Convexoid and generalized derivations Original Research Article

Let E,F be two Banach spaces and let S be a symmetric norm ideal of L(E,F). For Aset membership, variantL(F) and Bset membership, variantL(E) the generalized derivation δS,A,B is the operator on S that sends X to AX−XB. A bounded linear operator is said to be convexoid if its (algebraic) numerical range coincides with the convex hull of its spectrum. We show that δS,A,B is convexoid if and only if A and B are convexoid.