The 3-Isometric Lifting Theorem

An operator T on Hilbert space is a 3-isometry if T ∗ n T n = I + n B 1 + n 2 B 2 is quadratic in n . An operator J is a Jordan operator if J  =  U  +  N where U is unitary, N 2  = 0 and U and N commute. If T is a 3-isometry and c > 0 , then I - c - 2 B 2 + s B 1 + s 2 B 2 is positive semidefinit...

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Bibliographic Details
Published inIntegral equations and operator theory Vol. 84; no. 1; pp. 69 - 87
Main Authors McCullough, Scott, Russo, Benjamin
Format Journal Article
LanguageEnglish
Published Cham Springer International Publishing 01.01.2016
Subjects
Analysis
Mathematics
Mathematics and Statistics
Dilation theory
Wiener–Hopf factorization
non-normal spectral theory
3-isometric operators
complete positivity
34B24 (Secondary)
3-symmetric operators
47A20 (Primary)
47A45
47B99
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Summary:An operator T on Hilbert space is a 3-isometry if T ∗ n T n = I + n B 1 + n 2 B 2 is quadratic in n . An operator J is a Jordan operator if J  =  U  +  N where U is unitary, N 2  = 0 and U and N commute. If T is a 3-isometry and c > 0 , then I - c - 2 B 2 + s B 1 + s 2 B 2 is positive semidefinite for all real s if and only if it is the restriction of a Jordan operator J  =  U  +  N with the norm of N at most c . As a corollary, an analogous result for 3-symmetric operators, due to Helton and Agler, is recovered.
ISSN:0378-620X
1420-8989
DOI:10.1007/s00020-015-2240-7