Re: Adjoint of Integral operator.
- From: israel@xxxxxxxxxxx (Robert Israel)
- Date: 22 Sep 2005 17:07:40 GMT
In article <1127370447.074213.125950@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
Stephen <stevethemathsman@xxxxxxxxx> wrote:
>I have come across the following operator in my work (written in
>LaTeX),
>\int_t^\infty \tau^{\mu} f(\tau ) d \tau, where f is zero at
>infinity.
>I was wondering if anyone knows how to go about finding the adjoint of
>this operator.
I assume what you really mean is the following: you're working on the
Hilbert space H = L^2(0,infty), mu is a nonnegative constant, and your
operator is T where Tf(t) is given by your formula for all f in H
with compact support. Then for suitable g in H,
(g, Tf) = int_0^infty conjugate(g(t)) int_t^infty tau^mu f(tau) dtau dt
= int_0^infty tau^mu f(tau) int_0^tau conjugate(g(t)) dt dtau
= (T* g, f)
where T*g(tau) = tau^mu int_0^tau g(t) dt. The domain of T* is all
g in H for which tau^mu int_0^tau g(t) dt is in H (note in particular
that this will require int_0^infty g(t) dt = 0).
Robert Israel israel@xxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
.
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