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On normal derivations of Hilbert–Schmidt type

  • Fuad Kittaneh (a1)

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Let H denote a separable, infinite dimensional Hilbert space. Let B(H), C2 and C1 denote the algebra of all bounded linear operators acting on H, the Hilbert–Schmidt class and the trace class in B(H) respectively. It is well known that C2 and C1 each form a two-sided-ideal in B(H) and C2 is itself a Hilbert space with the inner product

where {ei} is any orthonormal basis of H and tr(.) is the natural trace on C1. The Hilbert–Schmidt norm of XC2 is given by ⅡX2=(X, X)½.

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References

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1.Anderson, J. H., On normal derivations, Proc. Amer. Math. Soc. 38 (1973), 135140.
2.Berger, C. A. and Shaw, B. I., Self-commutators of multicyclic hyponormal operators are always trace class, Bull. Amer. Math. Soc. 79 (1973), 11931199.
3.Fuglede, B., A commutativity theorem for normal operators, Proc. Nat. Acad. Sci. U.S.A. 36 (1950), 3540.
4.Kittaneh, Fuad, On generalized Fuglede–Putnam theorems of Hilbert–Schmidt type, Proc. Amer. Math. Soc. 88 (1983), 293298.
5.Kittaneh, Fuad, Commutators of Cp type, Thesis, (Indiana University, 1982).
6.Weiss, G., The Fuglede commutativity theorem modulo the Hilbert–Schmidt class and generating functions for matrix operators I. Trans. Amer. Math. Soc. 246 (1978), 193209.
7.Yoshino, T., Subnormal operators with a cyclic vector, Tohoku Math. J. 21 (1969), 4755.

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