We prove that if 0 < α < 1 and f is in the Hölder class Λα (R), then for arbitrary selfadjoint operators A and B with bounded A - B, the operator f (A) - f (B) is bounded and {norm of matrix} f (A) - f (B) {norm of matrix} ≤ const {norm of matrix} A - B {norm of matrix}α. We prove a similar result for functions f of the Zygmund class Λ1 (R): {norm of matrix} f (A + K) - 2 f (A) + f (A - K) {norm of matrix} ≤ const {norm of matrix} K {norm of matrix}, where A and K are selfadjoint operators. Similar results also hold for all Hölder-Zygmund classes Λα (R), α > 0. We also study properties of the operators f (A) - f (B) for f ∈ Λα (R) and selfadjoint operators A and B such that A - B belongs to the Schatten-von Neumann class Sp. We consider the same problem for higher order differences. Similar results also hold for unitary operators and for contractions. To cite this article: A. Aleksandrov, V. Peller, C. R. Acad. Sci. Paris, Ser. I 347 (2009).
Original language | English |
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Pages (from-to) | 483-488 |
Number of pages | 6 |
Journal | Comptes Rendus Mathematique |
Volume | 347 |
Issue number | 9-10 |
DOIs | |
State | Published - May 2009 |
Externally published | Yes |
ID: 5209600