Executive Summary : | Weighted spaces of holomorphic functions defined on a balanced open subset of a finite dimensional space have their origin in the study of growth conditions of such functions and have been investigated extensively after the appearance of D.L. Williams' work. Besides its intrinsic mathematical interest, these spaces have been proved useful in other fields of mathematics, for instance, in the approximation theory, distribution theory and representation of distributions as boundary values of holomorphic functions, Fourier analysis, partial differential equations, convolution equations, spectral theory, holomorphic calculus etc. The study of infinite-dimensional analogues of such spaces was initiated by P. Rueda while extending Banach-Dieudonne's theorem for spaces of holomorphic functions and further carried out by M. Beltran, D. Garcia, M. Gupta, M. Maestre, P Sevilla-Peris and several others. On the other hand, the study of composition and weighted composition operators lies at the interface of the function space theory and the operator theory; and provides examples/ counter examples of linear operators on function spaces which may be Hilbert/Banach/locally convex spaces. In the last decades a lot of research has been done studying the behaviour of these operators between weighted spaces of holomorphic functions defined on an open subset of C or in more general, on an open subset of C^n. Recently, D. Garcia, M. Maestre, and P. Sevilla-Peris have investigated these operators on weighted spaces of holomorphic functions defined on an open subset of an infinite dimensional space. Further investigations of the dynamical properties of composition and weighted composition operators on spaces of holomorphic functions defined on finite dimensional spaces has gained momentum in the recent years. But unfortunately much attention has not been paid to the infinite dimensional case. In this proposal, we propose to study the dynamical behaviour of composition and weighted composition operators on weighted spaces on holomorphic functions defined on open subsets of an infinite dimensional Banach space. Since the extensions of techniques available in finite dimensional case are commonly ineffective in the infinite dimensional case, this study requires new techniques in the infinite dimensional setting. Therefore, it is challenging and interesting to derive an appropriate theory and methods to achieve the proposed objectives. We will also be finding several new problems along the way for these function spaces and operators acting between them. |