Search

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., Comment: 28 pages, 7 figures, 6 tables

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.

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 M\"{o}bius 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 M\"{o}bius transformation $g_T(z)$, could be $U(1,1;\mathbb{H})$-similar to a diagonal matrix., Comment: 17 pages

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., Comment: 14 pages, 1 figure

Let $\mathcal{X}$ be a complex Banach space and $A\in\mathcal{L}(\mathcal{X})$ with $\sigma(A)=\{1\}$. 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., Comment: 7 pages

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 $C_{\varphi}$ with symbol $\varphi$ which is an analytic self-map on the closed unit disk. We prove that $C_{\varphi}$ acting on a weighted Hardy space of polynomial growth has closed range (semi-Fredholmness) if and only if $\varphi$ is a finite Blaschke product. Furthermore, it is obtained that $C_{\varphi}$ is Fredholm if and only if $\varphi$ is a disk automorphism., Comment: 25 pages, 2 figures

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., Comment: 25 pages, 3 figures

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 $M_{\varphi}$ for any analytic automorphism $\varphi$ 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., Comment: 30 pages

In this article, we focus on the left (translation) actions on noncommutative compact connected Lie groups ${\rm SU}(2) \times T^n$. We define the rotation vectors of the left actions induced by the elements in the maximal tori of ${\rm SU}(2) \times T^n$, and utilize rotation vectors to give the complete topologically conjugate classifications of left actions. Algebraic conjugacy and smooth conjugacy are also considered., Comment: 42 pages

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., Comment: v3: minor edits according to the referee's report. To appear in PAMS. 16 pages

In this paper, we give a new description of the group structure of the relative structure group of PL manifolds with boundary, and obtain a surgery exact sequence in the category of groups. Then we focus on the relative $L$-group of PL manifolds with boundary, and map it to the $K$-theory additively., Comment: 31 pages, 2 figures

Let $\mathcal{H}$ be a complex separable Hilbert space. We prove that if $\{f_{n}\}_{n=1}^{\infty}$ is a Schauder basis of the Hilbert space $\mathcal{H}$, then the angles between any two vectors in this basis must have a positive lower bound. Furthermore, we investigate that $\{z^{\sigma^{-1}(n)}\}_{n=1}^{\infty}$ can never be a Schauder basis of $L^{2}(\mathbb{T},\nu)$, where $\mathbb{T}$ is the unit circle, $\nu$ is a finite positive discrete measure, and $\sigma: \mathbb{Z} \rightarrow \mathbb{N}$ is an arbitrary surjective and injective map., Comment: 5 pages

In this article, we focus on the left translation actions on noncommutative compact connected Lie groups with topological dimension 3 or 4, consisting of ${\rm SU}(2),\,{\rm U}(2),\,{\rm SO}(3),\,{\rm SO}(3) \times S^1$ and ${{\rm Spin}}^{\mathbb{C}}(3)$. We define the rotation vectors (numbers) of the left actions induced by the elements in the maximal tori of these groups, and utilize rotation vectors (numbers) to give the topologically conjugate classification of the left actions. Algebraic conjugacy and smooth conjugacy are also considered. As a by-product, we show that for any homeomorphism $f:L(p, -1)\times S^1\rightarrow L(p, -1)\times S^1$, the induced isomorphism $(\pi\circ f\circ i)_*$ maps each element in the fundamental group of $L(p, -1)$ to itself or its inverse, where $i:L(p,-1)\rightarrow L(p, -1)\times S^1$ is the natural inclusion and $\pi:L(p, -1)\times S^1\rightarrow L(p, -1)$ is the projection., Comment: 50 pages. Accepted for publication in SCIENCE CHINA Mathematics

