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Bingzhe Hou

Publications and source records attributed to Bingzhe Hou.

At least 19 recordsLinked to original sources
arXiv2026

The $ϕ$-conjugation of quaternionic matrices and generalized Autonne-Takagi factorization

Let $ϕ$ be a quaternion of modulus $1$. In this article, we study some topics related to $ϕ$-conjugation for quaternionic matrices, including $ϕ$-Hermitian matrices, $ϕ$-conjugate normal matrices, unitary $ϕ$-congruence and $ϕ$-HSH decomposition (decomposition of a $ϕ$-Hermitian matrix and a skew $ϕ$-Hermitian matrix). In particular, we generalize the Autonne-Takagi factorization of quaternion $ϕ$-Hermitian matrices for all unit quaternion $ϕ$. This gives an affirmative answer to a problem proposed by R. Horn and F. Zhang in the paper ``A generalization of the complex Autonne-Takagi factorization to quaternion matrices, Linear Multilinear A. 60: 1239--1244, 2012''.

math.RA↗
arXiv2026

$\boldsymbol{i}$-conjugate for quaternionic matrices and related properties

Motivated by the result that a complex $n\times n$ matrix $A$ being unitarily equivalent to a real matrix, we extend the conclusion to the quaternion skew field in this paper, we present a necessary and sufficient condition for that a quaternion $n\times n$ matrix $A$ is unitarily equivalent to a complex matrix. To state the truth more clearly, we put forward the concept which we call $\boldsymbol{i}$-conjugate. Furthermore, we study the concepts related to $\boldsymbol{i}$-conjugate and their properties, such as unitary $\boldsymbol{i}$-congruence, $\boldsymbol{i}$-conjugate normality and $\boldsymbol{i}$-Hermicity in $M_{n}(\mathbb{H})$ as generalizations of the conventional unitary congruence, conjugate normality and Hermicity of matrices in $M_{n}(\mathbb{C})$. Finally, we present a new type of polar decomoposition of quaternion matrices.

math.RA↗
arXiv2026

The right invariant metric on the analytic automorphism group of the unit open disk induced by maximal modulus

In this paper, we study the right invariant metric $d_{H^{\infty}}$ on the analytic automorphism group $\rm{Aut}(\mathbb{D})$ of the unit open disk $\mathbb{D}$ induced by maximal modulus, that is, $d_{H^{\infty}}(φ, ψ)=\sup_{z\in\mathbb{D}}|φ(z)-ψ(z)|$ for any $φ, ψ\in \rm{Aut}(\mathbb{D})$. We give the explicit formula of the right invariant metric $d_{H^{\infty}}$ and characterize the almost regular Finsler geometric structure of $(\rm{Aut}(\mathbb{D}), d_{H^{\infty}})$.

math.CV↗
arXiv2026

G-LoG Bi-filtration for Medical Image Classification

Building practical filtrations on objects to detect topological and geometric features is an important task in the field of Topological Data Analysis (TDA). In this paper, leveraging the ability of the Laplacian of Gaussian operator to enhance the boundaries of medical images, we define the G-LoG (Gaussian-Laplacian of Gaussian) bi-filtration to generate the features more suitable for multi-parameter persistence module. By modeling volumetric images as bounded functions, then we prove the interleaving distance on the persistence modules obtained from our bi-filtrations on the bounded functions is stable with respect to the maximum norm of the bounded functions. Finally, we conduct experiments on the MedMNIST dataset, comparing our bi-filtration against single-parameter filtration and the established deep learning baselines, including Google AutoML Vision, ResNet, AutoKeras and auto-sklearn. Experiments results demonstrate that our bi-filtration significantly outperforms single-parameter filtration. Notably, a simple Multi-Layer Perceptron (MLP) trained on the topological features generated by our bi-filtration achieves performance comparable to complex deep learning models trained on the original dataset.

cs.CV↗
arXiv2025

Cowen-Douglas operators on quaternionic Hilbert spaces

In 1978, M. J. Cowen and R. G. Douglas introduced a class of geometric operators (known as Cowen-Douglas class of operators) and associated a Hermitian holomorphic vector bundle to such operators. In this paper, after giving some basic properties of $S$-spectrum and right eigenvalues of bounded right linear operators on separable quaternionic Hilbert spaces, we generalize the class of Cowen-Douglas operators to the quaternionic Hilbert space via the $S$-spectrum and denote this class as $B_n^s(Ω_q)$. Due to the lack of commutativity of quaternion multiplication, the quaternionic Cowen-Douglas operators are not trivial generalizations of the classical Cowen-Douglas operators. Each operator in $B_n^{s}(Ω_q)$ corresponds to an $n$-dimensional Hermitian right holomorphic quaternionic vector bundle. We first establish a rigidity theorem for Hermitian right holomorphic quaternionic vector bundles. It is then proven that two operators in $B_n^{s}(Ω_q)$ are quaternion unitarily equivalent if and only if the associate bundles are equivalent as Hermitian right holomorphic quaternionic vector bundles. In particular, we introduce canonical matrix representations of operators in $B_1^{s}(Ω_q)$ and furthermore, we give the quaternion unitarily equivalent classification of $B_1^{s}(Ω_q)$ by the canonical matrix representations. It is worth noting that curvature is a complete unitary invariant for the classical (complex) Cowen-Douglas operators, however, there exist two quaternionic Cowen-Douglas operators which have the same curvature but are not quaternion unitarily equivalent. In addition, we prove that the operators in $B_1^{s}(Ω_q)$ are quaternion unitarily equivalent if and only if their complex representations are unitarily equivalent. Some relevant examples of the above results are also provided.

math.FA↗
arXiv2025

Persistence modules induced by inner functions

As well-known, inner functions play an important role in the study of bounded analytic function theory. In recent years, persistence module theory, as a main tool applied to Topological Data Analysis, has received widespread attention. In this paper, we aim to use persistence module theory to study inner functions. We introduce the persistence modules arised from the level sets of inner functions. Some properties of these persistence modules are shown. In particular, we prove that the persistence modules (potentially not of locally finite type) induced by a class of inner functions have interval module decompositions. Furthermore, we demonstrate that the interleaving distance of the persistence modules is continuous with respect to the supremum norm for a class of Blaschke products, which could be used to discuss the path-connectedness of Blaschke products. As an example, we provide an explicit formula for the interleaving distance of the persistence modules induced by the Blaschke products with order two.

math.AT↗
arXiv2025

The invariant subspace problem and Rosenblum operators I

Let $T\in B(\mathcal{H})$ be an invertible operator. From the 1940's, Gelfand, Hille and Wermer investigated the invariant subspaces of $T$ by analyzing the growth of $\|T^n\|$, where $n\in \mathbb{Z}$. In this paper, we study the invariant subspaces of $T$ by estimating the growth of $\|T^n+λT^{-n}\|$, where $n\in \mathbb{N}$ and $λ$ is a nonzero complex constant. The key ingredient of our approach is introducing the notion of shift representation operators, which is based on the Rosenblum operators. In addition, by employing shift representation operators, we provide an equivalent of the Invariant Subspace Problem via the injectivity of certain Hankel operators.

