# Subgraph Clustering and Atom Learning
for Improved Image Classification

Aryan Singh  University of Limerick  Pepijn Van de Ven  University of Limerick
Ciarán Eising  University of Limerick  Patrick Denny  University of Limerick

###### Abstract

In this study, we present the Graph Sub-Graph Network (GSN), a novel hybrid
image classification model merging the strengths of Convolutional Neural
Networks (CNNs) for feature extraction and Graph Neural Networks (GNNs) for
structural modeling. GSN employs k-means clustering to group graph nodes into
clusters, facilitating the creation of subgraphs. These subgraphs are then
utilized to learn representative ‘atoms‘ for dictionary learning, enabling the
identification of sparse, class-distinguishable features. This integrated
approach is particularly relevant in domains like medical imaging, where
discerning subtle feature differences is crucial for accurate classification.

To evaluate the performance of our proposed GSN, we conducted experiments on
benchmark datasets, including PascalVOC and HAM10000. Our results demonstrate
the efficacy of our model in optimizing dictionary configurations across
varied classes, which contributes to its effectiveness in medical
classification tasks. This performance enhancement is primarily attributed to
the integration of CNNs, GNNs, and graph learning techniques, which
collectively improve the handling of datasets with limited labeled examples.
Specifically, our experiments show that the model achieves a higher accuracy
on benchmark datasets such as Pascal VOC and HAM10000 compared to conventional
CNN approaches.

Keywords: Dictionary Learning, GNN, Image Classification, Sub Graph
Clustering.

##  1 Introduction

CNNs have transformed image classification, showcasing impressive performance
by learning hierarchical feature representations through layers of
convolutions and pooling. Nevertheless, traditional CNNs may face challenges
in specific contexts, particularly when dealing with datasets characterized by
significant class overlap or sparsely labeled data, as often encountered in
medical imaging [Dinsdale et al., 2022].

A significant issue arises from the utilization of pre-trained CNNs, which are
typically trained on extensive datasets like ImageNet, having diverse classes
[Deng et al., 2009]. While these models excel in generalization, they may
encounter difficulties in scenarios requiring finer distinctions or handling
multi-label data. In such cases, the learned feature maps might exhibit
overlap or intertwining between representations of distinct objects, leading
to diminished classification accuracy, a concern heightened in medical imaging
due to the critical diagnostic implications of subtle feature variations
[Perone and Cohen-Adad, 2019].

To address these limitations, there has been a burgeoning interest in
amalgamating dictionary learning techniques with the feature extraction
capabilities of Deep Neural Networks (DNNs), particularly CNNs. Dictionary
learning aims to identify a set of representative elements, or ‘atoms‘, which,
when combined, can accurately reconstruct the original data. This set is known
as the ’dictionary’[Tariyal et al., 2016]. By learning a dictionary tailored
to the specific dataset, the model can more effectively isolate and represent
the diverse features present in images. This enhanced representation allows
the model to capture subtle nuances and variations that might be missed by
other feature extraction methods, leading to improved classification
performance [Zhang et al., 2012, Zhou et al., 2021a].

Our research extends upon these foundational concepts, advancing the
integration of CNNs, dictionary learning, and GNNs within the innovative
framework of Graph Sub-Graph Network (GSN). We propose a novel image
classification model tailored for scenarios characterized by multi-label
classes and sparsely labeled data. Our approach harnesses pre-trained CNN
features and segment nodes (image regions) based on similarity metrics and
learns ‘atoms‘ from these clusters. These atoms collectively form our
dictionary, resulting in superior classification performance compared to
traditional DNNs, particularly in the way the model’s performance decline is
less pronounced as training data is reduced.

##  2 Prior Art

The field of computer vision has experienced a profound transformation driven
by the remarkable advancements in CNNs and DNNs. Models such as ResNet18 have
achieved groundbreaking success in image classification, demonstrating the
immense potential of deep learning for visual understanding. Deep Neural
Networks (using convolutions or transformers) have become the dominant
paradigm in computer vision, enabling solutions to a wide array of challenging
problems. Their impact extends far beyond generic computer vision tasks,
revolutionizing the field of medical imaging. CNNs and DNNs are now
instrumental in tasks like medical image analysis, disease detection, and
treatment planning. Their ability to learn complex patterns from medical scans
has led to significant improvements in accuracy, efficiency, and early
diagnosis in various medical domains. This section explores work done in the
field of computer vision using DNNs and Dictionary learning.

###  2.1 CNN

CNNs[Lecun et al., 1998] get their name from the ‘convolution‘ operation, a
core building block of these networks. Imagine a small window (called a
kernel) sliding over an image, focusing on a small section at a time. At each
position, the kernel multiplies its own values with the corresponding image
pixels and sums up the results. This creates a single number for that specific
location, which gets stored in a new output map. The key goal of convolution
is to take image data and gradually distill it down, helping the network
understand larger-scale features and patterns within the image. Convolution
works like a series of intelligent filters that analyze an image at multiple
levels of detail. Early on, it identifies simple patterns like edges and lines
based on how neighboring pixels relate. As the process goes deeper into the
network, the filters build upon these basic patterns, detecting more complex
shapes and objects. The result is a layered understanding of what is in the
image, from small details to overall concepts. Finally, all this rich
information is condensed into a streamlined form. This compressed version can
then be used for all sorts of tasks, like deciding what the image shows,
pinpointing objects, or highlighting distinct areas.

ResNet [He et al., 2015] is a breakthrough deep neural network architecture
designed to overcome the challenges of training extremely deep
networks[Philipp et al., 2017]. Unlike traditional networks, ResNets introduce
‘residual blocks‘. These blocks use skip connections [Wang et al., 2018] to
add the original input directly to the output of a group of convolutional
layers. This helps the network focus on learning the changes between the input
and desired output, rather than trying to re-learn features from scratch.

ResNet achieved groundbreaking success on the ImageNet dataset [Deng et al.,
2009], significantly advancing the state-of-the-art in image classification at
the time. Its innovative architecture has become the building block of
numerous computer vision tasks, demonstrating its effectiveness as a powerful
and adaptable tool.

ResNet’s abilities have proven invaluable in medical image analysis. For
example, it has been used to successfully detect skin cancer from images.
Additionally, ResNet architectures have aided in the diagnosis of brain
diseases using MRI scans [Lu et al., 2020], and in the segmentation of lung
tumors in CT images [Farheen et al., 2021]. These examples highlight ResNet’s
versatility in extracting complex patterns from medical images, contributing
to improved diagnostic tools.

However, there are limitations in transferring knowledge directly from a large
dataset like ImageNet to medical image analysis tasks. While pre-trained
ResNets offer a strong foundation, medical images often contain unique
features and subtle class distinctions compared to natural images. This can
lead to the model overfitting to the general features learned on ImageNet,
hindering its ability to differentiate between fine-grained medical classes.

These limitations have led to the exploration of advanced techniques to
further enhance the performance of deep learning models in medical image
analysis [Zhou et al., 2021b]. Attention models have proven particularly
effective, allowing models to focus on the most relevant regions of an image.
Additionally, autoencoders can be used to learn compact, informative
representations tailored to specific medical imaging tasks. Another promising
approach to address these challenges is that of dictionary learning in
computer vision, which offers new solutions for extracting and representing
the subtle, distinguishing features crucial in medical image analysis.

###  2.2 Dictionary Learning

Dictionary learning is a powerful technique in representational learning aimed
at deriving a set of basis elements, known as ‘atoms‘ which can reconstruct
signals or images via sparse combinations [Yang et al., 2024]. This approach
provides a more flexible and richer representation than traditional bases like
Fourier or wavelet transforms. Its applications include the reduction of noise
in images, the filling in of missing image parts, and the enhancement of image
resolution [Maeda, 2022, Zheng et al., 2021].

Given a set of training samples
$Y=[\mathbf{y}_{1},\mathbf{y}_{2},\dots,\mathbf{y}_{N}]\in\mathbb{R}^{n\times
N}$, where each $\mathbf{y}_{i}$ is an $n$-dimensional column vector
representing a sample, the goal of dictionary learning is to learn a
dictionary matrix
$D=[\mathbf{d}_{1},\dots,\mathbf{d}_{k}]\in\mathbb{R}^{n\times k}$ and a
sparse coefficient matrix
$X=[\boldsymbol{\alpha}_{1},\dots,\boldsymbol{\alpha}_{N}]\in\mathbb{R}^{k\times
N}$ such that the training samples $Y$ can be approximated as a linear
combination of the atoms in $D$ with coefficients in $X$:

$Y\approx DX^{T}$ (1)

The dictionary learning process involves optimizing the following objective
function, subject to the specified constraints:

$\min_{D,X}\left\|Y-DX^{T}\right\|_{F}^{2}+\lambda\left\|X\right\|_{1}\quad\text{subject
to}\quad\left\|d_{k}\right\|_{2}=1,\text{ for }k=1,\ldots,K$ (2)

where: $\left\|Y-DX^{T}\right\|_{F}^{2}$ is the Frobenius norm (squared) of
the reconstruction error, measuring how well the dictionary and sparse
coefficients approximate the original data, $\left\|X\right\|_{1}$ is the
$\ell_{1}$ norm of the sparse coefficient matrix, promoting sparsity in the
representation, $\lambda$ is a regularization parameter that controls the
trade-off between reconstruction accuracy and sparsity,
$\left\|d_{k}\right\|_{2}=1$ is the constraint that enforces unit norm on each
atom in the dictionary.

