We extend the Fuglede-Putnam theorem modulo the Hilbert-Schmidt class to almost normal operators with finite Hilbert-Schmidt modulus of quasi-triangularity.

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...

Some results on quasi-affinity for bounded operators are extended to unbounded ones and normal extensions of an unbounded operator are discussed in connection with quasi-affinity.

The question whether a hyponormal weighted shift with trace class self-commutator is the compression modulo the Hilbert-Schmidt class of a normal operator, remains open. It is natural to ask whether Putinar's construction through which he proved that hyponormal operators are subscalar operators provides the answer to the above question. We show that the normal extension provided by Putinar's theory does not lead to the extension that would provide a positive answer to the question.

Let $L\left(H\right)$ denote the algebra of all bounded linear operators on a separable infinite dimensional complex Hilbert space $H$ into itself. Given $A\in L\left(H\right)$, we define the elementary operator ${\Delta }_A:L\left(H\right)⟶L\left(H\right)$ by ${\Delta }_A\left(X\right)=AXA-X$. In this paper we study the class of operators $A\in L\left(H\right)$ which have the following property: $ATA=T$ implies $A{T}^{*}A=T^{*}$ for all trace class operators $T\in C_1\left(H\right)$. Such operators are termed generalized quasi-adjoints. The main result is the equivalence between this character and the fact that the ultraweak closure of the range of ${\Delta }_A$ is closed under taking...

Let T be a spherical 2-expansive m-tuple and let $T^{}$ denote its spherical Cauchy dual. If $T^{}$ is commuting then the inequality $\genfrac{}{}{0pt}{}{{\sum }_{\left|\beta \right|=k}{(\beta !)}^{-1}{\left(T^{}\right)}^{\beta }\left(T^{}\right){*}^{\beta }\le (k+m-1}{k){\sum }_{\left|\beta \right|=k}{(\beta !)}^{-1}\left(T^{}\right){*}^{\beta }{\left(T^{}\right)}^{\beta }}$ holds for every positive integer k. In case m = 1, this reveals the rather curious fact that all positive integral powers of the Cauchy dual of a 2-expansive (or concave) operator are hyponormal.

We survey recent developments in operator theory and moment problems, beginning with the study of quadratic hyponormality for unilateral weighted shifts, its connections with truncated Hamburger, Stieltjes, Hausdorff and Toeplitz moment problems, and the subsequent proof that polynomially hyponormal operators need not be subnormal. We present a general elementary approach to truncated moment problems in one or several real or complex variables, based on matrix positivity and extension. Together...