposition 2.2).

######  Lemma 2.4.

If $f\in L^{\alpha}$ and $h\in\mathcal{L}^{{\alpha}^{\prime}}$, then $fh\in
L^{1}$. Further, $\|fh\|_{1}\leq{\alpha}(f){\alpha}^{\prime}(h)$.

###### Proof.

The proof is easy when $f$ and $h$ both belong to $L^{\infty}$. This can be
seen through the following calculation:

$\left\|\frac{f}{{\alpha}(f)}\;h\right\|_{1}=\frac{1}{{\alpha}(f)}\int_{{\mathbb{T}}}\lvert
fh\rvert dm\leq{\alpha}^{\prime}(h).$ (2.2)

The inequality in (2.2) follows by the definition of ${\alpha}^{\prime}$. It
can also be re-phrased as $\|fh\|_{1}\leq{\alpha}(f)\;{\alpha}^{\prime}(h)$.

We now show the validity of inequality (2.2) for any $f\in L^{\alpha}$ and
$h\in\mathcal{L}^{{\alpha}^{\prime}}$. Since $\lvert f\rvert$ and $\lvert
h\rvert$ are measurable functions on ${\mathbb{T}}$, there exist increasing
sequences of non-negative simple functions ${s_{n}}$ and ${r_{n}}$
respectively, such that $s_{n}\to\lvert f\rvert\;a.e.$ and $r_{n}\to\lvert
h\rvert\;a.e.$ By Proposition 2.2, we have ${\alpha}(s_{n})\to{\alpha}(\lvert
f\rvert)$ and ${\alpha}^{\prime}(r_{n})\to{\alpha}^{\prime}(\lvert h\rvert)$
as $n\to\infty$. Thus,

${\alpha}(s_{n})\;{\alpha}^{\prime}(r_{n})\to{\alpha}(\lvert
f\rvert)\;{\alpha}^{\prime}(\lvert h\rvert)\;\;\;\;\text{as}\;\;n\to\infty.$

Further, note that the sequence $s_{n}r_{n}\to\lvert fh\rvert\;a.e.$ So, by
the classical monotone convergence theorem, we have

$\|s_{n}r_{n}\|_{1}\to\|fh\|_{1}\;\;\;as\;\;n\to\infty.$

Applying the inequality (2.2), we get

$\|s_{n}r_{n}\|_{1}\;\leq\;{\alpha}(s_{n})\;{\alpha}^{\prime}(r_{n}).$

Finally, taking limit as $n\to\infty$, we conclude that for every $f\in
L^{\alpha}$ and $h\in\mathcal{L^{{\alpha}^{\prime}}}$:

$\|fh\|_{1}\;\leq{\alpha}(\lvert f\rvert)\;{\alpha}^{\prime}(\lvert
h\rvert)={\alpha}(f)\;{\alpha}^{\prime}(h).$

∎

######  Lemma 2.5.

Let ${\alpha}$ be a continuous $\|.\|_{1}$-dominating gauge norm. Suppose
$\mathcal{M}$ is a closed subspace of $H^{\alpha}$. Then, $\mathcal{M}$ is
invariant under $T_{B}$ if and only if $\mathcal{M}$ is invariant under the
algebra $H^{\infty}(B)$.

###### Proof.

The condition $H^{\infty}(B)\mathcal{M}\subset\mathcal{M}$ trivially implies
the invariance of $\mathcal{M}$ under the operator $T_{B}$. We now prove the
converse.

Suppose $T_{B}(\mathcal{M})\subset\mathcal{M}$. Then,
$(T_{B})^{n}\mathcal{M}\subset\mathcal{M}$ for all $n=1,2,3,\ldots$ This
implies that $\mathcal{M}$ is invariant under multiplication by all
polynomials in $B$. We shall show that $\mathcal{M}$ is also invariant under
multiplication by all functions in $H^{\infty}(B)$. Let $k\in H^{\infty}(B)$,
and $h\in\mathcal{M}$ be arbitrary. We can find a sequence of Cesaro means
$\\{\sigma_{n}(B)\\}$ converging to $k$ in the weak* topology. That is, for
every $f\in L^{1}$:

$\int_{{\mathbb{T}}}\sigma_{n}(B)f~{}dm\longrightarrow\int_{{\mathbb{T}}}kf~{}dm\;\;\;\text{as}\;\;n\rightarrow\infty.$
(2.3)

In view of Lemma 2.4 and equation (2.3) we conclude that for any non-zero
element $u\in\mathcal{L}^{{\alpha}^{\prime}}$:

$\int_{{\mathbb{T}}}\sigma_{n}(B)hu~{}dm\longrightarrow\int_{{\mathbb{T}}}khu~{}dm\;\;\;\text{as}\;\;n\rightarrow\infty.$

Note that $\sigma_{n}(B)$ are polynomials in $B$, and hence
$\sigma_{n}(B)h\in\mathcal{M}\subset L^{\alpha}$. Looking at the general form
of any bounded linear functional on $L^{\alpha}$ (Proposition 2.1), it follows
that $\sigma_{n}(B)h$ $\rightarrow$ $kh$ weakly in $L^{\alpha}$. Because all
closed subspaces are also weakly closed, we infer that $kh\in\mathcal{M}$.
This completes the proof. ∎

######  Remark 2.6.

The invariance of $\mathcal{M}$ under $T_{B^{2}}$ and $T_{B^{3}}$ but not
under $T_{B}$ is equivalent to the invariance of $\mathcal{M}$ under the
algebra $H^{\infty}_{1}(B)$ but not under $H^{\infty}(B)$.

We shall now state a few lemmas without proof which play a central role in
proving the main results of this paper.

######  Lemma 2.7 (Gamelin [10]).

Suppose $\\{f_{n}\\}$ is a sequence of $H^{\infty}$ functions which converges
to a $H^{\infty}$ function $f$ in the $H^{2}$ norm. Then, there exists a
sequence $\\{h_{n}\\}$ of uniformly bounded $H^{\infty}$ functions which
converges to 1 $a.e.$, such that $h_{n}f_{n}\to f$ in the weak* topology.

######  Lemma 2.8 (Sahni and Singh [22]).

Suppose $f\in H^{\infty}$ such that $f$ =
$\varphi_{1}g_{1}+\cdots+\varphi_{r}g_{r}$ for some $g_{1},\ldots,g_{r}$ $\in
H^{2}(B)$, and for some $\varphi_{1},\ldots,\varphi_{r}$ $B$-inner functions.
Then $g_{1},\ldots,g_{r}$ belongs to $H^{\infty}(B)$.

######  Lemma 2.9 (Lance and Stessin [17]).

Let $\varphi_{1},\ldots,\varphi_{r}$ ($r\leq n$) are $B$-inner functions and
for any $f\in H^{\infty}$ such that
$f(z)=\sum_{j=1}^{r}\varphi_{j}(z)f_{j}(B(z))$. Then there exist constants
$D_{j,p}$, such that $\|f_{j}\|_{p}\leq D_{j,p}\|f\|_{p}$ for each
$j=1,\ldots,r$ and $1\leq p\leq\infty$.

##  3\. Generalization of Invariant Subspace Theorem in $H^{\alpha}$ space

In [22], the closed subspaces of $H^{p}$ for $0<p\leq\infty$, that are
invariant under multiplication by all powers of $B$, except the first power
has been characterized. We extend the result for a far more general Hardy
spaces $H^{\alpha}$, where ${\alpha}$ is $\|.\|_{1}$-dominating normalized
gauge norm. Before stating our main result of this section, it is convenient
to first present the proof of the invariance theorem explicitly for
$H^{\infty}$ space. The idea of the proof is new and fairly different as seen
in [22].

We first prove the following decomposition for $H^{\infty}$:

$H^{\infty}=e_{00}H^{\infty}(B)\oplus e_{10}H^{\infty}(B)\oplus\cdots\oplus
e_{n-1,0}H^{\infty}(B).$ (3.1)

Note that

$H^{\infty}=H^{2}\cap H^{\infty}=(e_{00}H^{2}(B)\oplus
e_{10}H^{2}(B)\oplus\cdots\oplus e_{n-1,0}H^{2}(B))\cap H^{\infty}.$ (3.2)

We will establish that the right hand sides of (3.1) and (3.2) coincide.
It is trivial to see that the right hand side of (3.1) is contained in the
right hand side of (3.2). To establish the reverse containment, take any
$f\in\left(e_{00}H^{2}(B)\oplus e_{10}H^{2}(B)\oplus\cdots\oplus
e_{n-1,0}H^{2}(B)\right)\cap H^{\infty}$. This means that $f\in H^{\infty}$
and $f=e_{00}f_{1}+e_{10}f_{2}+\cdots+e_{n-1,0}f_{n}$ for some
$f_{1},\ldots,f_{n}\in H^{2}(B)$. Now in view of Lemma 2.8, we find that
$f_{1},\ldots,f_{n}\in H^{\infty}(B)$. This completes the argument.

The proofs of Theorem 3.3 and Theorem 3.4 shall make use of the descriptions
of common invariant subspaces established in [22] and the invariant subspaces
established in [24]. For the reader’s convenience, we record both these
results as under:

######  Theorem 3.1 (Sahni and Singh [22]).

Let $\mathcal{M}$ be a closed subspace of $H^{2}$ invariant under
${H_{1}}^{\infty}(B)$ but not under $H^{\infty}(B)$. Then there exist
$B$-inner functions $J_{1},\ldots,J_{r}$ ($r\leq n$) such that

$\mathcal{M}=\left(\sum_{i=1}^{k}\oplus\langle\varphi_{i}\rangle\right)\oplus\sum_{j=1}^{r}\oplus
B^{2}J_{j}H^{2}(B)$

where $k\leq 2r-1$, and $\varphi_{i}$ =
$(\beta_{1i}+\beta_{2i}B)J_{1}+(\beta_{3i}+\beta_{4i}B)J_{2}+\ldots+(\beta_{2r-1,i}+\beta_{2r,i}B)J_{r}$,
for all i = 1,…,k. Moreover,
$\lvert\beta_{1i}\rvert^{2}+\lvert\beta_{2i}\rvert^{2}+\ldots+\lvert\beta_{2r,i}\rvert^{2}=1$.

