Strictly positive definite functions on the complex Hilbert sphere

We write S∞ to denote the unit sphere of `, the set of all z in ` for which 〈z, z〉 = 1 (this set is often called the complex Hilbert sphere). A continuous complex-valued function f defined on the closed unit disc D := {ζ ∈ C : |ζ| ≤ 1} is said to be positive definite of order n on S∞ if, for any set of n distinct points z1, . . . , zn on S∞, the n×n matrix A with ij-entry given by f(〈zi, zj〉) is nonnegative definite, i.e., it is Hermitian and its eigenvalues are all nonnegative. If the matrix A in the above definition is positive definite, i.e., it is Hermitian and its eigenvalues are all positive, we say that f is strictly positive definite of order n on S∞. Note that an n × n Hermitian matrix A is positive definite if and only if c∗Ac > 0 whenever c is not the zero vector in C . If a function f is positive definite of order n for all n ∈ Z+ \ {0}, then f is said to be positive definite on S∞. If a function f is strictly positive definite of order n for all n ∈ Z+ \ {0}, then f is said to be strictly positive definite on S∞. 15