Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
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ON LOCAL SPECTRAL PROPERTIES OF RIESZ OPERATORS

  • Received : 2022.02.21
  • Accepted : 2023.01.06
  • Published : 2023.03.30
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Abstract

In this paper we show that if T ∈ L(X) and S ∈ L(X) is a Riesz operator commuting with T and XS(F) ∈ Lat(S), where F = {0} or F ⊆ ℂ ⧵ {0} is closed then T|XS(F) and T|XT(F) + S|XS(F) share the local spectral properties such as SVEP, Dunford's property (C), Bishop's property (𝛽), decomopsition property (𝛿) and decomposability. As a corollary, if T ∈ L(X) and Q ∈ L(X) is a quasinilpotent operator commuting with T then T is Riesz if and only if T + Q is Riesz. We also study some spectral properties of Riesz operators acting on Banach spaces. We show that if T, S ∈ L(X) such that TS = ST, and Y ∈ Lat(S) is a hyperinvarinat subspace of X for which 𝜎(S|Y ) = {0} then 𝜎*(T|Y + S|Y ) = 𝜎*(T|Y ) for 𝜎* ∈ {𝜎, 𝜎loc, 𝜎sur, 𝜎ap}. Finally, we show that if T ∈ L(X) and S ∈ L(Y ) on the Banach spaces X and Y and T is similar to S then T is Riesz if and only if S is Riesz.

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