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Let $G$ be a compact group, and $\pi : G \rightarrow \mathcal{U}(H)$ be a continuous unitary representation. Let $f \in L^{1}(G)$ be arbitrary.

By Riesz Representation Theorem we can find a bounded linear map $T: H \rightarrow H$ such that

$\langle T \xi, \eta \rangle = \int_{G} f(x)\langle \pi(x)\xi, \eta \rangle dx$,

Let $\tilde{\pi}(f) : = T$.

What is the norm of $\tilde{\pi}$?

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    $\begingroup$ As with most questions of the "what is" kind, the natural counter-question is: In what terms do you want the answer to be stated? Do you have reasons to believe that there is an answer stated in non-tautological terms? $\endgroup$
    – Iosif Pinelis
    Apr 7, 2023 at 19:27
  • $\begingroup$ There is no formula for the norm unless more information is given about the function $f$ and the representation $\pi$. I am voting to close as "question not well-defined" $\endgroup$
    – Yemon Choi
    Apr 8, 2023 at 12:54

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