Weyl quantization of symmetric spaces. I: Hyperbolic matrix domains. (English) Zbl 0736.47014
The classical Weyl quantization formula
\[
(W(f)u)(x)=\int_{R^ n}\int_{R^ n}f\left ({x+y\over 2},\zeta\right) e^{2\pi i(x-{y\over \zeta})}u(y)dyd\zeta
\]
is extended to the form
\[
W(f)=\int_ \Omega f(\zeta)W(s_ \zeta)d\mu_ \Omega(\zeta)
\]
where \(\Omega\) is a bounded symmetric domain with invariant measure \(\mu_ \Omega\) (the author specializes here on the hyperbolic matrix ball
\[
\Omega=\{z\in\mathbb{C}^{r\times r}: I-zz^*>0\}, r>0)
\]
and \(W\) is a unitary representation of the group of symmetries of \(\Omega\). The main result (Theorem 5.5) provides conditions on the symbol function \(f\) under which \(W(f)\) is a bounded Hilbert space operator.
Reviewer: K.N.Boyadzhiev (Ada)
MSC:
47A60 | Functional calculus for linear operators |
46L70 | Nonassociative selfadjoint operator algebras |
47G10 | Integral operators |
Keywords:
Weyl quantization formula; bounded symmetric domain; invariant measure; hyperbolic matrix ball; symbol function; bounded Hilbert space operator; unitary representation of the group of symmetriesReferences:
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