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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Convolution operators on Lebesgue spaces of the half-line


Author: Victor W. Daniel
Journal: Trans. Amer. Math. Soc. 164 (1972), 479-488
MSC: Primary 47A15; Secondary 47G05
MathSciNet review: 0291849
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Abstract: In this paper we determine the lattice of closed invariant subspaces for certain convolution operators on Lebesgue spaces $ {L^p}(d\sigma )$ where $ \sigma $ is a suitable weighted measure on the half-line. We exploit the rather close relationship between convolution operators and the collection of right translation operators $ {\{ {T_\lambda }\} _{\lambda \geqq 0}}$ on $ {L^p}(d\sigma )$. We show that a convolution operator K and the collection $ {\{ {T_\lambda }\} _{\lambda \geqq 0}}$ have the same lattice of closed invariant subspaces provided the kernel k of K is a cyclic vector. The converse also holds if we assume in addition that the closed span of $ {\{ {T_\lambda }k\} _{\lambda \geqq 0}}$ is all of $ {L^p}(d\sigma )$. We show that the lattice of closed right translation invariant subspaces of $ {L^p}(d\sigma )$ is totally ordered by set inclusion whenever $ \sigma $ has compact support. Thus in this case a convolution operator K is unicellular if and only if its kernel is a cyclic vector. Finally, we show for suitable weighted measures $ \sigma $ on the half-line that the convolution operators on $ {L^p}(d\sigma )$ are Volterra.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1972-0291849-4
PII: S 0002-9947(1972)0291849-4
Keywords: Convolution operator, right translation operator, strongly closed algebra, invariant subspace, Bochner integral, Volterra, quasinilpotent, unicellular, cyclic vector
Article copyright: © Copyright 1972 American Mathematical Society