Operator-Lipschitz functions in Schatten–von Neumann classes

Potapov, Denis; Sukochev, Fedor

doi:10.1007/s11511-012-0072-8

Cited by 104 publications

(98 citation statements)

References 9 publications

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“…This allowed the authors of [22] to deduce that for p 1 and ε > 0, the operator f (A) − f (B) belongs to S p+ε , whenever f is a Lipschitz function and A − B ∈ S p . The epsilon was removed later in [34] in the case 1 < p < ∞. It was shown in [34] that for p ∈ (1, ∞), the operator f (A) − f (B) belongs to S p , whenever A − B ∈ S p and f is a Lipschitz function.…”

Section: Introductionmentioning

confidence: 99%

Functions of operators under perturbations of class Sp

Aleksandrov

Peller

2010

Journal of Functional Analysis

136

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This is a continuation of our paper [2]. We prove that for functions f in the Hölder class Λ α (R) and 1 < p < ∞, the operator f (A) − f (B) belongs to S p/α , whenever A and B are self-adjoint operators with A − B ∈ S p . We also obtain sharp estimates for the Schatten-von Neumann norms f (A) − f (B) S p/α in terms of A − B S p and establish similar results for other operator ideals. We also estimate Schattenvon Neumann norms of higher order differences m j =0 (−1) m−j m j f (A + jK). We prove that analogous results hold for functions on the unit circle and unitary operators and for analytic functions in the unit disk and contractions. Then we find necessary conditions on f for f (A) − f (B) to belong to S q under the assumption that A − B ∈ S p . We also obtain Schatten-von Neumann estimates for quasicommutators f (A)R − Rf (B), and introduce a spectral shift function and find a trace formula for operators of the form f (A − K) − 2f (A) + f (A + K).

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Section: Introductionmentioning

confidence: 99%

Functions of operators under perturbations of class Sp

Aleksandrov

Peller

2010

Journal of Functional Analysis

136

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“…The following recent result of Sukochev and Potapov [35] settled a long outstanding conjecture of Krein for the index p in the range 1 < p < ∞.…”

Section: Theorem 12mentioning

confidence: 89%

Singular Bilinear Integrals in Quantum Physics

Jefferies

2015

Mathematics

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Bilinear integrals of operator-valued functions with respect to spectral measures and integrals of scalar functions with respect to the product of two spectral measures arise in many problems in scattering theory and spectral analysis. Unfortunately, the theory of bilinear integration with respect to a vector measure originating from the work of Bartle cannot be applied due to the singular variational properties of spectral measures. In this work, it is shown how "decoupled" bilinear integration may be used to find solutions X of operator equations AX − XB = Y with respect to the spectral measure of A and to apply such representations to the spectral decomposition of block operator matrices. A new proof is given of Peller's characterisation of the space L 1 ((P ⊗ Q) L(H) ) of double operator integrable functions for spectral measures P , Q acting in a Hilbert space H and applied to the representation of the trace of Λ×Λ ϕ d(P T P ) for a trace class operator T . The method of double operator integrals due to Birman and Solomyak is used to obtain an elementary proof of the existence of Krein's spectral shift function.

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“…Potapov and Sukochev [15] using a powerful technique of Banach space geometry and harmonic analysis proved that (4.12) extends to all S p ideals, if A = A * . For all normal A this was proved in [13].…”

Section: Norm Inequalitiesmentioning

confidence: 99%