Perturbation theory and higher order S p\mathcal{S}^{!p}Sp-differentiability of operator functions (original) (raw)
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Perturbation theory and higher order S^p-differentiability of operator functions
2019
We establish, for 1 < p < ∞, higher order S^p-differentiability results of the function φ : t∈R f(A+tK) - f(A) for selfadjoint operators A and K on a separable Hilbert space H with K element of the Schatten class S^p(H) and fn-times differentiable on R. We prove that if either A and f^(n) are bounded or f^(i), 1 ≤ i ≤ n are bounded, φ is n-times differentiable on R in the S^p-norm with bounded nth derivative. If f∈ C^n(R) with bounded f^(n), we prove that φ is n-times continuously differentiable on R. We give explicit formulas for the derivatives of φ, in terms of multiple operator integrals. As for application, we establish a formula and S^p-estimates for operator Taylor remainders for a more extensive class of functions. These results are the nth order analogue of the results of <cit.>. They also extend the results of <cit.> from S^2(H) to S^p(H) and the results of <cit.> from n-times continuously differentiable functions to n-times differentiable functions f.
Higher order mathcalSp\mathcal{S}^{p}mathcalSp-differentiability: The unitary case
arXiv (Cornell University), 2024
Consider the set of unitary operators on a complex separable Hilbert space H, denoted as U(H). Consider 1 < p < ∞. We establish that f is n times continuously Fréchet S p-differentiable at every point in U(H) if and only if f ∈ C n (T). Take U : R → U(H) such thatŨ : t ∈ R → U (t) − U (0) is n times continuously S p-differentiable on R. Consequently, for f ∈ C n (T), we prove that f is n times continuously Gâteaux S p-differentiable at U (t). We provide explicit expressions for both types of derivatives of f in terms of multiple operator integrals. In the domain of unitary operators, these results closely follow the nth order successes for self-adjoint operators achieved by the second author, Le Merdy, Skripka, and Sukochev. Furthermore, as for application, we derive a formula and S p-estimates for operator Taylor remainders for a broader class of functions. Our results extend those of Peller, Potapov, Skripka, Sukochev and Tomskova.
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