On the maximal numerical range of the bimultiplication M2,A,B
Abderrahim Baghdad, Chraibi Kaadoud Mohamed · Linear and Multilinear Algebra · 2021
Let B(H) denote the algebra of all bounded linear operators acting on a complex Hilbert space H. For A,B∈B(H), define the bimultiplication operator M2,A,B on the class of Hilbert–Schmidt operators by M2,A,B(X)=AXB. In this paper, we show that if B is normal, then co(W0(A)W0(B))⊆W0(M2,A,B),where co stands for the convex hull and W0(.) denotes the maximal numerical range. If in addition, A is hyponormal, this inclusion becomes an equality. Some remarks about the maximal numerical range of the generalized derivation δ2,A,B on the class of Hilbert–Schmidt operators are also given.