######  Theorem 3.2 (Singh and Thukral [24]).

Suppose $\mathcal{M}$ is a closed subspace of $H^{2}$ invariant under
$H^{\infty}(B)$. Then there exist $B$-inner functions $J_{1},\ldots,J_{r}$
($r\leq n$) such that

$\mathcal{M}=J_{1}H^{2}(B)\oplus J_{2}H^{2}(B)\oplus\cdots\oplus
J_{r}H^{2}(B).$

Also the above representation is unique in the following sense: Suppose

$\mathcal{M}=I_{1}H^{2}(B)\oplus I_{2}H^{2}(B)\oplus\cdots\oplus
I_{s}H^{2}(B)$

then $r=s$, and each $J_{i}=\sum\limits_{k=0}^{r}\phi_{ik}I_{k}$ for some
scalars $\phi_{ik}$ such that the matrix $(\phi_{ik})$ is unitary.

Now we state and prove the common invariant subspace characterization for
$H^{\infty}$.

######  Theorem 3.3.

Let $\mathcal{M}$ be a weak*-closed subspace of $H^{\infty}$ such that
$\mathcal{M}$ is invariant under ${H_{1}}^{\infty}(B)$, but not under
$H^{\infty}(B)$. Then there exist $B$-inner functions $J_{1},\ldots,J_{r}$
($r\leq n$) such that

$\mathcal{M}=\left(\sum_{i=1}^{k}\oplus\langle\varphi_{i}\rangle\right)\oplus\sum_{j=1}^{r}\oplus
B^{2}J_{j}H^{\infty}(B),$

where $k\leq 2r-1$, and $\varphi_{i}$ =
$(\beta_{1i}+\beta_{2i}B)J_{1}+(\beta_{3i}+\beta_{4i}B)J_{2}+\ldots+(\beta_{2r-1,i}+\beta_{2r,i}B)J_{r}$,
for all i = 1,…,k. We shall also see that
$\beta_{1i},\beta_{2i},\ldots,\beta_{2r,i}$ are complex numbers with
$\lvert\beta_{1i}\rvert^{2}+\lvert\beta_{2i}\rvert^{2}+\cdots+\lvert\beta_{2r,i}\rvert^{2}=1$.

###### Proof.

We first show that ${\overline{\mathcal{M}}}^{\|.\|_{2}}\cap
H^{\infty}=\mathcal{M}$. Note that
$\mathcal{M}\subset{\overline{\mathcal{M}}}^{\|.\|_{2}}$ and hence,
$\mathcal{M}\subset{\overline{\mathcal{M}}}^{\|.\|_{2}}\cap H^{\infty}$.

Conversely, start with any $f\in{\overline{\mathcal{M}}}^{\|.\|_{2}}\cap
H^{\infty}$. So, we can find a sequence $\\{f_{k}\\}$ in $\mathcal{M}$ such
that $f_{k}\longrightarrow f$ in the norm of $H^{2}$. Here $\\{f_{k}\\}$ and
$f$ are functions of $H^{\infty}$. Using the decomposition
$H^{\infty}=e_{00}H^{\infty}(B)\oplus e_{10}H^{\infty}(B)\oplus\cdots\oplus
e_{n-1,0}H^{\infty}(B)$, we can write

$f_{k}=e_{00}{f_{k}}^{(1)}+e_{10}{f_{k}}^{(2)}+\cdots+e_{n-1,0}{f_{k}}^{(n)},$
(3.3) $f=e_{00}{f}^{(1)}+e_{10}{f}^{(2)}+\cdots+e_{n-1,0}{f}^{(n)},$ (3.4)

where ${f_{k}}^{(i)}$ and $f^{(i)}$ belong to $H^{\infty}(B)$.
The following computation leads to the conclusion that $f_{k}^{(i)}\to
f^{(i)}$ in $H^{2}$:

$\displaystyle{\|f_{k}-f\|_{2}}^{2}$
$\displaystyle={\|e_{00}(f_{k}^{(1)}-f^{(1)})+e_{10}(f_{k}^{(2)}-f^{(2)})+\cdots+e_{n-1,0}(f_{k}^{(n)}-f^{(n)})\|_{2}}^{2}$
(3.5)
$\displaystyle={\|f_{k}^{(1)}-f^{(1)}\|_{2}}^{2}+{\|f_{k}^{(2)}-f^{(2)}\|_{2}}^{2}+\cdots+{\|f_{k}^{(n)}-f^{(n)}\|_{2}}^{2}.$

By Lemma 2.7, there exists a sequence $\\{{h_{k}}^{(i)}\\}$ of uniformly
bounded $H^{\infty}$ functions that converges to 1 $a.e.$ such that
$h_{k}^{(i)}f_{k}^{(i)}$ $\longrightarrow$ $f^{(i)}$ in the weak* topology.
Define

$g_{k}$ = $h_{k}^{(1)}(B^{2})h_{k}^{(2)}(B^{2})\ldots h_{k}^{(n)}(B^{2})$.

Observe that $g_{k}\in H^{\infty}(B^{2})$ and $h_{k}^{(i)}\to 1\;a.e.$ as
$k\to\infty$. Thus, $h_{k}^{(i)}(B^{2})$ is a uniformly bounded sequence which
converges to 1 $a.e.$ (because the composition operator on $H^{\infty}$
induced by $B^{2}$ is a contraction ,i.e,
$\|foB^{2}\|_{\infty}\leq\|f\|_{\infty}$). This forces $g_{k}\to 1\;a.e.$
For each $i$, $f_{k}^{(i)}\to f^{(i)}$ in $H^{2}$, there exists subsequence
$f_{k_{j}^{(i)}}\to f^{(i)}\;a.e.$ Without loss of generality, assume that
$f_{k}^{(i)}\to f^{(i)}\;a.e$. This gives $\\{g_{k}f_{k}^{(i)}\\}$ converges
to $f^{(i)}\;a.e$.
For any arbitrary $u\in L^{1}$, $\\{g_{k}f_{k}^{(i)}u\\}$ $\longrightarrow$
$f^{(i)}u\;a.e.$ and

$\lvert g_{k}f_{k}^{(i)}u-f^{(i)}u+f^{(i)}u\rvert$ $\leq$ $\lvert
g_{k}f_{k}^{(i)}u-f^{(i)}u\rvert+\lvert f^{(i)}u\rvert$.

In view of the convergence of $\\{g_{k}f_{k}^{(i)}u\\}$, we can find a natural
number $k_{0}$ such that

$\lvert g_{k}f_{k}^{(i)}u\rvert<1+\lvert f^{(i)}u\rvert\;\;a.e.\;\;\text{for
all}\;\;k\geq k_{0}.$

By the Dominated Convergence theorem,

$\int_{{\mathbb{T}}}g_{k}f_{k}^{(i)}udm$ $\longrightarrow$
$\int_{{\mathbb{T}}}f^{(i)}udm$.

Since $u$ was an arbitrary element of $L^{1}$, so $\\{g_{k}f_{k}^{(i)}\\}$
converges to $f^{(i)}$ for fixed $i=1,\ldots,n$ in the weak* topology.
As $e_{i0}u\in L^{1}$ whenever $u\in L^{1}$, $\\{e_{i0}g_{k}f_{k}\\}$
$\longrightarrow$ $e_{i0}f^{(i)}$ in the weak* topology.
Therefore,

$\sum_{i=0}^{n-1}e_{i0}g_{k}f_{k}^{(i+1)}\;\text{converges
weak}^{*}\;\text{to}\;\sum_{i=0}^{n-1}e_{i0}f^{(i+1)}.$

This implies that $\\{{g_{k}}{f_{k}}\\}$ converges weak* to $f$. Since
${g_{k}}\in H^{\infty}(B^{2})$ and $\mathcal{M}$ is invariant under the
operator $T_{B^{2}}$, we have $\\{{g_{k}}{f_{k}}\\}\in\mathcal{M}$. Further,
since $\mathcal{M}$ is weak*-closed, $f\in\mathcal{M}$ and hence
${\overline{\mathcal{M}}}^{\|.\|_{2}}\cap H^{\infty}\subset\mathcal{M}$.

Observe that $\overline{\mathcal{M}}^{\|.\|_{2}}$ is a closed subspace of
$H^{2}$ which is invariant under $H_{1}^{\infty}(B)$, but not under
$H^{\infty}(B)$. So by Theorem 3.1, we can write

$\overline{\mathcal{M}}^{\|.\|_{2}}=\left(\sum_{i=1}^{k}\oplus\langle\varphi_{i}\rangle\right)\oplus\sum_{j=1}^{r}\oplus
B^{2}J_{j}H^{2}(B)$ (3.6)

where $k\leq 2r-1$, and $\varphi_{i}$ =
$(\beta_{1i}+\beta_{2i}B)J_{1}+(\beta_{3i}+\beta_{4i}B)J_{2}+...+(\beta_{2r-1,i}+\beta_{2r,i}B)J_{r}$,
for all i = 1,2,…,k.
Therefore, we have

$\mathcal{M}=\overline{\mathcal{M}}^{\|.\|_{2}}\cap
H^{\infty}=\left(\left(\sum_{i=1}^{k}\oplus\langle\varphi_{i}\rangle\right)\oplus\sum_{j=1}^{r}\oplus
B^{2}J_{j}H^{2}(B)\right)\cap H^{\infty}.$

It remains to be shown that
$\mathcal{M}=\left(\sum_{i=1}^{k}\oplus\langle\varphi_{i}\rangle\right)\oplus\sum_{j=1}^{r}\oplus
B^{2}J_{j}H^{\infty}(B)$.
Note that

$H^{\infty}\cap\left[\left(\sum_{i=1}^{k}\oplus\langle\varphi_{i}\rangle\right)\oplus\sum_{j=1}^{r}\oplus
B^{2}J_{j}H^{\infty}(B)\right]$ $\subset$
$H^{\infty}\cap\left[\left((\sum_{i=1}^{k}\oplus\langle\varphi_{i}\rangle\right)\oplus\sum_{j=1}^{r}\oplus
B^{2}J_{j}H^{2}(B)\right]$.

