Operator Theory and Function Theory in Drury–Arveson Space and Its Quotients

Shalit, Orr

doi:10.1007/978-3-0348-0667-1_60

Cited by 13 publications

(4 citation statements)

References 103 publications

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“…(2) H 2 m is the Drury-Arveson space of analytic functions on B m , the one with the reproducing kernel .1 hz; wi/ 1 . (See [2], [34,Chapter 41].) It has the standard orthonormal basis .nŠ=jnjŠ/ 1=2 z n , n 2 N m .…”

Section: A Some Analytic Hilbert Spacesmentioning

confidence: 99%

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Perturbations of principal submodules in the Drury–Arveson space

Jabbari

Tang

2023

J. Noncommut. Geom.

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Section: A Some Analytic Hilbert Spacesmentioning

confidence: 99%

“…There have been many studies on Conjectures 1.1 and 1.2 in the last decades. A survey of results about these conjectures is given in [34,Chapter 41] and [21].…”

Section: Introductionmentioning

confidence: 99%

Perturbations of principal submodules in the Drury–Arveson space

Jabbari

Tang

2023

J. Noncommut. Geom.

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“…However, there are functions in A(B d ) which are not multipliers, and hence not uniformly bounded on CB d (see Sect. 3.7 "The strict containment M d H ∞ (B d )" in [25]).…”

Section: An Application To Uniform Continuity Of Noncommutative Funct...mentioning

confidence: 99%

“…We can now show that the answer to this question is positive. [25] ("Continuous multipliers versus A d "), where it was explained how this follows from the methods of [11].…”

Section: An Application To Uniform Continuity Of Noncommutative Funct...mentioning

confidence: 99%

von Neumann’s inequality for row contractive matrix tuples

2022

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We prove that for all $$n\in {\mathbb {N}}$$ n ∈ N , there exists a constant $$C_{n}$$ C n such that for all $$d \in {\mathbb {N}}$$ d ∈ N , for every row contraction T consisting of d commuting $$n \times n$$ n × n matrices and every polynomial p, the following inequality holds: $$\begin{aligned} \Vert p(T)\Vert \le C_{n} \sup _{z \in {\mathbb {B}}_d} |p(z)| . \end{aligned}$$ ‖ p ( T ) ‖ ≤ C n sup z ∈ B d | p ( z ) | . We apply this result and the considerations involved in the proof to several open problems from the pertinent literature. First, we show that Gleason’s problem cannot be solved contractively in $$H^\infty ({\mathbb {B}}_d)$$ H ∞ ( B d ) for $$d \ge 2$$ d ≥ 2 . Second, we prove that the multiplier algebra $${{\,\mathrm{Mult}\,}}({\mathcal {D}}_a({\mathbb {B}}_d))$$ Mult ( D a ( B d ) ) of the weighted Dirichlet space $${\mathcal {D}}_a({\mathbb {B}}_d)$$ D a ( B d ) on the ball is not topologically subhomogeneous when $$d \ge 2$$ d ≥ 2 and $$a \in (0,d)$$ a ∈ ( 0 , d ) . In fact, we determine all the bounded finite dimensional representations of the norm closed subalgebra $$A({\mathcal {D}}_a({\mathbb {B}}_d))$$ A ( D a ( B d ) ) of $${{\,\mathrm{Mult}\,}}({\mathcal {D}}_a({\mathbb {B}}_d))$$ Mult ( D a ( B d ) ) generated by polynomials. Lastly, we also show that there exists a uniformly bounded nc holomorphic function on the free commutative ball $$\mathfrak {C}\mathfrak {B}_d$$ C B d that is levelwise uniformly continuous but not globally uniformly continuous.

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An Invitation to the Drury–Arveson Space

Hartz

2023

Fields Institute Monographs

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Operator Theory and Function Theory in Drury–Arveson Space and Its Quotients

Cited by 13 publications

References 103 publications

Perturbations of principal submodules in the Drury–Arveson space

Perturbations of principal submodules in the Drury–Arveson space

von Neumann’s inequality for row contractive matrix tuples

An Invitation to the Drury–Arveson Space

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