A quantization of the Cartan domain BD I \((q=2)\) and operators on the light cone
type: Item
instance of: scholarly article
title: A quantization of the Cartan domain BD I \((q=2)\) and operators on the light cone
zbMATH Open document ID: 0632.58033
DOI: 10.1016/0022-1236(87)90090-5
author: Julianne Unterberger
author: André Unterberger
published in: Journal of Functional Analysis
publication date: 1987-01-01T00:00:00Z
review text: Let \(\pi =SO_ 0(2,n+1)/(SO(2)\times SO(n+1))\) be a symmetric space. For a real number \(\lambda\) let \(H_{\lambda}\) be the Hilbert space that consists of measurable complex-valued functions on the light cone \[ C=\{x=(x_ 0,x_ 1,...,x_ n)\in R^{n+1}| \quad x_ 0>0,\quad r(x)=x^ 2_ 0-x^ 2_ 1-...-x^ 2_ n>0\} \] satisfying \(\| u\|^ 2_{\lambda}=\int_{C}| u(t)|^ 2(r(t))^{-\lambda /2}dt<\infty\). The main result of the paper is a proof of the fact that for any reasonable function f on \(\pi\) one can associate it with a bounded linear operator op(f) on \(H_{\lambda}\). The authors point out that this \(H_{\lambda}\)-calculus is closely related to the Weyl calculus of pseudodifferential operators and the Fuchs calculus, and in some sense the \(H_{\lambda}\)-calculus is better than the Fuchs calculus.
Mathematics Subject Classification ID: 58J40
Mathematics Subject Classification ID: 35S05
zbMATH DE Number: 4028476
zbMATH Keywords: symbol calculus
zbMATH Keywords: symmetric space
zbMATH Keywords: Weyl calculus of pseudodifferential operators
zbMATH Keywords: Fuchs calculus
MaRDI profile type: Publication
full work available at URL: 0022-1236(87)90090-5
OpenAlex ID: W1985927942
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cites work: Q5734920
cites work: Q3897665
cites work: Q3338997
cites work: Q4102285
cites work: The weyl calculus of pseudo-differential operators
cites work: Q3736112
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cites work: Q3672190
cites work: Q5185404
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cites work: Q5517003
cites work: Symbols and orbits for 3-step nilpotent Lie groups
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description: scientific article; zbMATH DE number 4028476