TY - JOUR

T1 - von Neumann’s inequality for row contractive matrix tuples

AU - Hartz, Michael

AU - Richter, Stefan

AU - Shalit, Orr Moshe

N1 - Funding Information:
The collaboration leading to this paper was spurred by the presentation of the question behind Theorem 1.1 by one of the authors in the “Open Problems Session” that was held at the online conference OTWIA 2020. We wish to thank Meric Augat, the organizer of that session, as well as the organizers of the conference. We are also grateful to Łukasz Kosiński for helpful comments and for bringing [16] to our attention. Moreover, we are grateful to the editor Mikael de la Salle for bringing to our attention a theorem of Schur, as well [17], where an elementary proof can be found. This led to an improved upper bound in Lemma 4.5.
Funding Information:
The work of M.H. is partially supported by a GIF grant. The work of O.M. Shalit is partially supported by ISF Grants no. 195/16 and 431/20.
Publisher Copyright:
© 2022, The Author(s).

PY - 2022/8/1

Y1 - 2022/8/1

N2 - We prove that for all n∈ N, there exists a constant Cn such that for all d∈ N, for every row contraction T consisting of d commuting n× n matrices and every polynomial p, the following inequality holds: ‖p(T)‖≤Cnsupz∈Bd|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∞(Bd) for d≥ 2. Second, we prove that the multiplier algebra Mult(Da(Bd)) of the weighted Dirichlet space Da(Bd) on the ball is not topologically subhomogeneous when d≥ 2 and a∈ (0 , d). In fact, we determine all the bounded finite dimensional representations of the norm closed subalgebra A(Da(Bd)) of Mult(Da(Bd)) generated by polynomials. Lastly, we also show that there exists a uniformly bounded nc holomorphic function on the free commutative ball CBd that is levelwise uniformly continuous but not globally uniformly continuous.

AB - We prove that for all n∈ N, there exists a constant Cn such that for all d∈ N, for every row contraction T consisting of d commuting n× n matrices and every polynomial p, the following inequality holds: ‖p(T)‖≤Cnsupz∈Bd|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∞(Bd) for d≥ 2. Second, we prove that the multiplier algebra Mult(Da(Bd)) of the weighted Dirichlet space Da(Bd) on the ball is not topologically subhomogeneous when d≥ 2 and a∈ (0 , d). In fact, we determine all the bounded finite dimensional representations of the norm closed subalgebra A(Da(Bd)) of Mult(Da(Bd)) generated by polynomials. Lastly, we also show that there exists a uniformly bounded nc holomorphic function on the free commutative ball CBd that is levelwise uniformly continuous but not globally uniformly continuous.

KW - Gleason’s problem

KW - Noncommutative function theory

KW - Von Neumann type inequality

UR - http://www.scopus.com/inward/record.url?scp=85130583607&partnerID=8YFLogxK

U2 - 10.1007/s00209-022-03044-1

DO - 10.1007/s00209-022-03044-1

M3 - Article

AN - SCOPUS:85130583607

SN - 0025-5874

VL - 301

SP - 3877

EP - 3894

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

IS - 4

ER -