Convolution operators on Lebesgue spaces of the half-line

Daniel, Victor W.

doi:10.1090/S0002-9947-1972-0291849-4

Convolution operators on Lebesgue spaces of the half-line
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by Victor W. Daniel PDF

Trans. Amer. Math. Soc. 164 (1972), 479-488 Request permission

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.

References

Shmuel Agmon, Sur un problème de translations, C. R. Acad. Sci. Paris 229 (1949), 540–542 (French). MR 31110
Kenneth Hoffman, Banach spaces of analytic functions, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. MR 0133008
M. G. Kreĭn, Integral equations on the half-line with a kernel depending on the difference of the arguments, Uspehi Mat. Nauk 13 (1958), no. 5 (83), 3–120 (Russian). MR 0102721
Norbert Wiener, The Fourier integral and certain of its applications, Dover Publications, Inc., New York, 1959. MR 0100201
Kôsaku Yosida, Functional analysis, Die Grundlehren der mathematischen Wissenschaften, Band 123, Academic Press, Inc., New York; Springer-Verlag, Berlin, 1965. MR 0180824