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A Counterexample in the theory of derivations
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Let B(H) be the algebra of all bounded linear operators on a separable, infinite dimensional complex Hilbert space H. Let C2 and C1 denote respectively, the Hilbert–Schmidt class and the trace class operators in B(H). It is known that C2 and C1 are two-sided*-ideals in B(H) and C2 is a Hilbert space with respect to the inner product(where tr denotes the trace). For any Hilbert–Schmidt operator X let ∥X∥2=(X, X)½ be the Hilbert-Schmidt norm of X.For fixed A ∈ B(H) let δA be the operator on B(H) defined byOperators of the form (1) are called inner derivations and they (as well as their restrictions have been extensively studied (for example [1–3]). In [1], Fuad Kittaneh proved the following result.
Title: A Counterexample in the theory of derivations
Description:
Let B(H) be the algebra of all bounded linear operators on a separable, infinite dimensional complex Hilbert space H.
Let C2 and C1 denote respectively, the Hilbert–Schmidt class and the trace class operators in B(H).
It is known that C2 and C1 are two-sided*-ideals in B(H) and C2 is a Hilbert space with respect to the inner product(where tr denotes the trace).
For any Hilbert–Schmidt operator X let ∥X∥2=(X, X)½ be the Hilbert-Schmidt norm of X.
For fixed A ∈ B(H) let δA be the operator on B(H) defined byOperators of the form (1) are called inner derivations and they (as well as their restrictions have been extensively studied (for example [1–3]).
In [1], Fuad Kittaneh proved the following result.
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