H\"older functions are operator-H\"older

L. N. Nikol'skaya ; Yu. B. Farforovskaya

Algebra i analiz, Tome 22 (2010) no. 4, pp. 198-213.

Résumé

Let $A$ and $B$ be selfadjoint operators in a separable Hilbert space such that $A-B$ is bounded. If a function $f$ satisfies the Hölder condition of order $\alpha$, $0\alpha1$, i.e., $|f(x)-f(y)|\leq L|x-y|^\alpha$, then $\|f(A)-f(B)\|\leq CL\|A-B\|^\alpha$, where $C$ is a constant, specifically, $C=2^{1-\alpha}+2\pi\sqrt 8\frac1{(1-2^{\alpha-1})^2}$. This result is a consequence of a general inequality in which the norm of $f(A)-f(B)$ is controlled in terms of the continuity modulus of $f$. Similar results are true for the quasicommutators $f(A)K-Kf(B)$, where $K$ is a bounded operator.

Détail

Keywords: operator-Hölder functions, Adamar–Schur multipliers.

Bibliographie

[1] Farforovskaya Yu. B., “O svyazi metriki Kantorovicha–Rubinshteina dlya spektralnykh razlozhenii samosopryazhennykh operatorov s funktsiyami ot operatorov”, Vestn. Leningr. un-ta. Mat. mekh. astronom., 1968, no. 4, 94–97

[2] Aleksandrov A., Peller V., “Functions of perturbed operators”, C. R. Math. Acad. Sci. Paris, 347:9–10 (2009), 483–488 | MR | Zbl

[3] Farforovskaya Yu. B., Nikolskaya L., “Modulus of continuity of operator functions”, Algebra i analiz, 20:3 (2008), 224–242 | MR

[4] Nikolskaya L. N., Farforovskaya Yu. B., “Tëplitsevy i gankelevy matritsy kak multiplikatory Adamara–Shura”, Algebra i analiz, 15:6 (2003), 141–160 | MR | Zbl

[5] Matsaev V. I., “Ob odnom klasse vpolne nepreryvnykh operatorov”, Dokl. AN SSSR, 139:3 (1961), 548–551 | Zbl

[6] Bennett G., “Shure multiplieres”, Duke Math. J., 44 (1977), 603–639 | DOI | MR | Zbl

[7] Birman M. Sh., Solomyak M. Z., “Dvoinye operatornye integraly Stiltesa”, Probl. mat. fiz., 1, LGU, L., 1966, 33–67 | MR

[8] Birman M. Sh., Solomyak M., “Double operator integrals in a Hilbert space”, Integral Equations Operator Theory, 47:2 (2003), 131–168 | DOI | MR | Zbl