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

Abstract

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 M 2 , B , C the basic elementary operator on the class of Hilbert–Schmidt operators C 2 ( H ) , i.e. M 2 , B , C ( X ) = BXC for all X ∈ C 2 ( H ) . In this paper, we investigate the A -numerical range W A ( M 2 , B , ( C ♯ A ) ∗ ) and the A -maximal numerical range W max A ( M 2 , B , ( C ♯ A ) ∗ ) , where A = M 2 , 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: W A ( M 2 , B , ( C ♯ A ) ∗ ) ¯ = co ( W A ( B ) ¯ ⋅ W A ( C ) ¯ ) and W max A ( M 2 , B , ( C ♯ A ) ∗ ) = co ( W max A ( B ) ⋅ W max A ( C ) ) , where W A ( ⋅ ) , W max A ( ⋅ ) 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 ‖ M 2 , B , ( C ♯ A ) ∗ ‖ A = ‖ B ‖ A ‖ C ‖ A , where ‖ ⋅ ‖ A and ‖ ⋅ ‖ A designate the A -operator seminorm and the A-operator seminorm, respectively.