Year: 1980
Type: article
Abstract: For 0 < a = 1, let Lip(a) denote the space of functions / which are analytic on the open unit disk, continuous on the closed unit disk, and whose boundary values satisfy a Lipschitz condition of order a : \f(z) -f(w)\ ^ K\z -w\ a , for \z\ = \w\ = 1.For 0 < a < 1, let lip(a) denote thé space of functions / in Lip(a) such that \f(z) -f(w)\ -o(\z -w\ a ), as w -* z, \z\ = \w\ = 1.We prove that a function <p in Lip(a) (resp., lip(a)), with |<p(z)| ^ 1 for \z\ ^ 1, induces a composition operator on Lip(a) (resp., lip(a)) if and only if there exists a finite number M and a number r < 1 such that \<p(z)\ =r implies \<p'(z)\ = M.We also prove that a composition operator Cy on either Lip(a) or lip(a) is compact if and only if for each e > 0 there exists an r < 1 such that \<p(z)\ = r implies W(Z)\ ^ €.
Author: Raymond C. Roan
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: bronze
OpenAlex ID: W1973409264