Inverses of Toeplitz plus Hankel operators with generating matrix functions

The invertibility of Toeplitz plus Hankel operators \(T(\mathcal {A})+H(\mathcal {B})\), \(\mathcal {A},\mathcal {B}\in L^\infty _{d\times d}(\mathbb {T})\) acting on vector Hardy spaces \(H^p_d(\mathbb {T})\), \(1<p<\infty \), is studied. Assuming that the generating matrix functions \(\mathcal {A}\) and \(\mathcal {B}\) satisfy the equation

$$\begin{aligned} \mathcal {B}^{-1} \mathcal {A}= \widetilde{\mathcal {A}}^{-1}\widetilde{\mathcal {B}}, \end{aligned}$$

where \(\widetilde{\mathcal {A}}(t):=\mathcal {A}(1/t)\), \(\widetilde{\mathcal {B}}(t):=\mathcal {B}(1/t)\), \(t\in \mathbb {T}\), we establish sufficient conditions for the one-sided invertibility and invertibility of the operators mentioned and construct the corresponding inverses. If \(d=1\), the above equation reduces to the known matching condition, widely used in the study of Toeplitz plus Hankel operators with scalar generating functions.