Dictionary learning’s capability to capture essential image patterns and
textures underpins its success in computer vision, making it a critical tool
for extracting and representing subtle, distinguishing features in complex
datasets.

####  2.2.1 Dictionary Learning for Non-Euclidean Data

Traditional signal processing and dictionary learning techniques have
primarily been applied to data situated within Euclidean spaces. However, the
growing ubiquity of data embedded in complex structures like graphs—including
those representing social networks, sensor networks, and biological
interaction networks demands a paradigm shift. Unlike signals in Euclidean
spaces, the structure and topology of graphs deeply influence the support of
graph-based signals, revealing crucial information that extends beyond mere
signal values.

Dictionary learning has thus become a pivotal technique for achieving sparse
representations of signals in non-Euclidean spaces. This approach extensively
utilizes the graph Laplacian, a fundamental operator in spectral graph theory,
to tailor dictionaries to the unique spectral characteristics of graph
signals. The graph Laplacian $\mathbf{L}$ is defined as
$\mathbf{L}=\mathbf{D}-\mathbf{A}$, where $\mathbf{A}$ is the adjacency matrix
of the graph and $\mathbf{D}$ is the degree matrix, a diagonal matrix where
each element $D_{ii}$ equals the sum of the edges connected to vertex $i$,
facilitating an analysis that incorporates the graph’s inherent structure
[Zhang et al., 2012].

These learned dictionaries enable the efficient approximation of graph signals
using only a few significant components. This sparsity is vital for effective
analysis, compression, and understanding of data in these complex domains.

The adaptation of dictionary learning for graph-based signals underscores a
significant evolution in signal processing. It demonstrates the importance of
modifying classical methods to handle the diverse, non-Euclidean structures
found in modern data [Vincent-Cuaz et al., 2021]. These advancements pave the
way for analyzing and extracting meaningful information from the increasingly
intricate datasets encountered across a wide range of fields.

###  2.3 GNN

GNNs [Scarselli et al., 2009] are specialized neural networks designed for
processing data that can be represented as graphs. These networks handle
entities (nodes) and their connections (edges), capturing significant
structural and relational information. GNNs operate based on message passing,
aggregation, feature updates, and forward propagation.

  * •

Message Passing: Each node sends and receives messages based on its own
features and those of its adjacent nodes [Gilmer et al., 2017]. This process
allows information to flow through the graph, enabling nodes to gather
insights from their local neighborhoods.

  * •

Aggregation: Nodes aggregate the messages received from their neighbors using
functions such as sum, mean, or max, forming a single representative vector.
This synthesis is crucial for integrating information from multiple sources
within the graph.

  * •

Feature Update: Nodes update their own state by transforming the aggregated
vector through a neural network function, integrating the node’s existing
state with the newly aggregated information.

  * •

Forward Propagation: The updated node features are then processed through
additional network layers, transforming the features into higher-level
representations. This step mirrors operations in conventional neural networks
and is essential for feature learning.

The general update rule for a node $v$ in a GNN is given by the equation:

$h_{v}^{(l+1)}=f\left(h_{v}^{(l)},\bigoplus_{u\in\mathcal{N}(v)}g(h_{u}^{(l)},h_{v}^{(l)},e_{uv})\right),$
(3)

where $h_{v}^{(l)}$ is the node feature vector at layer $l$, $\mathcal{N}(v)$
denotes the set of neighboring nodes, $e_{uv}$ are the edge features, $f$ and
$g$ represent learnable neural network functions, and $\bigoplus$ is the
aggregation function.

Within this framework, Graph Convolutional Networks (GCNs) extend the concept
of convolution from regular grid data, such as images, to graph-structured
data [Kipf and Welling, 2016]. They leverage the spectral properties of
graphs, utilizing the eigendecomposition of the graph Laplacian $\mathbf{L}$.
The convolution operation in a GCN is defined as:

$H^{(l+1)}=\sigma\left(\mathbf{U}\mathbf{\Lambda}\mathbf{U}^{T}H^{(l)}W^{(l)}\right),$
(4)

where $H^{(l)}$ are the node features at layer $l$, $\mathbf{U}$ contains the
eigenvectors of $\mathbf{L}$, $\mathbf{\Lambda}$ is the diagonal matrix of
eigenvalues, $W^{(l)}$ is the weight matrix for that layer, and $\sigma$ is a
nonlinear activation function. Through these mechanisms, GCNs can effectively
capture and analyze the intricate relationships and structural nuances present
in graph data.

##  3 Methodology

At the core of our approach lies the integration of dictionary learning, GNNs,
and concepts from computer vision to achieve optimal class representation. Our
method integrates a series of components designed to optimally exploit the
structural nuances of graph data, including node selection, subgraph creation,
embedding generation, and classification. In our study, we have utilized
features extracted from a DNN and applied dictionary learning techniques to
learn sparse representations using GNN for optimal class representation.

This section elaborates on the methodology employed to represent and analyze
images through advanced graph-based techniques.

###  3.1 Graph Construction

The initial step in our methodology involves constructing a graph
representation of the image, which begins by segmenting the image into
superpixels. These superpixels serve as nodes within the graph. We extract
features from these nodes using a pre-trained DNN, specifically frozen
ResNet18 in our experiments. The extracted features are high-level descriptors
of the visual content within each superpixel, capturing detailed texture,
color, and structural information.

Once the nodes are defined with their corresponding features, we proceed to
establish edges between them. This is achieved through a K-nearest neighbors
(KNN) approach, where cosine similarity serves as the metric to determine the
connections between nodes. To leverage the grid structure of the image we have
also assign 4 adjacent nodes as neighbours. This graph structure highlights
the intricate interactions across the image, facilitating a deeper
understanding of the spatial and feature-based relationships within the image.

###  3.2 Graph Component Clustering

Following the graph construction, K-means clustering is employed to partition
the graph into several clusters. Each cluster groups nodes with similar
features, likely representing distinct patterns or regions within the image.
This step is crucial for identifying key areas of interest that are
significant to the subsequent classification task.

###  3.3 Subgraph Generation and Atom Learning

Each cluster identified from the previous step is then used to generate a
subgraph. These subgraphs are processed using a GCN. The GCN learns an
embedding, termed as an ‘atom‘ which encapsulates the spectral properties and
relational dynamics specific to the subgraph. These atoms provide a compact
yet comprehensive representation of distinct regions within the image.

###  3.4 Dictionary Construction

A dictionary of visual patterns is constructed by aggregating the atoms
learned from each significant graph cluster. These atoms serve as fundamental
building blocks for representing and understanding the image content. Each
atom embodies a compact yet informative embedding derived from a specific
subgraph, encapsulating the unique spectral and relational characteristics of
the corresponding image region.

The dictionary serves as a repository of diverse visual primitives, akin to a
visual vocabulary. By assembling atoms from various regions of the image, the
dictionary captures a wide range of local patterns and structures. This rich
collection of atoms enables the subsequent classification process to
accurately and efficiently characterize new images by identifying and
quantifying the presence of these learned patterns.

###  3.5 Classification by Feature Concatenation

Once the dictionary is constructed, the final classification is performed by
concatenating the features corresponding to different atoms. This concatenated
features is then fed into a classifier to assign the image to one of the
predefined categories.

Figure 2:  Graph Sub-Graph Network model architecture.

##  4 Results

We conducted an evaluation of our proposed model on two benchmark datasets
that are highly regarded in the image classification domain. Pascal VOC,
comprising 20 diverse classes [Everingham et al., 2010], and HAM10000, which
includes 7 skin cancer classes [Tschandl et al., 2018]. Our method is
implemented using PyTorch and PyTorch Geometric for graph convolution
operations. Model training is conducted with an initial learning rate of
0.001, employing ReduceLROnPlateau to dynamically adjust the learning rate
during training for a maximum of 100 epochs.

On the Pascal VOC dataset, our method achieves a peak accuracy of 88.63%. This
is a substantial improvement over the baseline ResNet-18 model, which achieves
an accuracy of 63%. Similarly, on the HAM10000 dataset, our model exceeds the
performance of popular ResNet models. Table 1, shows our GSN model’s accuracy
of 89.63% is slightly higher than the closest ResNet variant. By utilizing a
frozen ResNet18 model’s (which does not require training) features as
explained in Section 3.1, our approach excels across two diverse benchmark
datasets.

