Generalized Toeplitz plus Hankel operators: kernel structure and defect numbers

Generalized Toeplitz plus Hankel operators $T(a)+H_{\alpha}(b)$ generated by functions $a,b$ and a linear fractional Carleman shift $\alpha$ changing the orientation of the unit circle $\mathbb{T}$ are considered on the Hardy spaces $H^p(\mathbb{T})$, $1<p<\infty$. If the functions $a,b\in L^\infty(\mathbb{T})$ and satisfy the condition $$ a(t) a(\alpha(t))=b(t) b(\alpha(t)),\quad t\in \mathbb{T}, $$ the defect numbers of the operators $T(a)+H_{\alpha}(b)$ are established and an explicit description of the structure of the kernels and cokernels of the operators mentioned is given.