Some problems on composition operators. (English) Zbl 0908.47025

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].


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