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 = (T₁ ,…, 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.

Mathematics Subject Classification (2020):

• 47A12
• 47A63
• 47B65
• 46C05
• 47A10

Key words:

• Positive operator
• jointly A-normaloid operators
• joint A-spectral radius
• joint A-numerical radius
• joint A-seminorms
• joint approximate A-spectrum