Table 1: Model Accuracy Model (Parameters) | Pascal VOC | HAM10000
---|---|---
ResNet-18 (11.19M) | 63.43% | 88.85%
ResNet-34 (21.29M) | 66.03% | 84.54%
ResNet-50 (23.55M) | 84.01% | 88.76%
GSN (50.57K) |  88.63% |  89.63%

##  5 Conclusion

In this paper, we introduce the GSN, a novel image classification framework
that reimagines the integration of dictionary learning and GNNs. GSN
distinguishes itself by leveraging graph representations to capture the
intricate relationships between image features, combined with a sparse coding
approach for optimal class representation. Our experiments demonstrate that
GSN, leveraging features extracted from ResNet18, not only achieves
competitive performance on standard benchmarks like Pascal VOC but also
surpasses the performance of ResNet18, ResNet34, and ResNet50 on the
challenging HAM10000 medical dataset, showcasing the effectiveness of our
graph-based approach in utilizing these features. This success is achieved
with a significantly reduced parameter count compared to traditional DNNs,
highlights the potential of GSN to deliver both accuracy and efficiency in
image classification tasks. The unique fusion of graph-based learning and
sparse coding in GSN opens up new avenues for exploring the power of
structural representations in image understanding. Future research will focus
on extending GSN to a broader range of applications, refining graph
construction techniques, and delving deeper into the interpretability of the
learned graph representations.

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# Bi-isometries reducing the hyper-ranges
of the coordinates

Sameer Chavan and Md. Ramiz Reza  Department of Mathematics and Statistics
Indian Institute of Technology Kanpur, India<EMAIL_ADDRESS>Indian
Institute of Science Education and Research Thiruvananthapuram, India
<EMAIL_ADDRESS>
###### Abstract.

Let $(S_{1},S_{2})$ be a bi-isometry, that is, a pair of commuting isometries
$S_{1}$ and $S_{2}$ on a complex Hilbert space $\mathscr{H}.$ By the von
Neumann-Wold decomposition, the hyper-range
$\mathscr{H}_{\infty}(S_{1}):=\cap_{n=0}^{\infty}S^{n}_{1}\mathscr{H}$ of
$S_{1}$ reduces $S_{1}$ to a unitary operator. Although
$\mathscr{H}_{\infty}(S_{1})$ is an invariant subspace for $S_{2},$ in
general, $\mathscr{H}_{\infty}(S_{1})$ is not a reducing subspace for $S_{2}.$
We show that $\mathscr{H}_{\infty}(S_{1})$ reduces $S_{2}$ to an isometry if
and only if the subspaces $S_{2}(\ker S^{*}_{1})$ and
$\mathscr{H}_{\infty}(S_{1})$ of $\mathscr{H}$ are orthogonal. Further, we
describe all bi-isometries $(S_{1},S_{2})$ satisfying the orthogonality
condition mentioned above.

###### Key words and phrases:

commuting isometries, inner function, Hardy space, Wold-type decomposition

###### 2010 Mathematics Subject Classification:

Primary 47A13; Secondary 30H10, 30J05

##  1\. Motivation and the main theorem

For complex Hilbert spaces $\mathscr{H}$ and $\mathscr{K},$ let
$\mathscr{B}(\mathscr{H},\mathscr{K})$ denote the Banach space of bounded
linear transformations from $\mathscr{H}$ into $\mathscr{K}.$ We denote
$\mathscr{B}(\mathscr{H},\mathscr{H})$ simply by $\mathscr{B}(\mathscr{H}),$
the $C^{*}$-algebra of bounded linear operators on $\mathscr{H}.$ A closed
subspace $\mathscr{M}$ of $\mathscr{H}$ is invariant for
$T\in\mathscr{B}(\mathscr{H})$ if $T(\mathscr{M})\subseteq\mathscr{M}.$ We say
that $\mathscr{M}$ reduces $T$ if $\mathscr{M}$ and
$\mathscr{H}\ominus\mathscr{M}$ are invariant for $T.$ We say that
$\mathscr{M}$ essentially reduces $T$ if
$P_{\mathscr{M}}T|_{\mathscr{H}\ominus\mathscr{M}}\in\mathscr{B}(\mathscr{H}\ominus\mathscr{M},\mathscr{M})$
and
$P_{\mathscr{H}\ominus\mathscr{M}}T|_{\mathscr{M}}\in\mathscr{B}(\mathscr{M},\mathscr{H}\ominus\mathscr{M})$
are compact linear transformations, where $P_{\mathscr{K}}$ denotes the
orthogonal projection of $\mathscr{H}$ onto a closed subspace $\mathscr{K}$ of
$\mathscr{H}.$ The hyper-range $\mathscr{H}_{\infty}(T)$ of
$T\in\mathscr{B}(\mathscr{H})$ is defined as the space
$\cap_{n=0}^{\infty}T^{n}\mathscr{H}.$ An operator
$T\in\mathscr{B}(\mathscr{H})$ is said to be a contraction (resp. an isometry)
if $T^{*}T\leqslant I$ (resp. $T^{*}T=I$). An operator
$T\in\mathscr{B}(\mathscr{H})$ is completely nonunitary if there is no nonzero
closed subspace $L$ of $\mathscr{H}$ that reduces $T$ to a unitary.

A pair $(S_{1},S_{2})$ of operators $S_{1},S_{2}\in\mathscr{B}(\mathscr{H})$
is said to be commuting if $S_{1}S_{2}=S_{2}S_{1}.$ A commuting pair
$(S_{1},S_{2})$ is doubly commuting if $S^{*}_{1}S_{2}=S_{2}S^{*}_{1}.$ A
commuting pair $(S_{1},S_{2})$ of operators $S_{1}$ and $S_{2}$ in
$\mathscr{B}(\mathscr{H})$ is called a bi-contraction on $\mathscr{H}$ if
$S_{1}$ and $S_{2}$ are contractions. A commuting pair $(S_{1},S_{2})$ of
operators $S_{1}$ and $S_{2}$ in $\mathscr{B}(\mathscr{H})$ is a bi-isometry
on $\mathscr{H}$ if $S_{1}$ and $S_{2}$ are isometries. If, in addition,
$S_{1}$ and $S_{2}$ are unitaries, then we refer to $(S_{1},S_{2})$ as a bi-
unitary on $\mathscr{H}.$ Needless to say, there is considerable literature on
bi-contractions and related classes (see, for example, [2, 3, 4, 7, 8, 10, 11,
14]).

Let $\mathbb{D}$ denote the open unit disc centered at the origin in
$\mathbb{C}.$ For a complex Hilbert space $\mathscr{E},$ let
$\mathscr{H}^{2}_{\mathscr{E}}(\mathbb{D})$ denote the Hardy space of
${\mathscr{E}}$-valued holomorphic functions on $\mathbb{D}.$ For a complex
Banach space $X,$ let $\mathscr{H}^{\infty}_{X}(\mathbb{D})$ denote the Banach
space of bounded $X$-valued holomorphic functions on $\mathbb{D}.$ For
$\phi\in\mathscr{H}^{\infty}_{\mathscr{B}({\mathscr{E}})}(\mathbb{D}),$ let
$M_{\phi,{\mathscr{E}}}$ denote the operator of multiplication by $\phi$ on
$\mathscr{H}^{2}_{\mathscr{E}}(\mathbb{D}).$ If ${\mathscr{E}}=\mathbb{C},$ we
denote $\mathscr{H}^{\infty}_{\mathscr{B}({\mathscr{E}})}(\mathbb{D})$ (resp.
$M_{\phi,{\mathscr{E}}}$) simply by $\mathscr{H}^{\infty}(\mathbb{D})$ (resp.
$M_{\phi}$). The $\mathscr{B}({\mathscr{E}})$-valued Schur class of
$\mathbb{D}$ is the closed unit ball of
$\mathscr{H}^{\infty}_{\mathscr{B}({\mathscr{E}})}(\mathbb{D}).$ We say that
$\phi\in\mathscr{H}^{\infty}_{\mathscr{B}({\mathscr{E}})}(\mathbb{D})$ is an
inner function if
$\phi\in\mathscr{H}^{\infty}_{\mathscr{B}({\mathscr{E}})}(\mathbb{D})$ and
$\phi(\zeta)$ is an isometry for almost every $\zeta$ on the unit circle
$\mathbb{T}$ (see [2, 15] for the basic theory of the operator-valued Schur
class).

The starting point of the present investigation is the following theorem that
generalises the von Neumann-Wold decomposition (see [5, Theorem I.3.6] and
[15, Theorem I.3.2]).

######  Theorem 1.1.

Let $T\in\mathscr{B}(\mathscr{H})$ be a contraction and let $\mathscr{E}:=\ker
T^{*}.$ Consider the subspaces $\mathscr{H}_{u}(T)$ and $\mathscr{H}_{s}(T)$
of $\mathscr{H}$ given by

$\displaystyle\mathscr{H}_{u}(T)$ $\displaystyle=$
$\displaystyle\bigcap_{n=1}^{\infty}\big{\\{}h\in\mathscr{H}:\|T^{n}h\|=\|h\|=\|T^{*n}h\|\big{\\}},$
$\displaystyle\mathscr{H}_{s}(T)$ $\displaystyle=$
$\displaystyle\mathscr{H}\ominus\mathscr{H}_{u}(T).$

Then $\mathscr{H}_{u}(T)$ reduces $T$ to the unitary
$U=T|_{\mathscr{H}_{u}(T)},$ $S=T|_{\mathscr{H}_{s}(T)}$ is completely non-
unitary and the decomposition

$T=U\oplus
S~{}\mathrm{on}~{}\mathscr{H}=\mathscr{H}_{u}(T)\oplus\mathscr{H}_{s}(T)$

is uniquely determined.