math.FA↗
arXiv2025

Topologically conjugate classification of diagonal operators

Let $\ell^{p}$, $1\leq p<\infty$, be the Banach space of absolutely $p$-th power summable sequences and let $π_{n}$ be the natural projection to the $n$-th coordinate for $n\in\mathbb{N}$. Let $\mathfrak{W}=\{w_{n}\}_{n=1}^{\infty}$ be a bounded sequence of complex numbers. Define the operator $D_{\mathfrak{W}}: \ell^{p}\rightarrow\ell^{p}$ by, for any $x=(x_{1},x_{2},\ldots)\in \ell^p$, $π_{n}\circ D_{\mathfrak{W}}(x)=w_{n}x_{n}$ for all $n\geq1$. We call $D_{\mathfrak{W}}$ a diagonal operator on $\ell^{p}$. In this article, we study the topological conjugate classification of the diagonal operators on $\ell^{p}$. More precisely, we obtained the following results. $D_{\mathfrak{W}}$ and $D_{\vert\mathfrak{W}\vert}$ are topologically conjugate, where $\vert\mathfrak{W}\vert=\{\vert w_{n}\vert\}_{n=1}^{\infty}$. If $\inf_{n}\vert w_n\vert>1$, then $D_{\mathfrak{W}}$ is topologically conjugate to $2\mathbf{I}$, where $\mathbf{I}$ means the identity operator. Similarly, if $\inf_{n}\vert w_n\vert>0$ and $\sup_{n}\vert w_n\vert<1$, then $D_{\mathfrak{W}}$ is topologically conjugate to $\frac{1}{2}\mathbf{I}$. In addition, if $\inf_{n}\vert w_n\vert=1$ and $\inf_{n}\vert t_n\vert>1$, then $D_{\mathfrak{W}}$ and $D_{\mathfrak{T}}$ are not topologically conjugate.

math.DS↗
arXiv2025

The Jordan decomposition and Kaplansky's second test problem for Hermitian holomorphic vector bundles

In 1954, I. Kaplansky proposed three test problems for deciding the strength of structural understanding of a class of mathematical objects in his treatise "Infinite abelian groups", which can be formulated for very general mathematical systems. In this paper, we focus on Kaplansky's second test problem in a context of complex geometry. Let $H^2_β$ be a weighted Hardy space. The Cowen-Douglas operator theory tells us that each $h\in\textrm{Hol}(\overline{\mathbb{D}})$ induces a Hermitian holomorphic vector bundle on $H^2_β$, denoted by $E_{h(S_β)}(Ω)$, where $Ω$ is a domain. We show that the vector bundle $E_{h(S_β)}$ is a push-forwards Hermitian holomorphic vector bundle and study the similarity deformation problems. Our main theorem is that if $H^2_β$ is a weighted Hardy space of polynomial growth, then for any $f\in \textrm{Hol}(\overline{\mathbb{D}})$, there exists a unique positive integer $m$ and an function $h\in\textrm{Hol}(\overline{\mathbb{D}})$ inducing an indecomposable vector bundle $E_{h(S_β)}$, such that $E_{f(S_β)}$ is similar to $\bigoplus_1^m E_{h(S_β)}$, where $h$ is unique in the sense of analytic automorphism group action. That could be seemed as a Jordan decomposition theorem for the push-forwards Hermitian holomorphic vector bundles. Furthermore, we give the similarity classification of those push-forwards Hermitian holomorphic vector bundles induced by analytic functions, and give an affirmative answer to Kaplansky's second test problem for those objects. We also give an affirmative answer to the geometric version and generalized version of a problem proposed by R. Douglas in 2007, and obtain the $K_0$-group of the commutant algebra of a multiplication operator on a weighted Hardy space of polynomial growth. In addition, we give an example to show the setting of polynomial growth condition is necessary.

math.FA↗
arXiv2024

Mix-GENEO: A Flexible Filtration for Multiparameter Persistent Homology Detects Digital Images

Two important tasks in the field of Topological Data Analysis are building practical multifiltrations on objects and using TDA to detect the geometry. Motivated by the tasks, we build multiparameter filtrations by operators on images named multi-GENEO, multi-DGENEO and mix-GENEO, and we prove the stability of both the interleaving distance and multiparameter persistence landscape of multi-GENEO with respect to the pseudometric on bounded functions. We also give the estimations of upper bound for multi-DGENEO and mix-GENEO. In practical applications, we regard image as a discrete function space, and then we build multifiltrations on the discrete function space. Finally, we construct comparable experiment on MNIST dataset to demonstrate our bifiltrations are superior to 1-parameter filtrations including lower-star filtration and upper-star filtration. For instance, 6 and 9 can be distinguished by our bifiltrations, while they cannot be distinguished by 1-parameter filtrations. The experiment results demonstrate our bifiltrations have ability to detect geometric and topological differences of digital images.

