On (n,k)-quasiparanormal operators

We introduce the class of n-quasi-m-isometric operators on Hilbert space. This generalizes the class of m-isometric operators on Hilbert space introduced by Agler and Stankus. An operator T ∈ B(H) is said to be n-quasi-m-isometric if T∗n ∑ k=0m(−1)k m kT∗m−kTm−k Tn = 0. In this paper 2 × 2 matrix representation of a n-quasi-m-isometric operator is given. Using this representation we establish some basic properties of this class of operators.

On n-quasi-m-isometric operators

Continuity of the spectrum on a class of upper triangular operator matrices

Let T be a bounded linear operator on a complex Hilbert space ℌ. T is called a class A operator if |T|2 ≤ |2|, and (p, k)-quasihyponormal if T* k ((T*T) p − (TT*) p )T k ≥ 0. In this paper, we characterize the quasinilpotent part ℌ 0(T − λ) = {x ∈ ℌ: $ \mathop {\lim }\limits_{n \to \infty } \left\| {(T - \lambda )^n x} \right\|^{\tfrac{1} {n}} = 0 $} of a class A or (p, k)-quasihyponormal operator T.

Quasinilpotent Part of class A or (p, k)-quasihyponormal Operators

‎An operator $T\in B(H)$ is called quasi $*$-paranormal if‎ ‎$||T^{*}Tx||^{2}\leq||T^{3}x|||Tx||$ for all $x\in H$‎. ‎If $\mu$ is‎ ‎an isolated point of the spectrum of $T$‎, ‎then the Riesz idempotent‎ ‎$E$ of $T$ with respect to $\mu$ is defined by‎ ‎$$E:= {1\over {2\pi i}}\int_{\partial D}(\mu I‎ - ‎T)^{-1}d\mu,$$‎ ‎where $D$ is a closed disk centered at $\mu$ which contains no other‎ ‎points of the spectrum of $T$‎. ‎Stampfli [Trans‎. ‎Amer‎. ‎Math‎. ‎Soc.‎, ‎117 (1965)‎, ‎469-476]‎, ‎showed that if $T$ satisfies the growth‎ ‎condition $G_{1}$‎, ‎then $E$ is self-adjoint and $E(H)=N(T-\mu)$‎. ‎Recently‎, ‎Uchiyama and Tanahashi [Integral Equations and Operator‎ ‎Theory‎, ‎55 (2006)‎, ‎145-151] obtained Stampfli's result for‎ ‎paranormal operators‎. ‎In general even though $T$ is a paranormal‎ ‎operator‎, ‎the Riesz idempotent $E$ of $T$ with respect to $\mu \in ‎{\rm iso\,}\sigma(T)$ is not necessary self-adjoint‎. ‎In this paper‎ ‎$2\times 2$ matrix representation of a quasi $*$-paranormal operator‎ ‎is given‎. ‎Using this representation we show that if $E$ is the Riesz‎ ‎idempotent for a nonzero isolated point $\lambda_{0}$ of the‎ ‎spectrum of a quasi $*$-paranormal operator $T$‎, ‎then $E$ is‎ ‎self-adjoint if and only if the null space of $T-\lambda_{0}$‎ ‎satisfies $N(T-\lambda_{0})\subseteq‎ ‎N(T^{*}-\overline{\lambda_{0}})$‎. ‎Other related results are also‎ ‎given‎.

/pdf/on-quasi-paranormal-operators-iadvcknr1l.pdf

On quasi $*$-paranormal operators

On operators satisfying T*|T2|T ≥ T*|T|2T

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On operators satisfying T|T2|T ≥ T|T|2T