In particular, if $T$ is an isometry, then $S$ is unitarily equivalent to the
operator $M_{z,\mathscr{E}}$ of multiplication by $z$ on
$\mathscr{H}^{2}_{\mathscr{E}}(\mathbb{D}),$
$\mathscr{H}_{u}(T)=\mathscr{H}_{\infty}(T)$ and

$\displaystyle\mathscr{H}_{s}(T)=\bigvee_{w\in\mathbb{D}}\ker(S^{*}-w),\quad\mathscr{H}_{s}(T)=\bigoplus_{n=0}^{\infty}S^{n}(\ker
S^{*}).$

######  Remark 1.2.

Note that $\mathscr{H}_{u}(T)\subseteq\mathscr{H}_{\infty}(T).$ In general,
this inclusion is strict (for example, consider a contraction having a non-
unitary invertible orthogonal direct summand).

Let $T\in\mathscr{B}(\mathscr{H})$ be a contraction. The parts
$U=T|_{\mathscr{H}_{u}(T)}$ and $S=T|_{\mathscr{H}_{s}(T)}$ are called the
unitary part and the completely nonunitary part of $T$, respectively, and
$T=U\oplus S$ is called the canonical decomposition of $T.$ We say that $T$
has Wold-type decomposition if

$\displaystyle\mathscr{H}_{u}(T)=\mathscr{H}_{\infty}(T)~{}\mbox{and}~{}\mathscr{H}_{s}(T)=\bigvee_{n=0}^{\infty}S^{n}(\ker
S^{*})$ (1.1)

(cf. [12, Definition 1.1]).

######  Remark 1.3.

Let $T\in\mathscr{B}(\mathscr{H})$ be a left invertible operator. If $T$
satisfies $TT^{*}+(T^{*}T)^{-1}\leqslant 2I,$ then $T$ has the Wold-type
decomposition. This may be deduced from [12, Theorem 3.6].

The present note is motivated by the following question concerning the
structure of bi-isometries.

######  Question 1.4.

Let $(S_{1},S_{2})$ be a bi-isometry (resp. bi-contraction) on $\mathscr{H}.$
When $\mathscr{H}_{\infty}(S_{1})$ reduces $S_{2}$ to an isometry (resp. a
contraction)?

In case $S_{2}$ has finite dimensional cokernel, it has been shown in [4,
Theorem 2.3] that $\mathscr{H}_{\infty}(S_{1})$ reduces $S_{2}$ to an
isometry. In general, $\mathscr{H}_{\infty}(S_{1})$ is not a reducing subspace
for $S_{2}.$

######  Example 1.5.

Let $\varphi:\mathbb{D}\rightarrow\mathbb{C}$ belong to the Schur class of the
open unit disc $\mathbb{D}.$ Let $\mu$ denote the weighted arc-length measure
$w\frac{d\theta}{2\pi}$ with weight function $w=1-|\varphi|^{2}.$ Let
$\mathscr{G}:=L^{2}(\mathbb{T},\mu)$ and let
$V:\mathscr{G}\rightarrow\mathscr{G}$ be the operator of multiplication by the
coordinate function $\zeta$ in $\mathscr{B}(\mathscr{G}).$ For an integer
$n\geqslant 0,$ let $e_{n}\in\mathscr{G}$ be given by
$e_{n}(\zeta)=\zeta^{n},$ $\zeta\in\mathbb{T}.$ Define
$B:\mathscr{H}^{2}(\mathbb{D})\rightarrow\mathscr{H}^{2}_{\mathscr{G}}(\mathbb{D})$
by

$\displaystyle B(z^{n})=e_{n},\quad n\geqslant 0,$

which is extended linearly and continuously as a bounded linear operator to
the whole of $\mathscr{H}^{2}(\mathbb{D})$ (indeed,
$\|B\|\leqslant\mu(\mathbb{T})$). Note that

$M_{V}B(z^{n})=V(e_{n})=e_{n+1}=B(z^{n+1})=BM_{z}(z^{n}),\quad n\geqslant 0,$

and hence we obtain

$\displaystyle M_{V}B=BM_{z}.$ (1.2)

Define bounded linear operators $S_{1},S_{2}$ on
$\mathscr{H}^{2}_{\mathscr{G}}(\mathbb{D})\oplus\mathscr{H}^{2}(\mathbb{D})$
by

$\displaystyle S_{1}=\Big{[}\begin{smallmatrix}M_{V}&0\\\
0&M_{z}\end{smallmatrix}\Big{]},~{}S_{2}=\Big{[}\begin{smallmatrix}M_{z,{\mathscr{G}}}&B\\\
0&M_{\varphi}\end{smallmatrix}\Big{]}~{}\mbox{on}~{}\mathscr{H}^{2}_{\mathscr{G}}(\mathbb{D})\oplus\mathscr{H}^{2}(\mathbb{D}).$

Clearly, $S_{1}$ is an isometry. We check that $S_{2}$ is also an isometry.
Since $M^{*}_{z,{\mathscr{G}}}B(z^{n})=M^{*}_{z,{\mathscr{G}}}(e_{n})=0$ (as
$e_{n}\in{\mathscr{G}}$), we have $M^{*}_{z,{\mathscr{G}}}B=0,$ and hence

$\displaystyle
S^{*}_{2}S_{2}=\Big{[}\begin{smallmatrix}I&M^{*}_{z,{\mathscr{G}}}B\\\
B^{*}M_{z,{\mathscr{G}}}&B^{*}B+M^{*}_{\varphi}M_{\varphi}\end{smallmatrix}\Big{]}=\Big{[}\begin{smallmatrix}I&0\\\
0&B^{*}B+M^{*}_{\varphi}M_{\varphi}\end{smallmatrix}\Big{]}.$ (1.3)

Moreover, for any $f,g\in\mathscr{H}^{2}(\mathbb{D}),$

$\displaystyle\langle{(B^{*}B+M^{*}_{\varphi}M_{\varphi})f},\,{g}\rangle$
$\displaystyle=$ $\displaystyle\langle{Bf},\,{Bg}\rangle+\langle{\varphi
f},\,{\varphi g}\rangle$ $\displaystyle=$
$\displaystyle\int_{\mathbb{T}}f(\zeta)\overline{g(\zeta)}\,d\mu(\zeta)$
$\displaystyle+$
$\displaystyle\int_{\mathbb{T}}|\varphi(\zeta)|^{2}f(\zeta)\overline{g(\zeta)}\,d\zeta,$

which is equal to $\langle{f},\,{g}\rangle$ since
$d\mu(\zeta)=(1-|\varphi(\zeta)|^{2})d\zeta.$ It now follows from (1.3) that
$S_{2}$ is an isometry. Moreover,

$\displaystyle
S_{1}S_{2}=\Big{[}\begin{smallmatrix}M_{V}M_{z,{\mathscr{G}}}&M_{V}B\\\
0&M_{z}M_{\varphi}\end{smallmatrix}\Big{]},\quad
S_{2}S_{1}=\Big{[}\begin{smallmatrix}M_{z,{\mathscr{G}}}M_{V}&BM_{z}\\\
0&M_{\varphi}M_{z}\end{smallmatrix}\Big{]}.$

This, combined with (1.2), shows that $S_{1}$ and $S_{2}$ commute. Thus
$(S_{1},S_{2})$ is a bi-isometry. Note that

$\displaystyle\mbox{$\varphi$ is an inner
function}\,\Longleftrightarrow\,\mu=0\,\Longleftrightarrow\,\mu(\mathbb{T})=0\,\Longleftrightarrow\,{\mathscr{G}}=\\{0\\}.$

Since $B(1)=\mu(\mathbb{T})$ and
$\mathscr{H}_{\infty}(S_{1})=\mathscr{H}^{2}_{\mathscr{G}}(\mathbb{D}),$

$\mathscr{H}_{\infty}(S_{1})$ is a reducing subspace for $S_{2}$
$\displaystyle\Longleftrightarrow$ $\displaystyle B=0$
$\displaystyle\Longleftrightarrow$ $\varphi$ is an inner function.

Assume that $\varphi$ is not inner. Thus
$\mathscr{H}^{2}(\mathbb{D})\subseteq{\mathscr{G}},$ so that ${\mathscr{G}}$
is infinite dimensional. Moreover, since $B(1)\neq 0,$

$\displaystyle S_{2}(\ker S^{*}_{1})=\\{\big{[}\begin{smallmatrix}\alpha\\\
\alpha\varphi\end{smallmatrix}\big{]}:\alpha\in\mathbb{C}\\}.$

In particular, $S_{2}(\ker S^{*}_{1})$ and $\mathscr{H}_{\infty}(S_{1})$ are
not orthogonal.

If $S_{1}$ has the Wold-type decomposition, then it turns out that the
orthogonality of $S_{2}(\ker S^{*}_{1})$ and $\mathscr{H}_{\infty}(S_{1})$ is
the only decisive factor in answer to Question 1.4 (see Remark 1.7).