cs.CV↗
arXiv2024

Stable Similarity Comparison of Persistent Homology Groups

Classification in the sense of similarity is an important issue. In this paper, we study similarity classification in Topological Data Analysis. We define a pseudometric $d_{S}^{(p)}$ to measure the distance between barcodes generated by persistent homology groups of topological spaces, and we provide that our pseudometric $d_{S}^{(2)}$ is a similarity invariant. Thereby, we establish a connection between Operator Theory and Topological Data Analysis. We give the calculation formula of the pseudometric $d_{S}^{(2)}$ $(d_{S}^{(1)})$ by arranging all eigenvalues of matrices determined by barcodes in descending order to get the infimum over all matchings. Since conformal linear transformation is one representative type of similarity transformations, we construct comparative experiments on both synthetic datasets and waves from an online platform to demonstrate that our pseudometric $d_{S}^{(2)}$ $(d_{S}^{(1)})$ is stable under conformal linear transformations, whereas the bottleneck and Wasserstein distances are not. In particular, our pseudometric on waves is only related to the waveform but is independent on the frequency and amplitude. Furthermore, the computation time for $d_{S}^{(2)}$ $(d_{S}^{(1)})$ is significantly less than the computation time for bottleneck distance and is comparable to the computation time for accelerated Wasserstein distance between barcodes.

math.AT↗
arXiv2023

Analytic automorphism group and similar representation of analytic functions

In geometry group theory, one of the milestones is M. Gromov's polynomial growth theorem: Finitely generated groups have polynomial growth if and only if they are virtually nilpotent. Inspired by M. Gromov's work, we introduce the growth types of weighted Hardy spaces. In this paper, we focus on the weighted Hardy spaces of polynomial growth, which cover the classical Hardy space, weighted Bergman spaces, weighted Dirichlet spaces and much broader. Our main results are as follows. $(1)$ We obtain the boundedness of the composition operators with symbols of analytic automorphisms of unit open disk acting on weighted Hardy spaces of polynomial growth, which implies the multiplication operator $M_z$ is similar to $M_φ$ for any analytic automorphism $φ$ on the unit open disk. Moreover, we obtain the boundedness of composition operators induced by analytic functions on the unit closed disk on weighted Hardy spaces of polynomial growth. $(2)$ For any Blaschke product $B$ of order $m$, $M_B$ is similar to $\bigoplus_{1}^m M_z$, which is an affirmative answer to a generalized version of a question proposed by R. Douglas in 2007. $(3)$ We also give counterexamples to show that the composition operators with symbols of analytic automorphisms of unit open disk acting on a weighted Hardy space of intermediate growth could be unbounded, which indicates the necessity of the setting of polynomial growth condition. Then, the collection of weighted Hardy spaces of polynomial growth is almost the largest class such that Douglas's question has an affirmative answer. $(4)$ Finally, we give the Jordan representation theorem and similarity classification for the analytic functions on the unit closed disk as multiplication operators on a weighted Hardy space of polynomial growth.

math.FA↗
arXiv2023

Some invariants of $U(1,1;\mathbb{H})$ and diagonalization

Denote by $\mathbb{H}$ the set of all quaternions. We are interested in the group $U(1,1;\mathbb{H})$, which is a subgroup of $2\times 2$ quaternionic matrix group and is sometimes called $Sp(1,1)$. As well known, $U(1,1;\mathbb{H})$ corresponds to the quaternionic Möbius transformations on the unit ball in $\mathbb{H}$. In this article, some similar invariants on $U(1,1;\mathbb{H})$ are discussed. Our main result shows that each matrix $T\in U(1,1;\mathbb{H})$, which corresponds to an elliptic quaternionic Möbius transformation $g_T(z)$, could be $U(1,1;\mathbb{H})$-similar to a diagonal matrix.

math.RA↗
arXiv2023

Continuity of inner-outer factorization and cross sections from invariant subspaces to inner functions

