## Multivariable Operator Theory

#### Event Title

Extremal multipliers of the Drury-Arveson space

Crow 206

#### Start Date

7-18-2016 3:00 PM

#### End Date

18-7-2016 3:20 PM

#### Description

(This is joint work with Robert T. W. Martin.) We introduce a family of multipliers on the Drury-Arveson space $H^2_d$ which we call {\it quasi-extreme}. To each contractive multiplier $b$ is associated a de Branges-Rovnyak space $\mathcal{H}(b)$ with kernel $$k^b(z,w)=\frac{1-b(z)b(w)^*}{1-zw^*}$$ In one variable, the theory of $\mathcal{H}(b)$ spaces splits into two streams, one for $b$ which are extreme points of the unit ball of $H^\infty(\mathbb D)$, 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-quasi-extreme 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 $H^2_d$, and conjecture that the converse holds. A key tool is the analysis of contractive $d$-tuples of operators which solve the Gleason problem in $\mathcal{H}(b)$.

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Jul 18th, 3:00 PM Jul 18th, 3:20 PM

Extremal multipliers of the Drury-Arveson space

Crow 206

(This is joint work with Robert T. W. Martin.) We introduce a family of multipliers on the Drury-Arveson space $H^2_d$ which we call {\it quasi-extreme}. To each contractive multiplier $b$ is associated a de Branges-Rovnyak space $\mathcal{H}(b)$ with kernel $$k^b(z,w)=\frac{1-b(z)b(w)^*}{1-zw^*}$$ In one variable, the theory of $\mathcal{H}(b)$ spaces splits into two streams, one for $b$ which are extreme points of the unit ball of $H^\infty(\mathbb D)$, 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-quasi-extreme 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 $H^2_d$, and conjecture that the converse holds. A key tool is the analysis of contractive $d$-tuples of operators which solve the Gleason problem in $\mathcal{H}(b)$.