Modulus of continuity of an operator function
by
Ludmila Nikolskaia
Institut de Mathématiques de Bordeaux, Université Bordeaux-1, 351 cours de la Libération, 33405 Talence, France
Coauthors: Yu. B. Farforovskaya, Mathematics Department, St.Petersburg University of Electrical Engineering, St.Petersburg, Russia, email rabk@sut.ru

Theorem 1 Let A and B be bounded selfadjoint operators on the separable Hilbert space, f be a continuous function on the interval [a, b] containing the spectra of both A and B. Denote w_f the modulus of continuiy of the function f. Then

||f(A)-f(B)|| ≤ 4[log( b-a ||A-B|| +1)+1]2·wf(||A-B||).

In case of unbounded operators we suppose that function f on R satisfies the following conditions

(1) There exist b > 2 and M > 0 such that |f(t)| ≤ M|t|, for |t| > b.

(2) For 0 < d ≤ 1,  w_{f, t}(d) ≤ c₁[(w_{f, 0}(d))/((log|t|)³)],   c₁ > 0. Here w_{f, 0} is the modulus of continuity of f on the interval [-b, b] and w_{f, t} is the modulus of continuity of f on the set {x ∈ R:|x| ≥ t > b}.

Theorem 2 Let A and B be unbounded selfadjoint operators on a separable Hilbert space such that the operator A-B is bounded and suppose that the function f on the real line satisfies the conditions (1)-(2). Then the operator f(A)-f(B) is bounded and there is a positive constant c₂ such that

||f(A)-f(B)|| ≤ c2 log2(1+ 1 ||A-B|| ) wf, 0(||A-B||).

To proof Theorem 1 we prove some lemmas concerned Hadamard-Schur multipliers. In particular

Lemma 3 Let T_k=([1/(k+|i-j|)])_{i, j ≥ 0},   k > 0 be symmetric Toeplitz matrix determined by the sequence (t_m)_{m ∈ Z}: t_m=[1/(k+|m|)]. Then the matrix T_k is a Hadamard-Schur multiplier and the multiplier norm of T_k is : ||T_k||_H=[1/k] .

Date received: June 9, 2008