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Science Forum Index » Physics - Research Forum » Uniform boundedness of the functional calculus...
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 Posted: Sat Jun 14, 2008 1:45 am |
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In deriving the functional calculus for a (possibly unbounded) self-
adjoint operator A on a Hilbert space, one typically proceeds by
defining f(A) for a nice algebra of functions, and showing that the
map f -> f(A) is bounded when the function algebra is given the
supremum (L^\infinity) norm so that it can be extended to a larger
algebra by continuity. I am interested in knowing whether there are
simple yet general criteria for determining the boundedness of f ->
f(A).
To be more specific, I would like to choose the original function
algebra to be the Schwartz space of rapidly-decreasing functions,
which is uniformly dense in the space of continuous functions
vanishing at infinity, and define f(A) via the Fourier transform. I
do have a quick proof of boundedness in this case (using an idea from
E.B. Davies, "Spectral Theory and Differential Operators", proof of
Lemma 2.2. , but it feels as though it relies more on luck than on
general principles. The problem is really one of extending a *-
algebra homomorphism from a dense subalgebra of a C*-algebra to the
whole algebra, or of finding criteria for the boundedness of a
homomorphism on such a dense subalgebra, and I assume such criteria
are known. My immediate interest in the problem is that I will be
teaching an introductory graduate course in functional analysis this
Fall--hence the request for "simple" criteria.
Thank in advance for any help,
Kevin. |
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