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 a...

The Lifshits–Krein spectral shift function is considered for the pair of operators H0 = (−4)l, l > 0 and H = H0 + V in L2(R), d ≥ 1; here V is a multiplication operator. The estimates for this spectral shift function ξ(λ;H,H0) are obtained in terms of the spectral parameter λ > 0 and the integral norms of V . These estimates are in a good agreement with the ones predicted by the classical phase...

Fundamental problems of quantum field theory related to the representation problem of canonical commutation relations are discussed within a gauge field version of a van Hove-type model. The Coulomb field generated by a static charge distribution is described as a formal superposition of time-like pseudo-photons in Fock space with a Krein structure. In this context, a generalization of operator...

In a series of papers, we have shown that from the representation theory of a compact groupoid one can reconstruct the groupoid using the procedure similar to the Tannaka-Krein duality for compact groups. In this part we study continuous representations of compact groupoids. We show that irreducible representationshave finite dimensional fibres. We prove the Schur’s lemma, Gelfand-Raikov theore...

The aim of the present paper is to introduce the asymmetric locally convex spaces and to prove some basic properties. Among these I do mention the analogs of the EidelheitTuckey separation theorems, of the Alaoglu-Bourbaki theorem on the weak compactness of the polar of a neighborhood of 0, and a Krein-Milman-type theorem. These results extend those obtained by Garcı́a-Raffi et al. (2003) and Co...

The spectral shift function for a trace class perturbation of a self-adjoint (unitary) operator plays a very important role in perturbation theory. It was introduced in a special case by I.M. Lifshitz [L] and in the general case by M.G. Krein [Kr1]. He showed that for a pair of self-adjoint (not necessarily bounded) operators A and B satisfying B −A ∈ S1 there exists a unique function ξ ∈ L 1(R...

We discuss magnetic Schrödinger operators perturbed by measures from the generalized Kato class. Using an explicit Krein-like formula for their resolvent, we prove that these operators can be approximated in the strong resolvent sense by magnetic Schrödinger operators with point potentials. Since the spectral problem of the latter operators is solvable, one in fact gets an alternative way to ca...

In this note we prove a relation between the Riemann Zeta function, ζ and the ξ function (Krein spectral shift) associated with the Harmonic Oscillator in one dimension. This gives a new integral representation of the zeta function and also a reformulation of the Riemann hypothesis as a question in L1(R). ∗Part of talk presented at the Conference on Harmonic Analysis, 13-15 March 1997, Ramanuja...

L. Terry Spencer, Gregorino Paone, Peter M. Krein, Farshid N. Rouhani, Jesus Rivera-Nieves, and Mark L. Brantly Division of Pulmonary Diseases, Department of Pediatrics, and Division of Pulmonary and Critical Care Medicine, Department of Medicine, University of Florida College of Medicine, Gainesville, Florida 32610; and Clinical Studies Section, Pulmonary-Critical Care Medicine Branch, Nationa...

Abstract. In a series of papers, we have shown that from the representation theory of a compact groupoid one can reconstruct the groupoid using the procedure similar to the Tannaka-Krein duality for compact groups. In this part we study the Fourier and Fourier-Plancherel transforms and prove the Plancherel theorem for compact groupoids. We also study the central functions in the algebra of squa...