Composition operator on F(p,q,s) spaces in the unit ball of

Abstract

In this paper, the authors characterize those φ, holomorphic self-maps of B, for which the composition operator $C_{\phi }$ is bounded (or compact) on F(p,q,s) in some cases. Moreover, the following results are given.

1. If 1<p<q + n + 1 and $C_{\phi }$ satisfies some boundedness conditions, then $C_{\phi }$ is compact on F(p,q,s) if and only if $\underset{|z|\to 1^{-}}{lim}\frac{1-|z{|}^2}{1-|\phi \left(z\right){|}^2}=0.$

2. If $1\le p<\mathrm{\infty }$ and $q+n+1<p<\mathrm{\infty }$, then $C_{\phi }$ is a compact operator on F(p,q,s) if and only if $\Vert \phi {\Vert }_{\mathrm{\infty }}<1$ and ${\phi }_l\in F(p,q,s)$ for all $l\in \{1,2,\dots ,n\}$.

Keywords:

• Composition operator
• F(p
• q
• s) space
• boundedness
• compactness
• unit ball

AMS Subject Classifications:

• 47B38
• 32A37

Disclosure statement

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

Additional information

Funding

The research is supported by the National Natural Science Foundation of China (No. 11942109), the Natural Science Foundation of Hunan Province (No. 2022JJ30369), the Science Research Project of Education Department of Hunan Province of China (No. 21C0575) and the Science Research Project of Hunan Institute of Engineering (No. 22RC019).