Spectra of Some Composition Operators

✍ Scribed by C.C. Cowen; B.D. Maccluer

• Publisher: Elsevier Science
• Year: 1994
• Tongue: English
• Weight: 817 KB
• Volume: 125
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1994.1123

✦ Synopsis

If (\mathscr{H}) is a Hilbert space of holomorphic functions on the unit ball (B_{N}) in (\mathbf{C}^{N}) and (\varphi) is a non-constant holomorphic map of the unit ball into itself, the composition operator (C_{\varphi}) is the operator on (\mathscr{H}) defined by (C_{\varphi} f=f \circ \varphi). In this paper, we give spectral information 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 (\mathscr{H}) is the usual Hardy space or a standard weighted Bergman space on the unit disk, this information shows that the spectrum of the composition operator is the disk centered at 0 whose radius is the essential spectral radius of the operator together with some isolated eigenvalues. 1994 Academic Press, Inc.