Some problems on composition operators. (English) Zbl 0908.47025

Cowen, Carl C.; MacCluer, Barbara D.

Jafari, Farhad (ed.) et al., Studies on composition operators. Proceedings of the Rocky Mountain Mathematics Consortium, Laramie, WY, USA, July 8–19, 1996. Providence, RI: American Mathematical Society. Contemp. Math. 213, 17-25 (1998).

The authors’ purpose is to highlight some open problems concerning composition operators, i.e., operators of the form \(C_\phi(f)= f\circ\phi\), where \(f\) is in a Banach or Hilbert space of analytic functions on a domain \(\Omega\) (in \(\mathbb{C}\) or \(\mathbb{C}^N\)), and \(\phi:\Omega\to \Omega\) is analytic.
Open questions concerning the norm and the essential norm of \(C_\phi\), the explicit form of the adjoint \(C^*_\phi\), the spectrum of \(C_\phi\) and its parts, the hyponormality of \(C_\phi\) or of its adjoint, the equivalence (similarity and unitary equivalence) of two composition operators, the description of the commutant of a composition operator and the automorphism invariance (i.e., the invariance of the basic space under \(C_\phi\) when \(\phi\) is an automorphism) are discussed, examples and counterexamples are given and some conjectures are stated.
For the entire collection see [Zbl 0880.00042].

Reviewer: F.H.Vasilescu (Villeneuve d’Ascq)

MSC:

47B33	Linear composition operators
47B20	Subnormal operators, hyponormal operators, etc.
00A27	Lists of open problems

Keywords:

essential norm; similarity; composition operators; adjoint; hyponormality; commutant of a composition operator; automorphism invariance