Extremal multipliers of the Drury-Arveson space
نویسندگان
چکیده
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 characterizations of quasi-extremity, and prove that if b is quasi-extreme then b is an extreme point of the unit ball of the multiplier algebra of H d , and conjecture that the converse holds. A key tool is the analysis of contractive d-tuples of operators which solve the Gleason problem in H(b).
منابع مشابه
Multipliers 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 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 کاملDuality, Convexity and Peak Interpolation in the Drury-arveson Space
We consider the closed algebra Ad generated by the polynomial multipliers on the Drury-Arveson space. We identify Ad as a direct sum of the preduals of the full multiplier algebra and of a commutative von Neumann algebra, and establish analogues of many classical results concerning the dual space of the ball algebra. These developments are deeply intertwined with the problem of peak interpolati...
متن کاملInterpolation Problems for Schur Multipliers on the Drury-Arveson Space: from Nevanlinna- Pick to Abstract Interpolation Problem
We survey various increasingly more general operator-theoretic formulations of generalized left-tangential Nevanlinna-Pick interpolation for Schur multipliers on the Drury-Arveson space. An adaptation of the methods of Potapov and Dym leads to a chain-matrix linear-fractional parametrization for the set of all solutions for all but the last of the formulations for the case where the Pick operat...
متن کامل