Let $H^{\infty}$ be the Banach algebra of bounded analytic functions on the unit open disc $\mathbb{D}$ equipped with the supremum norm. As well known, inner functions play an important role of in the study of bounded analytic functions. In this paper, we are interested in the study of inner functions. Following by the canonical inner-outer factorization decomposition, define $Q_{inn}$ and $Q_{out}$ the maps from $H^{\infty}$ to $\mathfrak{I}$ the set of inner functions and $\mathfrak{F}$ the set of outer functions, respectively. In this paper, we study the $H^{2}$-norm continuity and $H^{\infty}$-norm discontinuity of $Q_{inn}$ and $Q_{out}$ on some subsets of $H^{\infty}$. On the other hand, the Beurling theorem connects invariant subspaces of the multiplication operator $M_z$ and inner functions. We show the nonexistence of continuous cross section from some certain invariant subspaces to inner functions in the supremum norm. The continuity problem of $Q_{inn}$ and $Q_{out}$ on $\textrm{Hol}(\overline{\mathbb{D}})$, the set of all analytic functions in the closed unit disk, are also considered.

math.CV↗
arXiv2023

Scaled Homology and Topological Entropy

In this paper, we build up a scaled homology theory, $lc$-homology, for metric spaces such that every metric space can be visually regarded as "locally contractible" with this newly-built homology. We check that $lc$-homology satisfies all Eilenberg-Steenrod axioms except exactness axiom whereas its corresponding $lc$-cohomology satisfies all axioms for cohomology. This homology can relax the smooth manifold restrictions on the compact metric space such that the entropy conjecture will hold for the first $lc$-homology group.

math.AT↗
arXiv2023

A new version of the Gelfand-Hille theorem

Let $\mathcal{X}$ be a complex Banach space and $A\in\mathcal{L}(\mathcal{X})$ with $σ(A)=\{1\}$. We prove that for a vector $x\in \mathcal{X}$, if $\|(A^{k}+A^{-k})x\|=O(k^N)$ as $k \rightarrow +\infty$ for some positive integer $N$, then $(A-\mathbf{I})^{N+1}x=0$ when $N$ is even and $(A-\mathbf{I})^{N+2}x=0$ when $N$ is odd. This could be seemed as a new version of the Gelfand-Hille theorem. As a corollary, we also obtain that for a quasinilpotent operator $Q\in\mathcal{L}(\mathcal{X})$ and a vector $x\in\mathcal{X}$, if $\|\cos(kQ)x\|=O(k^N)$ as $k \rightarrow +\infty$ for some positive integer $N$, then $Q^{N+1}x=0$ when $N$ is even and $Q^{N+2}x=0$ when $N$ is odd.

math.FA↗
arXiv2023

Composition operators on weighted Hardy spaces of polynomial growth

In the present paper, we study the composition operators acting on weighted Hardy spaces of polynomial growth, which are concerned with norms, spectra and (semi-)Fredholmness. Firstly, we estimate the norms of the composition operators with symbols of disk automorphisms. Secondly, we discuss the spectra of the composition operators with symbols of disk automorphisms. In particular, it is proven of that the spectrum of a composition operator with symbol of any parabolic disk automorphism is always the unit circle. Thirdly, we consider the Fredholmness of the composition operator $C_φ$ with symbol $φ$ which is an analytic self-map on the closed unit disk. We prove that $C_φ$ acting on a weighted Hardy space of polynomial growth has closed range (semi-Fredholmness) if and only if $φ$ is a finite Blaschke product. Furthermore, it is obtained that $C_φ$ is Fredholm if and only if $φ$ is a disk automorphism.

math.FA↗
arXiv2023

A note on connectedness of Blaschke products

Consider the space $\mathcal{F}$ of all inner functions on the unit open disk under the uniform topology, which is a metric topology induced by the $H^{\infty}$-norm. In the present paper, a class of Blaschke products, denoted by $\mathcal{H}_{SC}$, is introduced. We prove that for each $B\in\mathcal{H}_{SC}$, $B$ and $zB$ belong to the same path-connected component of $\mathcal{F}$. It plays an important role of a method to select a fine subsequence of zeros. As a byproduct, we obtain that each Blaschke product in $\mathcal{H}_{SC}$ has an interpolating and one-component factor.

math.CV↗