# Strict singularity of Volterra type operators on Hardy spaces

## TL;DR

This paper characterizes the boundedness, compactness, and spectrum of Volterra type operators on Hardy spaces, showing that non-compactness implies the operator fixes copies of ^p and ^2, thus linking strict singularity with compactness.

## Contribution

It provides a complete characterization of the spectral and compactness properties of Volterra type operators on Hardy spaces, establishing their strict singularity equivalence with compactness.

## Key findings

- The spectrum of the operator is explicitly determined.
- Non-compact operators fix isomorphic copies of ^p and ^2.
- Strict singularity coincides with compactness for these operators.

## Abstract

In this paper, we first characterize the boundedness and compactness of Volterra type operator $S_gf(z) = \int_0^z f'(\zeta)g(\zeta)d\zeta, \ z \in \mathbb{D},$ defined on Hardy spaces $H^p, \, 0< p <\infty$. The spectrum of $S_g$ is also obtained. Then we prove that $S_g$ fixes an isomorphic copy of $\ell^p$ and an isomorphic copy of $\ell^2$ if the operator $S_g$ is not compact on $H^p (1\leq p<\infty)$. In particular, this implies that the strict singularity of the operator $S_g$ coincides with the compactness of the operator $S_g$ on $H^p$. At last, we post an open question for further study.