A remark on averaging operators on homogeneous spaces (Q1599475)

Let \(G\) be a connected compact Lie group and \(H\) a proper closed subgroup of \(G\). Given a symmetric sequence \(S=\{g_{1},g_{2},\dots ,g_{d},g_{1}^{-1},g_{2}^{-1},\dots ,g_{d}^{-1}\}\) of elements of \(G\), Lubotzky, Phillips and Sarnak introduced the operator \(T_{S}:L^{2}(G/H)\rightarrow L^{2}(G/H)\) defined by: \[ (T_{S}f)(x)=\frac{1}{2d}\sum_{1}^{d}(f(g_{j}x)+f(g_{j}^{-1}x)) \] [see \textit{A. Lubotzky, R. Phillips} and \textit{P. Sarnak}, Commun. Pure Appl. Math. 39, Suppl., S149--S186 (1986; Zbl 0619.10052)]. Now observe that if \(f\) is a constant function then one has \(T_{S}f=f\). In particular, it is clear that the (ortho)complement \(L^{2}_{0}(G/H)\), in \(L^{2}(G/H)\), to the space of constant functions is invariant under \(T_{S}\). In the case where \(G=SO(3)\) and \(H=SO(2)\), i.e. \(SO(3)/SO(2)\) is the unit sphere \(S^{2}\), Lubotzky, Phillips and Sarnak produced, by computing explicit trace formulas, a lower bound for the \(L^{2}\)-norm of \(T_{S}\) in \(L^{2}_{0}(SO(3)/SO(2))\) (see the paper cited above). In the paper under review, the author proves, by using asymptotic behavior of group characters, a lower bound for the \(L^{2}\)-norm of the averaging operator \(T_{S}\) in \(L^{2}_{0}(G/H)\), when \(G\) is a connected compact Lie group and \(H\) a proper closed subgroup of \(G\). Moreover, a condition for this lower bound to be realized is also given, in terms of the subgroup of \(G\) generated by the elements \(\{g_{1},g_{2},\dots ,g_{d}\}\).