Local properties of powers of operators (original) (raw)

Generalized Browder's and Weyl's theorems for Banach space operators

Journal of Mathematical Analysis and Applications, 2007

We find necessary and sufficient conditions for a Banach space operator T to satisfy the generalized Browder's theorem, and we obtain new necessary and sufficient conditions to guarantee that the spectral mapping theorem holds for the B-Weyl spectrum and for polynomials in T . We also prove that the spectral mapping theorem holds for the B-Browder spectrum and for analytic functions on an open neighborhood of σ(T ). As applications, we show that if T is algebraically M -hyponormal, or if T is algebraically paranormal, then the generalized Weyl's theorem holds for f (T ), where f ∈ H((T )), the space of functions analytic on an open neighborhood of σ(T ). We also show that if T is reduced by each of its eigenspaces, then the generalized Browder's theorem holds for f (T ), for each f ∈ H(σ(T )).

On analytic families of operators

Israel Journal of Mathematics, 1969

The classical Riesz-Thorin interpolation theorem [6] was extended by Hirschman [2] and Stein [5] to analytic families of operators. We recall the notions: Let F(z), z = x+iy, be analytic in 0< Re z< 1 and continuous in 0 =< Re z _< 1. F(z) is said to be of admissible growth iff Sup log iF(x + iY) I < Ae~Iyl where a < 7z. O<_x~l The significance of this notion is in the following lemma due to Hirschman [2]: LEMMA. lf F(z) is of admissible growth and ifloglF(it) l ~ ao(t), log IF(i+ it) I <= a~(t) then log I F(0) ]_<f_% Po(0, t)ao(t)dt + f 2~ P~(O, t)a~(t)dt where P~(O, t) are the values of the Poisson kernel Jor the strip, on Rez = 0, Rez = 1. We next define analytic families of linear operators: Let (M,/~) (N, v) be two measure spaces. Let {~} be a family of linear operators indexed by z, 0 ~ Re z ~ 1 so that for each z, Tz is a mapping of simple functions on M to measurable functions on N. {T~} is called an analytic family iff for any measurable set E of M of finite measure, for almost every y 6 N, the function qSr(z) = T~(X~)(y) is analytic in 0 < Re z < 1, continuous in 0 ~ Re z __< 1. The analytic family is of admissible growth iff for almost every y ~ N, ~by(z) is of admissible growth. We finally recall the notion of L(p, q) spaces. An exposition of these spaces can be found in Hunt [3]. Let f be a complex valued measurable function defined on a ~-finite measure space (M,/~). # is assumed to be non-negative. We assume that f is finite valued a.e., and denoting Ey = {x/If(x)] > Y}, 2r(y) = /~(Ey), we assume also that for some y > 0, 2;(y)< oo. We define f*(t) = Inf{y > Oily(y) < t}.

On the Phragmén-Lindelöf principle for entire power series on a Banach algebra

Communications in Mathematics and Applications, 2015

Many authors have managed to successfully extend the classical theory of analytic functions to functions defined on more abstract spaces (see for example [3, 8, 9]). The purpose of this paper is to provide a bit in this direction. In this paper an extension of the Phragmen-Lindelof principles of the classical theory to the functions expressed as power series on a Banach algebra not necessarily commutative, of course, involving concepts such as harmonic and subharmonic, is introduced.

Generalised Weyl's Theorem for A Class of Operators Satisfying A Norm Condition II

Mathematical Proceedings of the Royal Irish Academy, 2006

For a Banach space operator T ∈ B(X), it is proved that if either T is an algebraically, totally hereditarily normaloid operator and the Banach space X is separable, or T satisfies the property that its quasinilpotent part H 0 (T − λ) = (T − λ) −p (0) for all complex numbers λ and some integer p ≥ 1, then f (T) satisfies generalized Weyl's theorem for every non-constant function f that is analytic on an open neighborhood of σ(T).

Some characterizations of operators satisfying a-Browder's theorem

Journal of Mathematical Analysis and Applications, 2005

We characterize the bounded linear operators T defined on Banach spaces satisfying a-Browder's theorem, or a-Weyl's theorem, by means of the discontinuity of some maps defined on certain subsets of C. Several other characterizations are given in terms of localized SVEP, as well as by means of the quasi-nilpotent part, the hyper-kernel or the analytic core of λI − T . 531 denote the class of all upper semi-Fredholm operators, and let Φ − (X) := T ∈ L(X): β(T ) < ∞ denote the class of all lower semi-Fredholm operators. The class of all semi-Fredholm operators is defined by

Functional Analysis, Operator Theory and Applications

We give necessary and sufficient conditions for the sequence of operators j A on a Hilbert space to have a bounded  H -calculus on a vertical strip symmetric to the imaginary axis. From this, a characterization of group generators on Hilbert spaces is obtained yielding recent results of Liu and Zwart as corollaries by Haase [9].