## 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).
[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)

### MSC:

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

### Citations:

Zbl 0704.00024; Zbl 0469.47021