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

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. [ABSTRACT FROM AUTHOR]