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Strictly Hermitian positive definite functions. (English) Zbl 1074.43004

Summary: Let \(H\) be any complex inner product space with inner product \(\langle\cdot,\cdot\rangle\). We say that \(f: \mathbb{C}\to\mathbb{C}\) is Hermitian positive definite on \(H\) if the matrix \[ (f(\langle{\mathbf z}^r,{\mathbf z}^s\rangle))^n_{r,s=1}\tag{\(*\)} \] is Hermitian positive definite for any choice of \({\mathbf z}_1,\dots,{\mathbf z}^n\) in \(H\) for all \(n\). It is strictly Hermitian positive definite if the matrix \((*)\) is also non-singular for any choice of distinct \({\mathbf z}_1,\dots,{\mathbf z}^n\) in \(H\). In this article, we prove that if \(\dim H\geq 3\), then \(f\) is Hermitian positive definite on \(H\) if and only if \[ f(z)= \sum^\infty_{k,\ell= 0} b_{k,\ell} z^k\overline z^\ell,\tag{\(**\)} \] where \(b_{k,\ell}\geq 0\) for all \(k\), \(\ell\) in \(\mathbb{Z}_+\), and the series converges for all \(z\) in \(\mathbb{C}\). We also prove that \(f\) of the form \((**)\) is strictly Hermitian positive definite on any \(H\) if and only if the set \[ J= \{(k,\ell): b_{k,\ell}> 0\} \] is such that \((0, 0)\in J\), and every arithmetic sequence in \(\mathbb{Z}\) intersects the values \(\{k-\ell:(k,\ell)\in J\}\) an infinite number of times.

MSC:

43A35 Positive definite functions on groups, semigroups, etc.
15B57 Hermitian, skew-Hermitian, and related matrices
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