CyberSec Research

In this paper, structural properties of lower semi-frames in separable Hilbert spaces are explored with a focus on transformations under linear operators (may be unbounded). Also, the direct sum of lower semi-frames, providing necessary and sufficient conditions for the preservation of lower semi-frame structure, is examined.

In this paper, we investigate a spline frame generated by oversampling against the well-known Battle-Lemari\'e wavelet system of nonnegative integer order, $n$. We establish a characterization of the Besov and Triebel-Lizorkin (quasi-) norms for the smoothness parameter up to $s < n+1$, which includes values of $s$ where the Battle-Lemari\'e system no longer provides an unconditional basis; we, additionally, prove a result for the endpoint case $s=n+1$. This builds off of earlier work by G. Garrig\'os, A. Seeger, and T. Ullrich, where they proved the case $n=0$, i.e. that of the Haar wavelet, and work of R. Srivastava, where she gave a necessary range for the Battle-Lemari\'e system to give an unconditional basis of the Triebel-Lizorkin spaces.

This seminal paper marks the beginning of our investigation into on the spectral theory based on $S$-spectrum applied to the Dirac operator on manifolds. Specifically, we examine in detail the cases of the Dirac operator $\mathcal{D}_H$ on hyperbolic space and the Dirac operator $\mathcal{D}_S$ on the spherical space, where these operators, and their squares $\mathcal{D}_H^2$ and $\mathcal{D}_S^2$, can be written in a very explicit form. This fact is very important for the application of the spectral theory on the $S$-spectrum. In fact, let $T$ denote a (right) linear Clifford operator, the $S$-spectrum is associated with a second-order polynomial in the operator $T$, specifically the operator defined as $ Q_s(T) := T^2 - 2s_0T + |s|^2. $ This allows us to associate to the Dirac operator boundary conditions that can be of Dirichlet type but also of Robin-like type. Moreover, our theory is not limited to Hilbert modules; it is applicable to Banach modules as well. The spectral theory based on the $S$-spectrum has gained increasing attention in recent years, particularly as it aims to provide quaternionic quantum mechanics with a solid mathematical foundation from the perspective of spectral theory. This theory was extended to Clifford operators, and more recently, the spectral theorem has been adapted to this broader context. The $S$-spectrum is crucial for defining the so-called $S$-functional calculus for quaternionic and Clifford operators in various forms. This includes bounded as well as unbounded operators, where suitable estimates of sectorial and bi-sectorial type for the $S$-resolvent operator are essential for the convergence of the Dunford integrals in this setting.

The famous Koecher-Vinberg theorem characterises the finite dimensional formally real Jordan algebras among the finite dimensional order unit spaces as the ones that have a symmetric cone. An alternative characterisation of symmetric cones was obtained by Walsh who showed that the symmetric cones correspond exactly to the finite dimensional order unit spaces for which there exists a gauge-reversing map from the interior of the cone to itself. In this paper we prove an infinite dimensional version of this characterisation of symmetric cones.

We investigate the spectrum of the generator induced on the space of Hilbert-Schmidt operators by a Gaussian quantum Markov semigroup with a faithful normal invariant state in the general case, without any symmetry or quantum detailed balance assumptions. We prove that the eigenvalues are entirely determined by those of the drift matrix, similarly to classical Ornstein-Uhlenbeck semigroups. This result is established using a quasi-derivation property of the generator. Moreover, the same spectral property holds for the adjoint of the induced generator. Finally, we show that these eigenvalues constitute the entire spectrum when the induced generator has a spectral gap.

In this paper, we investigate the properties of disjoint Ces$\grave{a}$ro-hypercyclic operators. First, the definition of disjoint Ces$\grave{a}$ro-hypercyclic operators is provided, and disjoint Ces$\grave{a}$ro-Hypercyclicity Criterion is proposed. Later, two methods are used to prove that operators satisfying this criterion possess disjoint Ces$\grave{a}$ro-hypercyclicity. Finally, this paper further investigates weighted shift operators and provides detailed characterizations of the weight sequences for disjoint Ces$\grave{a}$ro-hypercyclic unilateral and bilateral weighted shift operators on sequence spaces.

