Analysis of Non-normal Operators via Aluthge Transformation

Abstract

Let T be a bounded linear operator on a complex Hilbert space H. In this paper, we show that T has Bishop's property (β) if and only if its Aluthge transformationT has property (β). As applications, we can obtain the following results. Every w-hyponormal operator has property (β). Quasi-similar w-hyponormal operators have equal spectra and equal essential spectra. Moreover, in the last section, we consider Chō's problem that whether the semi-hyponormality of T implies the spectral mapping theorem Re σ(T ) = σ(Re T ) or not. (2000). Primary 47B20; Secondary 47A10, 47B40. Mathematics Subject Classification

Spectrum of Class wF(p,r,q) Operators

“…This result is a generalization of [18]. Lemma 3.4 and the relations between T and its transformation T(p,r) are important:…”

“…If T has SVEP, then for each λ ∈ Ꮿ, λ − T is invertible if and only if it is surjective (cf. [29,18]). When T has Bishop's property (β) at each λ ∈ Ꮿ, simply say that T has property (β).…”

“…Lemma 3.4 (see [18]). Let G be open subset of complex plane Ꮿ and let f n ∈ H(G) be functions such that μ f n (μ) → 0 uniformly on every compact subset of G, then f n (μ) → 0 uniformly on every compact subset of G.…”

“…We cite [8,[14][15][16][17][18][19][20][21][22]. In this paper, we will discuss several spectral properties of class wF(p,r, q) for p > 0, r > 0, p + r ≤ 1, and q ≥ 1.…”

SOME PROPERTIES OF CLASS A(k) OPERATORS AND THEIR HYPONORMAL TRANSFORMS

“…To prove that a class A(k) operator T has property (β) we need the following Theorem which is a modified form of [14,Lemma 2.5]. Theorem R 7 [14].…”