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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.

##### MSC:
 47B38 Linear operators on function spaces (general)