Finite-dimensional Toeplitz kernels and nearly-invariant subspaces (Q2835226)

Let \(H^p(\mathbb D),\, 1<p<\infty\), denote the Hardy space on the unit disc \(\mathbb D\), and let \(S^\ast\) denote the backward shift operator. A closed subspace \(\mathcal E\) of \(H^2(\mathbb D)\) is nearly \(S^\ast\)-invariant if \(z^{-1}\mathcal E\cap H^2 \subset \mathcal E\). This concept was introduced by \textit{D. Hitt} who in [Pac. J. Math. 134, No. 1, 101--120 (1988; Zbl 0662.30035)] characterized the nearly \(S^\ast\)-invariant subspaces of \(H^2\). In the paper under review, the authors consider finite dimensional nearly \(S^\ast\)-invariant subspaces in the Hardy spaces \(H_p^+, 1<p<\infty\), on the upper half-plane. One of the main results of the paper is the following. Suppose that \(\mathcal E\subset H_p^+\) and \(\dim \mathcal E=N\). Then \(\mathcal E\) is nearly \(S^\ast\)-invariant if and only if: NEWLINE{\parindent=7mm\begin{itemize}\item[(1)] \(\mathcal E\) contains at least one function that does not vanish at \(i\), and NEWLINE\item[(2)] the quotient of any two functions in \(\mathcal E\) is equal to a quotient of two polynomials of degree at most \(N-1\). NEWLINENEWLINE\end{itemize}} NEWLINEIn [Proc. Am. Math. Soc. 110, No. 2, 441--448 (1990; Zbl 0716.30028)], \textit{E. Hayashi} characterized the nontrivial closed subspaces of \(H^2(\mathbb D)\) which are kernels of Toeplitz operators. In the paper under review, an analogue of Hayashi's theorem is given for finite dimensional Toeplitz kernels on \(H^+_p\).