Functional Differential Operators and Equations
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This book deals with linear functional differential equations and operator theory methods for their investigation. The main topics are: the equivalence of the input-output stability of the equation Lx = &mathsf; and the invertibility of the operator L in the class of casual operators; the equivalence of input-output and exponential stability; the equivalence of the dichotomy of solutions for the homogeneous equation Lx = 0 and the invertibility of the operator L; the properties of Green's function; the independence of the stability of an equation from the norm on the space of solutions; shift invariant functional differential equations in Banach space; the possibility of the reduction of an equation of neutral type to an equation of retarded type; special full subalgebras of integral and difference operators, and operators with unbounded memory; and the analogue of Fredholm's alternative for operators with almost periodic coefficients where one-sided invertibility implies two-sided invertibility. Audience: This monograph will be of interest to students and researchers working in functional differential equations and operator theory and is recommended for graduate level courses.
PrefaceCh. IFunctional analysis preliminaries11.1Banach spaces and bounded linear operators11.2Diagrams71.3Lower norms141.4Banach algebras201.5Spaces of continuous and measurable functions311.6Infinite matrices441.7Tensor products611.8Conjugate spaces81Ch. IIThe initial value problem972.1The causal invertibility982.2The causal spectrum1062.3Spaces of smooth functions and distributions1122.4The unique solubility1272.5The evolutionary solubility1362.6Criteria for evolutionary solubility140Ch. IIIStability1503.1Algebraic preliminaries1503.2Input-output stability: discrete time1573.3Input-output stability: continuous time1623.4Exponential stability: discrete time1713.5Exponential stability: continuous time1793.6Exponential dichotomy: continuous time196Ch. IVShift invariant operators and equations2094.1Algebras of bounded measures2094.2The Fourier transform2214.3The Laplace transform2394.4Convolution operators2474.5Invertibility and causal invertibility267Ch. VOperators with varying coefficients2855.1Multiplication operators2855.2Difference operators2965.3Smoothing operators3075.4Integral operators3155.5Operators with locally fading memory3225.6Operators with continuous coefficients333Ch. VIDifferential difference equations3436.1The local Fredholm property3436.2The solubility for derivatives3556.3The independence from the choice of a norm3666.4Green's function3726.5Almost periodic operators382Comments403Bibliography411Index429Notation Index433
