Spectral Stieltjes-Type Integration and Some Applications

This paper presents Stieltjes-type integration for operator-valued functions with respect to spectral families. The relation between Riemann-Stieltjes integrals associated with some classes of spectral families including, in particular, those that arise in the context of the Stone representation theorem for Banach spaces is established. The developed approach is applied to the structural analysis of the operators commuting with a one-parameter representation satisfying the Generalized Stone theorem.Comment: This paper has been withdrawn by the author. This paper has been substantially rewritte

Multiple operator integrals and spectral shift

Multiple scalar integral representations for traces of operator derivatives are obtained and applied in the proof of existence of the higher order spectral shift functions.Comment: 18 pages, to appear in Illinois J. Mat

Trace formulas for tuples of commuting contractions

This paper extends the trace formulas of [5] with perturbations in normed ideals of $B(H)$ to multivariate functions of commuting contractions admitting a dilation to commuting normal contractions.Comment: 20 pages; several corrections made, abstract improved, references adde

About integrated invariance, arising at resonance of oscillations in some types non-conservative mechanical systems

For system of two ordinary differential equations of the second order representing autonomous non-conservative holonomic mechanical system, in case of dynamics such as one-frequency periodical oscillations, is found integrated invariant of movement.Comment: LaTeX2e, 3 page

Asymptotic expansions for trace functionals

We obtain Taylor approximations for functionals $V\mapsto Tr(f(H_0+V))$ defined on the bounded self-adjoint operators, where $H_0$ is a self-adjoint operator with compact resolvent and $f$ is a sufficiently nice scalar function, relaxing assumptions on the operators made in [17], and derive estimates and representations for the remainders of these approximations.Comment: 19 pages, J. Funct. Ana

About new quasilinear model of heat conduction with finite velocity of heat front movement

A new quasilinear mathematical model of heat conduction with finite velocity of heat front movement is offered.Comment: LaTeX2.09, 3 pages, 1 figur

On single commutators in II_1-factors

We investigate the question of whether all elements of trace zero in a II_1-factor are single commutators. We show that all nilpotent elements are single commutators, as are all normal elements of trace zero whose spectral distributions are discrete measures. Some other classes of examples are considered.Comment: 10 pages. version 2 modifies somewhat the discussion of Tucci's operators and corrects some minor mistakes. It will appear in Proc. Amer. Math. Soc version 3 adds references and makes minor correction

Perturbation formulas for traces on normed ideals

We prove perturbation results for traces on normed ideals in semifinite von Neumann algebra factors. This includes the case of Dixmier traces. In particular, we establish existence of spectral shift measures with initial operators being dissipative or bounded, and show that these measures can have singular components in the case of Dixmier traces. We also establish a linearization formula for a Dixmier trace applied to perturbed operator functions, a result that does not typically hold for normal traces.Comment: 33 pages, typos corrected; to appear in Comm. Math. Phy

On a perturbation determinant for accumulative operators

For a purely imaginary sign-definite perturbation of a self-adjoint operator, we obtain exponential representations for the perturbation determinant in both upper and lower half-planes and derive respective trace formulas.Comment: 15 page

Higher order spectral shift for contractions

We derive strong estimates for Schatten norms of operator derivatives along paths of contractions and apply them to prove existence of higher order spectral shift functions for pairs of contractions.Comment: 28 page