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

Najla Altwaijry (), Kais Feki and Nicuşor Minculete
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Najla Altwaijry: Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
Kais Feki: Faculty of Economic Sciences and Management of Mahdia, University of Monastir, Mahdia 5111, Tunisia
Nicuşor Minculete: Department of Mathematics and Computer Science, Transilvania University of Brasov, 500091 Brasov, Romania

Mathematics, 2023, vol. 11, issue 3, 1-21

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.

Keywords: positive operator; A -adjoint operator; A -joint operator seminorm; A -hyponormal operator; A -joint spectral radius; A -joint numerical radius (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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