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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Polynomials and the limit point condition

Author: Robert M. Kauffman
Journal: Trans. Amer. Math. Soc. 201 (1975), 347-366
MSC: Primary 47E05; Secondary 34B20
MathSciNet review: 0358438
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Abstract: An $ n$th order, possibly nonselfadjoint, ordinary differential expression $ L$ is said to be in the limit point condition if the maximal operator $ {L_M}$ in $ {L_2}[0,\infty )$ is an $ n$-dimensional extension of the minimal operator $ {L_0}$. If range $ {L_0}$ is closed, this definition is equivalent to the assertion that nullity $ {L_M} +$   nullity$ {({L^ + })_M} = n$, where $ {L^ + }$ is the formal adjoint of $ L$. It also implies that any operator $ T$ such that $ {L_0} \subseteq T \subseteq {L_M}$ is the restriction of $ {L_M}$ to a set of functions described by a boundary condition at zero. In this paper, we discuss the question of when differential expressions involving complex polynomials in selfadjoint expressions are in the limit point condition.

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Keywords: Limit point condition, minimal operator, maximal operator, Fredholm operator, perturbation theory, product of operators, complex polynomials in a formally selfadjoint expression
Article copyright: © Copyright 1975 American Mathematical Society