Therefore,

$\left(\sum_{i=1}^{k}\oplus\langle\varphi_{i}\rangle\right)\oplus\sum_{j=1}^{r}\oplus
B^{2}J_{j}H^{\infty}(B)\subset\mathcal{M}.$ (3.7)

For the reverse inclusion, write an arbitrary $f\in\mathcal{M}$ as

$f$ =
$\left(\sum_{i=1}^{k}a_{i}\varphi_{i}\right)+\sum_{j=1}^{r}B^{2}J_{j}f_{j}$

where $a_{i}$’s are complex numbers and $f_{1},\ldots,f_{r}$ $\in H^{2}(B)$.
Since the function $h=\sum_{j=1}^{r}B^{2}J_{j}f_{j}\in H^{\infty}$ and
$J_{1},\ldots,J_{r}$ are $B$-inner, so by Lemma 2.8, the functions
$B^{2}f_{1},\ldots,B^{2}f_{r}$ are in $H^{\infty}(B)$. Hence
$f_{1},\ldots,f_{r}$ are also in $H^{\infty}(B)$. This forces
$f\in\left(\sum_{i=1}^{k}\oplus\langle\varphi_{i}\rangle\right)\oplus\sum_{j=1}^{r}\oplus
B^{2}J_{j}H^{\infty}(B)$. This completes the proof. ∎

Now, we obtain the $B$-invariant subspaces for $H^{\infty}$ using elementary
arguments which completely avoids the general inner-outer factorization of
$H^{p}$ functions in terms of $B$-inner functions and outer functions. For
details, the reader may refer to Proposition $6$ in [17]. Our proof is on
similar lines as the proof of the decomposition (3.1) and is even more
elementary than the proof presented in [22].

######  Theorem 3.4.

Let $\mathcal{M}$ be a weak*-closed subspace of $H^{\infty}$ such that
$\mathcal{M}$ is invariant under $T_{B}$. Then there exist $B$-inner functions
$J_{1},\ldots,J_{r}$ ($r\leq n$) such that

$\mathcal{M}=J_{1}H^{\infty}(B)\oplus\cdots\oplus J_{r}H^{\infty}(B).$

###### Proof.

Using the arguments similar to those in the proof of Theorem 3.3, we see that
$\mathcal{M}={\overline{\mathcal{M}}}^{\|.\|_{2}}\cap H^{\infty}$. Again,
since ${\overline{\mathcal{M}}}^{\|.\|_{2}}$ is a $T_{B}$-invariant subspace
of $H^{2}$, so by Theorem 3.2 we can write:

$\overline{\mathcal{M}}^{\|.\|_{2}}=J_{1}H^{2}(B)\oplus\cdots\oplus
J_{r}H^{2}(B)$

for some $B$-inner functions $J_{1},\ldots,J_{r}$ ( $r\leq n$).

This implies

$\mathcal{M}=\overline{\mathcal{M}}^{\|.\|_{2}}\cap
H^{\infty}=\left(J_{1}H^{2}(B)\oplus\cdots\oplus J_{r}H^{2}(B)\right)\cap
H^{\infty}.$

Lastly, we can show that $\left(J_{1}H^{2}(B)\oplus\cdots\oplus
J_{r}H^{2}(B)\right)\cap H^{\infty}=J_{1}H^{\infty}(B)\oplus\cdots\oplus
J_{r}H^{\infty}(B)$ on the same lines as in the proof of the decomposition in
equation (3.1). ∎

For the convenience of the reader, we now state the main results of this paper
which generalize the Lax-Halmos type results in [22]. Before proving these, we
discuss the road map of the proof briefly.

######  Theorem 3.5.

Let ${\alpha}$ be a continuous $\|.\|_{1}$-dominating normalized gauge norm
and $\mathcal{M}$ be an ${\alpha}$-closed linear subspace of $H^{\alpha}$.
Suppose $\mathcal{M}$ is invariant under $T_{B^{2}}$ and $T_{B^{3}}$, but not
under $T_{B}$. Then,

$\mathcal{M}=\left(\sum_{i=1}^{k}\oplus\langle\varphi_{i}\rangle\right)\oplus
B^{2}\;[(J_{1}M_{\alpha}(B)\oplus\cdots\oplus J_{r}M_{\alpha}(B))]_{\alpha}$

where $M_{\alpha}(B):=[H^{\infty}(B)]_{\alpha}$, $J_{1},\ldots,J_{r}$ ($r\leq
n$) are $B$-inner functions and $k\leq 2r-1$. Furthermore, $\varphi_{i}$ =
$(\beta_{1i}+\beta_{2i}B)J_{1}+(\beta_{3i}+\beta_{4i}B)J_{2}+\ldots+(\beta_{2r-1,i}+\beta_{2r,i}B)J_{r}$
for $i=1,\ldots,k$. The complex numbers $\beta_{ji}$ satisfy
$\lvert\beta_{1i}\rvert^{2}+\lvert\beta_{2i}\rvert^{2}+\cdots+\lvert\beta_{2r,i}\rvert^{2}=1$.
Here $[H^{\infty}(B)]_{\alpha}$ stands for the ${\alpha}$-closure of
$H^{\infty}(B)$.

######  Theorem 3.6.

Suppose ${\alpha}$ to be a continuous $\|.\|_{1}$-dominating normalized gauge
norm and $\mathcal{M}$ be an ${\alpha}$-closed linear subspace of
$H^{\alpha}$. Suppose $\mathcal{M}$ is invariant under $T_{B}$. Then,

$\mathcal{M}=[J_{1}M_{\alpha}(B)\oplus\cdots\oplus
J_{r}M_{\alpha}(B)]_{\alpha}$

for some $B-$inner functions $J_{1},...,J_{r}$ $(r\leq n)$.

The proof of Theorem 3.5 comprises of the following key steps: First, we
identify $\mathcal{M}\cap H^{\infty}$ to be a non-trivial weak*-closed
subspace of $H^{\infty}$ which has the same invariance properties as that of
$\mathcal{M}$. Next, we use Theorems 3.3 and 3.4 to describe $\mathcal{M}\cap
H^{\infty}$. Lastly, we finish off the proof by establishing that
$\mathcal{M}\cap H^{\infty}$ is ${\alpha}$-dense in $M$. It is worth pointing
out that our arguments to show the ${\alpha}$-denseness of $\mathcal{M}\cap
H^{\infty}$ do not rely on any factorization of $L^{\alpha}$ functions as is
the case in [3]. Instead, we work with the classical inner-outer factorization
of $H^{2}$ functions.

The following two lemmas will be needed to prove that $\mathcal{M}\cap
H^{\infty}$ is a weak*-closed subspace of $H^{\infty}$ (Lemma 3.9).

######  Lemma 3.7 (Krein-Smulian Theorem, [6]).

Let $X$ be a Banach space. A convex set in $X^{\\#}$ is weak*-closed if and
only if its intersection with the closed unit ball $\\{\phi\in
X^{\\#}:\|\phi\|\leq 1\\}$ of $X^{\\#}$ is weak*-closed.

######  Lemma 3.8 (Chen [3]).

Let ${\alpha}$ be a continuous $\|.\|_{1}$-dominating normalized gauge norm.
Let $\mathbb{B}(L^{\infty})=\\{f\in L^{\infty}:\|f\|_{\infty}\leq 1\\}$ denote
the closed unit ball on $L^{\infty}$. Then
$(1)$ The ${\alpha}$-topology and the $\|.\|_{2}$-topology coincide on
$\mathbb{B}(L^{\infty})$.
$(2)$ The closed unit ball $\mathbb{B}(L^{\infty})$ is ${\alpha}$-closed.

######  Lemma 3.9.

Suppose ${\alpha}$ is a continuous $\|.\|_{1}$-dominating normalized gauge
norm and $\mathcal{M}$ be a closed subspace of $H^{\alpha}$. Then
$\mathcal{M}\cap H^{\infty}$ is weak*-closed in $H^{\infty}$.

###### Proof.

Let $\mathbb{B}=\\{\phi\in(L^{1})^{\\#}:\|\phi\|\leq 1\\}$ be the closed unit
ball in $(L^{1})^{\\#}$. Since $(L^{1})^{\\#}$ can be identified with
$L^{\infty}$, so $\mathbb{B}$ can be identified with
$\mathbb{B}(L^{\infty})=\\{f\in L^{\infty}({\mathbb{T}}):\|f\|_{\infty}\leq
1\\}$.
We shall thus consider $\mathcal{M}\cap H^{\infty}\cap\mathbb{B}(L^{\infty})$.
Note that $\mathcal{M}\cap H^{\infty}\cap\mathbb{B}(L^{\infty})$ simplifies to
$\mathcal{M}\cap\mathbb{B}(L^{\infty})$. This follows by observing that any
$f\in\mathcal{M}\cap\mathbb{B}(L^{\infty})$ also belongs to $H^{1}\cap
L^{\infty}=H^{\infty}$.
Thus, we now only need to prove that $\mathcal{M}\cap\mathbb{B}(L^{\infty})$
is weak*-closed in $L^{\infty}$.
By Lemma 3.8, we see that $\mathcal{M}\cap\mathbb{B}(L^{\infty})$ is closed in
$L^{\alpha}$. This in turn implies that
$\mathcal{M}\cap\mathbb{B}(L^{\infty})$ is closed in the $L^{2}$ norm.
Let $\\{f_{\lambda}\\}$ be a net in $\mathcal{M}\cap\mathbb{B}(L^{\infty})$
such that $f_{\lambda}$ converges weak* to $f$. This means that for every
$h\in L^{1}$, $\int_{{\mathbb{T}}}(f_{\lambda}-f)hdm$ $\to$ 0 . In particular,
$\int_{{\mathbb{T}}}(f_{\lambda}-f)hdm$ $\to$ 0, for every $h\in L^{2}$. This
implies $f_{\lambda}$ $\to$ $f$ weakly in $L^{2}$. Since a closed convex set
in a Banach space is weakly closed, so $\mathcal{M}\cap\mathbb{B}(L^{\infty})$
is also weakly closed in $L^{2}$. Hence
$f\in\mathcal{M}\cap\mathbb{B}(L^{\infty})$. ∎

######  Remark 3.10.

Note that the collection $\mathcal{M}\cap\mathbb{B}(L^{\infty})\neq{[0]}$
whenever $\mathcal{M}\cap H^{\infty}\neq{[0]}$. In case $\mathcal{M}$ is
invariant under any power of $B$, then we can guarantee that $\mathcal{M}\cap
H^{\infty}\neq{[0]}$.

######  Lemma 3.11.

Let ${\alpha}$ be a continuous $\|.\|_{1}$-dominating normalized gauge norm
and $\mathcal{M}$ be a closed subspace of $H^{\alpha}$ which is invariant
under $T_{B}$. Then, $\mathcal{M}\cap H^{\infty}$ is ${\alpha}$-dense in
$\mathcal{M}$.

###### Proof.

Since $\mathcal{M}$ is an ${\alpha}$-closed subspace of $H^{\alpha}$, it
follows that $[{\mathcal{M}\cap H^{\infty}}]_{\alpha}$ $\subset$
$\mathcal{M}$.

