Compact composition operators on BMOA and the Bloch space (original) (raw)
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Compact Composition Operators on BMOA
Transactions of the American Mathematical Society
We characterize the compact composition operators on BMOA, the space consisting of those holomorphic functions on the open unit disk U U that are Poisson integrals of functions on ∂ U \partial U , that have bounded mean oscillation. We then use our characterization to show that compactness of a composition operator on BMOA implies its compactness on the Hardy spaces (a simple example shows the converse does not hold). We also explore how compactness of the composition operator C ϕ : BMOA → BMOA C_\phi : \operatorname {BMOA}\rightarrow \operatorname {BMOA} relates to the shape of ϕ ( U ) \phi (U) near ∂ U \partial U , introducing the notion of mean order of contact. Finally, we discuss the relationships among compactness conditions for composition operators on BMOA, VMOA, and the big and little Bloch spaces.
Estimates of essential norms of weighted composition operators between Bloch-type spaces
Journal of Mathematical Analysis and …, 2012
We estimate the essential norm of weighted composition operators uCφuCφ acting on Bloch-type spaces BαBα in terms of the analytic function u:D→Cu:D→C and the nn-th power of the analytic selfmap φφ of the open unit disc DD. We obtain new characterizations for boundedness and compactness of uCφ:Bα→BβuCφ:Bα→Bβ for all 0<α,β<∞0<α,β<∞, thus answering an open problem of Manhas and Zhao concerning the case α=1α=1.
Journal of Function Spaces, 2020
In this paper, we obtain some characterizations of composition operators Cφ, which are induced by an analytic self-map φ of the unit disk Δ, from hyperbolic Bloch type space βμ∗ into hyperbolic type space QK,p,q∗.