Index formula for convolution type operators with data functions in alg(SO,PC)

Abstract

We establish an index formula for the Fredholm convolution type operators A = m k=1 a k W 0 (b k ) acting on the space L 2 (R), where a k , b k belong to the C * -algebra alg(SO, P C) of piecewise continuous functions on R that admit finite sets of discontinuities and slowly oscillate at ±∞, first in the case where all a k or all b k are continuous on R and slowly oscillating at ±∞, and then assuming that a k , b k ∈ alg(SO, P C) satisfy an extra Fredholm type condition. The study is based on a number of reductions to operators of the same form with smaller classes of data functions a k , b k , which include applying a technique of separation of discontinuities and eventually lead to the so-called truncated operators A r = m k=1 a k,r W 0 (b k,r ) for sufficiently large r > 0, where the functions a k,r , b k,r ∈ P C are obtained from a k , b k ∈ alg(SO, P C) by extending their values at ±r to all ±t ≥ r, respectively. We prove that ind A = lim r→∞ ind A r although A = s-lim r→∞ A r only.