Conversely, any $f\in\mathcal{M}\subset H^{\alpha}\subset H^{1}$ can be
factorized as $f=IO$, where $I$ is an inner function and $O$ is an outer
function.
We can further write $f$ as a product of three $H^{2}$ functions:

$f$ = $I$ $O^{\frac{1}{2}}$ $O^{\frac{1}{2}}.$

For the sake of convenience, put $f_{1}=I$, $f_{2}=f_{3}=O^{\frac{1}{2}}$.
Using the decomposition of $H^{2}$ in equation (2.1), we can express each
$f_{k}$ as

$f_{k}=e_{00}g_{1}^{(k)}+e_{10}g_{2}^{(k)}+\cdots+e_{n-1,0}g_{n}^{(k)}$ (3.8)

where $g_{1}^{(k)},g_{2}^{(k)},\ldots,g_{n}^{(k)}\in H^{2}(B)$. It is well
known that the operator $T:H^{2}\rightarrow H^{2}$, defined by $T(f)=f(B(z))$,
is an isometry and its range is $H^{2}(B)=\\{foB:f\in H^{2}\\}$ (see [7]).
Hence, each $g_{i}^{(k)}\in H^{2}(B)$ can be written as
$g_{i}^{(k)}=s_{i}^{(k)}(B(z))$ for some $s_{i}^{(k)}\in H^{2}$.
Thus equation (3.8) becomes

$f_{k}=e_{00}s_{1}^{(k)}(B(z))+e_{10}s_{2}^{(k)}(B(z))+\cdots+e_{n-1,0}s_{n}^{(k)}(B(z)).$
(3.9)

Define

$q_{m}^{(ik)}(z)$ = $exp\left(\dfrac{-\lvert
s_{i}^{(k)}(z)\rvert^{\frac{1}{2}}\;-i\;(\lvert
s_{i}^{(k)}(z)\rvert^{\frac{1}{2}})^{\sim}}{m}\right).$

Here $(\lvert s_{i}^{(k)}\rvert^{\frac{1}{2}})^{\sim}$ denotes for the
harmonic conjugate of the harmonic extension of the boundary function $\lvert
s_{i}^{(k)}(e^{i\theta})\rvert^{\frac{1}{2}}$ onto the unit disc (this is
possible for all $L^{p}$ functions [15]).

So, $\\{q_{m}^{(ik)}(z)\\}$ is a sequence of analytic functions such that
$\lvert q_{m}^{(ik)}(z)\rvert\leq 1$ for each $m\geq 1$. Next, for each $k$,
construct a sequence $h_{m}^{(k)}(z)$ = $q_{m}^{(1k)}(z)$
$q_{m}^{(2k)}(z)\ldots q_{m}^{(nk)}(z)$, then $\\{h_{m}^{(k)}(z)\\}\in
H^{\infty}$ and

$h_{m}^{(k)}(B(z))f_{k}(z)=h_{m}^{(k)}(B(z))\left(e_{00}s_{1}^{(k)}(B(z))+\cdots+e_{n-1,0}s_{n}^{(k)}(B(z))\right)$
$\hskip
71.13188pt=e_{00}h_{m}^{(k)}(B(z))s_{1}^{(k)}(B(z))+\cdots+e_{n-1,0}h_{m}^{(k)}(B(z))s_{n}^{(k)}(B(z)).$

Moreover,
$\lvert h_{m}^{(k)}(B(z))f_{k}\rvert\leq\lvert q_{m}^{(1k)}(B(z))\rvert\lvert
s_{1}^{(k)}(B(z))\rvert+\cdots+\lvert q_{m}^{(nk)}(B(z))\rvert\lvert
s_{n}^{(k)}(B(z))\rvert$
$=exp\left(\dfrac{-\lvert s_{1}^{(k)}(z)\rvert^{\frac{1}{2}}}{m}\right)\lvert
s_{1}^{(k)}(B(z))\rvert+\cdots+exp\left(\dfrac{-\lvert
s_{n}^{(k)}(z)\rvert^{\frac{1}{2}}}{m}\right)\lvert s_{n}^{(k)}(B(z))\rvert$
$\leq C$ for $C$ to be any constant.
Hence $h_{m}^{(k)}(B(z))f_{k}\in H^{\infty}$ for each $k=1,2,3$.
Note that $h_{m}^{(1)}(B(z))\;h_{m}^{(2)}(B(z))\;h_{m}^{(3)}(B(z))$ $\to$ 1
$a.e.$ Therefore
$t_{m}(B(z)):=h_{m}^{(1)}(B(z))\;h_{m}^{(2)}(B(z))\;h_{m}^{(3)}(B(z))\to
1\;a.e.$ as $m\to\infty$, and $\lvert t_{m}(B(z))-1\rvert\to 0\;a.e.$, implies

$\|t_{m}(B(z))-1\|_{\infty}$ $\longrightarrow$ 0 as $m\to\infty$.

We can see that ${\alpha}(t_{m}(B(z))f-f)$ = ${\alpha}(f\;(t_{m}(B(z))-1))$
$\leq{\alpha}(f)\;\|t_{m}(B(z))-1\|_{\infty}$, i.e,

${\alpha}(t_{m}(B(z))f-f)$ $\longrightarrow$ 0 as $m\to\infty$.

Since $h_{m}^{(k)}(B(z))f_{k}\in H^{\infty}$, so $\\{t_{m}(B(z))f\\}$
$=h_{m}^{(1)}(B(z))f_{1}$ $h_{m}^{(2)}(B(z))f_{2}$ $h_{m}^{(3)}(B(z))f_{3}$
belongs to $H^{\infty}$.
Lastly, the invariance of $\mathcal{M}$ under $H^{\infty}(B)$ gives
$\\{t_{m}(B(z))f\\}$ $\in\mathcal{M}$. So, there is a sequence
$\\{t_{m}(B(z))f\\}$ in $\mathcal{M}\cap H^{\infty}$ such that
${\alpha}(t_{m}(B(z))f-f)$ $\to$ 0 as $m\to\infty$, which implies that
$f\in[\mathcal{M}\cap H^{\infty}]_{\alpha}$. Therefore, $\mathcal{M}$ =
$[\mathcal{M}\cap H^{\infty}]_{\alpha}$. ∎

We now revert to the proof of Theorem 3.5.

###### Proof of Theorem 3.5.

We first show that $\mathcal{M}\cap H^{\infty}\neq[0]$. Let $0\neq
f\in\mathcal{M}$ then as argued previously in Lemma 3.11, we can express $f$
as

$f$ = $f_{1}$ $f_{2}$ $f_{3}$

for some $f_{1},f_{2},f_{3}\in H^{2}$. We follow the same argument to express
each $f_{k}$ as

$f_{k}=e_{00}s_{1}^{(k)}(B^{2}(z))+e_{10}s_{2}^{(k)}(B^{2}(z))+\cdots+e_{2n-1,0}s_{2n}^{(k)}(B^{2}(z))$

for some $s_{1}^{(k)}(B^{2}(z))$, $s_{2}^{(k)}(B^{2}(z))$,…,
$s_{2n}^{(k)}(B^{2}(z))\in H^{2}(B^{2})$. Define

$q_{ik}(z)$ = $exp\left(\dfrac{-\lvert
s_{i}^{(k)}(z)\rvert^{\frac{1}{2}}\;-i\;(\lvert
s_{i}^{(k)}(z)\rvert^{\frac{1}{2}})^{\sim}}{2}\right)$

( $\sim$ denotes the harmonic conjugate and defined in a similar fashion as
above).
Furthermore, $\lvert q_{ik}(z)\rvert\in H^{\infty}$ and $\lvert
q_{ik}(z)\rvert\leq 1$.
Define $h_{k}(z)=q_{1k}(z)\;q_{2k}(z)\ldots q_{2n,k}(z)\in H^{\infty}$. Note
that,
$h_{k}(B^{2}(z))f_{k}(z)=h_{k}(B^{2}(z))\left(e_{00}s_{1}^{(k)}(B^{2}(z))+\cdots+e_{2n-1,0}s_{2n}^{(k)}(B^{2}(z))\right)$

$=e_{00}h_{k}(B^{2}(z))s_{1}^{(k)}(B^{2}(z))+\cdots+e_{2n-1,0}h_{k}(B^{2}(z))s_{2n}^{(k)}(B^{2}(z))$
and hence $h_{k}(B^{2}(z))f_{k}(z)\in H^{\infty}$ for each $k=1,2,3$.
This implies $h_{1}(B^{2}(z))h_{2}(B^{2}(z))h_{3}(B^{2}(z))f$ is in
$H^{\infty}$. By Lemma 2.5 we see that
$h_{1}(B^{2}(z))h_{2}(B^{2}(z))h_{3}(B^{2}(z))f\in\mathcal{M}\cap H^{\infty}$.
This establishes that $\mathcal{M}\cap H^{\infty}\neq[0]$ whenever
$\mathcal{M}\neq[0]$.

