No abstract

An operator T on Hilbert space is a 3-isometry if T * n T n = I + nB1 + n 2 B2 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 B2 + sB1 + s 2 B2 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.Mathematics Subject Classification. 47A20 (Primary), 47A45, 47B99, 34B24 (Secondary).

Abstract. An operator T is called a 3-isometry if there exists operators B 1 (T * ,T ) andfor 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.Mathematics subject classification (2010): Primary: 47A20; Secondary: 47A45, 47B99, 34B24.

No abstract

The goal of motion tomography is to recover the description of a vector flow field using information about the trajectory of a sensing unit. In this paper, we develop a predictor corrector algorithm designed to recover vector flow fields from trajectory data with the use of occupation kernels developed by Rosenfeld et al.[1], [2]. Specifically, we use the occupation kernels as an adaptive basis; that is, the trajectories defining our occupation kernels are iteratively updated to improve the estimation on the next stage. We show for a simulated example we have good accuracy in recovering the flow-field using a simple metric. We also apply our algorithm to real world data first presented in [3].