On A-normaloid d-tuples of operators and related questions

Abstract

Let ß _A (ℍ) denote the algebra of bounded linear operators on a complex Hilbert space ℍ that admit A-adjoint operators, where A is a non-zero positive semi-definite operator on ℍ. A commuting operator tuple T = (T ₁ ,…, T _d ) ∈ ß _A (ℍ) ^d is called jointly A-normaloid if r_A (T) = ∥T∥ _A , where r_A (T) and ∥T∥ _A represent the joint A-spectral radius and the joint operator A-seminorm of T, respectively. This paper aims to investigate this new class of operators and provides several examples. Furthermore, a characterization of A-normaloidity is established. Additionally, the joint Euclidean A-seminorm of a d-tuple of A-bounded operators T, denoted by , is examined. Specifically, for all positive integers n, we prove that the following equivalence holds for any commuting operator tuple . Here . Finally, several related questions are explored.