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=∑k=1makW0(bk) acting on the space L2(R), where ak, bk belong to the C⁎-algebra alg(SO,PC) of piecewise continuous functions on R that admit finite sets of discontinuities and slowly oscillate at ±∞, first in the case where all ak or all bk are continuous on R and slowly oscillating at ±∞, and then assuming that ak,bk∈alg(SO,PC) 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 ak, bk, which include applying a technique of separation of discontinuities and eventually lead to the so-called truncated operators Ar=∑k=1mak,rW0(bk,r) for sufficiently large r>0, where the functions ak,r,bk,r∈PC are obtained from ak,bk∈alg(SO,PC) by extending their values at ±r to all ±t≥r, respectively. We prove that indA=limr→∞⁡indAr although A=s-limr→∞Ar only.