We prove that for normal operators $N_1,N_2\in ℒ\left(\mathscr{H}\right),$ the generalized commutator $[N_1,N_2;X]$ approaches zero when $[N_1,N_2;[N_1,N_2;X]]$ tends to zero in the norm of the Schatten-von Neumann class ${𝒞}_p$ with $p>1$ and $X$ varies in a bounded set of such a class.

We show that every class A operator has a scalar extension. In particular, such operators with rich spectra have nontrivial invariant subspaces. Also we give some spectral properties of the scalar extension of a class A operator. Finally, we show that every class A operator is nonhypertransitive.

A review of known decompositions of pairs of isometries is given. A new, finer decomposition and its properties are presented.

The purpose of the paper is to introduce and study a new class of operators on semi-Hilbertian spaces, i.e. spaces generated by positive semi-definite sesquilinear forms. Let $\mathscr{H}$ be a Hilbert space and let $A$ be a positive bounded operator on $\mathscr{H}$. The semi-inner product ${\langle h\mid k\rangle }_A:=\langle Ah\mid k\rangle$, $h,k\in \mathscr{H}$, induces a semi-norm ${\parallel ·\parallel }_A$. This makes $\mathscr{H}$ into a semi-Hilbertian space. An operator $T\in {ℬ}_A\left(\mathscr{H}\right)$ is said to be $(n,m)$-$A$-normal if $[T^n,{\left(T^{{♯}_A}\right)}^m]:=T^n{\left(T^{{♯}_A}\right)}^m-{\left(T^{{♯}_A}\right)}^m{T}^n=0$ for some positive integers $n$ and $m$.

Let T be a bounded linear operator on a complex Hilbert space . For positive integers n and k, an operator T is called (n,k)-quasiparanormal if $\left|\right|T^{1+n}\left(T^{k}x\right){\left|\right|}^{1/(1+n)}\left|\right|T^k{x\left|\right|}^{n/(1+n)}\ge \left|\right|T\left(T^{k}x\right)\left|\right|$ for x ∈ . The class of (n,k)-quasiparanormal operators contains the classes of n-paranormal and k-quasiparanormal operators. We consider some properties of (n,k)-quasiparanormal operators: (1) inclusion relations and examples; (2) a matrix representation and SVEP (single valued extension property); (3) ascent and Bishop’s property (β); (4) quasinilpotent...

The Cauchy dual operator T’, given by $T{(T*T)}^{-1}$, provides a bounded unitary invariant for a closed left-invertible T. Hence, in some special cases, problems in the theory of unbounded Hilbert space operators can be related to similar problems in the theory of bounded Hilbert space operators. In particular, for a closed expansive T with finite-dimensional cokernel, it is shown that T admits the Cowen-Douglas decomposition if and only if T’ admits the Wold-type decomposition (see Definitions 1.1 and 1.2 below)....

We introduce and discuss a class of operators, to be referred to as operators close to isometries. The Bergman-type operators, 2-hyperexpansions, expansive p-isometries, and certain alternating hyperexpansions are main examples of such operators. We establish a few decomposition theorems for operators close to isometries. Applications are given to the theory of p-isometries and of hyperexpansive operators.