Define the operator $$A:E\rightarrow E$$ as $$A(a)=\int_G\pi(x)a dx$$ where the integral has to be interpreted in the weak sense i.e., $$<A(a),b>=\int_G<\pi(x)a,b>$$ fotr all $a,b\in E.$
Best regards
Shravan
--- On Sat, 25/4/09, maslouhi mostafa <maslouhi_mostafa@...> wrote:
From: maslouhi mostafa <maslouhi_mostafa@...> Subject: [harmonic] A question To: harmonicanalysis@yahoogroups.com Date: Saturday, 25 April, 2009, 9:28 PM
Dear members,
I don't see how to prove the following:
Let $G$ be a compact group and $(\pi, E)$ a finite linear representation of $G$. We consider a a hermitian form where $<,>$ on $E$ and set $(a,b)=\int_ G <\pi_x(a),\pi_ x(b)> dx$, $a,b\in G$, where $dx$ is a Haar measure on $G$.
The question is: Show that there exists an invertible operator $A:E\to E$ such that $(a,b)=<A(a),A(b)> $ for all $a,b\in G$.
Thanks in advance,
Best regards,
Mostafa MASLOUHI.
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