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A New Seminorm for d -Tuples of A -Bounded Operators and Their Applications

Author

Listed:
  • Najla Altwaijry

    (Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
    These authors contributed equally to this work.)

  • Kais Feki

    (Faculty of Economic Sciences and Management of Mahdia, University of Monastir, Mahdia 5111, Tunisia
    Laboratory Physics-Mathematics and Applications (LR/13/ES-22), Faculty of Sciences of Sfax, University of Sfax, Sfax 3018, Tunisia
    These authors contributed equally to this work.)

  • Nicuşor Minculete

    (Department of Mathematics and Computer Science, Transilvania University of Brasov, 500091 Brasov, Romania
    These authors contributed equally to this work.)

Abstract

The aim of this paper was to introduce and investigate a new seminorm of operator tuples on a complex Hilbert space H when an additional semi-inner product structure defined by a positive (semi-definite) operator A on H is considered. We prove the equality between this new seminorm and the well-known A -joint seminorm in the case of A -doubly-commuting tuples of A -hyponormal operators. This study is an extension of a well-known result in [Results Math 75, 93(2020)] and allows us to show that the following equalities r A ( T ) = ω A ( T ) = ∥ T ∥ A hold for every A -doubly-commuting d -tuple of A -hyponormal operators T = ( T 1 , … , T d ) . Here, r A ( T ) , ∥ T ∥ A , and ω A ( T ) denote the A -joint spectral radius, the A -joint operator seminorm, and the A -joint numerical radius of T , respectively.

Suggested Citation

  • Najla Altwaijry & Kais Feki & Nicuşor Minculete, 2023. "A New Seminorm for d -Tuples of A -Bounded Operators and Their Applications," Mathematics, MDPI, vol. 11(3), pages 1-21, January.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:3:p:685-:d:1050424
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