Let D′+ be the space of Schwartz distributions with support on the closed positive half-line [0, +∞). We give a generalization of the Paley-Wiener theorem to the case of the distributions in D′+.

A generalization of the cross-correlation operation of the Schwartz distributions is considered, and some properties of this operation are established.

Let  stand for Roumieu ultradistributions with supports in the positive cone . Throughout  denotes the algebra of continuous scalar polynomials on the space . We investigate the dual pair  generated by the algebra  and by its strong dual . Properties of the polynomially extended operational calculus and the semigroups of shifts along the cone  are considered.

We construct a functional calculus for generators of analytic semigroups of operators on a Banach space. The symbol class of the calculus consists of hyperfunctions with a compact support in $[0,\infty)$. Domain of constructed calculus is dense in the Banach space.

Let S₊ and S’₊ be the Schwartz spaces of rapidly decreasing functions and tempered distributions on R+, respectively. Let P(S’₊) be the space of continuous polynomials over S’₊ and P’(S’₊) be its strong dual. These spaces have representations in the form of Fock type spaces Γ(S₊) and Γ(S’₊), respectively. In the paper the Gâteaux differentiability of the elements of the spaces P(S’₊), P’(S’₊), Γ(S₊) and Γ(S’₊) is investigated. It is established connection of Gâteaux derivative with the creation and annihilation operators on the Fock type spaces as well as with differentiations on Γ(S₊) and Γ(S’₊).

For generators of n-parameter strongly continuous operator semigroups in a Banach space, we construct a Hille-Phillips type functional calculus, the symbol class of which consists of analytic functions from the image of the Laplace transform of the convolution algebra of temperate distributions supported by the positive cone $ℝ^n_+$ . The image of such a calculus is described with the help of the commutant of the semigroup of shifts along the cone. The differential properties of the calculus and some examples are presented.

Using operators of cross-correlation with ultradistributions supported by a positive cone, we describe a commutative algebra of shift-invariant continuous linear operators, commuting with contraction multi-parameter semigroups over a Banach space. Thereby, we generalize classic Schwartz’s and Hormander’s theorems on shift-invariant operators.

We describe the cross-correlation operator over the space of real-analytic functions and generalize classic Schwartz’s theorem on shift-invariant operators.

In this paper we develop a functional calculus for a countable system of generators of contraction strongly continuous semigroups. As a symbol class of such calculus we use the algebra of polynomial tempered distributions. We prove a differential property of constructed calculus and describe its image with the help of the commutant of polynomial shift semigroup. As an application, we consider a function of countable set of second derivative operators.

We use the generalized n-dimensional Laplace transform of tempered distributions whose supports are located in a positive n-dimensional cone to construct functional calculus for the commutative collections of injective generators of n-parameter analytic semigroups of operators acting in a Banach space.

In this paper we construct an operator calculus over the symmetric Fock space for countable set of noncommuting generators of strongly continuous groups, acting on a Hilbert space. As a symbol class of the calculus we use some algebra of functions of infinitely many variables. This algebra is described as the image of the space of polynomial ultradifferentiable functions under Fourier-Laplace transformation.