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Functional differential equations: I. $$C^*$$-theory. (English) Zbl 0799.34001
Pitman Monographs and Surveys in Pure and Applied Mathematics. 70. Harlow: Longman Scientific & Technical. New York, NY: John Wiley & Sons Inc.. 504 p. (1994).
The book is an extensive exposition of a modern branch of mathematics lying between differential equations and $$C^*$$-algebras, intersecting both these theories. The objects treated in the monography are weighted shift operators, i.e. linear and bounded operators $$aT$$, $$a\in A$$, acting in a subalgebra $$A$$ of a Banach algebra $$L(E)$$ over a Banach space $$E$$. Here $$a\in A$$ and $$T: E\to E$$ is an invertible isometry such that $$TAT^{-1}=A$$. Numerous examples of weighted shift operators from different branches of mathematics are considered. A special kind of functional-differential equations with weighted shift operators treated in the monography are $$(Lf)(x)= g(x)$$, where $$(Lf)(x)$$ are of the following types: $$(Lf)(x)= \sum_{i=1}^ k a_ i(x) f(\alpha_ i(x))$$ or $$(Lf)(x)= \sum_{j=1}^ n \sum_{i=1}^ m a_{ij}(x) f^{(j)} (\alpha_ i(x))$$. The main objects considered in the book are abstract $$C^*$$-dynamical systems associated with $$C^*$$-algebras.
Here a normal algebra $$A$$ with an involution mapping * such that $$\| aa^*\|= \| a\|^ 2$$ is called a $$C^*$$-algebra and a triple $$(A,G,\alpha)$$ consisting of a $$C^*$$-algebra, a group $$G$$ and a homeomorphism $$\alpha ; G\to \text{Aut}(A)$$ is called a $$C^*$$-dynamical system. The algebra of all the linear bounded operators in a Hilbert space $$H$$ is the $$C^*$$-algebra.
For functional-differential operators invertibility, Fredholmity and spectral properties are investigated. Here an operator $$A\sigma L(E,F)$$ is called Fredholm if $$\dim \text{ Ker } A<\infty$$, $$\dim \text{ Ker } A^*<\infty$$ and $$\text{Im } A$$ is closed. In the first theoretical part of the book (Chapters 1, 2) besides preliminaries and general formulae for the spectral radius, spectral properties and invertibility conditions for weighted shift operators are obtained.
Some equivalent Fredholmity conditions are demonstrated. It was possible to construct a specified $$C^*$$-apparatus for the weighted shift operators which enable to bring to light the correlation between the invertibility of elements and the hyperbolic and trajectorial decomposition. Also the problem of approximation of dynamical systems is investigated. This part contains important $$C^*$$-algebra results, especially the isomorphism theorem and the invertibility conditions of elements in these algebras.
The second part (Chapters 3, 4) is devoted to numerous applications of weighted shift operators and related topics. For various classes of functional-differential equations a symbolic calculus and Fredholmity conditions are developed. The considered theory of functional- differential equations includes equations with continuous and discontinuous coefficients. Equations with singular integral operators Toeplitz operators, pseudodifferential operators and operators with transformations of the argument, and also non-local boundary value problems, convolution-type equations and equations generated by local type operators are presented.
The last part deals with non-Hilbert and non-$$C^*$$-algebra functional operators in $$L^ p$$-spaces of smooth functions. Functional operators generated by non-vertible transformations of an argument are discussed. In this case an isomorphism theorem is constructed and this enables a symbolic calculus for the examined equations. At the end of the chapters vast historical notes and remarks are added. The bibliography contains up to date sources connected strictly with the monography.

##### MSC:
 34-02 Research exposition (monographs, survey articles) pertaining to ordinary differential equations 47-02 Research exposition (monographs, survey articles) pertaining to operator theory 47C10 Linear operators in $${}^*$$-algebras 34K05 General theory of functional-differential equations 34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument)