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Second order trace formulae

Arup Chattopadhyay, Soma Das, Chandan Pradhan

Abstract

Koplienko \cite{Ko} found a trace formula for perturbations of self-adjoint operators by operators of Hilbert-Schmidt class $\mathcal{B}_2(\mathcal{H})$. Later, Neidhardt introduced a similar formula in the case of pair of unitaries $(U,U_0)$ via multiplicative path in \cite{NH}. In 2012, Potapov and Sukochev \cite{PoSu} obtained a trace formula like the Koplienko trace formula for pairs of contractions by answering an open question posed by Gesztesy, Pushnitski, and Simon in \cite[Open Question 11.2]{GePu}. In this article, we supply a new proof of the Koplienko trace formula in the case of pair of contractions $(T,T_0)$, where the initial operator $T_0$ is normal, via linear path by reducing the problem to a finite-dimensional one as in the proof of Krein's trace formula by Voiculescu \cite{Voi}, Sinha and Mohapatra \cite{MoSi94,MoSi96}. Consequently, we obtain the Koplienko trace formula for a class of pairs of contractions using the Schäffer matrix unitary dilation. Moreover, we also obtain the Koplienko trace formula for a pair of self-adjoint operators and maximal dissipative operators using the Cayley transform. At the end, we extend the Koplienko-Neidhardt trace formula for a class of pair of contractions $(T,T_0)$ via multiplicative path.

Second order trace formulae

Abstract

Koplienko \cite{Ko} found a trace formula for perturbations of self-adjoint operators by operators of Hilbert-Schmidt class . Later, Neidhardt introduced a similar formula in the case of pair of unitaries via multiplicative path in \cite{NH}. In 2012, Potapov and Sukochev \cite{PoSu} obtained a trace formula like the Koplienko trace formula for pairs of contractions by answering an open question posed by Gesztesy, Pushnitski, and Simon in \cite[Open Question 11.2]{GePu}. In this article, we supply a new proof of the Koplienko trace formula in the case of pair of contractions , where the initial operator is normal, via linear path by reducing the problem to a finite-dimensional one as in the proof of Krein's trace formula by Voiculescu \cite{Voi}, Sinha and Mohapatra \cite{MoSi94,MoSi96}. Consequently, we obtain the Koplienko trace formula for a class of pairs of contractions using the Schäffer matrix unitary dilation. Moreover, we also obtain the Koplienko trace formula for a pair of self-adjoint operators and maximal dissipative operators using the Cayley transform. At the end, we extend the Koplienko-Neidhardt trace formula for a class of pair of contractions via multiplicative path.

Paper Structure

This paper contains 9 sections, 22 theorems, 125 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Trace formula in finite dimension
  4. Reduction to finite dimension
  5. Existence of shift function in linear path
  6. Trace formula for contractions
  7. Trace formula for self-adjoint operators
  8. Trace formula for maximal dissipative operators
  9. Extension of Koplienko-Neidhardt trace formula

Key Result

Theorem 1.1

(see PoSu) Let $U$ and $U_0$ be two contractions in an infinite dimensional separable Hilbert space $\mathcal{H}$ such that $V:=U-U_0\in\mathcal{B}_2(\mathcal{H})$. Denote $U_s=U_0+sV, ~s\in[0,1]$. Then for any complex polynomial $p(\cdot)$, and there exists an $L^1(\mathbb{T})$-function $\eta$ (unique up to an analytic term) such that Moreover, for every given $\epsilon >0$, we can choose the f

Theorems & Definitions (46)

  • Theorem 1.1
  • Lemma 2.1
  • Theorem 2.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • Theorem 3.3
  • proof
  • Theorem 3.4
  • proof
  • ...and 36 more