## Banach Journal of Mathematical Analysis

### The associated weight and the essential norm of weighted composition operators

#### Abstract

For an almost radial and typical weight $v$ and any weight $w$, we characterize the continuity, compactness and we estimate the essential norm of weighted composition operators $u C_{\varphi}$, acting from the weighted Banach spaces of analytic functions $H_{v}^{\infty}$ into $H_{w}^{\infty}$, in terms of the quotients of the $w$-norm of the product of $u$ with $\varphi^n$ and the $v$-norm of the $n$th power of the identity function on $\Bbb D$, where $u: \mathbb{D} \to \mathbb{C}$ and $\varphi: \mathbb{D} \to \mathbb{D}$ are analytic. As a consequence, we estimate the essential norm of composition operators $C_\varphi$ (in terms of $\varphi^n$) acting on $\mu$-Bloch spaces, for very general weights $\mu$. We also characterize continuity and compactness of weighted composition operators $uC_\varphi$ acting on $\log$-Bloch space.

#### Article information

Source
Banach J. Math. Anal., Volume 9, Number 1 (2015), 144-158.

Dates
First available in Project Euclid: 19 December 2014

https://projecteuclid.org/euclid.bjma/1419000584

Digital Object Identifier
doi:10.15352/bjma/09-1-12

Mathematical Reviews number (MathSciNet)
MR3296092

Zentralblatt MATH identifier
1310.47038

Subjects
Primary: 46B33
Secondary: 47B38: Operators on function spaces (general) 30H30: Bloch spaces

#### Citation

Malavé-Ramìrez, Marìa T.; Ramos-Fernández, Julio C. The associated weight and the essential norm of weighted composition operators. Banach J. Math. Anal. 9 (2015), no. 1, 144--158. doi:10.15352/bjma/09-1-12. https://projecteuclid.org/euclid.bjma/1419000584