In view of Lemma 3.9, $\mathcal{M}\cap H^{\infty}$ is a weak*-closed subspace
of $H^{\infty}$ which is also invariant under $T_{B^{2}}$ and $T_{B^{3}}$, but
not under $T_{B}$. By Theorem 3.3,

$\mathcal{M}\cap
H^{\infty}=\left(\sum_{i=1}^{k}\oplus\langle\varphi_{i}\rangle\right)\oplus\sum_{j=1}^{r}\oplus
B^{2}J_{j}H^{\infty}(B)$

where $k\leq 2r-1$, and it follows from the Lemma 3.7 that

$\mathcal{M}=\left[\left(\sum_{i=1}^{k}\oplus\langle\varphi_{i}\rangle\right)\oplus\sum_{j=1}^{r}\oplus
B^{2}J_{j}H^{\infty}(B)\right]_{{\alpha}}$
$\;\;\;=\left(\sum_{i=1}^{k}\oplus\langle\varphi_{i}\rangle\right)\oplus\left[\sum_{j=1}^{r}\oplus
B^{2}J_{j}H^{\infty}(B)\right]_{{\alpha}}$ $\hskip
73.97733pt=\left(\sum_{i=1}^{k}\oplus\langle\varphi_{i}\rangle\right)\oplus
B^{2}\;\left[(J_{1}H^{\infty}(B)\oplus\cdots\oplus
J_{r}H^{\infty}(B))\right]_{{\alpha}}.$ (3.10)

We now claim that $[(J_{1}H^{\infty}(B)\oplus\cdots\oplus
J_{r}H^{\infty}(B))]_{\alpha}$ = $[(J_{1}M_{\alpha}(B)\oplus\cdots\oplus
J_{r}M_{\alpha}(B))]_{\alpha}$.
It can easily be shown $[(J_{1}H^{\infty}(B)\oplus\cdots\oplus
J_{r}H^{\infty}(B))]_{\alpha}$ $\subset$
$[(J_{1}M_{\alpha}(B)\oplus\cdots\oplus J_{r}M_{\alpha}(B))]_{\alpha}$ because
$H^{\infty}(B)$ $\subset$ $M_{\alpha}(B)$.
To show the other containment, consider $f\in
J_{1}M_{\alpha}(B)\oplus\cdots\oplus J_{r}M_{\alpha}(B)$, $\exists$
$f_{1},\ldots,f_{r}\in M_{\alpha}(B)$ such that
$f=J_{1}f_{1}+\cdots+J_{r}f_{r}$. Since
$M_{\alpha}(B)=\left[H^{\infty}(B)\right]_{{\alpha}}$, there exist the
sequence of functions $\\{f_{n}^{(1)}\\},\ldots,\\{f_{n}^{(r)}\\}$ in
$H^{\infty}(B)$ such that

${\alpha}(f_{n}^{(i)}-f_{i})\longrightarrow 0\;\;\text{ as}\;\;n\to\infty$

for $i=1,2,\ldots,r$.
Because $J_{i}$ is $B$-inner function, ${\alpha}(J_{i}f_{n}^{(i)}-J_{i}f_{i})$
$\leq$ $\|J_{i}\|_{\infty}\;{\alpha}(f_{n}^{(i)}-f_{i})$, we have

${\alpha}(J_{i}f_{n}^{(i)}-J_{i}f_{i})\longrightarrow 0\;\;\text{
as}\;\;n\to\infty$

which again gives

$J_{1}f_{n}^{(1)}+\cdots+J_{r}f_{n}^{(r)}\longrightarrow
J_{1}f_{1}+\cdots+J_{r}f_{r}\;\;\text{in}\;\;H^{\alpha}\;\;\text{as}\;\;n\longrightarrow\infty.$

Thus, $f\in\left[(J_{1}H^{\infty}(B)\oplus\cdots\oplus
J_{r}H^{\infty}(B))\right]_{{\alpha}}$ and so
$[J_{1}M_{\alpha}(B)\oplus\cdots\oplus J_{r}M_{\alpha}(B)]_{\alpha}$ $\subset$
$[(J_{1}H^{\infty}(B)\oplus\cdots\oplus J_{r}H^{\infty}(B))]_{\alpha}$.
Hence equation (3.10) becomes:

$\mathcal{M}=\left(\sum_{i=1}^{k}\oplus\langle\varphi_{i}\rangle\right)\oplus
B^{2}\;\left[(J_{1}M_{\alpha}(B)\oplus\cdots\oplus
J_{r}M_{\alpha}(B))\right]_{{\alpha}}.$

This completes the proof. ∎

######  Corollary 3.12.

Let $\mathcal{M}$ be a closed subspace of $H^{\alpha}$ which is invariant
under $H_{1}^{\infty}(z)$, but is not invariant under $H^{\infty}(z)$. Then
there exist scalars $\beta_{1}$ and $\beta_{2}$ in $\mathbb{C}$ such that
$|\beta_{1}|^{2}+|\beta_{2}|^{2}=1$, $\beta_{1}\neq 0$, and an inner function
$J$ such that

$\mathcal{M}=(\beta_{1}+\beta_{2}z)J\oplus z^{2}JH^{\alpha}.$

###### Proof.

Taking $B(z)=z$ in Theorem 3.5, we get:

$\mathcal{M}=(\beta_{1}+\beta_{2}z)J\oplus\left[z^{2}JH^{\alpha}\right]_{\alpha},$

where $J$ is an inner function, and $\beta_{1}$ and $\beta_{2}$ are scalars in
$\mathbb{C}$ such that $|\beta_{1}|^{2}+|\beta_{2}|^{2}=1$. Since $z^{2}$ and
$J$ are inner functions and the operator of multiplication by inner function
on $H^{\alpha}$ acts as an isometry, therefore

$\mathcal{M}=(\beta_{1}+\beta_{2}z)J\oplus z^{2}JH^{\alpha}.$

Moreover, $\beta_{1}\neq 0$ because if $\beta_{1}=0$ then $\mathcal{M}$
becomes invariant under $z$, which is clearly a contradiction. ∎

We now prove Theorem 3.6, which generalizes the Beurling type result proved by
Chen [3]. The proof is on the similar lines as Theorem 3.6, so we only provide
a broad sketch.

###### Proof of Theorem 3.6.

It can be shown as in the proof of Theorem 3.5 that $\mathcal{M}\cap
H^{\infty}$ is non-trivial and invariant under $T_{B}$. Further, by Lemma 3.9
it follows that $\mathcal{M}\cap H^{\infty}$ is a weak*-closed subspace of
$H^{\infty}$. Next, in view of Lemma 3.11 we conclude that
$\mathcal{M}=[\mathcal{M}\cap H^{\infty}]_{{\alpha}}$. By Theorem 3.4, there
exist $B$-inner functions $J_{1},\ldots,J_{r}$ (with $r\leq n$) such that

$\mathcal{M}\cap H^{\infty}=J_{1}H^{\infty}(B)\oplus\cdots\oplus
J_{r}H^{\infty}(B).$ (3.11)

Now, by taking ${\alpha}$-closure on both sides of equation 3.11,

$\mathcal{M}=\left[J_{1}H^{\infty}(B)\oplus\cdots\oplus
J_{r}H^{\infty}(B)\right]_{{\alpha}}.$

The claim that $\left[J_{1}H^{\infty}(B)\oplus\cdots\oplus
J_{r}H^{\infty}(B)\right]_{{\alpha}}$ =
$\left[J_{1}M_{\alpha}(B)\oplus\cdots\oplus
J_{r}M_{\alpha}(B)\right]_{\alpha}$ follows by the arguments identical to
those in Theorem 3.5. ∎

######  Corollary 3.13 (Chen [3]).

Suppose $\mathcal{M}$ is a closed subspace of $H^{\alpha}$ such that
$z\mathcal{M}\subset\mathcal{M}$. Then there exists an inner function $J$ such
that $\mathcal{M}=JH^{\alpha}$.

###### Proof.

Take $B(z)=z$ in Theorem 3.6, we have

$\mathcal{M}=[JH^{\alpha}]_{\alpha}$

where $J$ is an inner function. Also, the multiplication by an inner function
on $H^{\alpha}$ will be an isometry, which implies

$\mathcal{M}=JH^{\alpha}.$

∎

In the next section, we show that descriptions of the invariant subspaces of
Theorem 3.5 and 3.6 can be made sharper when ${\alpha}$ is a continuous
rotationally symmetric norm.

##  4\. Case of Continuous Rotationally Symmetric Norms

For a finite Blaschke factor $B(z)$, it is well known that the classical Hardy
spaces $H^{p}$ ($0<p\leq\infty$) can be decomposed as:

$H^{p}=e_{00}H^{p}(B)\oplus e_{10}H^{p}(B)\oplus\cdots\oplus
e_{n-1,0}H^{p}(B).$

Recall that $H^{p}(B)$ denotes the closed linear span of
$\\{B^{m}:m=0,1,2,\ldots\\}$ and $\oplus$ is an algebraic direct sum. This
direct sum becomes an orthogonal direct sum when $p=2$.
Consider $f\in H^{\alpha}$, and since $f\in H^{1}$, we can write $f(z)$ as a
Fourier series

$f(z)=\sum_{j=0}^{\infty}\hat{f}(j)z^{j}$

where $\hat{f}(j)$ = $\int_{\mathbb{T}}f(z)z^{-j}dm$ for $j=0,1,2,\ldots$

Consider the $n$th Cesaro means

$\sigma_{n}(f)=\frac{S_{0}(f)+S_{1}(f)+\cdots+S_{n}(f)}{n+1}$

where $S_{k}(f)$ = $\sum_{j=0}^{k}\hat{f}(j)z^{j}$ stands for the $k$th
partial sum.
The following lemma shows that any function in $H^{\alpha}$ converge to it’s
Cesaro means in ${\alpha}$-norm.

######  Lemma 4.1 (Chen [5]).

Let ${\alpha}$ be a continuous rotationally symmetric norm and $f\in
H^{\alpha}$. Then ${\alpha}(\sigma_{n}(f)-f)\rightarrow 0$ as $n\to\infty$.

We now obtain a similar decomposition for $H^{\alpha}$ in the context of the
Blaschke factor $B(z)=z^{n}$, when ${\alpha}$ is a continuous rotationally
symmetric norm.

######  Lemma 4.2.

Let ${\alpha}$ be a continuous rotationally symmetric norm. We can decompose
the Banach space $H^{\alpha}$ as

$H^{\alpha}=M_{\alpha}(z^{n})\oplus zM_{\alpha}(z^{n})\oplus\cdots\oplus
z^{n-1}M_{\alpha}(z^{n})$

where $M_{\alpha}(z^{n})$ = $\left[H^{\infty}(z^{n})\right]_{\alpha}$ and
$\oplus$ is the algebraic direct sum.

###### Proof.

Since $H^{\infty}(z^{n})\subset H^{\infty}\subset H^{\alpha}$, so we have
$M_{\alpha}(z^{n})\subset H^{\alpha}$. Observe that for any natural number $k$
: ${\alpha}(z^{k}f)={\alpha}(\lvert z^{k}f\rvert)={\alpha}(f)$ for every $f\in
H^{\alpha}$. Thus, multiplication by $z^{k}$ is an isometry on $H^{\alpha}$.
This implies

$M_{\alpha}(z^{n})\oplus zM_{\alpha}(z^{n})\oplus\cdots\oplus
z^{n-1}M_{\alpha}(z^{n})\subset H^{\alpha}$.

