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 [ SO,PC ] of the Banach algebra ℬ( L n p (ℝ)) of all bounded linear operators on the Lebesgue spaces L n p (ℝ) (1 < p < ∞, n ≥ 1) which contains all the convolution type operators W a,b = a ℱ −1 b ℱ with a ∈ [ SO, PC ] n × n and b ∈ [ SO p , PC p ] 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 [ SO p , PC p ] n × n is a Fourier multiplier analogue of [ SO, PC ] n × n on L p (ℝ). As a result, a Fredholm criterion for the operators A ∈ [ SO,PC ] is established. The study is based on the compactness of the commutators AW a,b − W a,b A where A ∈ [ SO,PC ] , a ∈ SO , and b ∈ SO p , on the Allan‐Douglas local principle, and on the two projections theorem. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)