Koplienko–Neidhardt trace formula for pairs of contraction operators and pairs of maximal dissipative operators

Abstract

We present a generalization of Koplienko-Neidhardt trace formula for pairs of Hilbert space operators (T, V ) with T contractive and V unitary such that T − V is a Hilbert-Schmidt operator. We extend the result to pairs of contractions and then, via Cayley transform, to pairs of maximal dissipative operators. where ξ is an integrable real function that depends only on the pair (U 1 , U 0 ) and is determined by (0.1) up to an additive constant. The identity (0.1) is usually referred to as the trace formula. A function ξ, as in the trace formula, is called a spectral shift function of the pair (U 1 , U 0 ). A distinguished spectral shift function ξ 0 is fixed by the conditionwhere the operator value of Log U 1 U −1 0 is chosen so that the spectrum of 1/iLogM. Sh. Birman and M. G. Kreȋn [2] established that the pair (U 1 , U 0 ) admits a family of unitary scattering matrices {S(e it )} such that the identityholds for almost every t ∈ [−π, π] with e it belonging to the spectrum of U 0 . The spectral shift function was introduced by I. M. Lifshits [10] in connection with a problem of quantum statistic and for pairs of self-adjoint operators (for which a similar trace formula holds). After the pioneering work of M. G. , various interesting problems concerning (0.1) and (0.2) have attracted the attention

Higher-order spectral shift for pairs of contractions via multiplicative path

“…In this direction of studies, Marcantognini and Morán obtained the Koplienko-Neidhardt trace formula for pairs of contraction operators and pairs of maximal dissipative operators via multiplicative path in [17]. The aim of the present article is to prove a higher order version of the Koplienko-Neidhardt trace formula for pairs of contraction operators and pairs of maximal dissipative operators via multiplicative path.…”

“…In other words, we prove the trace formula for pairs of contractions (T 0 , T 1 ) on H . The technique involved here is standard and similar to the idea mentioned in [17] with an appropriate modification, that means first we dilate (T 0 , T 1 ) to a pair of contractions (T, V ) with V is a unitary operator on the bigger space F containing H as a subspace and then use the existing trace formula for the pair (T, V ) obtained in our last section to get the required trace formula in this section. The following is the main result in this section.…”

“…The aim of the present article is to prove a higher order version of the Koplienko-Neidhardt trace formula for pairs of contraction operators and pairs of maximal dissipative operators via multiplicative path. We follow the same strategy as mentioned in [17] with an appropriate modification. In other words, first we consider a pair (T, V ), where V is a unitary operator and T is a contraction on H and then we prove the higher order version of the Koplienko-Neidhardt trace formula via multiplicative path corresponding to the pair (T, V ) under some additional hypotheses (see Theorem 3.2) by applying the existing Theorem 2.1 below and using dilation theory.…”

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In [17], Marcantognini and Morán obtained Koplienko-Neidhardt trace formula for pairs of contractions and pairs of maximal dissipative operators via multiplicative path. In this article, we prove the existence of higher-order spectral shift functions for pairs of contractions and pairs of maximal dissipative operators via multiplicative path by adapting the argument employed in [17].

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Second order trace formulae

“…Motivated from the work of Marcantognini and Morán [22] we have the following one of the main result in this section. Moreover, the function η satisfying (6.4) also satisfies the equation (5.12).…”