Properties ( BR ) and ( BgR ) for bounded linear operators
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Properties (BR) and (BgR) for bounded linear operators Anuradha Gupta1 · Ankit Kumar2 Received: 28 December 2018 / Accepted: 16 May 2019 © Springer-Verlag Italia S.r.l., part of Springer Nature 2019
Abstract In this paper we introduce the notion of property (B R) and property (Bg R) for bounded linear operators defined on an infinite-dimensional Banach space. These properties in connection with Weyl type theorems and in the frame of polaroid operators are investigated. Moreover, we study the stability of these properties under perturbations by commuting finite-dimensional, quasi-nilpotent, Riesz and algebraic operators. Keywords Property (B R) · Property (Bg R) · Weyl type theorems · SVEP · Polaroid operators · Perturbation theory Mathematics Subject Classification Primary 47A10 · 47A11; Secondary 47A53 · 47A55
1 Introduction and preliminaries Throughout this paper, X denotes an infinite-dimensional Banach space and let B(X ) be the Banach algebra of all bounded linear operators on X . For T ∈ B(X ), we denote the null space, range of T , adjoint of T by N (T ), T (X ) and T ∗ , respectively. Let σ (T ), σa (T ), iso σ (T ) and iso σa (T ) denote the spectrum of T , approximate point spectrum of T , isolated points of spectrum of T and isolated points of approximate point spectrum of T , respectively. Let α(T )= dim N (T ) and β(T )= codim T (X ) be the nullity of T and deficiency of T , respectively. An operator T ∈ B(X ) is called an upper semi-Fredholm operator if α(T ) < ∞ and T (X ) is closed. An operator T ∈ B(X ) is called a lower semi-Fredholm operator if β(T ) < ∞. The class of all upper semi-Fredholm operators (resp. lower semi-Fredholm operators) is denoted by φ+ (X ) (resp. φ− (X )). Let φ± (X ) := φ+ (X ) ∪ φ− (X ) be the class of all semi-Fredholm operators. For T ∈ φ± (X ) the index of T is defined by ind (T ) := α(T ) − β(T ). The class of all Fredholm operators is defined by φ(X ) := φ+ (X ) ∩ φ− (X ). The class of all upper semi-
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Ankit Kumar [email protected] Anuradha Gupta [email protected]
1
Department of Mathematics, Delhi College of Arts and Commerce, University of Delhi, Netaji Nagar, New Delhi 110023, India
2
Department of Mathematics, University of Delhi, New Delhi 110007, India
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A. Gupta, A. Kumar
Weyl operators (resp. lower semi-Weyl operators) is defined by W+ (X ) = {T ∈ φ+ (X ) : ind (T ) ≤ 0} (resp. W− (X ) = {T ∈ φ− (X ) : ind (T ) ≥ 0}). The set of all Weyl operators is defined by W (X ) := W+ (X ) ∩ W− (X ) = {T ∈ φ(X ) : ind (T ) = 0}. The upper semi-Weyl spectrum is defined by σuw (T ) := {λ ∈ C : λI − T ∈ / W+ (X )}. The lower semi-Weyl spectrum is defined by σlw (T ) := {λ ∈ C : λI − T ∈ / W− (X )} and the Weyl spectrum is defined by σw (T ) := {λ ∈ C : λI − T ∈ / W (X )}. For T ∈ B(X ), p(T ) be the ascent of T defined as the smallest non negative integer p such that N (T p ) = N (T p+1 ). If no such integer exists, then we set p(T ) = ∞. Similarly, q(T ) be the descent of T defined as the smallest non negative integer q such that T q (X ) = T q+1 (X ). If
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