Norm estimates for resolvents of linear operators in a Banach space and spectral variations

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Abstract

Let Pt (a ≤ t ≤ b) be a function whose values are projections in a Banach space. The paper is devoted to bounded linear operators A admitting the representation A = ∫ a b ϕ(t)dPt + V, where ϕ(t) is a scalar function and V is a compact quasi-nilpotent operator such that PtV Pt = V Pt (a ≤ t ≤ b). We obtain norm estimates for the resolvent of A and a bound for the spectral variation of A. In addition, the representation for the resolvents of the considered operators is established via multiplicative operator integrals. That representation can be considered as a generalization of the representation for the resolvent of a normal operator in a Hilbert space. It is also shown that the considered operators are Kreiss-bounded. Applications to integral operators in Lp are also discussed. In particular, bounds for the upper and lower spectral radius of integral operators are derived.

Original languageEnglish
Pages (from-to)113-139
Number of pages27
JournalAdvances in Operator Theory
Volume4
Issue number1
DOIs
StatePublished - 1 Jan 2019

Keywords

  • Banach space
  • Integral operator
  • Invariant chain of projections
  • Resolvent
  • Spectral variation

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