The 3-Isometric Lifting Theorem

Scott McCullough; Benjamin Russo

doi:10.1007/s00020-015-2240-7

The 3-Isometric Lifting Theorem

Scott McCullough, Benjamin Russo

Research output: Contribution to journal › Article › peer-review

8 Scopus citations

Abstract

An operator T on Hilbert space is a 3-isometry if T^*nTⁿ= I +n B₁ +n² B₂ is quadratic in n. An operator J is a Jordan operator if J = U + N where U is unitary, N² = 0 and U and N commute. If T is a 3-isometry and c > 0, then I-c^-2 B₂ + s²B₂ 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.

Original language	English
Pages (from-to)	69-87
Number of pages	19
Journal	Integral Equations and Operator Theory
Volume	84
Issue number	1
DOIs	https://doi.org/10.1007/s00020-015-2240-7
State	Published - Jan 1 2016
Externally published	Yes

Funding

S. McCullough was partially supported by NSF Grants DMS-1101137 and DMS-1361501.

Funders	Funder number
National Science Foundation	DMS-1101137, DMS-1361501

Keywords

34B24 (Secondary)
47A20 (Primary)
47A45
47B99

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10.1007/s00020-015-2240-7

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N2 - An operator T on Hilbert space is a 3-isometry if T*nTn= I +n B1 +n2 B2 is quadratic in n. An operator J is a Jordan operator if J = U + N where U is unitary, N2 = 0 and U and N commute. If T is a 3-isometry and c > 0, then I-c-2 B2 + s2B2 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.

AB - An operator T on Hilbert space is a 3-isometry if T*nTn= I +n B1 +n2 B2 is quadratic in n. An operator J is a Jordan operator if J = U + N where U is unitary, N2 = 0 and U and N commute. If T is a 3-isometry and c > 0, then I-c-2 B2 + s2B2 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.

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The 3-Isometric Lifting Theorem

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