Using [1], which is a local generalization of Gelfand's result for powerbounded operators, we first give a quantitative local extension of Lumer-Philips' result that states conditions under which a quasi-nilpotent dissipative operator vanishes. Secondly, we also improve Lumer-Phillips' theorem on strongly continuous semigroups of contraction operators.

In 1941, I. Gelfand proved that if a is a doubly power-bounded element of a Banach algebra A such that Sp(a) = 1, then a = 1. In [4], this result has been extended locally to a larger class of operators. In this note, we first give some quantitative local extensions of Gelfand-Hille’s results. Secondly, using the Bernstein inequality for multivariable functions, we give short and elementary proofs of two extensions of Gelfand’s theorem for m commuting bounded operators, $T_1,...,T_m$, on a Banach space X.

We give several conditions for (A,m)-expansive operators to have the single-valued extension property. We also provide some spectral properties of such operators. Moreover, we prove that the A-covariance of any (A,2)-expansive operator T ∈ ℒ(ℋ ) is positive, showing that there exists a reducing subspace ℳ on which T is (A,2)-isometric. In addition, we verify that Weyl's theorem holds for an operator T ∈ ℒ(ℋ ) provided that T is (T*T,2)-expansive. We next study (A,m)-isometric operators as a special...

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.

We characterize the bounded linear operators T satisfying generalized a-Browder's theorem, or generalized a-Weyl's theorem, by means of localized SVEP, as well as by means of the quasi-nilpotent part H₀(λI - T) as λ belongs to certain sets of ℂ. In the last part we give a general framework in which generalized a-Weyl's theorem follows for several classes of operators.

An operator T acting on a Banach space X has property (gw) if ${\sigma }_a\left(T\right)\setminus {\sigma }_{SBF₊¯}\left(T\right)=E\left(T\right)$, where ${\sigma }_a\left(T\right)$ is the approximate point spectrum of T, ${\sigma }_{SBF₊¯}\left(T\right)$ is the upper semi-B-Weyl spectrum of T and E(T) is the set of all isolated eigenvalues of T. We introduce and study two new spectral properties (v) and (gv) in connection with Weyl type theorems. Among other results, we show that T satisfies (gv) if and only if T satisfies (gw) and $\sigma \left(T\right)={\sigma }_a\left(T\right)$.

Let $B\left(X\right)$ be the algebra of all bounded linear operators in a complex Banach space $X$. We consider operators $T_1,T_2\in B\left(X\right)$ satisfying the relation ${\sigma }_{T_1}\left(x\right)={\sigma }_{T_2}\left(x\right)$ for any vector $x\in X$, where ${\sigma }_T\left(x\right)$ denotes the local spectrum of $T\in B\left(X\right)$ at the point $x\in X$. We say then that $T_1$ and $T_2$ have the same local spectra. We prove that then, under some conditions, $T_1-T_2$ is a quasinilpotent operator, that is $\parallel {(T_1-T_2)}^n{\parallel }^{1/n}\to 0$ as $n\to \infty$. Without these conditions, we describe the operators with the same local spectra only in some particular cases.

We extend and generalize some results in local spectral theory for upper triangular operator matrices to upper triangular operator matrices with unbounded entries. Furthermore, we investigate the boundedness of the local resolvent function for operator matrices.

There are a number of spectra studied in the literature which do not fit into the axiomatic theory of Żelazko. This paper is an attempt to give an axiomatic theory for these spectra, which, apart from the usual types of spectra, like one-sided, approximate point or essential spectra, include also the local spectra, the Browder spectrum and various versions of the Apostol spectrum (studied under various names, e.g. regular, semiregular or essentially semiregular).

We study the local spectral properties of both unilateral and bilateral weighted shift operators.

We study a new class of bounded linear operators which strictly contains the class of bounded linear operators with the decomposition property (δ) or the weak spectral decomposition property (weak-SDP). We treat general local spectral properties for operators in this class and compare them with the case of operators with (δ).

In this paper we study some properties of a totally $*$-paranormal operator (defined below) on Hilbert space. In particular, we characterize a totally $*$-paranormal operator. Also we show that Weyl’s theorem and the spectral mapping theorem hold for totally $*$-paranormal operators through the local spectral theory. Finally, we show that every totally $*$-paranormal operator satisfies an analogue of the single valued extension property for $W^2(D,H)$ and some of totally $*$-paranormal operators have scalar extensions....

We study some properties of w-hyponormal operators. In particular we show that some w-hyponormal operators are subscalar. Also we state some theorems on invariant subspaces of w-hyponormal operators.