We study left symmetric and right symmetric elements in the space $\ell_{\infty}(K, \mathbb{X}) $ of bounded functions from a non-empty set $K$ to a Banach space $\mathbb{X}.$ We prove that a non-zero element $ f \in\ell_{\infty}(K, \mathbb{X}) $ is left symmetric if and only if $f$ is zero except for an element $k_0 \in K$ and $f(k_0)$ is left symmetric in $\mathbb{X}.$ We characterize left symmetric elements in the space $C_0(K, \mathbb{X}),$ where $K$ is a locally compact perfectly normal space. We also study the right symmetric elements in $\ell_{\infty}(K, \mathbb{X}).$ Furthermore, we characterize right symmetric elements in $C_0(K, \mathbb{X}),$ where $K$ is a locally compact Hausdorff space and $\mathbb{X}$ is real Banach space. As an application of the results obtained in this article, we characterize the left symmetric and right symmetric operators on some special Banach spaces. These results improve and generalize the existing ones on the study of left and right symmetric elements in operator spaces.

We establish a convergence result for the vanishing discount problem in the context of nonlocal HJ equations. We consider a fairly general class of discounted first-order and convex HJ equations which incorporate an integro-differential operator posed on the $d$-dimensional torus, and we show that the solutions converge to a specific critical solution as the discount factor tends to zero. Our approach relies on duality techniques for nonlocal convex HJ equations, building upon Hahn-Banach separation theorems to develop a generalized notion of Mather measure. The results are applied to a specific class of convex and superlinear Hamiltonians.

By a theorem of Dixmier-Douady the unitary group of an infinite-dimensional separable Hilbert space $H$ in the strong operator topology is contractible. The Dixmier-Douady proof is based on an explicit construction of families of subspaces and operators in $H$ with rather special properties. Unfortunately, this proof leaves hidden the geometric meaning of the theorem. The first goal of this note is to give a direct geometric proof of this theorem. The second goal is to provide a geometic analogue of Dixmier-Douady construction.

We study the C$^*$ algebra generated by the composition operator $C_a$ acting on the Hardy space $H^2$ of the unit disk, given by $C_af=f\circ\varphi_a$, where $$ \varphi_a(z)=\frac{a-z}{1-\bar{a}z}, $$ for $|a|<1$. Also several operators related to $C_a$ are examined.

The dynamics of weighted translation operators on Lebesgue spaces, Orlicz spaces, and in general on solid Banach function spaces have been studied in numerous papers. Recently, the dynamics of weighted translations on weighted Orlicz spaces have also been studied by Chen and others, extending thus the previous results concerning the dynamics of weighted translations on classical Orlicz spaces. The main idea of this paper is to obtain a generalization of these results to the case of general weighted solid Banach function spaces. More precisely, in this paper, we characterize topologically transitive weighted composition operators on weighted solid Banach function spaces. This approach has applications in the dynamics of weighted translations on weighted Morrey spaces.

Let $\mathbb{D}=\{z\in\mathbb{C}: |z|<1\}$ and $\mathbb{T}=\{z\in\mathbb{C}: |z|=1\}$. For $a\in\mathbb{D}$, consider $\varphi_a(z)=\frac{a-z}{1-\bar{a}z}$ and $C_a$ the composition operator in $L^2(\mathbb{T})$ induced by $\varphi_a$: $$ C_a f=f\circ\varphi_a. $$ Clearly $C_a$ satisties $C_a^2=I$, i.e., is a non-selfadjoint reflection. We also consider the following symmetries (selfadjoint reflections) related to $C_a$: $$ R_a=M_{\frac{|k_a|}{\|k_a\|_2}}C_a \ \hbox{ and } \ W_a=M_{\frac{k_a}{\|k_a\|_2}}C_a, $$ where $k_a(z)=\frac{1}{1-\bar{a}z}$ is the Szego kernel. The symmetry $R_a$ is the unitary part in the polar decomposition of $C_a$. We characterize the eigenspaces $N(T_a\pm I)$ for $T_a=C_a, R_a$ or $W_a$, and study their relative positions when one changes the parameter $a$, e.g., $N(T_a\pm I)\cap N(T_b\pm I)$, $N(T_a\pm I)\cap N(T_b\pm I)^\perp$, $N(T_a\pm I)^\perp\cap N(T_b\pm I)$, etc., for $a\ne b\in\mathbb{D}$.

