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Complete Nevanlinna–Pick Kernels

✍ Scribed by Jim Agler; John E. McCarthy

Publisher
Elsevier Science
Year
2000
Tongue
English
Weight
126 KB
Volume
175
Category
Article
ISSN
0022-1236

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✦ Synopsis

We give a new treatment of Quiggin's and McCullough's characterization of complete Nevanlinna Pick kernels. We show that a kernel has the matrix-valued Nevanlinna Pick property if and only if it has the vector-valued Nevanlinna Pick property. We give a representation of all complete Nevanlinna Pick kernels, and show that they are all restrictions of a universal complete Nevanlinna Pick kernel.

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Let k be the reproducing kernel for a Hilbert space HðkÞ of analytic functions on B d , the open unit ball in C d , d51. k is called a complete NP kernel if k 0 1 and if 1 À 1=k l ðzÞ is positive definite on B d  B d . Let D be a separable Hilbert space, and consider HðkÞ D ffi Hðk; DÞ, and think o