Let μ be a finite positive Borel measure on [0,1). Let ${\mathscr{H}}_{\mu }={\left({\mu }_{n,k}\right)}_{n,k\ge 0}$ be the Hankel matrix with entries ${\mu }_{n,k}={\int }_{[0,1)}{t}^{n+k}d\mu \left(t\right)$. The matrix ${}_{\mu }$ induces formally an operator on the space of all analytic functions in the unit disc by the fomula ${\mathscr{H}}_{\mu }\left(f\right)\left(z\right)={\sum }_{n=0}^{\infty }i\left({\sum }_{k=0}^{\infty }{\mu }_{n,k}{a}_k\right)zⁿ$, z ∈ , where $f\left(z\right)={\sum }_{n=0}^{\infty }aₙzⁿ$ is an analytic function in . We characterize those positive Borel measures on [0,1) such that ${\mathscr{H}}_{\mu }\left(f\right)\left(z\right)={\int }_{[0,1)}f\left(t\right)/(1-tz)d\mu \left(t\right)$ for all f in the Hardy space H¹, and among them we describe those for which ${\mathscr{H}}_{\mu }$ is bounded and compact on H¹. We also study the analogous problem for the Bergman space A².

Let φ be a Jordan automorphism of an algebra . The situation when an element a ∈ satisfies $1/2(\phi \left(a\right)+{\phi }^{-1}\left(a\right))=a$ is considered. The result which we obtain implies the Kleinecke-Shirokov theorem and Jacobson’s lemma.

This article is dedicated to localization of the principal eigenvalue (PE) of the Stokes operator acting on solenoidal vector fields that vanish outside a large random domain modeling the pore space in a cubic block of porous material with disordered micro-structure. Its main result is an asymptotically deterministic lower bound for the PE of the sum of a low compressibility approximation to the Stokes operator and a small scaled random potential term, which is applied to produce a similar bound...

Motivated by the concept of separation between propositions in quantum logic, we introduce the so-called separation metric or Santos metric on the space of all projections in a Hilbert space. We show that the resulting metric space has only "nice" surjective isometries. On the nontrivial projections they are all unitarily or antiunitarily equivalent to the identity or to taking the orthogonal complement. We relate this result to Wigner's classical theorem on the form of quantum mechanical symmetry...

We present a change of variable method and use it to prove the equivalence to bundle shifts for certain analytic Toeplitz operators on the Banach spaces $H^p\left(G\right)(1\le p<\infty )$. In Section 2 we see this approach applied in the analysis of essential spectra. Some partial results were obtained in [9] in the Hilbert space case.

The aim of the paper is to present some initial results about a possible generalization of moment sequences to a so-called q-calculus. A characterization of such a q-analogue in terms of appropriate positivity conditions is also investigated. Using the result due to Maserick and Szafraniec, we adapt a classical description of Hausdorff moment sequences in terms of positive definiteness and complete monotonicity to the q-situation. This makes a link between q-positive definiteness and q-complete...

Let $\mathscr{H}$ be a separable infinite dimensional complex Hilbert space, and let $ℒ\left(\mathscr{H}\right)$ denote the algebra of all bounded linear operators on $\mathscr{H}$ into itself. Let $A=(A_1,A_2,\cdots ,A_n)$, $B=(B_1,B_2,\cdots ,B_n)$ be $n$-tuples of operators in $ℒ\left(\mathscr{H}\right)$; we define the elementary operators ${\Delta }_{A,B}ℒ\left(\mathscr{H}\right)\mapsto ℒ\left(\mathscr{H}\right)$ by ${\Delta }_{A,B}\left(X\right)={\sum }_{i=1}^n{A}_{i}X{B}_i-X.$ In this paper, we characterize the class of pairs of operators $A,B\in ℒ\left(\mathscr{H}\right)$ satisfying Putnam-Fuglede’s property, i.e, the class of pairs of operators $A,B\in ℒ\left(\mathscr{H}\right)$ such that ${\sum }_{i=1}^n{B}_{i}T{A}_i=T$ implies ${\sum }_{i=1}^n{A}_i^{*}T{B}_i^{*}=T$ for all $T\in {𝒞}_1\left(\mathscr{H}\right)$ (trace class operators). The main result is the equivalence between this property and the fact that...