Toeplitz operators with symbols generated by slowly oscillating and semi-almost periodic matrix functions

Abstract

We develop the Fredholm theory for Toeplitz operators, with symbols in the C*-algebra D = [SO, SAP]n, n generated by all slowly oscillating (SO) and semi-almost periodic (SAP) n × n matrix functions, on the Hardy spaces H n p (with 1 < p < ∞) over the upper half-plane. Using limit operator techniques, we get necessary Fredholm conditions for any operator in the Banach algebra alg(S, D) of singular integral operators with coefficients in D on the space [Lp (R)]n. Applying the Allan–Douglas local principle and the theory of Toeplitz operators with SAP matrix symbols, we also establish Fredholm criteria for Toeplitz operators with matrix symbols g ∈ D on the space H n p . An index formula for Fredholm Toeplitz operators with matrix symbols in D is obtained on the basis of an appropriate approximation of slowly oscillating components of the symbols. 2000 Mathematics Subject Classification 47B35 (primary), 47A53 (secondary).