RACSAM https://doi.org/10.1007/s13398-018-0555-3 ORIGINAL PAPER 1,2 1 Junli Shen · Guoxing Ji Received: 21 January 2018 / Accepted: 24 May 2018 © Springer-Verlag Italia S.r.l., part of Springer Nature 2018 Abstract In this paper we study the properties of property (t ), which is introduced by Rashid. We investigate the property (t ) in connection with Weyl type theorems, and establish sufﬁcient and necessary conditions for which property (t ) holds. Especially, we obtain the equivalence of a-Weyl’s theorem and property (t) without the condition that T is a-polaroid, which improves a corresponding result of Rashid (Mediterr J Math 11:1–16, 2014). We also study the stability of property (t ) under perturbations by nilpotent operators, by ﬁnite rank operators, by quasi-nilpotent operators and by Riesz operators commuting with T . Keywords Property (t) · Perturbation · Weyl type theorem Mathematics Subject Classiﬁcation Primary 47B20; Secondary 47A10 1 Introduction and basic results Throughout this paper, we denote X an inﬁnite dimensional complex Banach space and L(X ) the algebra of all bounded linear operators on X.For T ∈ L(X ), we denote the null space, the range, the spectrum, the point spectrum, the approximate point spectrum, the surjective spectrum, the isolated
Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas – Springer Journals
Published: Jun 5, 2018
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