2015/08/06 by Benjamin P. Russo, Russo, Benjamin
Mathematics · #(Primary). 47A45 #34B24 (Secondary) #47A20 #47B99 #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory
paper · pdf · doi:10.48550/arxiv.1508.01273
openalex publication_date 2015/08/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
An operator T is called a 3-isometry if there exists operators B1(T^*,T) and B2(T^*,T) such that Q(n)=T*nTn=1+nB1(T^*,T)+n2 B2(T^*,T) for all natural numbers n. An operator J is a Jordan operator of order 2 if J=U+N where U is unitary, N is nilpotent order 2, and U and N commute. An easy computation shows that J is a 3-isometry and that the restriction of J to an invariant subspace is also a 3-isometry. Those 3-isometries which are the restriction of a Jordan operator to an invariant subspace can be identified, using the theory of completely positive maps, in terms of a positivity condition on the operator pencil Q(s). In this article, we establish the analogous result in the multi-variable setting and show, by modifying an example of Choi, that an additional hypothesis is necessary. Lastly we discuss the joint spectrum of sub-Jordan tuples and derive results for 3-symmetric operators as a corollary.