Products of Nevanlinna-Pick kernels and operator colligations
- Cite this article as:
- Tomerlin, A.T. Integr equ oper theory (2000) 38: 350. doi:10.1007/BF01291719
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The Schur classSd of analytic functionsf ofd complex variables for whichf(T1,T2,...,Td) has norm at most one for everyd-tuple of commuting strict contractions is characterized by J. Agler  in terms of a Szego kernel factorization property and in terms of a transfer function of a certain type ofd-variable unitary colligation. Replacing the Szego kernel by positive kernels whose reciprocal has one positive square we can define a new Schur class in terms of a kernel factorization property. By using, in part, the approach found in Ball and Trent , we characterize this Schur class in terms of a transfer function of a certain type of unitary colligation. Further, with these results we establish a Nevanlinna-Pick interpolation theorem for our Schur class. Such interpolation theorems already exist in the literature for two special cases where  all kernels are taken to be the Szego kernel, and  d=1.