Abstract
Let G be a commutative group, written additively, with a neutral element 0, and let K be a finite group. Suppose that K acts on G via group automorphisms \({G \ni a \mapsto ka \in G}\) , \({k \in K}\) . Let \({{\mathfrak{H}}}\) be a complex Hilbert space and let \({{\mathcal L}({\mathfrak{H}})}\) be the algebra of all bounded linear operators on \({{\mathfrak{H}}}\) . A mapping \({u \colon G \to {\mathcal L}({\mathfrak{H}})}\) is termed a K-spherical function if it satisfies (1) \({|K|^{-1} \sum_{k\in K} u (a+kb)=u (a) u (b)}\) for any \({a,b\in G}\) , where |K| denotes the cardinality of K, and (2) \({u (0) = {\rm id}_{\mathfrak {H}},}\) where \({{\rm id}_{\mathfrak {H}}}\) designates the identity operator on \({{\mathfrak{H}}}\) . The main result of the paper is that for each K-spherical function \({u \colon G \to {\mathcal {L}}({\mathfrak {H}})}\) such that \({\| u \|_{\infty} = \sup_{a\in G} \| u (a)\|_{{\mathcal L}({\mathfrak{H}})} < \infty,}\) there is an invertible operator S in \({{\mathcal L}({\mathfrak{H}})}\) with \({\| S \| \, \| S^{-1}\| \leq |K| \, \| u \|_{\infty}^2}\) such that the K-spherical function \({{\tilde{u}} \colon G \to {\mathcal L}({\mathfrak{H}})}\) defined by \({{\tilde{u}}(a) = S u (a) S^{-1},\,a \in G,}\) satisfies \({{\tilde{u}}(-a) = {\tilde{u}}(a)^*}\) for each \({a \in G}\) . It is shown that this last condition is equivalent to insisting that \({{\tilde{u}}(a)}\) be normal for each \({a \in G}\) .
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Chojnacki, W. On uniformly bounded spherical functions in Hilbert space. Aequat. Math. 81, 135–154 (2011). https://doi.org/10.1007/s00010-010-0037-y
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DOI: https://doi.org/10.1007/s00010-010-0037-y