The dissertation [Mathematics Department of Dortmund University (1995)] is devoted to the spectral theoretic investigations of the left-definite differential equation, in particular of singular cases. The basic aim is to develop the most general spectral theory of left-definite differential equations. The problem was first initiated by Weyl with some strong restrictions on the coefficients, e.g. boundedness. The main objective of the author is to study the problem without these restrictions. The first chapter is on the introduction and further elaboration of the relevant differential equations. It first deals with a canonical system with locally integrable matrix coefficients, in a real domain, having certain symmetries. In the second section, on the basis of the left-definiteness a positive semi-definite scalar product and a semi-norm topology is expounded. In the next section, the solution of the inhomogeneous differential equation is constructed with the help of certain projection operators on its characteristic space. This leads to the Neumann formula and thus splits the solution space in a sum of the characteristic space with parameters \(i\) and \(-i\), and a degenerate subspace. In the second chapter, a class of boundary value problems is introduced which makes it equivalent to the concomittant eigenvalue problem of a selfadjoint operator in Hilbert space. After developing the material in these preliminary chapters, the author concentrates on the spectral theory of selfadjoint operators in Hilbert space, in the third chapter. The most important result is the expansion theorem which, as expected, is of the form given by Weyl. The expansion is developed with the concept of semi-norm. In the last section, a specific form of spectral representation is established for a class of boundary value problems, with the help of integral transformations. The fourth chapter, which is basically devoted to applications of the theory to higher order differential equations, starts with explicit treatment of selfadjoint linear differential equations. This section attempts to answer the question what are the conditions that these differential equations induce left-definite systems as investigated in the preceding chapters. The remaining sections deal with the Sturm-Liouville equation, which is the most important example with the classical orthogonal polynomials as eigenfunctions. It is pointed out that these can also be considered as left-definite problems. In the studies of fundamental space with Dirichlet integral, in the first chapter, two inequalities with topology of locally uniform convergence are stipulated. The first one forms the basis of the expansion theorem and the second of this inequalities, according to the authors should form the starting point for further development of the theory.
The presentation of the material is mostly in an orthodox manner, e.g. definitions, theorem followed by proof, occasionally inserted with notes. To each of the chapters, there is an adjunct with explanatory notes, often with historical reference and the relation to other work. In addition to an almost exhaustive list of references of literature pertaining to the problem, the book is supplemented with indexes of general notations used and further of usual notations of different types of spaces.