 

Sameer Chavan,
Shubham Jain, and
Paramita Pramanick
von Neumann's inequality for the Hartogs triangle
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Published: 
May 7, 2022. 
Keywords: 
Hartogs triangle, commuting tuple, von Neumann inequality. 
Subject: 
Primary 47A13; Secondary 32A10. 


Abstract
For a commuting pair T of bounded linear operators T_{1} and T_{2} on a Hilbert space H,
let D_{T} = T_{2}*T_{2} T_{1}*T_{1}. If
T_{2}* D_{T} T_{2} ≤ D_{T} and the Taylor spectrum of T is contained
in the Hartogs triangle Δ_{H}, then for any bounded holomorphic function φ on
Δ_{H}, φ(T) ≤ φ_{∞}. We deduce this fact from an analogue
of von Neumann's inequality for bounded domains in C^{d}. The proof of the latter closely follows
the model theory approach as developed in [1].


Acknowledgements
N/A


Author information
Sameer Chavan:
Department of Mathematics and Statistics
Indian Institute of Technology
Kanpur, India
chavan@iitk.ac.in
Shubham Jain:
Department of Mathematics and Statistics
Indian Institute of Technology
Kanpur, India
shubjain@iitk.ac.in
Paramita Pramanick:
School of Mathematics
HarishChandra Research Institute
Chhatnag Road, Jhunsi, Allahabad 211019, India
paramitapramanick@hri.res.in

