Composition operators belonging to operator ideals
Let \(D\) be the unit disc \(\{z\in\mathbb{C}:|z|< 1\}\) of the complex plane \(\mathbb{C}\), and let \(\mu\) be the normalized Lebesgue measure on \(D\). If \(1\leq p<\infty\), then the Bergman space \(A^p\) is the subspace of \(L^p(\mu)\) consisting of holomorphic functions on \(D\). The sequence space consisting of elements of the form \(a= (a_0,a_1,\dots, a_j,\dots)\) such that \(\|a\|^p_p= \sum^\infty_{j=0}|a_j|^p<\infty\) is denoted by \(\ell^p\). If \(\phi: D\to D\) is analytic (holomorphic), then the composition operator \(C_\phi\) is defined by \(C_\phi(f)(z)= f\circ\phi(z)= f(\phi(z))\), and the image measure \(\mu_\phi\) is defined by \(\mu_\phi(A)= \mu(\phi^{-1}(A))\). If \((D_k)\) is a partition of \(D\) consisting of polar rectangles defined in terms of a diadic lattice and \(v_k= \mu(Q_k)^{-1/p} \mu_\phi(Q_k)^{1/q}\), \(k= 0,1,2,\dots\), then the diagonal operator \(\underline D_v\) is defined by \(\underline D_v(a_k)= v_ka_k\), \(k= 0,1,2,\dots\)\ . In the main results of this paper, a quasi-Banach ideal is introduced as a method which assigns to each pair of Banach spaces \((X,Y)\) a bounded linear operator \({\mathcal A}(X,Y)\) from \(X\) into \(Y\), and it is shown in the main theorem that the composition operator \(C_\phi\) is in \({\mathcal A}(A^p,A^q)\), where \(1\leq p,q<\infty\), if and only if \(\underline D_v\) is in \({\mathcal A}(\ell^p, \ell^q)\). Some previously derived special cases of the general statements include results by \textit{W. W. Hastings} [Proc. Am. Math. Soc. 52, 237-241 (1975; Zbl 0304.31007)] and \textit{B. D. MacCluer} and \textit{J. H. Shapiro} [Can. J. Math. 38, 878-906 (1986; Zbl 0608.30050)], providing conditions for boundedness and compactness of \(C_\phi: A^p\to A^q\) in terms of finiteness or limits of sequences of the types defining \(v_k\), \(k= 0,1,2,\dots\)\ .
scientific article; zbMATH DE number 1348152
scientific article