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Lipschitz functions, Schatten ideals, and unbounded derivations. (English. Russian original) Zbl 1272.47024

Funct. Anal. Appl. 45, No. 2, 157-159 (2011); translation from Funkts. Anal. Prilozh. 45, No. 2, 93-96 (2011).
Summary: It is proved that, for any Lipschitz function \(f(t_1,\dots,t_n)\) of \(n\) variables, the corresponding map \(f_{\mathrm{op}}:(A_1,\dots,A_n)\mapsto f(A_1,\dots,A_n)\) on the set of all commutative \(n\)-tuples of Hermitian operators on a Hilbert space is Lipschitz with respect to the norm of each Schatten ideal \(\mathcal S^p\), \(p\in (1,\infty)\). This result is applied to the functional calculus of normal operators and contractions. It is shown that Lipschitz functions of one variable preserve domains of closed derivations with values in \(\mathcal S^p\). It is also proved that the map \(f_{\mathrm{op}}\) is Fréchet differentiable in the norm of \(\mathcal S^p\) if \(f\) is continuously differentiable.

MSC:

47A60 Functional calculus for linear operators
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
46G05 Derivatives of functions in infinite-dimensional spaces
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References:

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