WEIGHTED COMPOSITION OPERATORS ON THE MINIMAL MÖBIUS INVARIANT SPACE

Abstract

Abstract. We will characterize the boundedness and compactness ofweighted composition operators on the minimal Mobius invariant space. 1. IntroductionHere and henceforth, Dwill denote the open unit disk D:= {z ∈ C: |z| < 1}.The set of all conformal automorphisms of Dforms a group, called a Mo¨biusgroup and denoted by Aut(D). For any λ ∈ D, letα λ (z) =λ −z1−λzbe the Mo¨biustransformationof D. LetX be alinearspaceofanalyticfunctionson D. Then X is said to be Mo¨bius invariant if f ◦α ∈ X for all f ∈ X and allα ∈Aut(D). A typical example of Mo¨bius invariant spaces is the Besov space.For 1 < p < ∞, let B p be the space of analytic functions f on Dsuch thatZ D |f ′ (z)| p (1−|z|) p−2 dA(z) < ∞,where dA is the normalized Lebesgue area measure on D. Then B p is theBanach space with the normkfk B p = |f(0)|+Z D |f ′ (z)| p (1 −|z|) p−2 dA(z) 1/p .If p = 2, B 2 is the classical Dirichlet space that is minimal as Mo¨bius invariantHilbert space of analytic functions on D. The analytic Besov space B