Operator representation and biduals of weighted function spaces (Q2728726)

[See also \textit{K.-D. Bierstedt} and the author, Bull. Belg. Math. Soc., Simon Stevin 8, 577-589 (2001) and Result Math. 36, 9-20 (1999; Zbl 0942.46023).]NEWLINENEWLINENEWLINEFrom the introduction: In this work the following topics will be addressed:NEWLINENEWLINENEWLINE-- an operator representation for weighted spaces of vector valued holomorphic or continuous functions with \(O\)-growth conditions,NEWLINENEWLINENEWLINE-- conditions for an operator representation for weighted inductive limits of spaces of vector valued holomorphic functions,NEWLINENEWLINENEWLINE-- the biduality problem for weighted spaces of holomorphic functions on half-planes and strips in \(\mathbb{C}\), NEWLINENEWLINENEWLINE-- the biduality problem for weighted spaces of harmonic functions on \(D\) and \(\mathbb{C}\),NEWLINENEWLINENEWLINE-- an operator representation for weighted spaces of vector valued harmonic functions.NEWLINENEWLINENEWLINEChapter 2 starts by recalling the classical \(\varepsilon\)-product representation for weighted spaces with \(o\)-growth conditions. Then an operator representation for the more difficult case of weighted spaces of vector valued holomorphic functions satisfying \(O\)-growth conditions is derived (Theorem 2.2.3). Such representations are very useful since they can serve to reduce several problems for spaces of vector valued functions to the corresponding spaces of scalar functions by applying various inheritance properties. In the case of weighted spaces of vector valued continuous functions, an operator representation is also deduced, but it is more complicated and hence less applicable. Chapter 2 also provides an isomorphism of weighted spaces of holomorphic functions on product sets with weighted spaces of vector valued functions (Theorem 2.3.3), generalizing a result of W. Cutrer for \(H^\infty(D)\). This chapter ends by an operator representation for spaces of vector valued holomorphic functions defined on a locally convex space (Theorem 2.4.1).NEWLINENEWLINENEWLINEThe aim of Chapter 3 is an operator representation for weighted inductive limits of spaces of vector valued holomorphic functions (Theorem 3.2.7). The existence of such a representation in the inductive limit case is a totally different problem than the problem treated in Chapter 2, since it only holds if the operator space has strong locally convex properties, i.e., is a bornological space. Close connections to a special case of Grothendieck's ``problème des topologies'' and to the vector valued projective description problem are established. For some examples a recent results of A. Peris is used. Chapter 3 also contains a discussion of an operator representation for inductive limits of weighted spaces on product sets (Theorem 3.3.1).NEWLINENEWLINENEWLINEA problem raised in 1970 by Rubel and Shields is studied in Chapter 4: When does \(Hv_0(G)''= Hv (G)\) hold for non-balanced domain \(G\subset\mathbb{C}\) and non-radial weights \(v\) on \(G\)? This problem will be treated here for half-planes (Theorem 4.2.1) and strips in \(\mathbb{C}\) (Theorem 4.2.3) using classical geometrical complex analysis tools like the auxiliary functions in the proof of the Phragmén-Lindelöf Theorem. The biduality problem is also dealt with for the corresponding inductive limits.NEWLINENEWLINENEWLINEIn Chapter 5, the biduals of weighted spaces of real valued harmonic functions on \(D\) and \(\mathbb{C}\) (Theorem 5.1.2 and Theorem 5.1.3) are considered using Cesàro means of the series representation of the harmonic functions. An operator representation for weighted spaces of vector valued harmonic functions is also derived (Theorem 5.2.1). Since no analogue to Hartog's Theorem exists for harmonic functions, the result which follows from the slice-product technique is different from the holomorphic case (Theorem 5.2.3).