Numerical range of tensor product of operators in semi-Hilbert spaces

Abstract

Let A and B be two positive bounded linear operators acting on the complex Hilbert spaces H and K, respectively. In this paper, we study the (A⊗B)-numerical range WA⊗B(T⊗S) of the tensor product T⊗S for two bounded linear operators T and S on H and K, respectively. In the context of this work, we demonstrate that if either T is A-hyponormal or S is B-hyponormal, then WA⊗B(T⊗S)¯=coWA(T)¯⋅WB(S)¯,where WA(T) and WB(S) denote the A-numerical range of T and the B-numerical range of S, respectively. Here, co(⋅) and the over-line denote the convex hull and the closure, respectively. Moreover, we provide some (A⊗B)-numerical radius inequalities. © 2025 The Authors

Keywords

• bounded linear operators
• convex hull
• hyponormal operators
• numerical radius inequalities
• numerical range
• semi-hilbert spaces
• tensor product