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Published September 15, 2004 | public
Journal Article

A canonical factorization for meromorphic Herglotz functions on the unit disk and sum rules for Jacobi matrices

Abstract

We prove a general canonical factorization for meromorphic Herglotz functions on the unit disk whose notable elements are that there is no restriction (other than interlacing) on the zeros and poles for their Blaschke product to converge and there is no singular inner function. We use this result to provide a significant simplification in the proof of Killip–Simon (Ann. Math. 158 (2003) 253) of their result characterizing the spectral measures of Jacobi matrices, J, with J−J_0 Hilbert-Schmidt. We prove a nonlocal version of Case and step-by-step sum rules.

Additional Information

© 2003 Elsevier Inc. Received 8 July 2003, Revised 12 November 2003, Accepted 13 November 2003, Available online 23 January 2004. Supported in part by NSF Grant DMS-0140592.

Additional details

Created:
August 22, 2023
Modified:
October 25, 2023