Geodesic
A Counterexample in the theory of derivations
Glasgow mathematical journal, Tome 31 (1989) no. 2, pp. 161-163

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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)1⁄2 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. 
Wenying, Feng; Guoxing, Ji. A Counterexample in the theory of derivations. Glasgow mathematical journal, Tome 31 (1989) no. 2, pp. 161-163. doi: 10.1017/S0017089500007679
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[1] 1.Kittaneh, Fuad, On normal derivations of Hilbert-Schmidt type, Glasgow Math. J. 29 (1987), 245–248. Google Scholar | DOI

[2] 2.Anderson, J. H., On normal derivations, Proc. Amer. Math. Soc. 38 (1973), 135–140. Google Scholar

[3] 3.Fialkow, L. A., A note on norm ideals and the operator, X → AX-XB, Israel J. Math. 32, (1979), 331–348. Google Scholar | DOI

[4] 4.Williams, L. R., Quasisimilarity and hyponormal operators, j. Operator Theory 5 (1981), 127–139. Google Scholar

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