Closed range integral operators on Hardy, BMOA and Besov spaces

ABSTRACT

If $g\in H^{\mathrm{\infty }}$, the integral operator $S_g$ on $H^p$, BMOA and $B^p$(Besov) spaces, is defined as $S_{g}f(z)={\int }_0^z{f}^{\mathrm{\prime }}(w)g(w)\mathrm{d}w$. In this paper, we prove three necessary and sufficient conditions for the operator $S_g$ to have closed range on $H^p(1\le p<\mathrm{\infty })$, BMOA and $B^p(1<p<\mathrm{\infty })$.

AMS SUBJECT CLASSIFICATIONS:

• Primary 47B38
• Secondary 47G10
• 30D45

Acknowledgements

Many thanks to Prof. Michael Papadimitrakis for discussions about the mathematical content of this paper. His contribution was essential in order for it to take its final form. I also thank Prof. Petros Galanopoulos for helpful discussions.

Disclosure statement

No potential conflict of interest was reported by the author(s).