Perturbation theory and higher order \S^p-differentiability\n of operator functions

2019/06/13 by Clément Coine, Coine, Clément
Mathematics · #46L52 #47A55 #47B10 #47B49 #Advanced Operator Algebra Research #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1906.05585

openalex publication_date 2019/06/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We establish, for 1 < p < \∞, higher order\n\S^p-differentiability results of the function \φ : t\∈\n\ℝ \↦ f(A+tK) - f(A) for selfadjoint operators A and K on a\nseparable Hilbert space \H with K element of the Schatten class\n\S^p(\H) and f n-times differentiable on \ℝ.\nWe prove that if either A and f⁽ⁿ⁾ are bounded or f⁽ⁱ⁾, 1 \≤ i\n\≤ n are bounded, \φ is n-times differentiable on \ℝ in\nthe \S^p-norm with bounded nth derivative. If f\∈\nCⁿ(\ℝ) with bounded f⁽ⁿ⁾, we prove that \φ is n-times\ncontinuously differentiable on \ℝ. We give explicit formulas for the\nderivatives of \φ, in terms of multiple operator integrals. As for\napplication, we establish a formula and \S^p-estimates for operator\nTaylor remainders for a more extensive class of functions. These results are\nthe nth order analogue of the results of citeKPSS. They also extend the\nresults of citeCLSS from \S²(\H) to\n\S^p(\H) and the results of citeLMS from n-times\ncontinuously differentiable functions to n-times differentiable functions\nf.\n

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