Spectra of some composition operators. (English) Zbl 0814.47040

Cowen, Carl C.; MacCluer, Barbara D.

J. Funct. Anal. 125, No. 1, 223-251 (1994).

If \(\mathcal H\) is a Hilbert space of holomorphic functions on the unit ball \(B_ N\) in \({\mathbf C}^ N\) and \(\phi\) is a non-constant holomorphic map of the unit ball into itself, the composition operator \(C_ \varphi\) is the operator on \(\mathcal H\) defined by \(C_ \varphi f= f\circ \varphi\). The authors give spectral properties for bounded composition operators on some weighted Hardy spaces under the condition that \(\varphi\) is univalent and has a fixed point in the ball. When \(\mathcal H\) is the usual Hardy space or a standard Bergman space on the unit disk, these properties show that the spectrum of the composition operator is the disk centered at 0 whose radius is the essential spectral radius of the operator.

Reviewer: V.S.Rabinovich (Rostov-na-Donu)

MSC:

47B38

Linear operators on function spaces (general)

Keywords:

Hilbert space of holomorphic functions on the unit ball; holomorphic map; composition operator; spectral properties for bounded composition operators on some weighted Hardy spaces