Abstract
Generalized composition operators from area Nevanlinna spaces to Zygmund-type spaces
Highlights
Let C be the complex plane of the open unit disk, H(D) the space of all analytic function on D
The little Zygmund-type space Zμ,0 consists of those functions f in Zμ satisfying lim|z|→1− μ(|z|)|f (z)| = 0 and it is easy to see that Zμ,0 is a closed subspace of Zμ
Taking f (z) = z and f (z) = z2, and obviously each of them belongs to Nαp, and using the boundedness of the function φ(z), we get sup μ(|z|)|g (z)| < ∞
Summary
Let C be the complex plane of the open unit disk, H(D) the space of all analytic function on D. A closed set K in Zμ,0 is compact if and only if it is bounded and satisfies lim sup μ(|z|)|f (z)| = 0. Suppose that Cφg : Nαp →Zμ is bounded. Taking f (z) = z and f (z) = z2, and obviously each of them belongs to Nαp, and using the boundedness of the function φ(z), we get sup μ(|z|)|g (z)| < ∞.
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