Kato smoothness and functions of perturbed self-adjoint operators

Abstract

We consider the difference f (H 1 ) − f (H 0 ) for self-adjoint operators H 0 and H 1 acting in a Hilbert space. We establish a new class of estimates for the operator norm and the Schatten class norms of this difference. Our estimates utilise ideas of scattering theory and involve conditions on H 0 and H 1 in terms of the Kato smoothness. They allow for a much wider class of functions f (including some unbounded ones) than previously available results do. As an important technical tool, we propose a new notion of Schatten class valued smoothness and develop a new framework for double operator integrals.

Schatten Class Conditions for Functions of Schrödinger Operators

“…These conditions are given in terms of the smoothness of f and the exponent ρ in (1.2). This paper is a continuation of [7], where this problem was considered in the general operator theoretic context. It is also a further development of [5], where the trace class membership of D(f ) was considered.…”

“…here several terms vanish because of the assumption supp f ⊂ Λ. We estimate the "diagonal term" ½ Λ (H 1 )D(f )½ Λ (H 0 ) by directly applying the results of [7] and some variants of the limiting absorption principle. We estimate the "off-diagonal terms" (the second and third terms in the right side of (1.6)) by using rather standard Schatten class bounds for Schrödinger operators.…”

“…In [7], we also use the space CMO(R) (continuous mean oscillation) which can be characterised as the closure of C comp (R) ∩ BMO(R) in BMO(R). However, for a compactly supported function f , conditions f ∈ VMO and f ∈ CMO coincide.…”

“…Here we briefly recall (with minor simplifications) the relevant definitions and main results of [7]. To motivate what comes next, we should explain that we will factorise the potential V in the form V = (sign V )|V | 1−θ |V | θ with an appropriate exponent θ ∈ (0, 1).…”

“…To motivate what comes next, we should explain that we will factorise the potential V in the form V = (sign V )|V | 1−θ |V | θ with an appropriate exponent θ ∈ (0, 1). This corresponds to the "abstract" factorisation V = G * 1 G 0 of [7]. In [7], we consider the general case, where G 0 , G 1 are possibly unbounded operators from a Hilbert space H to another Hilbert space K, such that G 0 is H 0bounded and G 1 is H 1 -bounded.…”