The invertibility of Toeplitz plus Hankel operators with subordinated operators of even index

Toeplitz plus Hankel operators T(a) + H(b) acting on Hardy spaces H-P(T), p is an element of(1, infinity), where T is the unit circle, are studied. If the functions a, b is an element of L-infinity(T) satisfy the relation a(t)a(1/t) = b(t)b(1/t), t is an element of T and the Toeplitz operators T(ab(-1)) and T(a (b) over tilde (-1)), (b) over tilde (t) = b(1/t) have even indices, necessary and sufficient conditions for the invertibility of T(a) + H(b) are established and efficient formulas for their inverses are obtained. Moreover, it is shown that for any n is an element of N there are invertible operators T(a) + H(b) such that ind T(ab(-1)) = -2n and ind T(a (b) over tilde (-1)) = 2n. (C) 2019 Elsevier Inc. All rights reserved.