Double operator integral methods applied to continuity of spectral shift functions

Abstract

We derive two principal results in this note. To describe the first, assume that $A$, $B$, $A_n$, $B_n$, $n\in \mathbb{N}$, are self-adjoint operators in a complex, separable Hilbert space $\mathcal{H}$, and suppose that

and

for some $z_0\in \mathbb{C}\backslash \mathbb{R}$. Fix $m\in \mathbb{N}$, $m$ odd, $p\in [1,\infty )$, and assume that for all $a\in \mathbb{R}\backslash \{0\}$,

Then for any function $f$ in the class ${\mathfrak{F}}_m(\mathbb{R})\supset C_0^{\infty }(\mathbb{R})$ (cf. (1.1) for details),

Moreover, for each $f\in \mathfrak{F}\_m(\mathbb{R})$, $p\in [1,\infty )$, we prove the existence of constants $a_1,a_2\in \mathbb{R}\backslash \{0\}$ and $C=C(f,m,a_1,a_2)\in (0,\infty )$ such that

which permits the use of differences of higher powers $m\in \mathbb{N}$ of resolvents to control the $\Vert \cdot {\Vert }_{{\mathcal{B}}_p(\mathcal{H})}$-norm of the left-hand side $[f(A)-f(B)]$ for $f\in \mathfrak{F}\_m(\mathbb{R})$.

Our second result is concerned with the continuity of spectral shift functions $\xi (\cdot ;B,B_0)$ associated with a pair of self-adjoint operators $(B,B_0)$ in $\mathcal{H}$ with respect to the operator parameter $B$. For brevity, we only describe one of the consequences of our continuity results. Assume that $A_0$ and $B_0$ are fixed self-adjoint operators in $\mathcal{H}$, and there exists $m\in \mathbb{N}$, $m$ odd, such that, $[(B_0-z{I}_{\mathcal{H}}{)}^{-m}-(A_0-z{I}_{\mathcal{H}}{)}^{-m}]\in {\mathcal{B}}_1(\mathcal{H})$, $z\in \mathbb{C}\backslash \mathbb{R}$. For $T$ self-adjoint in $\mathcal{H}$ we denote by ${\Gamma }_m(T)$ the set of all self-adjoint operators $S$ in $\mathcal{H}$ for which the containment $[(S-z{I}_{\mathcal{H}}{)}^{-m}-(T-zI\_\mathcal{H}{)}^{-m}]\in {\mathcal{B}}_1(\mathcal{H})$, $z\in \mathbb{C}\backslash \mathbb{R}$, holds. Suppose that $B_1\in {\Gamma }_m(B_0)$ and let $\{B_{\tau }{\}}_{\tau \in [0,1]}\subset {\Gamma }_m(B_0)$ denote a continuous path (in a suitable topology on ${\Gamma }_m(B_0)$, cf. (1.3)) from $B_0$ to $B_1$ in ${\Gamma }_m(B_0)$. If $f\in L^{\infty }(\mathbb{R})$, then

The fact that higher powers $m\in \mathbb{N}$, $m\ge 2$, of resolvents are involved, permits applications of this circle of ideas to elliptic partial differential operators in ${\mathbb{R}}^n$, $n\in \mathbb{N}$. The methods employed in this note rest on double operator integral (DOI) techniques.