Double operator integral methods applied to continuity of spectral shift functions
Authors: Carey Alan L. , Gesztesy Fritz , Levitina Galina , Nichols Roger , Potapov Denis , Sukochev Fedor 
Published: December 2016

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

\text{s-}\!\!\lim_{n \to \infty} (A_n - z_0 I_{\mathcal{H}})^{-1}= (A - z_0 I_{\mathcal{H}})^{-1}

and

\text{s-}\!\!\lim_{n \to \infty} (B_n - z_0 I_{\mathcal{H}})^{-1}= (B - z_0 I_{\mathcal{H}})^{-1}

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\}

,

\begin{split}
T(a)&:= [( A -aiI_{\mathcal{H}})^{-m} - ( B -ai I_{\mathcal{H}})^{-m}] \in \mathcal{B}_p(\mathcal{H}), \\
T_n(a) &:= [( A_n -aiI_{\mathcal{H}})^{-m} - ( B_n -aiI_{\mathcal{H}})^{-m}] \in \mathcal{B}_p(\mathcal{H}),
\end{split}

\lim_{n \rightarrow \infty} \|T_n(a) - T(a)\|_{\mathcal{B}_p(\mathcal{H})} =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),

\lim_{n \rightarrow \infty} \| [f(A_n) - f(B_n)] - [f(A)- f(B)]\|_{\{\mathcal{B}\_p(\mathcal{H})\}}=0.

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

\begin{align*}
\|f(A)-f(B)\|_{\mathcal B_p(\mathcal H)}&\leq C(\|(A-a_1iI_{\mathcal H})^{-m} - (B-a_1iI\_{\mathcal H})^{-m}\|_{\mathcal B_p(\mathcal H)}\\
&\qquad+\|(A-a_2iI_{\mathcal H})^{-m} - (B-a_2iI_{\mathcal H})^{-m}\|_{\mathcal B_p(\mathcal H)}),
\end{align*}

which permits the use of differences of higher powers

m \in \mathbb{N}

of resolvents to control the

\| \cdot \|_{\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 - z I\_{\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

\lim_{\tau\to 0^+} \|\xi(\, \cdot \, ; B_{\tau}, A_0) f - \xi(\, \cdot \, ; B_0, A_0) f\|_{L^1(\mathbb{R}; (|\nu|^{m+1} + 1)^{-1}d\nu)} = 0.

The fact that higher powers

m\in\mathbb{N}

,

m \geq 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.