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...

It is well known that the only proper non-trivial norm closed ideal in the algebra L(X) for $X={\ell }_p$ (1 ≤ p < ∞) or X = c₀ is the ideal of compact operators. The next natural question is to describe all closed ideals of $L\left({\ell }_p\oplus {\ell }_q\right)$ for 1 ≤ p,q < ∞, p ≠ q, or equivalently, the closed ideals in $L({\ell }_p,{\ell }_q)$ for p < q. This paper shows that for 1 < p < 2 < q < ∞ there are at least four distinct proper closed ideals in $L({\ell }_p,{\ell }_q)$, including one that has not been studied before. The proofs use various methods from Banach...

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.