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On the differentiability of functions of an operator. (English) Zbl 0837.47013

Azéma, J. (ed.) et al., Séminaire de probabilités XXIX. Berlin: Springer-Verlag. Lect. Notes Math. 1613, 218-219 (1995).
Let \(f\) be a continuous function on \(\mathbb{R}\). Then it is well-known how to define \(f(A)\) when \(A\) is a bounded selfadjoint operator on a Hilbert space \(\mathcal H\), using the spectral decomposition of \(A\). But if \(A\), \(H\) are two non-commuting selfadjoint operators, no explicit computation of \(f(A+ tH)\) is known. Our problem here is to study the regularity of \(f(A+ tH)\) under some regularity assumptions on \(f\). We will assume that \(\mathcal H\) is finite-dimensional. This note is a complement to “Some operator inequalities” in volume XXVIII [Lect. Notes Math. 1583, 316-333 (1994; Zbl 0820.47020)], and answers a question of P. A. Meyer.
For the entire collection see [Zbl 0826.00027].

MSC:

47A60 Functional calculus for linear operators
47A63 Linear operator inequalities

Citations:

Zbl 0820.47020