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We develop a complete Fredholm and invertibility theory for Toeplitz+Hankel operators T (a) + H(b) on the Hardy space Hp(T), 1 < p < ∞, with piecewise continuous functions a, b defined on the unit circle which are subject to the condition a(t)a(t−1) = b(t)b(t−1), |t| = 1. In particular, in the case of Fredholmness, formulas for the defect numbers are established. The results are applied to several important examples.

Fredholm and invertibility theory for a special class of Toeplitz + Hankel operators