Search

We study some basic properties of the class of universal operators on Hilbert spaces, and provide new examples of universal operators and universal pairs. [ABSTRACT FROM AUTHOR]

We survey recent results about composition operators induced by analytic self-maps of the unit disk in the complex plane on various Banach spaces of analytic functions taking values in infinitedimensional Banach spaces. We mostly concentrate on the research line into qualitative properties such as weak compactness, initiated by Liu, Saksman and Tylli (1998), and continued in several other papers. We discuss composition operators on strong, respectively weak, spaces of vector-valued analytic functions, as well as between weak and strong spaces. As concrete examples, we review more carefully and present some new observations in the cases of vector-valued Hardy and BMOA spaces, though the study of composition operators has been extended to a wide range of spaces of vector-valued analytic functions, including spaces defined on other domains. Several open problems are stated. [ABSTRACT FROM AUTHOR]

Let ϕ be an analytic map taking the unit disk D into itself. We establish that the class of composition operators f ↦ C ϕ ( f ) = f ∘ ϕ exhibits a rather strong rigidity of non-compact behaviour on the Hardy space H p , for 1 ≤ p < ∞ and p ≠ 2 . Our main result is the following trichotomy, which states that exactly one of the following alternatives holds: (i) C ϕ is a compact operator H p → H p , (ii) C ϕ fixes a (linearly isomorphic) copy of ℓ p in H p , but C ϕ does not fix any copies of ℓ 2 in H p , (iii) C ϕ fixes a copy of ℓ 2 in H p . Moreover, in case (iii) the operator C ϕ actually fixes a copy of L p ( 0 , 1 ) in H p provided p > 1 . We reinterpret these results in terms of norm-closed ideals of the bounded linear operators on H p , which contain the compact operators K ( H p ) . In particular, the class of composition operators on H p does not reflect the quite complicated lattice structure of such ideals. [ABSTRACT FROM AUTHOR]