We establish two Frequent Hypercyclicity Criteria for unbounded operators, inspired by the frameworks of Bayart Grivaux and deLaubenfels Emamirad Grosse Erdmann. These criteria simplify the verification and construction of frequently hypercyclic operators.

We verify that for all $n \geq 3$ and $2 \leq k \leq n+1$, the standard $k$-bubble clusters, conjectured to be minimizing total perimeter in $\mathbb{R}^n$, $\mathbb{S}^n$ and $\mathbb{H}^n$, are stable -- an infinitesimal regular perturbation preserving volume to first order yields a non-negative second variation of area modulo the volume constraint. In fact, stability holds for all standard $\textit{partitions}$, in which several cells are allowed to have infinite volume. In the Gaussian setting, any partition in $\mathbb{G}^n$ ($n\geq 2$) obeying Plateau's laws and whose interfaces are all $\textit{flat}$, is stable. Our results apply to non-standard partitions as well - starting with any (regular) flat Voronoi partition in $\mathbb{S}^n$ and applying M\"obius transformations and stereographic projections, the resulting partitions in $\mathbb{R}^n$, $\mathbb{S}^n$ and $\mathbb{H}^n$ are stable. Our proof relies on a new conjugated Brascamp-Lieb inequality on partitions with conformally flat umbilical boundary, and the construction of a good conformally flattening boundary potential.

We refine the well-known Blanco-Koldobsky-Turn\v{s}ek Theorem which states that a norm one linear operator defined on a Banach space is an isometry if and only if it preserves orthogonality at every element of the space. We improve the result for Banach spaces in which the set of all smooth points forms a dense $G_{\delta}$-set by proving that a norm one linear operator that preserves orthogonality on a dense subset of the space is an isometry. We further demonstrate that if such an operator preserves orthogonality on a hyperplane not passing through the origin then it is an isometry. In the context of finite-dimensional Banach spaces, we prove that preserving orthogonality on the set all extreme points of the unit ball forces the operator to be an isometry, which substantially refines Blanco-Koldobsky-Turn\v{s}ek theorem. Finally, for finite-dimensional polyhedral spaces, we establish the significance of the set of all $k$-smooth points for any possible $k,$ in the study of isometric theory.

We characterize the amenability of unitary representations (in the sense of Bekka) via the existence of an orthonormal basis supporting an invariant probability charge. Based on this, we explore several natural notions of amenable near representations on Hilbert spaces. Establishing an analogue of a theorem about sofic groups by Elek and Szab\'o, we prove that a group is hyperlinear if and only if it admits an essentially free amenable near representation. This answers a question raised by Pestov and Kwiatkowska. For comparison, we also provide characterizations of Kirchberg's factorization property, as well as Haagerup's property, along similar lines.

On real metric manifolds admitting a co-dimension one foliation, sectorial operators are introduced that interpolate between the generalized Laplacian and the d'Alembertian. This is used to construct a one-parameter family of analytic semigroups that remains well-defined into the near Lorentzian regime. In the strict Lorentzian limit we identify a sense in which a well-defined Schr\"{o}dinger evolution group arises. For the analytic semigroups we show in addition that: (i) they act as integral operators with kernels that are jointly smooth in the semigroup time and both spacetime arguments. (ii) the diagonal of the kernels admits an asymptotic expansion in (shifted) powers of the semigroup time whose coefficients are the Seeley-DeWitt coefficients evaluated on the complex metrics.

In this paper, we investigate properties of a reproducing kernel Hilbert space of a group action. In particular, we introduce an equivalence relation on a compact Hausdorff space $X$, and consequently establish three equivalent definitions for when two elements are related. We also see how the equivalence classes of $X$ correspond to subgroups of the group acting transitively on $X$, which we aptly refer to as relation stabilizers.