NUMERICAL RANGE AND SOT-CONVERGENCY

Abstract

Abstract. AsequenceofcompositionoperatorsonHardyspaceiscon-sidered. Weprovethat,bynumericalrangeproperties,itisSOT-conver-gencebutnotconverge. 1. introductionLet ϕ be a holomorphic self-map of the unit disc U := {z ∈ C : |z| < 1}.The function ϕ induces the composition operator C ϕ , defined on the space ofholomorphic functions on Uby C ϕ f = f ◦ ϕ. The restriction of C ϕ to variousBanach spaces of holomorphic functions on U has been an active subject ofresearch for more than three decades and it will continue to be for decadesto come (see [11], [12] and [4]). Let H 2 denote the Hardy space of analyticfunctions on the open unit disc with square summable Taylor coefficients. Inrecent yearsthe study ofcomposition operatorson the Hardy space has receivedconsiderable attention.In this paper we consider the numerical range of elliptic composition opera-tors on H 2 . The numerical range of a bounded linear operator A on a Hilbertspace H is the set of complex numbersW(A) := {hAx,xi : x ∈ H,kxk = 1},where h·,· denotes the inner product in H.In [9] V. Matache determined the shape W(C