Inverses of Toeplitz plus Hankel operators with generating matrix functions

The invertibility of Toeplitz plus Hankel operators \(T(\mathcal {A})+H(\mathcal {B})\) $T(\mathcal{A})+H(\mathcal{B})$ , \(\mathcal {A},\mathcal {B}\in L^\infty _{d\times d}(\mathbb {T})\) $\mathcal{A},\mathcal{B}\in L_{d\times d}^{\infty }(\mathbb{T})$ acting on vector Hardy spaces \(H^p_d(\mathbb {T})\) $H_d^p(\mathbb{T})$ , \(1<p<\infty \) $1<p<\infty$ , is studied. Assuming that the generating matrix functions \(\mathcal {A}\) $\mathcal{A}$ and \(\mathcal {B}\) $\mathcal{B}$ satisfy the equation \(\begin{aligned} \mathcal {B}^{-1} \mathcal {A}= \widetilde{\mathcal {A}}^{-1}\widetilde{\mathcal {B}}, \end{aligned}\) $$\begin{array}{c}\hfill {\mathcal{B}}^{-1}\mathcal{A}={\tilde{\mathcal{A}}}^{-1}\tilde{\mathcal{B}},\end{array}$$ where \(\widetilde{\mathcal {A}}(t):=\mathcal {A}(1/t)\) $\tilde{\mathcal{A}}(t):=\mathcal{A}(1/t)$ , \(\widetilde{\mathcal {B}}(t):=\mathcal {B}(1/t)\) $\tilde{\mathcal{B}}(t):=\mathcal{B}(1/t)$ , \(t\in \mathbb {T}\) $t\in \mathbb{T}$ , we establish sufficient conditions for the one-sided invertibility and invertibility of the operators mentioned and construct the corresponding inverses. If \(d=1\) $d=1$ , the above equation reduces to the known matching condition, widely used in the study of Toeplitz plus Hankel operators with scalar generating functions.