Advances in Operator Theory最新文献

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)).

Let (L=-{Delta }_{{mathbb {G}} }+V) be a Schrödinger operator on the stratified Lie group ({mathbb {G}},) where ({Delta }_{{mathbb {G}} }) is the sub-Laplacian and the nonnegative potential V belongs to the reverse Hölder class (B_{q}, qge {mathcal {Q}}/2,) in which ({mathcal {Q}}) is the homogeneous dimension of ({mathbb {G}}.) In this article, we firstly study the fractional heat semigroups ({e^{-tL^{alpha }}}_{t>0}) with (alpha >0) associated with L. Subsequently, the regularities of the fractional heat semigroup is estimated with the use of the subordinative formula. Furthermore, in terms of application, we establish the (BMO_{L}^{gamma }({mathbb {G}}))-boundedness of the maximal function and the Littlewood–Paley ({mathfrak {g}})-functions related with the Schrödinger operator L by T1 theorem, respectively.

An operator (Tin {{mathcal {L}}}({{mathcal {H}}})) is called conjugation-quasinormal if there is a conjugation C on an infinite-dimensional complex Hilbert space ({{mathcal {H}}}) such that ([CT,T^{*}T]=0) where ([R,S]:=RS-SR). In particular, if such a conjugation C is specified, an operator T is said to be C-quasinormal. In this paper, we study various properties of conjugation-quasinormal operators. Especially, we provide several characterizations of conjugation-quasinormal operators. Moreover, if T is a partial isometry, we show that T is C-hyponormal if and only if T is C-quasinormal. Finally, we consider conjugation-operator transforms.

We consider random linear continuous operators (Omega rightarrow {mathcal {L}}({mathcal {X}}, {mathcal {X}})) on a Banach space ({mathcal {X}}.) For example, such random operators may be random quantum channels. The Law of Large Numbers is known when ({mathcal {X}}) is a Hilbert space, in the form of the usual Law of Large Numbers for random operators, and in some other particular cases. Instead of the sum of i.i.d. variables, there may be considered the composition of random semigroups (e^{A_i t/n}.) We obtain the Strong Law of Large Numbers in Strong Operator Topology for random semigroups of bounded linear operators on a uniformly smooth Banach space. We also develop another approach giving the SLLN in Weak Operator Topology for all Banach spaces.

For (phi ) and (psi ) inner functions that fix the origin on the open unit disk (mathbb {D}) in the complex plane, we consider the question of whether the associated linear operator (C_{phi }^*C_{psi }) can be compact or finite-rank on (H^2(mathbb {D})). We show that (C_{phi }^*C_{psi }) cannot be rank-one when (phi ) has purely atomic Aleksandrov–Clark measure and (psi ) extends continuously to the boundary of (mathbb {D}). When (phi ) and (psi ) are finite Blaschke products each with two distinct factors, we show (C_{phi }^*C_{psi }) cannot be compact. Finally, following work of Cowen and MacCluer, we characterize the range of (C_{phi }^*) when (phi ) is a Blaschke product.

We prove a Lie–Trotter type formula for q-exponential operators:

where E is a Banach space, U, V bounded linear operators on E and (qin mathbb {R}), (qne 1).

New characterizations of left and right generalized Drazin inverses are presented in Banach algebra using tripotents. We explore Cline’s formula for those inverses. Also, we find the conditions under which the product of two left (or right) generalized Drazin invertible elements has the appropriate one-sided generalized Drazin inverse. Consequently, we study when the sum of two zero product elements has a one-sided generalized Drazin inverse. Perturbation results for the left (or right) generalized Drazin inverse are also investigated. Applying our results, we obtain new properties of the left (or right) Drazin inverse and recover known results about the generalized Drazin inverse.

In this work, we introduce and study generalized entropy numbers for sets and operators acting on Banach spaces. The classical notion of Hausdorff entropy numbers becomes a particular case of the given definition. We also provide several other examples of generalized entropy numbers for sets and operators. We prove several properties for the general case.

In this review paper we study non-compact operators and embeddings between function spaces, highlighting interesting phenomena and the significance of Bernstein numbers. In particular, we demonstrate that for non-compact maps the usual s-numbers (e.g., approximation, Kolmogorov, and entropy numbers) fail to reveal finer structural properties, and one must instead consider concepts such as strict singularity and Bernstein numbers.