For the reverse inclusion, take an arbitrary $f\in H^{\alpha}$. We can write
$f(z)$ as a Fourier series

$f(z)=\sum_{j=0}^{\infty}\hat{f}(j)z^{j}$

where $\hat{f}(j)=\int_{\mathbb{T}}f(z)z^{-j}\;dm$ for $j=0,1,2,\ldots$
Let $\omega=e^{2\pi i/n}$. For each $j=1,2,\ldots,n$, define

$g_{n-j}(z)=\frac{f(z)+\omega^{j}f(\omega
z)+\omega^{2j}f(\omega^{2}z)+\cdots+\omega^{(n-1)j}f(\omega^{n-1}z)}{n}.$

Since ${\alpha}$ is a rotationally symmetric norm, so $f(\omega^{k}z)\in
H^{\alpha}$. Thus, $g_{n-j}\in H^{\alpha}$.
For each $k=0,1,\ldots,n-1$:

$f(\omega^{k}z)=\sum_{j=0}^{\infty}\hat{f}(j)\;\omega^{kj}\;z^{j}.$

Next, a simple computation shows that the coefficient of $z^{m}$ in the
expansion of $n\;g_{n-j}$ is given by

$\hat{f}(m)\;\left[\frac{\omega^{n(j+m)}-1}{\omega^{j+m}-1}\right]\\\
=\left\\{\begin{array}[]{lr}0\;\;,&n\nmid{j+m}\\\ \hat{f}(m)\;n\;,&n\mid
j+m\;.\end{array}\right.$ (4.1)

Equation (4.1) is valid for all $m=0,1,2,\ldots$ and all $j=1,2,\ldots,n$.
Therefore

$n\;g_{n-j}=n\hat{f}(n-j)z^{n-j}+n\hat{f}(2n-j)z^{2n-j}+\cdots.$

Thus

$g_{n-j}=z^{n-j}h_{n-j}(z),$

where $h_{n-j}(z)=\hat{f}(n-j)+\hat{f}(2n-j)z^{n}+\cdots$. Because $g_{n-j}\in
H^{\alpha}$, we can see that $h_{n-j}\in H^{\alpha}$.

We now claim that $h_{n-j}\in M_{\alpha}(z^{n})$.
Since $f(\omega^{k}z)\in H^{\alpha}$ for all $k=0,1,\ldots,n-1$ and
$\sigma_{l}(f(\omega^{k}z))$ be the $l$-th Cesaro mean for $f(\omega^{k}z)$.
By Lemma 4.1,

${\alpha}(\sigma_{l}(f(\omega^{k}z))-f(\omega^{k}z))\longrightarrow
0\;\;\;\text{as}\;\;l\to\infty.$

A straightforward calculation shows that the sum

$\sigma_{l}(f(z))+\omega^{j}\sigma_{l}(f(\omega
z))+\omega^{2j}\sigma_{l}(f(\omega^{2}z))+\cdots+\omega^{(n-1)j}\sigma_{l}(f(\omega^{n-1}z)$
(4.2)

is of the form $z^{n-j}\kappa_{l}(z^{n})$, where $\kappa_{l}(z^{n})$ is a
polynomial in $z^{n}$ which belong to $M_{\alpha}(z^{n})$.
Further, the sum in equation (4.2) converges in $H^{\alpha}$ to
$f(z)+\omega^{j}f(\omega z)+\cdots+\omega^{(n-1)j}f(\omega^{n-1}z)$ as
$l\to\infty$. Therefore, $z^{n-j}\kappa_{l}(z^{n})\to nz^{n-j}h_{n-j}$ as
$l\to\infty$ in $H^{\alpha}$.
That is,

$\frac{1}{n}\kappa_{l}(z^{n})\longrightarrow
h_{n-j}\;\text{as}\;l\to\infty\;\text{in }\;H^{\alpha}.$

Hence the claim follows. Therefore, $f=h_{0}+zh_{1}+\cdots+z^{n-1}h_{n-1}$
belongs to $M_{\alpha}(z^{n})\oplus zM_{\alpha}(z^{n})\oplus\cdots\oplus
z^{n-1}M_{\alpha}(z^{n})$. This completes the proof. ∎

The proof of Lemma 4.2 relies on the fact that every function $f\in
H^{\alpha}$ can be written as $f=f_{0}+zf_{1}+\cdots+z^{n-1}f_{n-1}$, and in
turn we can express each $f_{k}$ in terms of the original $f$. This
manipulation has been figured out only for $B(z)=z^{n}$. Hence, we make the
following conjecture:

######  Conjecture 4.3.

Let ${\alpha}$ be a continuous rotationally symmetric norm and $B$ be a finite
Blaschke factor. Then is it true that $H^{\alpha}=e_{00}M_{\alpha}(B)\oplus
e_{10}M_{\alpha}(B)\oplus\cdots\oplus e_{n-1,0}M_{\alpha}(B)\;?$

We are unaware whether the decomposition in Lemma 4.2 is valid for general
$\|.\|_{1}$-dominating normalized gauge norms. So, we make the following
conjecture:

######  Conjecture 4.4.

Let $\alpha$ be a continuous $\|.\|_{1}$-dominating normalized gauge norm.
Then is it true that $H^{\alpha}=M_{\alpha}(z^{n})\oplus
zM_{\alpha}(z^{n})\oplus\cdots\oplus z^{n-1}M_{\alpha}(z^{n})\;?$

As we have noticed that the class of rotationally symmetric norm is contained
in the class of all $||.||_{1}-$dominating normalized gauge norm. The Theorem
3.5 and Theorem 3.6 are stands true in this case. Our main result of this
section is to show that the decomposition which we have obtained for
$H^{\alpha}$ is useful to sharpen the expression of all closed subspaces of
$H^{\alpha}$ invariant under $H^{\infty}_{1}(z^{n})$, but is not under
$H^{\infty}(z^{n})$.

######  Theorem 4.5.

Suppose ${\alpha}$ is a continuous rotationally symmetric norm and
$\mathcal{M}$ be an ${\alpha}$-closed linear subspace of $H^{\alpha}$. Suppose
$\mathcal{M}$ is invariant under $H^{\infty}_{1}(z^{n})$, but not under
$H^{\infty}(z^{n})$. Then,

$\mathcal{M}=\left(\sum_{i=1}^{k}\oplus\langle\varphi_{i}\rangle\right)\oplus
z^{2n}\;(J_{1}M_{\alpha}(z^{n})\oplus\cdots\oplus J_{r}M_{\alpha}(z^{n}))$

where $M_{\alpha}(z^{n})=\left[H^{\infty}(z^{n})\right]_{{\alpha}}$ ,
$J_{1},...,J_{r}$ ($r\leq n$) are $n$-inner functions and $k\leq 2r-1$.
Furthermore, for all i = 1,2,…,k , $\varphi_{i}$ =
$(\beta_{1i}+\beta_{2i}z^{n})J_{1}+(\beta_{3i}+\beta_{4i}z^{n})J_{2}+\cdots+(\beta_{2r-1,i}+\beta_{2r,i}z^{n})J_{r}$.

###### Proof.

For $B(z)=z^{n}$, we have by Theorem 3.5 that there exist $n$-inner functions
$J_{1},\ldots,J_{r}$ such that $\mathcal{M}$ is of the form:

$\mathcal{M}=\left(\sum_{i=1}^{k}\oplus\langle\varphi_{i}\rangle\right)\oplus
z^{2n}\;\left[J_{1}M_{\alpha}(z^{n})\oplus\cdots\oplus
J_{r}M_{\alpha}(z^{n})\right]_{{\alpha}}$ (4.3)

where the natural number $k$ and the functions $\varphi_{i}$ satisfy the
hypothesis of Theorem 4.5. Consider the closed subspace
$W=\left[J_{1}M_{\alpha}(z^{n})\oplus\cdots\oplus
J_{r}M_{\alpha}(z^{n})\right]_{{\alpha}}$.

We first claim that

$W\cap H^{\infty}=J_{1}H^{\infty}(z^{n})\oplus\cdots\oplus
J_{r}H^{\infty}(z^{n}).$ (4.4)

Note that $W=\left[J_{1}H^{\infty}(z^{n})\oplus\cdots\oplus
J_{r}H^{\infty}(z^{n})\right]_{{\alpha}}$. For any $g\in W\cap H^{\infty}$,
there exists a sequence $\\{g_{m}\\}\subset
J_{1}H^{\infty}(z^{n})\oplus\cdots\oplus J_{r}H^{\infty}(z^{n})$ such that
$g_{m}\to g$ in $H^{\alpha}$. We can write
$g_{m}=J_{1}g_{m}^{(1)}+\cdots+J_{r}g_{m}^{(r)}$ where $g_{m}^{(i)}\in
H^{\infty}(z^{n})$.

By Lemma 2.9, we conclude that ${g_{m}^{(i)}}$ is a Cauchy sequence in $H^{1}$
and hence converges to some $g^{(i)}\in H^{1}$. Consequently, $g_{m}\to
J_{1}g^{(1)}+\cdots+J_{r}g^{(r)}$ in $H^{1}$. Since $g\in H^{\infty}$, Lemma
2.9 implies $g^{(i)}\in H^{\infty}(z^{n})$. So, $g\in
J_{1}H^{\infty}(z^{n})\oplus\cdots\oplus J_{r}H^{\infty}(z^{n})$. This means
that $W\cap H^{\infty}\subset J_{1}H^{\infty}(z^{n})\oplus\cdots\oplus
J_{r}H^{\infty}(z^{n})$. The reverse containment is trivial.

