Spectral radius of semi-Hilbertian space operators and its applications

Abstract

In this paper, we aim to introduce the notion of the spectral radius of bounded linear operators acting on a complex Hilbert space \(\mathcal {H}\), which are bounded with respect to the seminorm induced by a positive operator A on \(\mathcal {H}\). Mainly, we show that \(r_A(T)\le \omega _A(T)\) for every A-bounded operator T, where \(r_A(T)\) and \(\omega _A(T)\) denote respectively the A-spectral radius and the A-numerical radius of T. This allows to establish that \(r_A(T)=\omega _A(T)=\Vert T\Vert _A\) for every A-normaloid operator T, where \(\Vert T\Vert _A\) is denoted to be the A-operator seminorm of T. Moreover, some characterizations of A-normaloid and A-spectraloid operators are given.

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Correspondence to Kais Feki.

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Communicated by Takeaki Yamazaki.

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Feki, K. Spectral radius of semi-Hilbertian space operators and its applications. Ann. Funct. Anal. 11, 929–946 (2020). https://doi.org/10.1007/s43034-020-00064-y

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Keywords

  • Positive operator
  • Semi-inner product
  • Spectral radius
  • Numerical radius
  • Normaloid operator
  • Spectraloid operator

Mathematics Subject Classification

  • 46C05
  • 47A12
  • 47B65
  • 47B15
  • 47B20