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Published in:
2022 | OriginalPaper | Chapter
8. Discrete Case: Infinite Leslie Operators
Authors : Marinus A. Kaashoek, Sjoerd M. Verduyn Lunel
Published in: Completeness Theorems and Characteristic Matrix Functions
Publisher: Springer International Publishing
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Abstract
In this chapter we discuss a class of operators acting on the Hilbert space \(\ell _+^2({\mathbb C})\). The operators are infinite versions of Leslie matrices, and therefore they are called Leslie operators. These operators are again of the form finite rank perturbation of a Volterra operator. The chapter consists of four sections. The first two present the definition of a Leslie operator and describe the connection with boundary value systems. The third section has an auxiliary character. In the fourth section we present a class of Leslie operators T with special completeness properties, namely T
∗ has a complete span of eigenvectors and generalised eigenvectors, while the latter is not true for the operator T. In the final section we consider completeness for a generalised Leslie operator which is a finite rank perturbation of a non-compact quasi-nilpotent operator, i.e., the role of Volterra operators is taken over by quasi-nilpotent operator, as in Sect. 6.3.