Anna Skripka

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.

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.

THE BIRMAN-SCHWINGER PRINCIPLE IN VON NEUMANN ALGEBRAS OF FINITE TYPE

Abstract We introduce a relative index for a pair of dissipative operators in a von Neumann algebra of finite type and prove an analog of the Birman–Schwinger principle in this setting. As an application of this result, revisiting the Birman–Krein formula in the abstract scattering theory, we represent the de la Harpe–Skandalis determinant of the characteristic function of dissipative operators in the algebra in terms of the relative index.

Taylor Approximations of Operator Functions

This survey on approximations of perturbed operator functions addresses recent advances and some of the successful methods.

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.

Upper Triangular Toeplitz Matrices and Real Parts of Quasinilpotent Operators

We show that every self--adjoint matrix B of trace 0 can be realized as B=T+T^* for a nilpotent matrix T of norm no greater than K times the norm of B, for a constant K that is independent of matrix size. More particularly, if D is a diagonal, self--adjoint n-by-n matrix of trace 0, then there is a unitary matrix V=XU_n, where X is an n-by-n permutation matrix and U_n is the n-by-n Fourier matrix, such that the upper triangular part, T, of the conjugate V^*DV of D has norm no greater than K times the norm of D. This matrix T is a strictly upper triangular Toeplitz matrix such that T+T^*=V^*DV. We apply this and related results to give partial answers to questions about real parts of quasinilpotent elements in finite von Neumann algebras.