In order to finish off the proof, it remains to be established that
$W=J_{1}M_{\alpha}(z^{n})\oplus\cdots\oplus J_{r}M_{\alpha}(z^{n})$.
It is easy to see that $J_{1}M_{\alpha}(z^{n})\oplus\cdots\oplus
J_{r}M_{\alpha}(z^{n})\subset W$. We now establish the reverse containment.
Let $f\in W$. As demonstrated in the proof of Theorem 3.5, we can construct an
outer function $O\in H^{\infty}(z^{2n})$ such that $Of\in W\cap H^{\infty}$.
By equation (4.4), there exist functions $g_{1},\ldots,g_{r}\in
H^{\infty}(z^{n})\subset M_{\alpha}(z^{n})$ such that:

$Of=J_{1}g_{1}+J_{2}g_{2}+\cdots+J_{r}g_{r}.$ (4.5)

By Lemma 4.2, we can express $f$ as

$f=f_{1}+zf_{2}+\cdots+z^{n-1}f_{n}$

for some $f_{1},...,f_{n}\in M_{\alpha}(z^{n})$. So,

$Of=Of_{1}+zOf_{2}+\cdots+z^{n-1}Of_{n}.$ (4.6)

For the tuple $(J_{1},\ldots,J_{r})$ of $n$-inner functions, we have a matrix
$A=(\varphi_{ij})_{n\times r}$ of functions in $H^{\infty}(z^{n})$ such that
$A^{*}A=I$.
The $n$-inner functions $J_{j}$ can thus be written as

$J_{j}=\sum\limits_{i=0}^{n-1}z^{i}\varphi_{ij}\;\;,\;\;1\leq j\leq r.$ (4.7)

From equations (4.5), (4.6), and (4.7), we conclude that for each
$i=1,2,\ldots,n$:

$Of_{i}$ =
$\varphi_{i-1,1}g_{1}+\varphi_{i-1,2}g_{2}+\cdots+\varphi_{i-1,r}g_{r}.$

Therefore,

$\begin{pmatrix}Of_{1}\\\ \vdots\\\
Of_{n}\end{pmatrix}\;=\;A\;\begin{pmatrix}g_{1}\\\ \vdots\\\
g_{r}\end{pmatrix}.$ (4.8)

Taking the conjugate transpose, we have

$\begin{pmatrix}\overline{Of_{1}}&\cdots&\overline{Of_{n}}\end{pmatrix}\;=\;\begin{pmatrix}\overline{g_{1}}&\cdots&\overline{g_{r}}\end{pmatrix}\;A^{*}.$
(4.9)

Multiplying equations (4.7) and (4.8) yields

$\lvert Of_{1}\rvert^{2}+\cdots+\lvert Of_{n}\rvert^{2}=\lvert
g_{1}\rvert^{2}+\cdots+\lvert g_{r}\rvert^{2}.$

Therefore, for $1\leq i\leq r$,

$\left\lvert\frac{g_{i}}{O}\right\rvert\leq\lvert f_{1}\rvert+\cdots+\lvert
f_{n}\rvert.$

Since $\lvert f_{1}\rvert+\cdots+\lvert f_{n}\rvert\in L^{\alpha}$, we have,
${\alpha}\left(\dfrac{g_{i}}{O}\right)={\alpha}\left(\left\lvert\dfrac{g_{i}}{O}\right\rvert\right)$
$\leq{\alpha}(\lvert f_{1}\rvert+\cdots+\lvert f_{n}\rvert)<\infty.$ So,
$\dfrac{g_{i}}{O}\in L^{\alpha}$.
Further, $\dfrac{g_{i}}{O}\in H^{1}$ because $O$ is outer. Hence,
$\dfrac{g_{i}}{O}\in H^{\alpha}$. By Lemma 4.1, the Cesaro means
$\sigma_{l}\left(\dfrac{g_{i}}{O}\right)$ converge to $\dfrac{g_{i}}{O}$ in
$H^{\alpha}$. Since $\sigma_{l}\left(\dfrac{g_{i}}{O}\right)$ is a polynomial
in $z^{n}$, so $\dfrac{g_{i}}{O}\in M_{\alpha}(z^{n})$.
Hence,
$f=J_{1}\dfrac{g_{1}}{O}+J_{2}\dfrac{g_{2}}{O}+\cdots+J_{r}\dfrac{g_{r}}{O}$
$\in J_{1}M_{\alpha}(z^{n})\oplus\cdots\oplus J_{r}M_{\alpha}(z^{n})$. This
establishes that $W=J_{1}M_{\alpha}(z^{n})\oplus\cdots\oplus
J_{r}M_{\alpha}(z^{n})$.
Hence equation (4.3) can be written as:

$\mathcal{M}=\left(\sum_{i=1}^{k}\oplus\langle\varphi_{i}\rangle\right)\oplus
z^{2n}\;(J_{1}M_{\alpha}(z^{n})\oplus\cdots\oplus J_{r}M_{\alpha}(z^{n})).$

∎

We now derive the $T_{z^{n}}$-invariant subspaces of $H^{\alpha}$, when
${\alpha}$ is a continuous rotationally symmetric norm. The proof proceeds on
similar lines as that of Theorem 4.5, so we only present a sketch.

######  Theorem 4.6.

Let ${\alpha}$ be a continuous rotationally symmetric norm and $\mathcal{M}$
be a closed subspace of $H^{\alpha}$. Suppose $\mathcal{M}$ is invariant under
the operator $T_{z^{n}}$. Then there exist $n$-inner functions
$J_{1},\ldots,J_{r}$ ($r\leq n$) such that

$\mathcal{M}=J_{1}M_{\alpha}(z^{n})\oplus\cdots\oplus J_{r}M_{\alpha}(z^{n}).$

###### Proof.

On the same lines as in the proof of Theorem 3.5, we can show that
$\mathcal{M}\cap H^{\infty}$ is a non-trivial weak*-closed subspace of
$H^{\infty}$ which is invariant under $T_{z^{n}}$.
By Theorem 3.4, there exist $n$-inner functions $J_{1},\ldots,J_{r}$ (with
$r\leq n$ ) such that

$\mathcal{M}\cap H^{\infty}=J_{1}H^{\infty}(z^{n})\oplus\cdots\oplus
J_{r}H^{\infty}(z^{n}).$

Therefore,

$[\mathcal{M}\cap
H^{\infty}]_{\alpha}=[J_{1}H^{\infty}(z^{n})\oplus\cdots\oplus
J_{r}H^{\infty}(z^{n})]_{\alpha}.$

Further, in view of Lemma 3.11 it can be deduced that $[\mathcal{M}\cap
H^{\infty}]_{\alpha}=\mathcal{M}$. Also, it has been established in Theorem
3.5 that $[J_{1}H^{\infty}(z^{n})\oplus\cdots\oplus
J_{r}H^{\infty}(z^{n})]_{\alpha}=[J_{1}M_{\alpha}(z^{n})\oplus\cdots\oplus
J_{r}M_{\alpha}(z^{n})]_{\alpha}$.
So, $\mathcal{M}$ takes the form:

$\mathcal{M}=[J_{1}M_{\alpha}(z^{n})\oplus\cdots\oplus
J_{r}M_{\alpha}(z^{n})]_{\alpha}.$

In order to show that $\mathcal{M}$ takes the desired form it suffices to
establish that

$\mathcal{M}\subset J_{1}M_{\alpha}(z^{n})\oplus\cdots\oplus
J_{r}M_{\alpha}(z^{n}).$

Note that for any $f\in M$, there exists an outer function $O$ such that $Of$
takes the form:

$Of=J_{1}g_{1}+\cdots+J_{r}g_{r}~{}~{}~{}(\;g_{i}\in H^{\infty}(z^{n})\;).$

Now, proceeding as in the proof of Theorem 4.5, we find that
$f=J_{1}\dfrac{g_{1}}{O}+J_{2}\dfrac{g_{2}}{O}+\cdots+J_{r}\dfrac{g_{r}}{O}$.
This completes the proof because $\dfrac{g_{i}}{O}\in M_{\alpha}(z^{n})$. ∎

##  5\. General Inner-Outer Factorization in $H^{\alpha}$

In this section, we derive the general inner-outer factorization of
$H^{\alpha}$ functions. The definition of inner and $n-$inner functions have
been discussed beforehand. The outer function in $H^{\alpha}$ is studied by
Chen in [5]. Let $f\in H^{\alpha}$ is outer if $f$ is a cyclic vector for
$H^{\infty}$ acting on $H^{\alpha}$ which means $[f\cdot
H^{\infty}]_{\alpha}=H^{\alpha}$. We now define an $n$-outer function in
$H^{\alpha}$.

######  Definition 5.1.

Suppose ${\alpha}$ is continuous rotationally symmetric norm and $f\in
H^{\alpha}$. We say $f$ is $n$-outer function in $H^{\alpha}$ if:

$[f\cdot H^{\infty}(z^{n})]_{\alpha}=\;p(z)\;M_{\alpha}(z^{n})$ (5.1)

where $p(z)$ is a polynomial of degree less than $n$ and
$M_{\alpha}(z^{n})=[H^{\infty}(z^{n})]_{\alpha}$.

######  Remark 5.2.

Observe that for any $f\in H^{\alpha}$, $[f\cdot
H^{\infty}(z^{n})]_{\alpha}=\bigvee\limits_{k=0}^{\infty}\\{f\cdot z^{nk}\\}$
where $\bigvee\limits_{k=0}^{\infty}\\{f\cdot z^{nk}\\}$ is an
${\alpha}$-closed linear span of $\\{f\cdot z^{nk}:k=0,1,\ldots\\}$.

It is trivial to show that $\bigvee\limits_{k=0}^{\infty}\\{f\cdot
z^{nk}\\}\subset[f\cdot H^{\infty}(z^{n})]_{\alpha}$.
For the reverse inclusion, let $g\in H^{\infty}(z^{n})$. Then, the sequence of
Cesaro means $\\{\sigma_{m}(g)\\}$ converges to $g$ in the weak* topology.
Therefore,

$\int_{{\mathbb{T}}}\sigma_{m}(g)hdm\longrightarrow\int_{{\mathbb{T}}}ghdm~{}~{}~{}~{}\forall~{}~{}h\in
L^{1}.$

Thus, for any non-zero element $u\in\mathcal{L}^{{\alpha}^{\prime}}$, we see
that

$\int_{{\mathbb{T}}}\sigma_{m}(g)fudm\longrightarrow\int_{{\mathbb{T}}}gfudm.$

Therefore, sequence $\\{\sigma_{m}(g)f\\}$ converge weakly to $gf$ in
$L^{\alpha}$. But each $\sigma_{m}(g)$ is a polynomial in $z^{n}$, so
$\sigma_{m}(g)f\in\bigvee\limits_{k=0}^{\infty}\\{f\cdot z^{nk}\\}$. Since
$\bigvee\limits_{k=0}^{\infty}\\{f\cdot z^{nk}\\}$, being a closed subspace of
$H^{\alpha}$ is also weakly closed, and hence $g\cdot
f\in\bigvee\limits_{k=0}^{\infty}\\{f\cdot z^{nk}\\}$.

######  Remark 5.3.

Note that when $f$ is $n$-outer function in $H^{\alpha}$ such that $f\in
M_{\alpha}(z^{n})$ then

$[f\cdot H^{\infty}(z^{n})]_{\alpha}=M_{\alpha}(z^{n}).$

This can be seen as follows:
Since $f$ is $n$-outer in $H^{\alpha}$, so we can write $f(z)=p(z)g(z)$ for
some $g\in M_{\alpha}(z^{n})$. Since the $deg\;p(z)<n$, we can write
$p(z)=a_{0}+a_{1}z+\cdots+a_{n-1}z^{n-1}$. Therefore,

$f(z)=a_{0}g(z)+a_{1}zg(z)+\cdots+a_{n-1}z^{n-1}g(z).$

But $f\in M_{\alpha}(z^{n})$. This is possible only when
$a_{1},a_{2},\ldots,a_{n-1}$ to be 0. Therefore $p(z)$ is a constant
polynomial and the result follows.

