IJPAM: Volume 52, No. 2 (2009)

THE ALGEBRA OF SMOOTH FUNCTIONS
OF RAPID DESCENT

Paul O. Oleche$^1$, N. Omolo-Ongati$^2$, John O. Agure$^3$
$^{1,2,3}$Department of Mathematics
Maseno University
P.O. Box 333, Maseno, KENYA
$^1$e-mail: poleche@maseno.ac.ke
$^2$e-mail: omolo_ongati@yahoo.com
$^2$e-mail: johnagure@maseno.ac.ke


Abstract.A bounded operator with the spectrum lying in a compact set $V\subset$ $\fld{R}$, has $C^\infty(V)$ functional calculus. On the other hand, an operator $H$ acting on a Hilbert space H, admits a $C(\fld{R})$ functional calculus if $H$ is self-adjoint. So in a Banach space setting, we really desire a large enough intermediate topological algebra A, with $C_0^\infty(\fld{R}) \subset \algebra{A} \subseteq C(\fld{R})$ such that spectral operators or some sort of their restrictions, admit a A functional calculus.

In this paper we construct such an algebra of smooth functions on R that decay like $(\sqrt{1+x^2})^\beta$ as $\abs{x}\to \infty$, for some $\beta < 0$. Among other things, we prove that $C_c^\infty(\fld{R})$ is dense in this algebra. We demonstrate that important functions like $x\mapsto \Exp x$ are either in the algebra or can be extended to functions in the algebra. We characterize this algebra fully.

Received: February 21, 2009

AMS Subject Classification: 46J15

Key Words and Phrases: Banach algebra, smooth function, extension

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2009
Volume: 52
Issue: 2