منابع مشابه
On fully operator Lipschitz functions
Let A(D) be the disc algebra of all continuous complex-valued functions on the unit disc D holomorphic in its interior. Functions from A(D) act on the set of all contraction operators (‖A‖ 1) on Hilbert spaces. It is proved that the following classes of functions from A(D) coincide: (1) the class of operator Lipschitz functions on the unit circle T; (2) the class of operator Lipschitz functions...
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The spectral shift function for pairs of selfadjoint operators was introduced in the paper by I.M. Lifshits [17]. In the same paper a trace formula for the difference of functions of the perturbed operator and the unperturbed operator was established. Ideas by Lifshits were developed in the paper by M.G. Krein [14], in which the spectral shift function ξ in L1(R) was defined for arbitrary pairs...
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Suppose that f is a Lipschitz function on R with ‖f‖Lip ≤ 1. Let A be a bounded self-adjoint operator on a Hilbert space H. Let p ∈ (1,∞) and suppose that x ∈ B(H) is an operator such that the commutator [A, x] is contained in the Schatten class Sp. It is proved by the last two authors, that then also [f(A), x] ∈ Sp and there exists a constant Cp independent of x and f such that ‖[f(A), x]‖p ≤ ...
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In a recent paper by H. X. Cao, J. H. Zhang and Z. B. Xu an α-Lipschitz operator from a compact metric space into a Banach space A is defined and characterized in a natural way in the sence that F : K → A is a α-Lipschitz operator if and only if for each σ ∈ X∗ the mapping σ ◦ F is a α-Lipschitz function. The Lipschitz operators algebras Lα(K,A) and lα(K,A) are developed here further, and we st...
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Given any positive integers m and d, we say the a sequence of points (xi)i∈I in Rm is Lipschitz-d-controlling if one can select suitable values yi (i ∈ I) such that for every Lipschitz function f : Rm → Rd there exists i with |f(xi)−yi| < 1. We conjecture that for every m ≤ d, a sequence (xi)i∈I ⊂ Rm is d-controlling if and only if sup n∈N |{i ∈ I : |xi| ≤ n}| nd =∞. We prove that this conditio...
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ژورنال
عنوان ژورنال: Russian Mathematical Surveys
سال: 2016
ISSN: 0036-0279,1468-4829
DOI: 10.1070/rm9729