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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Toeplitz operators and differential equations on a half-line


Author: J. W. Moeller
Journal: Proc. Amer. Math. Soc. 36 (1972), 531-534
MSC: Primary 47E05; Secondary 34G05, 47B35
MathSciNet review: 0315517
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Abstract: Let $ \mathcal{K}$ be a separable Hilbert space, let $ {A_0},{A_1}, \cdots ,{A_n}$ denote bounded linear operators from $ \mathcal{K}$ into $ \mathcal{K}$, and let $ \mathcal{D}$ represent the set of all functions in $ {L^2}(0,\infty ;\mathcal{K})$ whose first n derivatives belong to $ {L^2}(0,\infty ;\mathcal{K})$. Suppose further that the space $ \mathcal{D}$ is equipped with an inner product inherited from $ {L^2}(0,\infty ;\mathcal{K})$. The main result of this note states that the differential operator

$\displaystyle L = {A_n}\frac{{{d^n}}}{{d{t^n}}} + {A_{n - 1}}\frac{{{d^{n - 1}}}}{{d{t^{n - 1}}}} + \cdots + {A_1}\frac{d}{{dt}} + {A_0}$

acting on $ \mathcal{D}$ is continuously invertible if and only if the operator

$\displaystyle P(\sigma ) = \sum {A_k^ \ast } {\sigma ^k}\quad (0 \leqq k \leqq n)$

acting on the Hilbert space $ \mathcal{K}$ has a uniformly bounded inverse everywhere in the open half-plane $ \Re \sigma < 0$.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1972-0315517-0
PII: S 0002-9939(1972)0315517-0
Keywords: Separable Hilbert space, differential operator, Laguerre function, Toeplitz operator
Article copyright: © Copyright 1972 American Mathematical Society