Abstract
Some properties of the linear continuous operator and separation of convex subsets are investigated in this paper and a dual space for a subspace of a reflexive Banach space with a strictly convex norm is constructed. Here also an existence theorem and fixed-point theorem for general mappings are obtained. Moreover, certain remarks on the problem of existence of invariant subspaces of a linear continuous operator are given.
Highlights
The separation of convex sets in a real reflexive Banach space are investigated, existence of a fixed-point theorem for a general mapping acting in a Banach space and the obtained results are applied to study certain properties of continuous linear operators
Here is proved the solvability theorem for an inclusion with sufficiently general mapping
We prove results about the separation of convex sets in an infinite-dimensional space which resemble the results in the finite-dimensional case, provided that the space has a geometry satisfying some complementary conditions
Summary
The separation of convex sets in a real reflexive Banach space are investigated, existence of a fixed-point theorem for a general mapping acting in a Banach space and the obtained results are applied to study certain properties of continuous linear operators. It should be noted that many works are devoted to the problem of type of the existence of an invariant subspace of the linear operator (see, e.g., [15,16,17,18], etc.) and one of the essential results is obtained in [16] (see, [17]) In these papers, the connection of the considered linear operator with a completely continuous operator played a basic role as in [16] (see, [17, 18]). We conduct a result about existence of an invariant subspace of a linear bounded operator without using a completely continuous operator
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