Abstract. We present a new and simple approach to the theory of multiple operator integrals that applies to unbounded operators affiliated with general von Neumann algebras. For semifinite von Neumann algebras we give applications to the Fréchet differentiation of operator functions that sharpen existing results, and establish the Birman-Solomyak representation of the spectral shift function of M. G. Krein in terms of an average of spectral measures in the type II setting. We also exhibit a surprising connection between the spectral shift function and spectral flow.

At the 1974 International Congress, I. M. Singer proposed that eta invariants and hence spectral flow should be thought of as the integral of a one form. In the intervening years this idea has lead to many interesting developments in the study of both eta invariants and spectral flow. Using ideas of [24] Singer's proposal was brought to an advanced level in [16] where a very general formula for spectral flow as the integral of a one form was produced in the framework of noncommutative geometry. This formula can be used for computing spectral flow in a general semifinite von Neumann algebra as described and reviewed in [5]. In the present paper we take the analytic approach to spectral flow much further by giving a large family of formulae for spectral flow between a pair of unbounded self-adjoint operators D and D + V with D having compact resolvent belonging to a general semifinite von Neumann algebra N and the perturbation V ∈ N . In noncommutative geometry terms we remove summability hypotheses. This level of generality is made possible by introducing a new idea from [3]. There it was observed that M. G. Krein's spectral shift function (in certain restricted cases with V trace class) computes spectral flow. The present paper extends Krein's theory to the setting of semifinite spectral triples where D has compact resolvent belonging to N and V is any bounded selfadjoint operator in N . We give a definition of the spectral shift function under these hypotheses and show that it computes spectral flow. This is made possible by the understanding discovered in the present paper of the interplay between spectral shift function theory and the analytic theory of spectral flow. It is this interplay that enables us to take Singer's idea much further to create a large class of one forms whose integrals calculate spectral flow. These advances depend critically on a new approach to the calculus of functions of non-commuting operators discovered in [3] which generalizes the double operator integral formalism of [8,9,10]. One surprising conclusion that follows from our results is that the Krein spectral shift function 1 2 is computed, in certain circumstances, by the Atiyah-Patodi-Singer index theorem [2].

We show the existence of a spectral shift function in the sense of Krein for bounded trace class perturbations of a self-adjoint operator affiliated with a semifinite von Neumann algebra Mathematics Subject Classification (2000). Primary 46E30; Secondary 46L51, 46L52.

The spectral flow is a classical notion of functional analysis and differential geometry which was given different interpretations as Fredholm index, Witten index, and Maslov index. The classical theory treats spectral flow outside the essential spectrum. Inside essential spectrum, the spectral shift function could be considered as a proper analogue of spectral flow, but unlike the spectral flow, the spectral shift function is not an integer-valued function.In this paper it is shown that the notion of spectral flow admits a natural extension for a.e. value of the spectral parameter inside essential spectrum too and appropriate theory is developed. The definition of spectral flow inside essential spectrum given in this paper applies to the classical spectral flow and thus gives one more new alternative definition of it.One of the results of this paper is the following Theorem. Let H0 be a self-adjoint operator and let V be a trace class self-adjoint operator acting on a separable Hilbert space. Let Hs = H0 + sV, s ∈ C. The following four functions are equal for a.e. λ; their common value is the spectral flow inside essential spectrum by definition.

It has been shown recently that spectral flow admits a natural integer-valued extension to essential spectrum. This extension admits four different interpretations; two of them are singular spectral shift function and total resonance index. In this work we study resonance index outside essential spectrum.Among results of this paper are the following. 1. Total resonance index satisfies Robbin-Salamon axioms for spectral flow.2. Direct proof of equality "total resonance index = intersection number". 3. Direct proof of equality "total resonance index = total Fredholm index". 4. (a) Criteria for a perturbation V to be tangent to the resonance set at a point H, where the resonance set is the infinite-dimensional variety of self-adjoint perturbations of the initial self-adjoint operator H0 which have λ as an eigenvalue. (b) Criteria for the order of tangency of a perturbation V to the resonance set.5. Investigation of the root space of the compact operator (H0This analysis gives a finer information about behaviour of discrete spectrum compared to spectral flow.Finally, many results of this paper are non-trivial even in finite dimensions, in which case they can be and were tested in numerical experiments.2000 Mathematics Subject Classification. Primary 47A40, 47A55 . Key words and phrases. spectral flow, resonance index, Robbin-Salamon axioms. 1 55 6.1. Topology of the vector space of directions 56 6.2. Continuous dependence of P λ and V P λ on simple directions 57 6.3. Homotopy stability of total resonance index 58 6.4. Robbin-Salamon axioms for spectral flow and resonance index 61 6.5. Geometric properties of the resonance set 62 7. Resonance index and Fredholm index 63 7.1. Resonance matrix as direction reduction 63 7.2. Resonance index and Fredholm index 66 8. Resonance index and spectral shift function 68 References 71In such diagram each square represents a resonance vector from a Jordan basis, and the height of the square is the order of the vector. The number of squares is N, the width is m, and the height is d. Each Jordan basis defines a direct sum decomposition of the resonance vector space Υ z (r z ), the νth summand of which we denote by ΥThe vector space Υ 1 z (z; H 0 , V ) depends only on the operator H rz = H 0 + r z V and does not depend on V, but the vector spaces Υ k z (r z ; H 0 , V ), k 2, depend on both H rz and V. A complex number r z is a resonance point iff