Higher order S2-differentiability and application to Koplienko trace formula

Abstract

Let A be a selfadjoint operator in a separable Hilbert space, K a selfadjoint Hilbert-Schmidt operator, and f ∈ C n (R). We establish that ϕ(t) = f (A + tK) − f (A) is n-times continuously differentiable on R in the Hilbert-Schmidt norm, provided either A is bounded or the derivatives f (i) , i = 1, . . . , n, are bounded. This substantially extends the results of [3] on higher order differentiability of ϕ in the Hilbert-Schmidt norm for f in a certain Wiener class. As an application of the second order S 2 -differentiability, we extend the Koplienko trace formula from the Besov class B 2 ∞1 (R) [20] to functions f for which the divided difference f [2] admits a certain Hilbert space factorization.

Higher Order Differentiability of Operator Functions in Schatten Norms

“…, X n,m be as in Lemma 2.3. By Theorem 3.6 and Remark 3.5, f is n times Fréchet S p -differentiable at A m and given m ∈ N, there exists δ m,ǫ > 0 such By letting t = 0 in (3.43)-(3.45) we obtain [4] (3.46)…”

“…. , f (n) are bounded" (the latter property is a necessary condition for n times Gâteaux S p -differentiability, see Proposition 3.9), extending the result of [4] from S 2 to the general S p and the consequence of [19]…”

“…The crucial point in the construction leading to Definition 2.5 is the w * -continuity of Γ A 1 ,...,A n+1 , which allows to reduce various computations to elementary tensor product manipulations. See [4] for illustrations. Proposition 2.6.…”

“…Proof. If p = 2, then (3.13) follows from [4,Corollary 4.4] because the transformations T and Γ given by Definitions 2.1 and 2.5 coincide on S 2 × · · · × S 2 (see Proposition 2.6). If 1 < p < 2, then S p ⊂ S 2 , so (3.13) holds for all X 1 , .…”

“…, n + 1. It was proved in [4,Theorem 4.1] that a function f in C n (R) is n times Gâteaux S 2 -differentiable at every bounded self-adjoint operator and, under the additional assumption ' f ( j) is bounded, j = 0, . .…”