IJPAM: Volume 52, No. 2 (2009)
OF RAPID DESCENT
Department of Mathematics
P.O. Box 333, Maseno, KENYA
Abstract.A bounded operator with the spectrum lying in a compact set , has functional calculus. On the other hand, an operator acting on a Hilbert space H, admits a functional calculus if is self-adjoint. So in a Banach space setting, we really desire a large enough intermediate topological algebra A, with 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
, for some . Among other things,
we prove that
is dense in
this algebra. We demonstrate that important functions like
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