Geodesic
Range inclusion for normal derivations
Glasgow mathematical journal, Tome 25 (1984) no. 2, pp. 255-262

Voir la notice de l'article provenant de la source Cambridge University Press

For a (bounded, linear) operator A in a (complex, infinite-dimensional, separable) Hilbert space H, the inner derivation DA as an operator on B(H), is defined by DAX = AX – XA. Johnson and Williams [4] showed that, when A is a normal operator, range inclusion DBB(H)⊆DA(H)⊆ is equivalent to the condition that B = f(A), where f is a Lipschitz function on σ(A) such that t(z, w)(f(z)–f(w))/(z–w) is a trace class kernel on L2(μ) whenever t(z, w) is such a kernel. (Here μ is the dominating scalar valued spectral measure of A constructed in multiplicity theory). This result is deep and its proof is difficult. In the present paper, we establish the following analogous result which is easier to prove: for a normal operator A, range inclusion DB℘2(H) holds if and only if B = f(A) for some Lipschitz function f on σ(A). Here ℘(H) stands for the Hilbert-Schmidt class of operators on H. As by-products of our argument, we generalize some results in [4], [8], [9] concerning the non-existence of a one-sided ideal contained in certain derivation ranges; for example, we show that if A is hyponormal and if the point spectrum σP(A*) of A* is empty, then DAB(H) does not contain any nonzero right ideal. 
Fong, C. K. Range inclusion for normal derivations. Glasgow mathematical journal, Tome 25 (1984) no. 2, pp. 255-262. doi: 10.1017/S001708950000567X
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