Perturbations and operator trace functions

We study the spectral functional for a suitable function f, a self-adjoint operator D having compact resolvent, and a certain class of bounded self-adjoint operators A. Such functionals were introduce by Chamseddine and Connes in the context of noncommutative geometry. Motivated by the physical applications of these functionals, we derive a Taylor expansion of them in terms of Gâteaux derivatives. This involves divided differences of f evaluated on the spectrum of D, as well as the matrix coefficients of A in an eigenbasis of D. This generalizes earlier results to infinite dimensions and to any number of derivatives.     

Selberg's trace formula for an automorphic Schrodinger operator

Leningrad Division of V. A. Steklov Mathematics Institute, Academy of Sciences of the USSR. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 25, No. 2, pp. 26–37, April–June, 1991.     

A trace formula of a boundary value problem for the operator Sturm-Liouville equation

We obtain a regularized trace formula for the operator Sturm-Liouville equation with a boundary condition depending on a spectral parameter.     

About the spectrum and the trace formula for the operator Bessel equation

We study the asymptotics of the distribution function and compute the regularized trace of a boundary value problem for the operator-differential equation with the boundary value depending on a spectral parameter.     

A simple proof of the compactness of the trace operator on a Lipschitz domain

Archiv der Mathematik - We construct whole-space extensions of functions in a fractional Sobolev space of order $$sin (0,1)$$ and integrability $$pin (0,infty )$$ on an open set O which vanish...     

Inverse problem and the trace formula for the Hill operator, II

Consider the Hill operator on where is a 1-periodic real potential and The spectrum of T is absolutely continuous and consists of intervals separated by gaps . Let be the Dirichlet eigenvalue of the equation on the interval [0,1]. Introduce the vector with components and where the sign or for all . Using nonlinear functional analysis in Hilbert spaces we show, that the mapping is a real analytic isomorphism. In the second part a trace formula for is proved. Received December 22, 1997; in final form July 7, 1998     

Lipschitz and darbo conditions for the superposition operator in some non-ideal spaces of smooth functions

Let X be a Banach space of differentiable functions and A: XX be a superposition operator. We consider for A the conditions

On the trace of certain rational operator functions

In this paper operator functions of type $$L(\lambda ): = I - \sum\limits_{k = 1}^n {\lambda ^k } A_k + \sum\limits_{k = 1}^m {\frac{{\lambda ^{\varepsilon _k } }}{{(\lambda - a_k )^{\mu _k } }}H_k } $$ are considered. In the first part of the paper a linearization ofL is constructed, and it is shown that the geometric multiplicities and the null multiplicities of the eigenvalues λ ∈ χ ofL and the linearization coincide. In the second part of the paper trace and determinant formulas forL are derived under certain conditions for the coefficients ofL.     

An infinitesimal trace formula for the Laplace operator on compact Riemann surfaces

We derive an infinitesimal (or variational) version of the Selberg trace formula for compact Riemann surfaces, which gives information on the behaviour of the eigenvalues of the Laplace-Beltrami operator as the surface varies over the appropriate moduli space.     

Mean value formula for the root vector functions of the Dirac operator

We consider the one-dimensional Dirac operator on an arbitrary interval and obtain a mean value formula for the root vector functions of this operator.     

A formula for the generalized resolvent of an isometric operator for the exterior of the unit disk

In this note we establish the relationship between characteristic functions and generalized resolvents of isometric operators for the exterior of the unit disk. A certain sufficient condition is given in terms of characteristic functions that the form of the formula for the generalized resolvent be preserved across the unit circle.Translated from Matematicheskie Zametki, Vol. 13, No. 6, pp. 849–856, June, 1973.     

Lipschitz and darbo conditions for the superposition operator in ideal spaces

Summary In this paper we give necessary and sufficient conditions for the superposition operator Fx(s)=f(s, x(s)) to satisfy a Lipschitz condition Fx₁ - Fx₂kx₁ - x₂ or a Darbo condition (FN)k(N) in ideal spaces of measurable functions, where is the Hausdorff measure of noncompactness. Moreover, we characterize a large class of spaces in which the above mentioned two conditions are equivalent.

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Sunto In questo lavoro diamo delle condizioni necessarie e sufficienti perchè l'operatore di sovrapposizione Fx(s)=f (s, x(s)) soddisfi alla condizione di Lipschitz Fx₁–Fx₂ kx₁–x₂ o quella di Darbo (FN)k(N) in spazi ideali di funzioni misurabili, ove è la misura di non compattezza di Hausdorff. Inoltre, caratterizziamo un'ampia classe di spazi in cui le suddette due condizioni sono equivalenti.