On the M 2, A,A -numerical range and the M 2, A,A -maximal numerical range of the basic elementary operator M 2, B,C
Zakaria Taki · Linear and Multilinear Algebra · 2024
Let A be a positive bounded operator acting on a complex Hilbert space H. For two bounded operators B and C on H, we denote by M2,B,C the basic elementary operator on the class of Hilbert–Schmidt operators C2(H), i.e. M2,B,C(X)=BXC for all X∈C2(H). In this paper, we investigate the A-numerical range WA(M2,B,(C♯A)∗) and the A-maximal numerical range WmaxA(M2,B,(C♯A)∗), where A=M2,A,A, C♯A is the reduced solution of the equation AX=C∗A and C∗ is the adjoint of C. Within this framework, we show, under some A-hyponormality conditions, the following two equalities: WA(M2,B,(C♯A)∗)¯=co(WA(B)¯⋅WA(C)¯)and WmaxA(M2,B,(C♯A)∗)=co(WmaxA(B)⋅WmaxA(C)),where WA(⋅), WmaxA(⋅) and co(⋅) denote respectively the A-numerical range, the A-maximal numerical range and the convex hull. Here, the bar stands for the closure. The first equality allows us to establish that ‖M2,B,(C♯A)∗‖A=‖B‖A‖C‖A,where ‖⋅‖A and ‖⋅‖A designate the A-operator seminorm and the A-operator seminorm, respectively.