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Let T be the group {e it ∣0≤t<2π}. We consider T with its euclidean topology. Our main results are as follows: 1) Theorem 1 which establishes a connection between spectrums of a continuous linear representation of T in a reflexive Banach space and its conjugate linear representation; 2) Theorem 11 which gives a description of all left (right) simple actions of T in Banach algebras; 3) Theorem 18 which establishes an analog of the theorem on integral for continuous linear representations of T in Banach spaces.

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Let D be the infinitesimal generator of a strongly continuous periodic one-parameter group of linear operators in a Banach space. Main results: An analog of the resolvent operator (= quasi-resolvent operator of D) is defined for points of the spectrum of D and its evident form is given. The theorem on integral for the operator D, theorems on the existence of periodic solutions of a linear differential equation of the nth order with constant coefficients and systems of linear differential equations with constant coefficients in Banach spaces are obtained. In the case of the existence of periodic solutions, evident forms of all periodic solutions of a linear differential equation of the nth order with constant coefficients and systems of linear differential equations with constant coefficients in Banach spaces are given in terms of resolvent and quasi-resolvent operators of D.
MSC:
42A, 43, 47D.

An approach to a definition of an integral, which differs from definitions of Lebesgue and Henstock–Kurzweil integrals, is considered. We use trigonometrical polynomials instead of simple functions. Let V be the space of all complex trigonometrical polynomials without the free term. The definition of a continuous integral on the space V is introduced. All continuous integrals are described in terms of norms on V. The existence of the widest continuous integral is proved, the explicit form of its norm is obtained and it is proved that this norm is equivalent to the Alexiewicz norm. It is shown that the widest continuous integral is wider than the Lebesgue integral. An analog of the fundamental theorem of calculus for the widest continuous integral is given.

a# , p# , r # }. We show that in the case when the spaces (1) are linear, they are Banach spaces. Conditions are obtained for the superreflexivity and separability of such spaces. We determine relations between the spaces # W # {a# , p# , r # } and nuclear spaces via Frechet projective limits of sequences of spaces of finite order. We prove that the Schauder bases in # W # {a# , p# , r # } are unconditional. We establish functional properties of the dual spaces. 1. Introduction of the norms in # W # {a# , p# , r # }. Let (2) V# (#) = {u # #<F9

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