Sharyn S. Algebras of polynomial distributions on infinite dimensional spaces and their application to operator calculus

A new approach to investigation of the dual pair <P'(X'), P(X')> is developed, where P(X') is the space of continuous polynomials over X', P'(X') is the space of polynomial distributions, which is strong dual space of P(X'), where X and X' are linear locally convex nuclear $(F)$ or $(DF)$ spaces. It is extended the Fourier transformation on the spaces of polynomial rapidly decreasing test functions and polynomial tempered distributions; properties of the transformation are investigated. It is also extended the Fourier-Laplace and Laplace transformations on the spaces of polynomial Gevrey ultradifferentiable functions and polynomial Roumieu ultradistributions; properties of these transformations are investigated. Herewith the image of the test space with respect to Fourier-Laplace transformation is described; this image is presented as a class of entire functions of exponential type. Paley-Wiener type theorems for polynomial ultradifferentiable functions and polynomial ultradistributions are proved. Some structure theorems for shift-invariant operators are proved. In particular, we describe a commutative algebra of shift-invariant continuous linear operators, commuting with contraction multiparameter semigroups over a Banach space. Structure theorem for locally convex Fock type space is proved, namely it is shown that this space may be represented as a commutant of polynomial shift semigroup. The Gateaux differentiability of the elements of the spaces of polynomial rapidly decreasing test functions and polynomial tempered distributions is investigated. Hille-Phillips type functional calculi for commuting sets of generators of strongly continuous semigroups of operators, acting in a Banach space, are constructed; the symbol classes of such calculi are spaces of analytic functions of finite or countable quantity of variables. It is constructed a functional calculus for countable noncommuting set of generators of strongly continuous groups of operators, acting in a Hilbert space; the symbol class of such calculus is a space of entire analytic functions of countable quantity of variables. We found a solution of infinite dimensional heat equation, generated by Gross Laplacian. It is constructed the semigroup, which is generated by Gross Laplacian. It is described homomorphisms from the algebra of analytic functions of bounded type on an infinite dimensional Banach space into some commutative Banach algebra. It is shown that not every such homomorphism is defined by a functional calculus.