On tuples of commuting operators in positive semidefinite inner product spaces

This paper is concerned with the study of certain properties of operator tuples on a complex Hilbert space H when a semi-inner product induced by a positive operator A on H is considered. In particular, we show that r A ( T ) ≤ ω A ( T ) for every commuting operator tuple T = ( T 1 , … , T d ) such that each T k admits an A-adjoint operator, where r A ( T ) and ω A ( T ) denote respectively the A-joint spectral radius and the A-joint numerical radius of T. This study allows to establish that r A ( T ) = ω A ( T ) = ‖ T ‖ A for every A-normal commuting tuple of operators T, where ‖ T ‖ A is denoted to be the A-joint operator seminorm of T. In addition, the A-joint spectral radius of an ( A , m ) -isometric tuple of commuting operators is studied.