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

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 rA(T)=ωA(T)=∥T∥A hold for every A-doubly-commuting d-tuple of A-hyponormal operators T=(T1,…,Td). Here, rA(T),∥T∥A, and ωA(T) denote the A-joint spectral radius, the A-joint operator seminorm, and the A-joint numerical radius of T, respectively.