######  Theorem 1.6.

Let $(S_{1},S_{2})$ be a bi-isometry on $\mathscr{H}$ and let
${\mathscr{E}}:=\ker S^{*}_{1}.$ The following statements are equivalent$:$

  1. $\mathrm{(i)}$

$\mathscr{H}_{\infty}(S_{1})$ reduces $S_{2}$ to an isometry,

  2. $\mathrm{(ii)}$

$\mathscr{H}_{\infty}(S_{1})$ reduces $(S_{1},S_{2})$ to a doubly commuting
bi-isometry,

  3. $\mathrm{(iii)}$

$S_{2}({\mathscr{E}})$ and $\mathscr{H}_{\infty}(S_{1})$ are orthogonal
subspaces of $\mathscr{H},$

  4. $\mathrm{(iv)}$

$P_{\infty}\bigvee\\{S^{*k}_{1}S_{2}(x):k\geqslant 1\\}$ is finite-dimensional
for every $x\in{\mathscr{E}},$ where $P_{\infty}$ denotes the orthogonal
projection of $\mathscr{H}$ onto $\mathscr{H}_{\infty}(S_{1}),$

  5. $\mathrm{(v)}$

$\mathscr{H}_{\infty}(S_{1})$ essentially reduces $S_{2}$ to an isometry.

If $\mathrm{(i)}$ holds, then there exist an inner function
$\phi:\mathbb{D}\rightarrow\mathscr{B}({\mathscr{E}}),$ a Hilbert space
${\mathscr{F}},$ a constant unitary operator-valued function
$\psi:\mathbb{D}\rightarrow\mathscr{B}({\mathscr{F}}),$ a closed subspace
$\mathscr{H}_{uu}$ of $\mathscr{H}_{\infty}(S_{1})$ and a bi-unitary
$(V_{1},V_{2})$ on $\mathscr{H}_{uu}$ such that $(S_{1},S_{2})$ is unitarily
equivalent to

$\displaystyle(V_{1},V_{2})\oplus(M_{\psi,{\mathscr{F}}},M_{z,{\mathscr{F}}})\oplus(M_{z,{\mathscr{E}}},M_{\phi,{\mathscr{E}}})\,\mbox{on}\,\mathscr{H}_{uu}\oplus\mathscr{H}^{2}_{{\mathscr{F}}}(\mathbb{D})\oplus\mathscr{H}^{2}_{{\mathscr{E}}}(\mathbb{D}).$
(1.4)

######  Remark 1.7.

An examination of the proof of Theorem 1.6, as presented in Section 2, shows
that if $(S_{1},S_{2})$ be a bi-contraction on $\mathscr{H}$ and $S_{1}$ has
the Wold-type decomposition, then $\mathscr{H}_{\infty}(S_{1})$ reduces
$S_{2}$ to a contraction if and only if $S_{2}({\mathscr{E}})$ and
$\mathscr{H}_{\infty}(S_{1})$ are orthogonal subspaces of $\mathscr{H}.$

##  2\. Proof of Theorem 1.6

Let $(S_{1},S_{2})$ be a bi-contraction on $\mathscr{H}.$ Throughout this
section, we use the following notations$:$

  1. $\bullet$

${\mathscr{E}}=\ker S^{*}_{1}$

  2. $\bullet$

$P_{\infty}$ the orthogonal projection of $\mathscr{H}$ onto
$\mathscr{H}_{\infty}(S_{1})$

The proof of Theorem 1.6, as presented below, requires several lemmas.

######  Lemma 2.1.

Let $(S_{1},S_{2})$ be a bi-contraction on $\mathscr{H}$. Consider the closed
subspace $\mathscr{M}=\bigvee_{\lambda\in\mathbb{C}}\ker(S_{1}-\lambda)$ of
$\mathscr{H}.$ Then the following are valid$:$

  1. $\mathrm{(i)}$

for every $\lambda\in\mathbb{C},$ $\ker(S_{1}-\lambda)$ reduces
$(S_{1},S_{2})$ and

$\displaystyle\mathscr{M}=\bigoplus_{\lambda\in\mathbb{T}}\ker(S_{1}-\lambda).$
(2.1)

  2. $\mathrm{(ii)}$

$\mathscr{M}$ is a subspace of $\mathscr{H}_{u}(S_{1})$ such that

$\displaystyle\mathscr{H}_{u}(S_{1})=\mathscr{M}\oplus\mathscr{H}_{u}({S_{1}}|_{\mathscr{H}\ominus\mathscr{M}}).$

###### Proof.

(i) Since $S_{1}|_{\mathscr{H}_{u}(S_{1})}$ is unitary (see Theorem 1.1) and
$\ker(S_{1}-\mu),$ $\mu\in\mathbb{T},$ is a subspace of
$\mathscr{H}_{u}(S_{1}),$ the spaces $\ker(S_{1}-\lambda)$ and
$\ker(S_{1}-\mu)$ are orthogonal for any $\lambda\in\mathbb{T}$ provided
$\lambda\neq\mu.$ Since the eigenvalues of any isometry on a Hilbert space lie
on the unit circle, we obtain (2.1). Let $\lambda\in\mathbb{T}.$ Clearly,
$\ker(S_{1}-\lambda)$ is an invariant subspace of $S_{1}.$ Also, for any
$x\in\ker(S_{1}-\lambda),$ $S_{1}(S_{2}x)=\lambda S_{2}x$ (since
$S_{1}S_{2}=S_{2}S_{1}$), and hence $S_{2}x\in\ker(S_{1}-\lambda).$ Thus
$\ker(S_{1}-\lambda)$ is invariant for $S_{2}.$ To see that
$\ker(S_{1}-\lambda)$ reduces the pair $(S_{1},S_{2}),$ let
$x\in\ker(S_{1}-\lambda).$ By [15, Proposition 1.3.1],
$S^{*}_{1}x=\overline{\lambda}x,$ and hence $S^{*}_{1}x\in\ker(S_{1}-\lambda)$
showing that $\ker(S_{1}-\lambda)$ reduces $S_{1}.$ This also shows that
$S^{*}_{2}(S^{*}_{1}x-\overline{\lambda}x)=0$ or equivalently,
$S^{*}_{1}(S^{*}_{2}x)=\overline{\lambda}S^{*}_{2}x.$ Another application of
[15, Proposition 1.3.1] shows that $S_{1}(S^{*}_{2}x)=\lambda S^{*}_{2}x$ or
$S^{*}_{2}x\in\ker(S_{1}-\lambda)$ completing the verification of (i).

(ii) Clearly, $\mathscr{M}$ is a subspace of $\mathscr{H}_{u}(S_{1})$ such
that $S_{1}\mathscr{M}=\mathscr{M}.$ Thus
$\mathscr{H}_{u}({S_{1}}|_{\mathscr{M}})=\mathscr{M},$ and hence by (i),
$\mathscr{H}_{u}(S_{1})=\mathscr{M}\oplus\mathscr{H}_{u}({S_{1}}|_{\mathscr{H}\ominus\mathscr{M}}).$
∎

A particular case of part (i) below (when $(S_{1},S_{2})$ is a bi-isometry)
has been obtained in [14, Lemma 1] by an entirely different method. Also, the
space $\mathscr{N}_{x},$ as given in (ii) below, appears in the proof of [4,
Theorem 2.3].

######  Lemma 2.2.

Let $(S_{1},S_{2})$ be a bi-contraction on $\mathscr{H}.$ Assume that $S_{1}$
is a left invertible operator that has the Wold-type decomposition
$S_{1}=U\oplus S$ on
$\mathscr{H}=\mathscr{H}_{u}(S_{1})\oplus\mathscr{H}_{s}(S_{1}).$ Then the
following statements are valid$:$

  1. $\mathrm{(i)}$

$\mathscr{H}_{u}(S_{1})$ is an invariant subspace for $S_{2},$

  2. $\mathrm{(ii)}$

if $\mathscr{N}_{x}:=P_{\infty}\bigvee\\{S^{*k}_{1}S_{2}(x):k\geqslant 1\\}$
is finite-dimensional for every $x\in{\mathscr{E}},$ then
$\mathscr{H}_{u}(S_{1})$ is a reducing subspace for $S_{2},$

  3. $\mathrm{(iii)}$

if $\mathscr{H}_{u}(S_{1})$ reduces $S_{2},$ then
$({S_{1}}|_{\mathscr{H}_{u}(S_{1})},{S_{2}}|_{\mathscr{H}_{u}(S_{1})})$ is a
doubly commuting bi-contraction.

