Nevanlinna algebras (Q2759178)

This paper discusses certain Nevanlinna algebras \(N^\alpha_p\) of analytic functions on the unit disc \(U\) in the complex plane which are \(L^p\), \(1< p\leq\infty\), variants of the weighted area Nevanlinna classes defined by the measure \(d\sigma_\alpha(z)= (\alpha+ 1)(1-|z|^2)^{1/\alpha}d\sigma(z)\), \(\alpha> -1\), \(\sigma\) being the normalized area measure on \(U\). These algebras are the ``logarithmic annalogs'' of the weighted Bergman spaces \(A^\alpha_p:= H(U)\cap L^p(d\sigma_\alpha)\). The \(\infty\) case are also treated and play an important role. The algebras \(N^\alpha_p\) admit dense maximal ideals, hence they are not \(Q\)-algebras if \((\alpha+2)\leq p\). Moreover, the invertible elements of \(N^\alpha_p\) are the exponential functions in \(A^\alpha_p\). A theorem on multipliers proved by \textit{H. Jarchow}, \textit{V. Montesinos}, \textit{K. J. Wirths} and \textit{J. Xiao} [Proc. Edinb. Math. Soc. II. Ser. 44, No. 3, 571-583 (2001; Zbl 1001.46014)], is of fundamental importance in the rest of the paper. The space \(N^\alpha_p\) is an \(F\)-algebra which is not locally pseudoconvex, but it has a separating dual. The authors describe the Fréchet envelope of \(N^\alpha_p\) (i.e., the completion of this space with respect to the finest locally convex topology which is coarser than the original one). Two algebras \(N^\alpha_p\) and \(N^\beta_q\) generate the same Fréchet envelope if and only if \((\alpha+ 2)/p= (\beta+ 2)/q\). Results about boundedness or compactness of composition operators between \(N^\alpha_p\) can be translated to corresponding ones for the associated Bergman spaces \(A^\alpha_p\). The composition operators which are order bounded are also characterized. These operators are closely related to absolutely summing operators. Interesting consequences of these characterizations for embeddings are obtained. Consequences for natural locally convex spaces which appear as intersections or unions of the spaces studied in the article are derived. They strengthen recent results of \textit{H. Jarchow} and \textit{J. Xiao} [to appear].