Schur Multipliers and Spherical Functions on Homogeneous Trees

Let X be a homogeneous tree of degree q + 1 (2 ≤ q ≤ ∞) and let ψ : X × X → C be a function for which ψ(x, y) only depend on the distance between x, y ∈ X. Our main result gives a necessary and sufficient condition for such a function to be a Schur multiplier on X×X. Moreover, we find a closed expression for the Schur norm ‖ψ‖S of ψ. As applications, we obtain a closed expression for the completely bounded Fourier multiplier norm ‖ ·‖M0A(G) of the radial functions on the free (non-abelian) group FN on N generators (2 ≤ N ≤ ∞) and of the spherical functions on the p-adic group PGL2(Qq) for every prime number q. Introduction Let Y be a non-empty set. A function ψ : Y × Y → C is called a Schur multiplier if for every operator A = (ax,y)x,y∈Y ∈ B(l(Y )) the matrix (ψ(x, y)ax,y)x,y∈Y again represents an operator from B(l (Y )) (this operator is denoted byMψA). If ψ is a Schur multiplier it follows easily from the closed graph theorem that Mψ ∈ B(B(l(Y ))), and one referrers to ‖Mψ‖ as the Schur norm of ψ and denotes it by ‖ψ‖S. The following result, which gives a characterization of the Schur multipliers, is essentially due to Grothendieck, cf. [Pis01, Theorem 5.1] for a proof. Partially supported by the Danish Natural Science Research Council. Partially supported by the Ph.D.-school OP–ALG–TOP–GEO. Supported by European Commission Marie Curie Host Fellowship for the Transfer of Knowledge “Harmonic Analysis, Nonlinear Analysis and Probability” MTKD-CT-2004013389 and by MNiSW Grant N201 054 32/4285.