###### Proof.

Write $S_{2}=\big{[}\begin{smallmatrix}A&B\\\ C&D\end{smallmatrix}\big{]}$
relative to the orthogonal decomposition
$\mathscr{H}=\mathscr{H}_{u}(S_{1})\oplus\mathscr{H}_{s}(S_{1}).$ Since
$S_{2}$ commutes with $S_{1}=\big{[}\begin{smallmatrix}U&0\\\
0&S\end{smallmatrix}\big{]},$ a routine calculation shows that

$\displaystyle UA=AU,~{}CU=SC,~{}UB=BS,~{}SD=DS.$ (2.2)

(i) We adopt an argument from [9, Theorem 2.1] to the present situation. It
follows from (2.2) that for any $f\in\ker(S^{*}-w),$ $w\in\mathbb{D},$

$\displaystyle{w}C^{*}f=C^{*}S^{*}f=U^{*}C^{*}f,$

and hence $(U^{*}-{w}I)C^{*}f=0.$ Since $\sigma(U^{*})$ is a subset of the
unit circle, $C^{*}f=0$ for any $f\in\ker(S^{*}-w)$ and every
$w\in\mathbb{D}.$ Thus it suffices to check that

$\displaystyle\mathscr{H}_{s}(S_{1})=\bigvee_{w\in\mathbb{D}}\ker(S^{*}-w).$
(2.3)

Since $S=S_{1}|_{\mathscr{H}_{s}(S_{1})}$ is an analytic left invertible
operator, by the Shimorin’s analytic model (see [12, p. 154]), $S$ is
unitarily equivalent to the operator $M_{z}$ of multiplication by $z$ on a
reproducing kernel Hilbert space $\mathscr{H}_{\kappa}$ for some reproducing
kernel
$\kappa:\mathbb{D}_{r}\times\mathbb{D}_{r}\rightarrow\mathscr{B}(\mathscr{E}),$
where $\mathbb{D}_{r}$ denote the disc centred at the origin and of radius
$r>0.$ In view of

$\mathscr{H}_{\kappa}=\bigvee\\{\kappa(\cdot,w):w\in\mathbb{D}_{r}\\}=\bigvee_{w\in\mathbb{D}_{r}}\ker(M^{*}_{z}-w),$

we obtain (2.3).

(ii) Because of Lemma 2.1, after replacing $\mathscr{H}$ by
$\mathscr{H}\ominus\mathscr{M},$ if required, we may assume without loss of
generality that

$S_{1}$ has no eigenvalues. (2.4)

Assume that $\dim\mathscr{N}_{x}<\infty$ for every $x\in{\mathscr{E}}.$ We
claim that $B=0.$ Since $S_{1}$ has Wold-type decomposition (see (1.1)),
$\mathscr{H}_{s}(S_{1})=\bigvee_{n=0}^{\infty}S^{n}(\ker S^{*}).$ Thus, it
suffices to check that $BS^{n}|_{\ker S^{*}}=0$ for every integer $n\geqslant
0.$ However, by (2.2), $BS^{n}=U^{n}B,$ and hence it is enough to check that
$B|_{\ker S^{*}}=0.$ Let $x\in\ker S^{*}.$ Since
$\mathscr{H}_{u}(S_{1})=\mathscr{H}_{\infty}(S_{1}),$
$U={S_{1}}|_{\mathscr{H}_{u}(S_{1})}$ commutes with $P_{\infty},$ and hence
$U^{*k}=P_{\infty}{S^{*k}_{1}}|_{\mathscr{H}_{u}(S_{1})},$ $k\geqslant 1.$
Since $S_{2}=\big{[}\begin{smallmatrix}A&B\\\ 0&D\end{smallmatrix}\big{]},$ it
follows that

$\displaystyle\mathscr{N}_{x}=\bigvee\\{U^{*k}Bx:k\geqslant 1\\}.$

Since $\mathscr{N}_{x}$ is finite dimensional,
$U^{*}\mathscr{N}_{x}\subseteq\mathscr{N}_{x}$ and $U$ is unitary, the
operators $U^{*}$ and (equivalently) $U$ have an eigenvalue provided
$\mathscr{N}_{x}\neq\\{0\\}.$ In view of (2.4), this is not possible forcing
that $\mathscr{N}_{x}=\\{0\\}.$ It follows that $Bx=0$ completing the
verification of (ii).

(iii) Since ${S_{2}}|_{\mathscr{H}_{u}(S_{1})}$ commutes with the unitary
operator ${S_{1}}|_{\mathscr{H}_{u}(S_{1})},$ this is immediate. ∎

To prove Theorem 1.6, we need another lemma, which describes all doubly
commuting bi-isometries (see Corollary 2.4 below for a finer version).

######  Lemma 2.3.

Let $(S_{1},S_{2})$ be a bi-isometry on $\mathscr{H}.$ If the pair
$(S_{1},S_{2})$ is doubly commuting, then there exist an isometry
$V\in\mathscr{B}(\mathscr{H}_{\infty}(S_{1}))$ and a constant inner function
$\phi:\mathbb{D}\rightarrow\mathscr{B}({\mathscr{E}})$ such that

$\displaystyle(S_{1},S_{2})\cong(U,V)\oplus(M_{z,{\mathscr{E}}},M_{\phi,{\mathscr{E}}})~{}\mbox{on}~{}\mathscr{H}_{\infty}(S_{1})\oplus\mathscr{H}^{2}_{{\mathscr{E}}}(\mathbb{D}),$
(2.5)

where $U={S_{1}}|_{\mathscr{H}_{\infty}(S_{1})}$ and $\cong$ denotes the
unitary equivalence. If, in addition, $S_{2}$ is unitary, then $\phi$ is a
constant unitary operator-valued function.

###### Proof.

Note that $\mathscr{H}_{u}(S_{1})=\mathscr{H}_{\infty}(S_{1})$ (see Theorem
1.1). Assume that $(S_{1},S_{2})$ is doubly commuting. Thus,
$S^{*k}_{1}S_{2}(x)=0$ for every $x\in{\mathscr{E}},$ and hence by Lemma
2.2(ii), $\mathscr{H}_{\infty}(S_{1})$ is a reducing subspace for $S_{2}$ and
$V:={S_{2}}|_{\mathscr{H}_{\infty}(S_{1})}$ is an isometry. By [15, Lemma
V.3.2] and [15, Proposition V.2.2], there exists an inner function $\phi\in
H^{\infty}_{\mathscr{B}({\mathscr{E}})}(\mathbb{D})$ such that
${S_{2}}|_{H\ominus\mathscr{H}_{\infty}(S_{1})}\cong M_{\phi,{\mathscr{E}}}.$
Thus (2.5) holds provided we check that $\phi$ is constant. Since
$(S_{1},S_{2})$ is doubly commuting,

$\displaystyle
M^{*}_{\phi,{\mathscr{E}}}M_{z,{\mathscr{E}}}=M_{z,{\mathscr{E}}}M^{*}_{\phi,{\mathscr{E}}}.$
(2.6)

Let $P_{0}$ denote the orthogonal projection of
$\mathscr{H}^{2}_{{\mathscr{E}}}(\mathbb{D})$ onto the space ${\mathscr{E}}$
of constant polynomials. Note that
$M_{z,{\mathscr{E}}}M^{*}_{z,{\mathscr{E}}}=I-P_{0},$ and hence

$\displaystyle M^{*}_{\phi,{\mathscr{E}}}(I-P_{0})\overset{\eqref{m-phi-
m-z}}{=}M_{z,{\mathscr{E}}}M^{*}_{\phi,{\mathscr{E}}}M^{*}_{z,{\mathscr{E}}}.$
(2.7)

By [1, Exercise 2.66],

$\displaystyle
M^{*}_{\phi,{\mathscr{E}}}\kappa(\cdot,w)f=\kappa(\cdot,w)\phi(w)^{*}f,\quad
f\in{\mathscr{E}},~{}w\in\mathbb{D},$ (2.8)

where $\kappa(z,w)=\frac{I_{{\mathscr{E}}}}{1-z\overline{w}},$
$z,w\in\mathbb{D}$ with $I_{{\mathscr{E}}}$ denoting the identity operator on
$\mathscr{E}.$ Thus for any $f,g\in{\mathscr{E}}$ and $u,v\in\mathbb{D},$

$\displaystyle\langle{M^{*}_{\phi,{\mathscr{E}}}(I-P_{0})\kappa(\cdot,v)f},\,{\kappa(\cdot,u)g}\rangle$
$\displaystyle\overset{\eqref{text-ex}}{=}$
$\displaystyle\langle{\kappa(\cdot,v)\phi(v)^{*}f},\,{\kappa(\cdot,u)g}\rangle-\langle{\phi(0)^{*}f},\,{\kappa(\cdot,u)g}\rangle,$

and also

$\displaystyle\langle{M_{z,{\mathscr{E}}}M^{*}_{\phi,{\mathscr{E}}}M^{*}_{z,{\mathscr{E}}}\kappa(\cdot,v)f},\,{\kappa(\cdot,u)g}\rangle\overset{\eqref{text-
ex}}{=}u\overline{v}\langle{\kappa(\cdot,v)\phi(v)^{*}f},\,{\kappa(\cdot,u)g}\rangle.$

This combined with the reproducing property

$\langle{h},\,{\kappa(\cdot,u)f}\rangle=\langle{h(u)},\,{f}\rangle,\quad
h\in\mathscr{H}^{2}_{{\mathscr{E}}}(\mathbb{D}),~{}f\in{\mathscr{E}},$

and (2.7) yields $\phi(v)=\phi(0)$ for every $v\in\mathbb{D}.$ To see the
remaining part, one may argue as above to show that

$\displaystyle
M_{\phi,{\mathscr{E}}}M^{*}_{\phi,{\mathscr{E}}}=I~{}\Rightarrow\phi(0)\phi(0)^{*}=I,$

which completes the proof. ∎

###### Proof of Theorem 1.6.

By Theorem 1.1, $S_{1}$ has the Wold-type decomposition $U\oplus S$ on
$\mathscr{H}=\mathscr{H}_{\infty}(S_{1})\oplus\mathscr{H}_{s}(S_{1}).$ By
Lemma 2.2(i), $S_{2}$ decomposes as $\big{[}\begin{smallmatrix}A&B\\\
0&D\end{smallmatrix}\big{]}$ on
$\mathscr{H}=\mathscr{H}_{u}(S_{1})\oplus\mathscr{H}_{s}(S_{1}).$ In view of
the proof of Lemma 2.2, we may assume that (2.2) holds. Clearly, (i) holds if
and only if $B=0.$

(i)$\Rightarrow$(ii) This follows from Lemma 2.2(iii).

