Advances in Operator Theory最新文献

We consider a class of Markov interval maps (fin mathcal {M}(I)) with I an interval, whose associated transition matrix (A_f) is necessarily primitive. Then we search for subdynamics, i.e., a subset (Jsubset I) and (g=fvert _{J} in mathcal { M}([J])), with [J] the minimal closed interval containing J. The transition matrix (A_g) of g is obtained through successive state splittings of (A_f), followed by the removal of appropriate row(s) and column(s). We prove the existence of such J ensuring (gin mathcal {M}([J])). We also consider the Cuntz–Krieger algebra (mathcal {O}_{A_{f}}) representation (pi _{f,x}) on the Hilbert space associated to the f-orbit of each point (xin J). We similarly obtain a representation (pi _{g,x}) of (mathcal {O}_{A_{g}}). We prove that (pi _{g,x}(mathcal {O}_{A_{g}})) is a subalgebra of (pi _{f,x}(mathcal {O}_{A_{f}})). By exploring this further, we show that in fact (mathcal {O}_{A_{g}}) is a corner algebra of (mathcal {O}_{A_{f}}) by finding a projection (p_J) such that (mathcal {O}_{A_{g}}=p_{J},mathcal {O}_{A_{f}}p_{J}). We explicitly enumerate such corner algebras for each splitting state. This method provides a systematic way to construct concrete examples of specific Cuntz–Krieger corner algebras.

A unified approach to construct collections of K-spaces based on paramater couples with certain strong properties is formulated. This approach includes in particular the collections of K-spaces and small k-spaces.

Given a bounded positive linear operator A on a Hilbert space (mathcal {X}), this operator induces a semi-Hilbertian structure on (mathcal {X}). For a bounded linear operator T defined on the corresponding semi-Hilbertian space, we introduce a new definition of a Fredholm operator that is compatible with the semi-Hilbertian structure. Based on this definition, we proceed to define the essential spectra of T and establish their fundamental properties. Within this framework, we consider the bounded off-diagonal operator matrix (mathbb {T} = begin{pmatrix} 0 & M N & 0 end{pmatrix}) acting on the semi-Hilbertian space and demonstrate that the essential spectra of (mathbb {T}) are entirely determined by the essential spectra of the products MN and NM. Finally, an illustrative example is provided to substantiate the theoretical conclusions.

We prove some results concerning the finitely additive, vector integrals of Bochner and Pettis and their representation over a countably additive probability space. An application to the non compact Choquet theorem is also provided.

The notion of a self-similar ultragraph ((G,mathcal {U},varphi )) and its (C^*)-algebra (mathcal {O}_{G,mathcal {U}}) were introduced in our recent work, where we proposed inverse semigroup and groupoid models for such (C^*)-algebras as well. In this paper, we investigate minimality and effectiveness of the groupoid of a self-similar ultragraph ((G,mathcal {U},varphi )). In particular, we obtain a result for simplicity of the (C^*)-algebras (mathcal {O}_{G,mathcal {U}}) in a certain case.

In this paper, we study property ((UW_E)) for hypercyclic and supercyclic operators. The stability of variants of Weyl type theorems under compact perturbations for Toeplitz operators on the Bergman space is also studied. We also provide some examples of Toeplitz operators satisfying Weyl type theorems on the Bergman space and the harmonic Bergman space.

In this paper, we consider analytic Hardy spaces associated with fractal domains in the complex plane. In 2013, Dong et al. obtained a positive solution to the Cantor set conjecture in (Adv Math 232:543–570, 2013) for the Cauchy transforms on the Sierpinski triangle. In this paper, we achieve a deeper understanding of these Cauchy transforms by showing that they do not belong to any Hardy space on (triangle ^*,) where (triangle ^*= widehat{{mathbb {C}}}setminus triangle ) and (triangle ) is the compact regular triangle with vertexes ({varepsilon _k=e^{2kpi i/3}, k=0,1,2}.) Along the way, a Hardy–Littlewood-type theorem for general domains, which is of independent interests, is established.

This paper considers the existence and multiplicity of normalized solutions for the following Schrödinger–Poisson equation involving p-Laplacian operator and Hardy term

where (1<p<3), (p+frac{p^{2}}{3}<q<p^{*}:=frac{3p}{3-p}), (0le mu <bar{mu }:=left( frac{3-p}{p}right) ^p), (lambda ) is a Lagrange multiplier and (kappa >0) is a parameter. We prove the existence of normalized solution by using the Pohozaev manifold and obtain the infinitely many radial solutions by a fountain theorem type argument. Moreover, we explore the asymptotic behavior of normalized solutions as (mu rightarrow 0) and (kappa rightarrow 0).

This paper is concerned with the extension to semigroups two interesting results by Vijayaraju that guarantee the existence of a fixed point for asumptotically nonexpansive and uniformly asymptotically regular mappings on a star-shaped set in a separated locally convex space. We prove that those results are extensible to the class of (left) reversible topological semigroups S and can be improved significantly by dropping some conditions, and study some related results in connection with amenability of AP(S) and WAP(S).

In this paper, we study Hilbert numbers of embedding of Sobolev space of mixed smoothness (H^{s,r}_{textrm{mix}}({{mathbb {T}}}^d)) on the torus ({{mathbb {T}}}^d) into (L_infty ({{mathbb {T}}}^d)) and the Wiener class (mathcal {A}({{mathbb {T}}}^d)). We obtain the exact asymptotic order of Hilbert numbers of these embeddings and the asymptotic constant for the embedding into (mathcal {A}({{mathbb {T}}}^d)). We also obtain the asymptotic constant of Hilbert numbers of embedding of Gaussian weighted Sobobev space (H^s({{mathbb {R}}}^d,gamma )) into (mathcal {A}({{mathbb {R}}}^d,gamma )) which is a counterpart of the Wiener class (mathcal {A}({{mathbb {T}}}^d)).