Composition operators on Hardy-Sobolev spaces with bounded reproducing kernels

2021/01/11 by Cao, Guangfu, He, Li, Huang, Sui
#30H05 #47B91 #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2101.03659

Abstract

For any real β let H²_β be the Hardy-Sobolev space on the unit disc \mathbbD. H²_β is a reproducing kernel Hilbert space and its reproducing kernel is bounded when β>1/2. In this paper, we characterize that for a non-constant analytic function φ:\mathbbD→\mathbbD, when the composition operator C_φ on H²_β is Fredholm. For 1/2<β<1, we also prove that C_φ has dense range in H_β² if and only if the polynomials are dense in a certain Dirichlet space of the domain φ(\mathbbD). It follows that if the range of C_φ is dense in H_β², then φ is a weak-star generator of H^∞, although the conclusion is false for the classical Dirichlet space \mathfrakD. Moreover, we study the relation between the density of the rang of C_φ and the cyclic vector of the multiplier M_φ^β.

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