CONTENTS Introduction..............................................................................5 1. Preliminaries........................................................................5 2. Spaces of smooth elements.................................................8 3. Spaces of D-analytic elements...........................................21 4. Spaces of D-paraanalytic elements....................................32 5. Characterization of .D-paraanalytic elements by means of property (c) and canonical mappings.............39 6. D-R paraanalytic functions .................................................47 7. Smooth elements in Leibniz D-algebras..............................59 8. Weak E-solutions................................................................63 9. Linear systems with scalar coefficients...............................79 10. Linear boundary value problems .....................................82 References.............................................................................97
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In 1950 N. Jacobson proved that if u is an element of a ring with unit such that u has more than one right inverse, then it has infinitely many right inverses. He also mentioned that I. Kaplansky proved this in another way. Recently, K. P. Shum and Y. Q. Gao gave a new (non-constructive) proof of the Kaplansky-Jacobson theorem for monoids admitting a ring structure. We generalize that theorem to monoids without any ring structure and we show the consequences of the generalized Kaplansky-Jacobson theorem for the theory of linear operators, and even for the classical Calculus. In order to do that, we recall some multiplicative systems, called pseudocategories, very useful in the algebraic theory of perturbations of linear operators. In the second part of the paper, basing on the Kaplansky-Jacobson theorem, we show how to use the above mentioned structures for building Algebraic Analysis of linear operators over a class of linear spaces. We also define (non-linear) logarithmic and antilogarithmic mappings on these structures.
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Let X be a Leibniz algebra with unit e, i.e. an algebra with a right invertible linear operator D satisfying the Leibniz condition: D(xy) = xDy + (Dx)y for x,y belonging to the domain of D. If logarithmic mappings exist in X, then cosine and sine elements C(x) and S(x) defined by means of antilogarithmic mappings satisfy the Trigonometric Identity, i.e. $[C(x)]^2 + [S(x)]^2 = e$ whenever x belongs to the domain of these mappings. The following question arises: Do there exist non-Leibniz algebras with logarithms such that the Trigonometric Identity is satisfied? We shall show that in non-Leibniz algebras with logarithms the Trigonometric Identity does not exist. This means that the above question has a negative answer, i.e. the Leibniz condition in algebras with logarithms is a necessary and sufficient condition for the Trigonometric Identity to hold.
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We consider nonlinear equations in linear spaces and algebras which can be solved by a "separation of variables" obtained due to Algebraic Analysis. It is shown that the structures of linear spaces and commutative algebras (even if they are Leibniz algebras) are not rich enough for our purposes. Therefore, in order to generalize the method used for separable ordinary differential equations, we have to assume that in algebras under consideration there exist logarithmic mappings. Section 1 contains some basic notions and results of Algebraic Analysis. In Section 2 we consider equations in linear spaces. Section 3 contains results for commutative Leibniz algebras. In Section 4 basic notions and facts concerning logarithmic and antilogarithmic mappings are collected. Section 5 is devoted to separable nonlinear equations in commutative Leibniz algebras with logarithms.