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

Abstract

Toeplitz plus Hankel operators T(a)+H(b) acting on Hardy spaces Hp(T), p∈(1,∞), where T is the unit circle, are studied. If the functions a,b∈L∞(T) satisfy the relation a(t)a(1/t)=b(t)b(1/t), t∈T and the Toeplitz operators T(ab−1) and T(ab˜−1), b˜(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∈N there are invertible operators T(a)+H(b) such that indT(ab−1)=−2n and indT(ab˜−1)=2n.