(ii)$\Rightarrow$(iii) Since $B=P_{\infty}S_{2}|_{\mathscr{H}_{s}(S_{1})}$ and
${\mathscr{E}}\subseteq\mathscr{H}_{s}(S_{1}),$ (ii) implies that
$P_{\infty}S_{2}({\mathscr{E}})=\\{0\\},$ and hence (iii) holds.

(iii)$\Rightarrow$(iv) By (iii), $P_{\infty}S_{2}({\mathscr{E}})=\\{0\\}.$
Since $P_{\infty}$ commutes with $S_{1},$ for every $x\in{\mathscr{E}},$
$P_{\infty}\bigvee\\{S^{*k}_{1}S_{2}(x):k\geqslant 1\\}=\\{0\\}.$

(iv)$\Rightarrow$(i) This follows from Lemma 2.2(ii).

(i)$\Leftrightarrow$(v) Clearly, it is enough to verify that
(v)$\Rightarrow$(i). In turn, it suffices to check that if
$B=P_{\infty}S_{2}|_{\mathscr{H}_{u}(S_{1})}$ is compact, then $B=0.$ By
(2.2), $UB=BS,$ and hence

$\displaystyle(UB)^{*}UB=(BS)^{*}BS~{}\Rightarrow~{}B^{*}B=S^{*}B^{*}BS.$

Thus $B^{*}B$ is $S$-Toeplitz operator in the sense of [6]. Now, if $B$ is
compact, then $B^{*}B$ is a compact $S$-Toeplitz operator, and since $S$ does
not have eigenvalues, by [6, Theorem 3.3], $B^{*}B=0$ or equivalently, $B=0.$

Assume now that (i) holds. Then

$\displaystyle S_{2}=A\oplus
D~{}\mbox{on}~{}\mathscr{H}=\mathscr{H}_{\infty}(S_{1})\oplus{\mathscr{H}}_{s}(S_{1}),$
(2.9)

where $A$ and $D$ are contractions such that

$\displaystyle AU=UA,~{}DS=SD.$ (2.10)

This yields

$\displaystyle(S_{1},S_{2})=(U,A)\oplus(S,D)~{}\mbox{on}~{}\mathscr{H}=\mathscr{H}_{\infty}(S_{1})\oplus\mathscr{H}_{s}(S_{1}).$
(2.11)

An application of [15, Lemma V.3.2] together with (2.10) shows that there
exists $\phi\in\mathscr{H}^{\infty}_{\mathscr{B}({\mathscr{E}})}(\mathbb{D})$
such that $D=M_{\phi,{\mathscr{E}}}.$ Since $D$ is an isometry, by [15,
Proposition V.2.2], $\phi$ is an inner function. It now follows from (2.11)
and Theorem 1.1 that

$\displaystyle(S_{1},S_{2})\cong(U,A)\oplus(M_{z},M_{\phi})~{}\mbox{on}~{}\mathscr{H}=\mathscr{H}_{\infty}(S_{1})\oplus\mathscr{H}^{2}_{{\mathscr{E}}}(\mathbb{D}).$

Moreover, by Lemma 2.2(iii), $(U,A)$ is a doubly commuting bi-isometry. Since
$U$ is unitary, one may now apply Lemma 2.3 (applied to $S_{1}=A$ and
$S_{2}=U$) to obtain (1.4). ∎

We conclude this section with two consequences of Theorem 1.6. The first one
recovers [14, Theorem 2].

######  Corollary 2.4 (M. Słociński).

Let $(S_{1},S_{2})$ be a doubly commuting bi-isometry on $\mathscr{H}.$ There
exist Hilbert spaces ${\mathscr{E}_{j}},$ $j=1,\ldots,3,$ constant unitary
operator-valued functions
$\phi:\mathbb{D}\rightarrow\mathscr{B}({\mathscr{E}_{1}})$ and
$\psi:\mathbb{D}\rightarrow\mathscr{B}({\mathscr{E}_{2}}),$ a closed subspace
$\mathscr{H}_{uu}$ of $\mathscr{H}_{\infty}(S_{1})$ and a bi-unitary
$(V_{1},V_{2})$ on $\mathscr{H}_{uu}$ such that

$\displaystyle(S_{1},S_{2})\cong(V_{1},V_{2})\oplus(M_{\phi,{\mathscr{E}_{1}}},M_{z,{\mathscr{E}_{1}}})\oplus(M_{z,{\mathscr{E}_{2}}},M_{\psi,{\mathscr{E}_{2}}})\oplus(M_{z,{\mathscr{E}_{3}}},M_{z,{\mathscr{E}_{3}}})$
$\displaystyle~{}\mbox{on~{}}\mathscr{H}_{uu}\oplus\mathscr{H}^{2}_{{\mathscr{E}_{1}}}(\mathbb{D})\oplus\mathscr{H}^{2}_{{\mathscr{E}_{2}}}(\mathbb{D})\oplus\mathscr{H}^{2}_{{\mathscr{E}_{3}}}(\mathbb{D}).$

###### Proof.

Since $(S_{1},S_{2})$ is a doubly commuting bi-isometry, condition (iv) of
Theorem 1.6 is satisfied. Hence, by Theorem 1.6, there exist Hilbert space
${\mathscr{F}},$ an inner function
$\phi:\mathbb{D}\rightarrow\mathscr{B}({\mathscr{E}}),$ a constant unitary
operator-valued function
$\psi:\mathbb{D}\rightarrow\mathscr{B}({\mathscr{F}}),$ a closed subspace
$\mathscr{H}_{uu}$ of $\mathscr{H}_{\infty}(S_{1})$ and a bi-unitary
$(V_{1},V_{2})$ on $\mathscr{H}_{uu}$ such that $(S_{1},S_{2})$ admits the
decomposition (1.4). Apply now Lemma 2.3 to the doubly commuting bi-isometry
$(M_{\phi,{\mathscr{E}}},M_{z,{\mathscr{E}}})$ to get the desired
decomposition. ∎

The following result provides some finiteness conditions ensuring the
orthogonal decomposition (1.4). In particular, it generalises [4, Theorem
2.3].

######  Corollary 2.5.

Let $(S_{1},S_{2})$ be a bi-isometry on $\mathscr{H}.$ Assume that any one of
the following conditions holds$:$

  1. $\mathrm{(a)}$

$\dim\,\ker(P_{\infty}{S^{*}_{2}}|_{\mathscr{H}_{\infty}(S_{1})})<\infty,$

  2. $\mathrm{(b)}$

$\dim\,P_{\infty}\ker(S^{*}_{2})<\infty,$

  3. $\mathrm{(c)}$

$\mathrm{card}\,\sigma({S_{1}}|_{\mathscr{H}_{\infty}(S_{1})})<\infty,$ where
$\mathrm{card}$ denotes the cardinality.

Then there exist a Hilbert space ${\mathscr{F}},$ an inner function
$\phi:\mathbb{D}\rightarrow\mathscr{B}({\mathscr{E}}),$ a constant unitary
operator-valued function
$\psi:\mathbb{D}\rightarrow\mathscr{B}({\mathscr{F}}),$ a closed subspace
$\mathscr{H}_{u}$ of $\mathscr{H}_{\infty}(S_{1})$ and a bi-unitary
$(V_{1},V_{2})$ on $\mathscr{H}_{u}$ such that $(S_{1},S_{2})$ admits the
decomposition (1.4).

###### Proof.

By Lemma 2.2(i), $S_{2}=\big{[}\begin{smallmatrix}A&B\\\
0&D\end{smallmatrix}\big{]},$ where $A,B,D$ satisfy (2.2). After replacing
$\mathscr{H}$ by $\mathscr{H}\ominus\mathscr{M},$ if required, we may assume
that (2.4) holds.

Assume that (a) holds. A routine calculation using $S^{*}_{2}S_{2}=I$ shows
that

$\displaystyle A^{*}A=I,\quad A^{*}B=0.$ (2.12)

By (2.2), $UA=AU.$ Since $U$ is unitary, $UA^{*}=A^{*}U.$ Thus, $\ker A^{*}$
is a reducing subspace for $U.$ If $\ker A^{*}\neq\\{0\\},$ then $U$ has an
eigenvalue (since by (a), $\ker A^{*}$ is finite dimensional), and hence
$S_{1}$ has eigenvalue, which is not possible in view of (2.4). Thus $\ker
A^{*}=\\{0\\},$ and hence by (2.12), $B=0.$

To see the remaining assertions, for $x\in\mathscr{H},$ let

$\displaystyle\mathscr{N}_{x}:=\bigvee\\{U^{*k}P_{\infty}S_{2}(x):k\geqslant
1\\}.$ (2.13)

In view of Lemma 2.2(ii), it suffices to check that $\mathscr{N}_{x}$ is
finite-dimensional for any $x\in{\mathscr{E}}$ provided either of (b) and (c)
holds.

