Linear equations and bornology. (Linejnye uravneniya i bornologiya). (Russian) Zbl 0534.46004

Radyno, Ya. V.

Minsk: Izdatel’stvo Belorusskogo Gósudarstvennogo Universiteta im. V. I. Lenina. 200 p. R. 1.40 (1982).

This booklet consists of two parts: In the first a description of the theory of bornological vector spaces in the spirit of Waelbroeck and Hogbe-Nlend is given, following very closely the Lecture Note Math. 277 [Séminaire Banach (1972; Zbl 0239.46075)]. In the second half of the book the spectral theory of multiplicatively convex bornological algebras (Gelfand-theory, holomorphic functional calculus) is developed and applied to concrete problems of analysis: Volterra integral equations, ordinary and partial differential equations, Cauchy- and Goursat-problem. Here the equicontinuous bornology of the space of operators of a locally convex space is used systematically and the rôle of regular operators is emphasized (for such an operator A we have estimations of the type \(p(A^ nx)\leq M^ nq(x)\), p a given, q a seminorm to be found, \(M>0,n\in {\mathbb{N}})\). Roughly one can say, that for regular operators the classical Banach space theory goes through.
A fairly balanced presentation of parts of ”soft” and ”hard” analysis.

Reviewer: J.Lorenz

MSC:

46-02	Research exposition (monographs, survey articles) pertaining to functional analysis
47-02	Research exposition (monographs, survey articles) pertaining to operator theory
46A08	Barrelled spaces, bornological spaces
46H30	Functional calculus in topological algebras
46E40	Spaces of vector- and operator-valued functions
46H05	General theory of topological algebras
47A60	Functional calculus for linear operators
47A10	Spectrum, resolvent
47D03	Groups and semigroups of linear operators
47E05	General theory of ordinary differential operators
34G10	Linear differential equations in abstract spaces
47F05	General theory of partial differential operators
47Gxx	Integral, integro-differential, and pseudodifferential operators
45D05	Volterra integral equations

Keywords:

Cauchy problem; bornological vector spaces; multiplicatively convex bornological algebras; Gelfand-theory; holomorphic functional calculus; Volterra integral equations; Goursat-problem; equicontinuous bornology of the space of operators of a locally convex space; regular operators

Citations:

Zbl 0239.46075