Composition operators on a space of operators. (English) Zbl 0732.47034

Let B(H) denote the full operator algebra of a Hilbert space and let \(B_ 0(H)\) denote the Banach space of bounded maps from H into H with respect to the sup-norm. The authors consider the set of composition operators \(C_ T: B(H)\to B_ 0(H)\), \(C_ T(S)=S\circ T\), induced by maps T: \(H\to H\). They prove that \(C_ T\) is continuous iff T is bounded. Moreover they show that a continuous linear operator C: B(H)\(\to B_ 0(H)\) is a composition operator iff \(C(A\circ B)=A\circ C(B)\) for all A,B\(\in B(H)\), and that these make up a closed subspace of the Banach space of all continuous linear operators.


47B38 Linear operators on function spaces (general)