Aleksandrov-clark Theory for Drury-arveson Space
نویسنده
چکیده
Recent work has demonstrated that Clark’s theory of unitary perturbations of the backward shift restricted to a deBranges-Rovnyak subspace of Hardy space on the disk has a natural extension to the several variable setting. In the several variable case, the appropriate generalization of the Schur class of contractive analytic functions is the closed unit ball of the Drury-Arveson multiplier algebra and the Aleksandrov-Clark measures are necessarily promoted to positive linear functionals on a symmetrized subsystem of the Cuntz-Toeplitz operator system A + A∗, where A is the non-commutative disk algebra. We continue this program for vector-valued Drury-Arveson space by establishing the existence of a canonical ‘tight’ extension of any Aleksandrov-Clark map to the full Cuntz-Toeplitz operator system. We apply this tight extension to generalize several earlier results and we characterize all extensions of the Aleksandrov-Clark maps.
منابع مشابه
Clark Theory in the Drury-arveson Space
We extend the basic elements of Clark’s theory of rank-one perturbations of backward shifts, to row-contractive operators associated to de Branges-Rovnyak type spaces H(b) contractively contained in the Drury-Arveson space on the unit ball in C. The Aleksandrov-Clark measures on the circle are replaced by a family of states on a certain noncommutative operator system, and the backward shift is ...
متن کاملOn the Problem of Characterizing Multipliers for the Drury-arveson Space
Let H n be the Drury-Arveson space on the unit ball B in C , and suppose that n ≥ 2. Let kz, z ∈ B, be the normalized reproducing kernel for H n. In this paper we consider the following rather basic question in the theory of the Drury-Arveson space: For f ∈ H n, does the condition sup|z|<1 ‖fkz‖ < ∞ imply that f is a multiplier of H n? We show that the answer is negative. We further show that t...
متن کاملCommutators and Localization on the Drury-arveson Space
Let f be a multiplier for the Drury-Arveson space H n of the unit ball, and let ζ1, ..., ζn denote the coordinate functions. We show that for each 1 ≤ i ≤ n, the commutator [M∗ f ,Mζi ] belongs to the Schatten class Cp, p > 2n. This leads to a localization result for multipliers.
متن کاملMultipliers and Essential Norm on the Drury-arveson Space
It is well known that for multipliers f of the Drury-Arveson space H n, ‖f‖∞ does not dominate the operator norm of Mf . We show that in general ‖f‖∞ does not even dominate the essential norm of Mf . A consequence of this is that there exist multipliers f of H n for which Mf fails to be essentially hyponormal, i.e., if K is any compact, self-adjoint operator, then the inequality M∗ f Mf −MfM f ...
متن کاملExtremal multipliers of the Drury-Arveson space
In one variable, the theory of H(b) spaces splits into two streams, one for b which are extreme points of the unit ball of H∞(D), and the other for non-extreme points. We show that there is an analogous splitting in the Drury-Arveson case, between the quasi-extreme and non-quasiextreme cases. (In one variable the notions of extreme and quasi-extreme coincide.) We give a number of equivalent cha...
متن کامل