Completeness theorems and characteristic matrix functions. Applications to integral and differential operators (Q2139902)

This book focuses attention on the completeness issue of the eigenvectors and generalized eigenvectors (or root vectors) of compact linear operators on Banach spaces, along with the interesting history in the Hilbert space setting regarding non-selfadjoint compact operators. Classical results usually present sufficient conditions for completeness, the problem to give necessary and sufficient conditions for completeness is more difficult to analyze. In the present book, for the completeness or non-completeness of a given operator \(T\) acting on a Banach space \(X\), an important role is played by \(T\)-invariant linear subspaces. The necessary and sufficient conditions for completeness of the linear span of eigenvectors and generalized eigenvectors for operators that admit a characteristic matrix function in a Banach space setting are obtained. The main results are based on sharp resolvent estimates near infinity, using the theory of entire functions of completely regular growth and the Phragmén and Lindelöf theorem. The obtained completeness results allow to derive the Keldysh type results of non-selfadjoint operators in the Hilbert space setting. The book has essentially two parts: the first part provides the theoretical research, in which new completeness and non-completeness results for operators that are a finite rank perturbation of a quasi-nilpotent operator are given; the second part provides some important applications of completeness results for concrete models of Banach space operators that are finite rank perturbations of Volterra operators. The book has fourteen chapters. Chapters 1 and 2 presents some basic notation about compact linear operators on a Banach space. The main results are included in Chapters 3 to 13. Chapter~14 presents elements from the theory of entire functions that are used throughout the book. Chapter 1 is introductory. The generalized eigenspace \(M_T\) (also called spectral subspace) and the analytic subspace \(S_T\) of a compact linear operator \(T\) are introduced, the notion of completeness for a compact linear operator on Banach space is defined. By the singular value of a compact linear operator on Hilbert space, the finite-order compact operators are defined, including trace-class operators and Hilbert-Schmidt operators. Chapter 2 includes three completeness theorems for non-selfadjoint and finite-order compact operators on Hilbert space. The first theorem gives a sufficient condition guaranteeing completeness, based on the Phragmén-Lindelöf theorem. Using it, the famous Keldysh completeness theorem is derived. The second and third theorems refine the conditions in the first theorem. Chapter 2 brings together a number of classical results on completeness and recovers some classical and new completeness results for trace class and Hilbert-Schmidt operators. In particular, as an example, the completeness issue of rank one perturbations of a Volterra operator is discussed. Chapter 3 collects and further develops some basic properties of compact Hilbert space operators of order one; such operators are Hilbert-Schmidt operators and not necessarily trace-class operators. The main result is Theorem 3.4.1 that gives a completeness result for this class of operators. We note that the Fredholm resolvent of the compact operator \(T\) is a meromorphic function of finite-exponential type, i.e., \((I-zT)^{-1}=P(z)/q(z)\), in which $q$ is a scalar exponential-type entire function. Chapter 4 discusses completeness for a class of Banach space operators. A~bounded linear operator \(T\) on Banach space \(X\) is said to be a Riesz operator if its nonzero spectral point are isolated eigenvalues. Chapter~4 extends the result of order one compact in Hilbert space to Banach space. The main result is given in Theorem 41.3. Suppose that \(T\) is a Riesz operator on a Banach space \(X\) and whose Fredholm resolvent is of the form \((I-zT)^{-1}=P(z)/q(z)\), where \(q(z)\) is a scalar entire function with finite nonzero order \(\rho\), and \(P(z)\) is an operator-valued entire function with order at most \(\rho\). Theorem 4.1.3 characterizes the closure of the generalised eigenspace of \(T\). Chapter 5 introduces the notion of a characteristic matrix function for a class of bounded linear operators on Banach space. For operators that admit a characteristic matrix function, the necessary and sufficient conditions for completeness are obtained. The main result of Chapter 5 is given in Theorem 5.2.6. Chapter 6 discusses the finite rank perturbation of a Volterra operator \(V\) that admits a characteristic matrix function. The main result is Theorem 6.2.1, which provides a sharp completeness result for this class of operators. Theorem 6.3.2 extends the result of Theorem 6.2.1 from Volterra operators to quasi-nilpotent operators provided that its resolvent has finite order \(\rho\). We mention that the result of Corollary 6.2.2 is true only when \(X^*\) is a separable space. Chapter 7 discusses the finite perturbations of operators of integration, which is a special form of Volterra operators. The different completeness results are given in the function space \(C[0,1] \) or in the Hilbert space \(L^2[0, 1]\), respectively. Chapter 8 discusses a class of operators acting on the Hilbert space \(\ell^2 (C)\). The operators are infinite versions of Leslie matrices, which are called infinite Leslie operators, and again of the form finite rank perturbations of a Volterra operator. The conditions of density of range and observability are given, In particular, Theorem 8.4.3 shows that \(T\) does not have completeness, but \(T^*\) has. The generalised Leslie operator is introduced. Chapter 9 studies completeness of a semi-separable operator \(T\) on \(\ell^2(\mathbb C^m)\) or \(L^2([0,1],\mathbb C^m)\). In the discrete, case, the operator \(T\) has a block matrix representation that can be written in the form of a finite rank perturbation of the Volterra operator. The completeness result is given in Theorem 9.1.6. In the semi-separable integral operator case, \(T\) can be regarded as an finite rank perturbation of the Volterra operator, the completeness of \(T\) is given in Theorem 9.2.4 for \(m=1\). Chapter 10 is a preliminary chapter on delay differential equations, in which the resolvent family (or fundamental solution operators) for delay is given. Chapter 11 discusses the completeness problem of the period map $T$ that associates with a periodic delay differential equation. In a first step, it is proven that the period map is a finite rank perturbation of a Volterra operator. Second, it is shown that completeness of the eigenvectors and generalised eigenvectors of the period map implies that, on bounded intervals, solutions of the periodic delay equation can be approximated by a sequence of elementary solutions. The invariant decomposition of delay space under the periodic maps \(T\) is given in Theorem 11.3.1. Two examples of scalar periodic delay equations and their completeness discussion are given. In particular, in the case of non-completeness, the existence of an infinite small solution of a delay equation is also presented. Chapter 12 extends the completeness result on compact linear operators to a general class of unbounded operators that admit a characteristic matrix function. The main result is Theorem 12.2.1 which proves sharp completeness theorems. Chapter 13 applies the results of Chapter 12 to some specified models with unbounded operators in infinite dimensional dynamical systems. The first class models concern dichotomous operators associated with mixed type neutral functional differential equations, three examples are given to show the completeness and non completeness; the second class models involve age-dependent population equations, and the third class is the so-called Zig-Zag process which is associated with a \(C_0\)- semigroup. Chapter 14 gives a number of properties of entire functions of completely regular growth that are used throughout in the proofs of the completeness theorems. Special attention is given to entire functions of the form \[ f (z) = p(z) + q(z)\int^a_{-a}e^{-zt}\phi(t) \,dt,\ z \in \mathbb{C}, \] where \( p\) and \(q\ne 0\) are polynomials, \(a\in (0,\infty) \), and \(\phi\) is a non-zero square integrable function on the interval \([-a,a]\). The theory of entire functions developed by Phragmén and Lindelöf, and by Paley and Wiener plays essential role in the present book..