Unitary invariants in multivariable operator theory

Unitary Invariants in Multivariable Operator Theory

The problems considered in this paper come as a natural continuation of our program to develop a free analogue of Sz.-Nagy–Foiaş theory, for row contractions. An n-tuple (T1, . . . , Tn) of operators acting on a Hilbert space is called row contraction if T1T ∗ 1 + · · ·+ TnT ∗ n ≤ I. In this study, the role of the unilateral shift is played by the left creation operators on the full Fock space ...

Constrained Multivariable Operator Theory Ii

On Bilateral Weighted Shifts in Noncommutative Multivariable Operator Theory

We present a generalization of bilateral weighted shift operators for the noncommutative multivariable setting. We discover a notion of periodicity for these shifts, which has an appealing diagramatic interpretation in terms of an infinite tree structure associated with the underlying Hilbert space. These shifts arise naturally through weighted versions of certain representations of the Cuntz C...

Subalgebras of C∗-algebras Iii: Multivariable Operator Theory

A d-contraction is a d-tuple (T1, . . . , Td) of mutually commuting operators acting on a common Hilbert space H such that ‖T1ξ1 + T2ξ2 + · · ·+ Tdξd‖ 2 ≤ ‖ξ1‖ 2 + ‖ξ2‖ 2 + · · ·+ ‖ξd‖ 2 for all ξ1, ξ2, . . . , ξd ∈ H. These are the higher dimensional counterparts of contractions. We show that many of the operator-theoretic aspects of function theory in the unit disk generalize to the unit ball...

J ul 2 00 5 CONSTRAINED MULTIVARIABLE OPERATOR THEORY

We develop a dilation theory for row contractions where P is a set of noncommutative polynomials. The model n-tuple is the universal row contraction [B1,. .. , Bn] satisfying the same constraints as T , which turns out to be, in a certain sense, the maximal constrained piece of the n-tuple [S1,. .. , Sn] of left creation operators on the full Fock space on n generators. The theory is based on a...