Toeplitz operators – an asymptotic quantization of symplectic cones. (English) Zbl 0735.47014

Stochastic processes and their applications in mathematics and physics, Proc. 3rd Symp., Bielefeld/Ger. 1985, Math. Appl. 61, 95-106 (1990).

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[For the entire collection see Zbl 0704.00024.]
Let \(X\) be a compact \(C^ \infty\) manifold and let \(H\) denote the Hilbert space \(L^ 2(X)\). \(H^ s(X)\) is the Hilbert space of distributions on \(X\) whose derivatives of order \(s\) are \(L^ 2\). The author constructs for \(X\) a Hilbert space \({\mathcal O}^ s(X)\) and an orthogonal projector \(S\) such that \({\mathcal O}^ s(X)=SH^ s(X)\) for all \(s\in R\), which is a Fourier integral operator with a complex phase. A Toeplitz operator on \(X\) of degree \(d\) is a linear continuous operator \({\mathcal O}^ s(X)\to{\mathcal O}^{s-d}(X)\) of the form \(u\to T_ Q(u)=S(Q(u))\) with a pseudo-differential operator of degree \(d\) on \(X\). The author gives a unitary operator from pseudo-differential operators of \(X\) into Toeplitz operators. The proofs and details of the construction are in [The spectral theory of Toeplitz operators. Ann. Math. Studies 99 (1981; Zbl 0469.47021)] by the present author and V. Guillemin.
Reviewer: T.Nakazi (Sapporo)


47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
47G30 Pseudodifferential operators