In order to study Li-Yorke chaos by the scalar perturbation for a given bounded linear operator $T$ on Banach spaces $X$, we introduce the Li-Yorke chaos translation set of $T$, which is defined by $S_{LY}(T)=\{\lambda\in\mathbb{C};\lambda+T \text{ is Li-Yorke chaotic}\}$. In this paper, some operator classes are considered, such as normal operator, compact operator, shift and so on. In particular, we show that the Li-Yorke chaos translation set of Kalisch operator on Hilbert space $\mathcal{L}^2[0,2\pi]$ is a simple point set $\{0\}$., Comment: 12 pages,1 figures

In this paper, we construct a homeomorphism on the unit closed disk to show that an invertible mapping on a compact metric space is Li-Yorke chaotic does not imply its inverse being Li-Yorke chaotic., Comment: 5 pages

In this paper, we introduce a new entropy-like invariant, named Hausdorff metric entropy, for finitely generated semigroups acting on compact metric spaces from a set-valued view and study its properties. We establish the relation between Hausdorff metric entropy and topological entropy of a semigroup defined by Bi\'s. Some examples with positive or zero Hausdorff metric entropy are given. Moreover, some notions of chaos are also well generalized for finitely generated semigroups from a set-valued view., Comment: 13 pages, 0 figures

In this paper, we study spectrums of Schauder operators. We show that we always can choose a Schauder operator in a given orbit such that the Schauder spectrum of it is empty.

In this paper, we firstly give a matrix approach to the bases of a separable Hilbert space and then correct a mistake appearing in both review and the English translation of the Olevskii's paper. After this, we show that even a diagonal compact operator may map an orthonormal basis into a conditional basis., Comment: 17 pages

The purpose of the present work is to answer an open problem which is raised by G.Costakis and A.Manoussos in their paper "J-class operators and hypercyclicity " accepted by J. Operator Theory. More precisely, we give the spectral description of the closure of the set of J-class operators acting on a separable Hilbert space., Comment: 8 pages, 0 figures

As well-known, the concept "hypercyclic" in operator theory is the same as the concept "transitive" in dynamical system. Now the class of hypercyclic operators is well studied. Following the idea of research in hypercyclic operators, we consider classes of operators with some kinds of chaotic properties in this article. First of all, the closures of the sets of all Li-Yorke chaotic operators or distributionally chaotic operators are discussed. We give a spectral description of them and prove that the two closures coincide with each other. Moreover, both the set of all Li-Yorke chaotic operators and the set of all distributionally chaotic operators have nonempty interiors which coincide with each other as well. The article also includes the containing relation between the closure of the set of all hypercyclic operators and the closure of the set of all distributionally chaotic operators. Finally, we get connectedness of the sets considered above., Comment: 20 pages, 1 figure

In our another recent article, we introduce a new dynamical property for linear operators called norm-unimodality which implies distributional chaos. In the present paper, we'll give a further discussion of norm-unimodality. It is showed that norm-unimodality is similar invariant and the spectra of norm-unimodal operator is referred to. As an application, in each nest algebra there exist distributional chaotic operators. Moreover, normal operators and compact operators with regard to norm-unimodality and Li-Yorke chaos are also be considered. Specially, a small compact perturbation of the unit operator could be distributionally chaotic., Comment: 11 pages

Let $\lambda B_{p}$, where $\lambda$ is a nonzero complex number, denote a constant-weighted backward shift operators on $l^{p}$ for $1\leq p<\infty$. In this article, we investigate, in topologically conjugacy, the complete classification for $\lambda B_{p}$., Comment: 6 pages

In this article, we provide a sufficient condition which gives Devaney chaos and distributional chaos for Cowen-Douglas operators. In fact, we obtain a distributionally chaotic criterion for bounded linear operators on Banach spaces., Comment: 8 pages

We discuss the relation between Devaney chaos in the base system and Devaney chaos in its induced hyperspace system. We show that the latter need not imply the former. We also argue that this implication is not true even in the strengthened condition. Additionally we give an equivalent condition for the periodically density in the hyperspace system., Comment: 9pages