PATH CONNECTED COMPONENTS IN THE SPACE OF WEIGHTED COMPOSITION OPERATORS ON THE DISK ALGEBRA

Abstract

Let H2 be the Hardy space and H°° the Banach space of bounded analytic functions on the open unit disk ID with the supremum norm || • Hoc. For / £ H°°, we denote by /* the radial limit of / defined a.e. on the boundary 3D of P. We write {|/*| = 1} = {el° £ 3D : \f*(ete)\ — 1}. Let m be the normalized Lebesgue measure on 3D. We denote by 5#* the set of analytic self-maps of D and Sh^.o — G Sh=c : Hvlloo = !}• For p £ Shthe composition operator C^ on H2 is defined by C^f = f o tp for / £ H2. Let C(H2) be the space of composition operators on H2 with the operator norm. The most interesting subject in the study of C(H2) is called the connected component problem, describing the connected component containing a given Cv. It was first studied by Berkson [2]. He showed that if p € 0, then is an isolated point in C(H2). Around 1990, MacCluer [13] and Shapiro and Sundberg [19] revisited the connected component problem on H2. Succeedingly, Bourdon [3], Gallardo-Gutierrez, Gonzalez, Nieminen and Saksman [6] and Moorhouse and Toews [16] followed. Still it remains unclear the connected component problem on H2 (see [4; 17; 18]). Similarly we have the space C(H°°) of composition operators on H°° with the