Symbol calculus and Fredholmness for a Banach algebra of convolution type operators with slowly oscillating and piecewise continuous data

A symbol calculus for the smallest Banach subalgebra A[SO,PC] of the Banach algebra B︁(Lnp(R)) of all bounded linear operators on the Lebesgue spaces Lnp(R) (1 < p < ∞, n ≥ 1) which contains all the convolution type operators Wa,b = aF−1bF with a ∈ [SO, PC]n×n and b ∈ [SOp, PCp]n×n is constructed. Here [SO, PC]n×n means the C*‐algebra generated by all slowly oscillating (SO) and all piecewise continuous (PC) n × n matrix functions, and [SOp, PCp]n×n is a Fourier multiplier analogue of [SO, PC]n×n on Lp(R). As a result, a Fredholm criterion for the operators A ∈ A[SO,PC] is established. The study is based on the compactness of the commutators AWa,b − Wa,bA where A ∈ A[SO,PC], a ∈ SO, and b ∈ SOp, on the Allan‐Douglas local principle, and on the two projections theorem. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)