Minimal invariant subspaces for composition operators
The research of this paper is motivated by a remarkable result by \textit{E. A. Nordgren}, \textit{P. Rosenthal} and \textit{F. S. Wintrobe} [C. R. Math. Acad. Sci., Soc. R. Can. 6, 279--283 (1984; Zbl 0599.47041)] stating that a positive answer to the invariant subspace problem for operators on complex separable Hilbert space is equivalent to the statement that every minimal invariant subspace for a composition operator \(C_\varphi\) induced by a hyperbolic automorphism \(\varphi\) on the unit disc \(\mathbb{D}\) in the Hardy space \(H^2\) is one-dimensional. If \(f \in H^2\), the cyclic subspace \(K_f\) generated by \(f\) is the closure in \(H^2\) of the orbit of \(f\). If \(\varphi\) is a hyperbolic automorphism, then \(K_f\) is the minimal closed \(C_\varphi\)-invariant subspace containing \(f\). In view of results due to Matache and Mortini in the mid 1990's, the authors focus on trying to understand the behaviour of functions in \(K_f\) tending to \(0\) near the attractive fixed point. Accordingly, they consider functions \(f\) for which the sequence \((f(\varphi_n(z_0))_n\) tends to \(0\) for a fixed point \(z_0\) in \(\mathbb{D}\). Assuming that \(f \in H^2\) is bounded in a neighbourhood of the attractive fixed point, they show that \(K_f\) contains a function of the form \(Bh\), where \(h \neq 0\) and \(B\) vanishes on the set of iterates \((\varphi_n(z_0))_n\). Moreover, they prove that, if \(f\) has a nonzero limit on the sequence of iterates \((\varphi_n(z_0))_n\) and \(K_f\) is minimal, then \(f\) is constant on the sequence. In Section 3, the following explicit construction is made: Let \(\varphi\) be a hyperbolic disc automorphism fixing the points \(1\) and \(-1\). If \(f\) is not an eigenvalue of \(C_\varphi\) and has radial limit \(0\) at \(1\) and \(-1\), then there exists \(g \in H^\infty \cap K_f\) such that \(g\) does not have radial limit \(0\) at \(-1\). In this way, the discussion on minimality is moved from a space with no interesting cluster points to one with interesting cluster points.
scientific article; zbMATH DE number 5860470