The invariant subspaces of the shift plus integer multiple of Volterra operator on Hardy spaces

TL;DR

This paper extends the characterization of invariant subspaces of the shift plus Volterra operator from Hilbert spaces to Hardy spaces $H^p$, providing a comprehensive understanding of their structure across different $p$ values.

Contribution

It offers a complete description of invariant subspaces for the shift plus a positive integer multiple of the Volterra operator on Hardy spaces $H^p$, generalizing previous results.

Findings

Characterization of invariant subspaces on $H^p$ spaces.

Extension of previous Hilbert space results to $H^p$.

Framework applicable for all $1 \,\leq\, p < \infty$.

Abstract

\v{C}u\v{c}kovi\'{c} and Paudyal recently characterized the lattice of invariant subspaces of the shift plus a complex Volterra operator on the Hilbert space $H^2$ on the unit disk. Motivated by the idea of Ong, in this paper, we give a complete characterization of the lattice of invariant subspaces of the shift operator plus a positive integer multiple of the Volterra operator on Hardy spaces $H^p$, which essentially extends their works to the more general cases when $1\leq p<\infty$.