First, assume that (b) holds. Since $S_{1}S_{2}=S_{2}S_{1}$ and
$S^{*}_{2}S_{2}=I,$ for any positive integer $k,$

$\displaystyle S^{*}_{2}\Big{(}S^{*k}_{1}S_{2}\big{[}\begin{smallmatrix}0\\\
x\end{smallmatrix}\big{]}\Big{)}=S^{*k}_{1}\big{[}\begin{smallmatrix}0\\\
x\end{smallmatrix}\big{]}=\big{[}\begin{smallmatrix}0\\\
S^{*k}x\end{smallmatrix}\big{]}=0.$

This combined with

$\displaystyle S^{*k}_{1}S_{2}\big{[}\begin{smallmatrix}0\\\
x\end{smallmatrix}\big{]}=\big{[}\begin{smallmatrix}U^{*k}&0\\\
0&S^{*k}\end{smallmatrix}\big{]}\big{[}\begin{smallmatrix}Bx\\\
Dx\end{smallmatrix}\big{]}=\big{[}\begin{smallmatrix}U^{*k}Bx\\\
S^{*k}Dx\end{smallmatrix}\big{]}$

shows that $\big{[}\begin{smallmatrix}U^{*k}Bx\\\
S^{*k}Dx\end{smallmatrix}\big{]}\in\ker S^{*}_{2}$ for every integer
$k\geqslant 1.$ Thus, by (2.13), $\mathscr{N}_{x}\subseteq P_{\infty}(\ker
S^{*}_{2}),$ and hence $\dim\mathscr{N}_{x}$ is finite for every
$x\in{\mathscr{E}}.$

Next, assume that (c) holds. Thus there exists a polynomial $p$ vanishing on
$\sigma(U),$ where $U:={S_{1}}|_{\mathscr{H}_{\infty}(S_{1})}$ is a unitary
operator. By the spectral mapping property, $\sigma(p(U))=\\{0\\}.$ Since
$p(U)$ is a normal operator, $p(U)=0.$ It is now easy to see that
$\dim\mathscr{N}_{x}$ is finite for every $x\in{\mathscr{E}}.$ ∎

It is worth noting that the conclusion of Corollary 2.5 extends naturally to
commuting $n$-tuples $(S_{1},\ldots,S_{n})$ of isometries for which the
dimension of $\ker\prod_{j=1}^{n}S^{*}_{j}$ is finite. The proof of this fact
can be given along the lines of proof of [4, Theorem 2.4]. We leave the
details to the reader.

##  3\. Bi-isometries not reducing hyper-ranges of the coordinates

In this short section, we discuss a class of bi-isometries $(S_{1},S_{2})$ on
$\mathscr{H}$ for which $\mathscr{H}_{\infty}(S_{1})$ is not a reducing
subspace of $S_{2}.$ We begin with the following consequence of the proof of
Theorem 1.6 (see Lemma 2.2(i), (2.2) and the discussion following (2.9)).

######  Proposition 3.1.

If $(S_{1},S_{2})$ is a bi-isometry on $\mathscr{H},$ then there exist a
unitary operator $U\in\mathscr{B}(\mathscr{H}_{\infty}(S_{1})),$
$\phi\in\mathscr{H}^{\infty}_{\mathscr{B}({\mathscr{E}})}(\mathbb{D})$ with
$\|\phi\|_{\infty}\leqslant 1,$ an isometry
$A\in\mathscr{B}(\mathscr{H}_{\infty}(S_{1}))$ and
$B\in\mathscr{B}(\mathscr{H}^{2}_{\mathscr{E}}(\mathbb{D}),\mathscr{H}_{\infty}(S_{1}))$
such that

$UA=AU,$ $UB=BM_{z,{\mathscr{E}}},$ $A^{*}B=0,$
$B^{*}B+M^{*}_{\phi,{\mathscr{E}}}M_{\phi,{\mathscr{E}}}=I$ and
$\displaystyle(S_{1},S_{2})\cong\Big{(}\Big{[}\begin{smallmatrix}U&0\\\
0&M_{z,{\mathscr{E}}}\end{smallmatrix}\Big{]},\Big{[}\begin{smallmatrix}A&B\\\
0&M_{\phi,{\mathscr{E}}}\end{smallmatrix}\Big{]}\Big{)}~{}\mbox{on}~{}\mathscr{H}_{\infty}(S_{1})\oplus\mathscr{H}^{2}_{\mathscr{E}}(\mathbb{D}),$

where $\cong$ denotes the unitary equivalence.

The previous proposition raises the following question$:$

######  Question 3.2.

With the notations as in Proposition 3.1, when does there exist a nonzero
$B\in\mathscr{B}(\mathscr{H}^{2}_{\mathscr{E}}(\mathbb{D}),\mathscr{H}_{\infty}(S_{1}))$
such that $UB=BM_{z,{\mathscr{E}}},$ $A^{*}B=0$ and
$B^{*}B+M^{*}_{\phi,{\mathscr{E}}}M_{\phi,{\mathscr{E}}}=I$?

Here is an answer to Question 3.2 in case of ${\mathscr{E}}=\mathbb{C}.$

######  Proposition 3.3.

Under the notations of Proposition 3.1 with ${\mathscr{E}}=\mathbb{C},$

$\displaystyle UB=BM_{z}~{}\Longleftrightarrow~{}B(p)=p(U)B(1)~{}\mbox{for
every polynomial}~{}p.$ (3.1)

If $UB=BM_{z},$ then the following statements are valid$:$

  1. $\mathrm{(i)}$

$A^{*}B=0$ if and only if $A^{*}B(1)=0,$

  2. $\mathrm{(ii)}$

if $\mu$ denotes the spectral measure of the unitary operator $U$ $($extended
trivially to $\mathbb{T}),$ then $B^{*}B+M^{*}_{\phi}M_{\phi}=I$ if and only
if

$\displaystyle
d\langle{\mu(\cdot)B(1)},\,{B(1)}\rangle=(1-|\phi|^{2})d\theta.$

###### Proof.

Since the polynomials are dense in $\mathscr{H}^{2}(\mathbb{D}),$ (3.1) is
immediate. To see (i) and (ii), assume that $UB=BM_{z}.$

(i) Assume that $A^{*}B(1)=0.$ By (3.1), $A^{*}B(p)=A^{*}p(U)B(1)$ for every
polynomial $p.$ However, since $AU=UA,$ by $A^{*}B(p)=p(U)A^{*}B(1)=0.$ Since
polynomials are dense in $\mathscr{H}^{2}(\mathbb{D}),$ $A^{*}B=0.$ The
converse is trivial.

(ii) Let $p,q$ be polynomials in $z.$ By (3.1),

$\displaystyle\langle{B(p)},\,{B(q)}\rangle+\langle{M_{\phi}(p)},\,{M_{\phi}(q)}\rangle$
$\displaystyle=$
$\displaystyle\langle{p(U)B(1)},\,{q(U)B(1)}\rangle+\langle{\phi p},\,{\phi
q}\rangle$ $\displaystyle=$
$\displaystyle\int_{\mathbb{T}}p(e^{i\theta})\overline{q(e^{i\theta})}d\langle{\mu(e^{i\theta})B(1)},\,{B(1)}\rangle+\int_{\mathbb{T}}p(e^{i\theta})\overline{q(e^{i\theta})}|\phi(e^{i\theta})|^{2}d\theta.$

Thus
$\langle{B(p)},\,{B(q)}\rangle+\langle{M_{\phi}(p)},\,{M_{\phi}(q)}\rangle=\langle{p},\,{q}\rangle$
if and only if

$\displaystyle\int_{\mathbb{T}}p(e^{i\theta})\overline{q(e^{i\theta})}d\langle{\mu(e^{i\theta})B(1)},\,{B(1)}\rangle=\int_{\mathbb{T}}p(e^{i\theta})\overline{q(e^{i\theta})}(1-|\phi(e^{i\theta})|^{2})d\theta.$

The desired conclusion in (ii) now follows from the uniqueness of the
trigonometric moment problem (see [13, Theorem 1.4]). ∎

######  Corollary 3.4.

Under the notations of Proposition 3.1 with ${\mathscr{E}}=\mathbb{C},$ if the
arc-length measure of $\sigma(U)$ is $0,$ then $\mathscr{H}_{\infty}(S_{1})$
reduces $(S_{1},S_{2})$ to a doubly commuting bi-isometry on
$\mathscr{H}_{\infty}(S_{1}),$

###### Proof.

By Proposition 3.1, there exists $\phi\in\mathscr{H}^{\infty}(\mathbb{D})$
with $\|\phi\|_{\infty}\leqslant 1$ such that $B^{*}B+M^{*}_{\phi}M_{\phi}=I.$
It now follows from Proposition 3.3(ii) and the assumption that the arc-length
measure of $\sigma(U)$ is $0$ that $\phi$ is an inner function. Thus
$M_{\phi}$ is an isometry or, equivalently, $B=0.$ One may now apply Theorem
1.6. ∎

It would be interesting to know if the conclusion of the previous corollary
holds in case ${\mathscr{E}}$ is of a dimension bigger than $1.$

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