We now derive a factorization of $H^{\alpha}$ functions into the sum of
products of $n$-inner and $n$-outer functions. We use Theorem 4.5 to obtain a
general inner-outer factorization of $H^{\alpha}$ functions.

######  Theorem 5.4.

Suppose ${\alpha}$ is continuous rotationally symmetric norm and $f\in
H^{\alpha}$ then

$f=J_{1}f_{1}+\cdots+J_{r}f_{r}\;\;(r\leq n)$

where, each $J_{i}$ is $n$-inner and $f_{i}\in M_{\alpha}(z^{n})$ is $n$-outer
for all $i=1,\ldots,r$.

###### Proof.

Consider $\mathcal{M}=[f\cdot H^{\infty}(z^{n})]_{\alpha}$. Then $\mathcal{M}$
is a closed subspace of $H^{\alpha}$ which is invariant under $T_{z^{n}}$. By
Theorem 4.5, there exist $n$-inner functions $J_{1},\ldots,J_{r}$ such that

$\mathcal{M}=J_{1}M_{\alpha}(z^{n})\oplus\cdots\oplus J_{r}M_{\alpha}(z^{n}).$

Since $f\in\mathcal{M}$ , there exists $f_{1},\ldots,f_{r}\in
M_{\alpha}(z^{n})$ such that

$f=J_{1}f_{1}+\cdots+J_{r}f_{r}.$

We claim that each $f_{i}$ is $n$-outer.
If possible assume that some $f_{i}$ is not $n$-outer function, then

$[f\cdot H^{\infty}(z^{n})]_{\alpha}\subsetneq M_{\alpha}(z^{n}).$

We can write

$J_{i}[f\cdot H^{\infty}(z^{n})]_{\alpha}\subsetneq J_{i}M_{\alpha}(z^{n}).$

Note that,

$\displaystyle\mathcal{M}$ $\displaystyle=[f\cdot H^{\infty}(z^{n})]_{\alpha}$
$\displaystyle\subset[J_{1}f_{1}H^{\infty}(z^{n})\oplus\cdots\oplus
J_{i}f_{i}H^{\infty}(z^{n})\oplus\cdots\oplus
J_{r}f_{r}H^{\infty}(z^{n})]_{\alpha}$ $\displaystyle\subset[J_{1}[f_{1}\cdot
H^{\infty}(z^{n})]_{\alpha}\oplus\cdots\oplus J_{i}[f_{i}\cdot
H^{\infty}(z^{n})]_{\alpha}\oplus\cdots\oplus J_{r}[f_{r}\cdot
H^{\infty}(z^{n})]_{\alpha}]_{\alpha}$
$\displaystyle\subsetneq[J_{1}M_{\alpha}(z^{n})\oplus\cdots\oplus
J_{i}M_{\alpha}(z^{n})\oplus\cdots\oplus J_{r}M_{\alpha}(z^{n})]_{\alpha}$
$\displaystyle=\mathcal{[M]_{\alpha}}$

i.e., $\mathcal{M}\subsetneq\mathcal{M}$ which is a contradiction. Therefore,
each $f_{i}$ is $n$-outer function and thus the proof is complete. ∎

######  Remark 5.5.

The factorization established in Theorem 5.4 is different to the one presented
in [17] for classical $H^{p}$ spaces. In [17], the authors decomposed every
$H^{p}$ function as a product of $z^{n}$-$p$-inner and $p$-outer functions.
Whereas, we are decomposing $H^{\alpha}$ in terms of $n$-inner and $n$-outer
functions.
A function $\varphi\in H^{\infty}$ is called a $z^{n}$-$p$-inner function if
$\int_{\mathbb{T}}|\varphi|^{p}z^{n}dm<\infty$. We have not attempted to
define a concept on the similar lines for $H^{\alpha}$.

######  Corollary 5.6 (Singh and Thukral [24]).

Suppose $f$ is in $H^{2}$ then

$f=\varphi_{1}f_{1}+\cdots+\varphi_{r}f_{r}\;\;(r\leq n)$

where for each $i$, $f_{i}\in H^{2}(z^{2})$ and is $n$-outer and $\varphi_{i}$
is $n$-inner.

######  Theorem 5.7.

Every $n$-outer function which belongs to $M_{\alpha}(z^{n})$ is an outer
function.

###### Proof.

Let $f\in M_{\alpha}(z^{n})$ be an $n$-outer function. By Remark 5.2, we have

$[f\cdot H^{\infty}(z^{n})]_{\alpha}=M_{\alpha}(z^{n}).$ (5.2)

In order to prove that $f$ is an outer function, we need to show that
$H^{\alpha}=[f\cdot H^{\infty}]_{\alpha}$.
It is trivial to note that $[f\cdot H^{\infty}]_{\alpha}\subset H^{\alpha}$.
So only the reverse inequality needs to be established.
Let $h\in H^{\alpha}$ and write $h$ as

$h=h_{1}+zh_{2}+\cdots+z^{n-1}h_{n}$

for some $h_{1},\ldots,h_{n}\in M_{\alpha}(z^{n})$.
By equation (5.1), we see that each $h_{i}\in[f\cdot
H^{\infty}(z^{n})]_{\alpha}$, hence there exists a sequence
$\\{fh_{k}^{(i)}\\}\subset f\cdot H^{\infty}(z^{n})$ such that

${\alpha}(fh_{k}^{(i)}-h_{i})\to 0\;\;\text{as}\;\;k\to\infty.$

Therefore,

${\alpha}(z^{i}fh_{k}^{(i)}-z^{i}h_{i})\to 0\;\;\text{as}\;\;k\to\infty.$

This means that

$f\cdot\left(h_{k}^{(1)}+zh_{k}^{(2)}+\cdots+z^{n-1}h_{k}^{(n)}\right)\longrightarrow
h\;\;\text{in}\;\;H^{\alpha}\;\;\text{as}\;\;k\to\infty.$

Thus, we have $h\in[f\cdot H^{\infty}]_{\alpha}$. So the result follows. ∎

### Acknowledgments

The authors would like to thank Prof. Sachi Srivastava, University of Delhi,
India for useful comments during the course of the present research.

## References

  * [1] Beurling, A.: On two problems concerning linear transformations in Hilbert space.  Acta Math. 81, 239–255 (1948)
  * [2] Chen, Y.: Function spaces based on symmetric norms.  ProQuest LLC, Ann Arbor, MI (2014).  Thesis (Ph.D.)–University of New Hampshire
  * [3] Chen, Y.: A general Beurling-Helson-Lowdenslager theorem on the disk.  Adv. in Appl. Math. 87, 1–15 (2017)
  * [4] Chen, Y., Hadwin, D., Zhang, Y.: A general vector-valued Beurling theorem.  Integral Equations Operator Theory 86(3), 321–332 (2016)
  * [5] Chen, Y.K.: Lebesgue and hardy spaces for symmetric norms i.  arXiv: Operator Algebras (2014)
  * [6] Conway, J.B.: A course in functional analysis, vol. 96.  Springer (2019)
  * [7] Cowen, C.C., MacCluer, B.D.: Composition operators on spaces of analytic functions.  Studies in Advanced Mathematics. CRC Press, Boca Raton, FL (1995)
  * [8] Davidson, K.R., Paulsen, V.I., Raghupathi, M., Singh, D.: A constrained Nevanlinna-Pick interpolation problem.  Indiana Univ. Math. J. 58(2), 709–732 (2009)
  * [9] De Branges, L., Rovnyak, J.: Square summable power series.  Courier Corporation, North Chelmsford (2015)
  * [10] Gamelin, T.W.: Uniform algebras.  Prentice-Hall, Inc., Englewood Cliffs, N.J. (1969)
  * [11] Halmos, P.R.: Shifts on Hilbert spaces.  J. Reine Angew. Math. 208, 102–112 (1961)
  * [12] Helson, H.: Lectures on invariant subspaces.  Academic Press, New York-London (1964)
  * [13] Helson, H., Lowdenslager, D.: Invariant subspaces.  In: Proc. Internat. Sympos. Linear Spaces (Jerusalem, 1960), pp. 251–262. Jerusalem Academic Press, Jerusalem; Pergamon, Oxford (1961)
  * [14] Hoffman, K.: Banach spaces of analytic functions.  Prentice-Hall Series in Modern Analysis. Prentice-Hall, Inc., Englewood Cliffs, N.J. (1962)
  * [15] Koosis, P., Paul, K.: Introduction to Hp spaces.  115\. Cambridge University Press (1998)
  * [16] Kumar, A., Sahni, N., Singh, D.: Invariance under bounded analytic functions: generalizing shifts.  New York J. Math. 22, 1249–1270 (2016)
  * [17] Lance, T.L., Stessin, M.I.: Multiplication invariant subspaces of Hardy spaces.  Canad. J. Math. 49(1), 100–118 (1997)
  * [18] Lata, S., Mittal, M., Singh, D.: A finite multiplicity Helson-Lowdenslager-de Branges theorem.  Studia Math. 200(3), 247–266 (2010)
  * [19] Lax, P.D.: Translation invariant spaces.  Acta Math. 101, 163–178 (1959)
  * [20] Paulsen, V.I., Singh, D.: A Helson-Lowdenslager-de Branges theorem in $L^{2}$.  Proc. Amer. Math. Soc. 129(4), 1097–1103 (2001)
  * [21] Raghupathi, M.: Nevanlinna-Pick interpolation for $\mathbb{C}+BH^{\infty}$.  Integral Equations Operator Theory 63(1), 103–125 (2009)
  * [22] Sahni, N., Singh, D.: Lax-Halmos type theorems of $H^{p}$ spaces.  Houston J. Math. 41(2), 571–587 (2015)
  * [23] Singh, D., Agrawal, S.: de Branges spaces contained in some Banach spaces of analytic functions.  Illinois J. Math. 39(3), 351–357 (1995)
  * [24] Singh, D., Thukral, V.: Multiplication by finite Blaschke factors on de Branges spaces.  J. Operator Theory 37(2), 223–245 (1997)
  * [25] Srinivasan, T.P.: Simply invariant subspaces.  Bull. Amer. Math. Soc. 69, 706–709 (1963)
  * [26] Srinivasan, T.P.: Doubly invariant subspaces.  Pacific J. Math. 14, 701–707 (1964)