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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Szegő limit theorems for Toeplitz operators on compact homogeneous spaces


Authors: I. I. Hirschman, D. S. Liang and E. N. Wilson
Journal: Trans. Amer. Math. Soc. 270 (1982), 351-376
MSC: Primary 47B35; Secondary 22C05, 43A85
DOI: https://doi.org/10.1090/S0002-9947-1982-0645321-6
MathSciNet review: 645321
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Abstract: Let $ f$ be a real valued integrable function on a compact homogeneous space $ M = K\backslash G$ and $ {M_f}$ the operator of pointwise multiplication by $ f$. The authors consider families of Toeplitz operators $ {T_{f,P}} = P{M_f}P$ as $ P$ ranges over a net of orthogonal projections from $ {L^2}(M)$ to finite dimensional $ G$-invariant subspaces. Necessary and sufficient conditions are given on the net in order that the distribution of eigenvalues of these Toeplitz operators is asymptotic to the distribution of values of $ f$ in the sense of Szegö's classical theorem for the circle. Explicit sequences satisfying these conditions are constructed for all compact Lie groups and for all Riemannian symmetric compact spaces.


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DOI: https://doi.org/10.1090/S0002-9947-1982-0645321-6
Keywords: Toeplitz operators, Szegö theorem, compact groups, homogeneous spaces, Lie groups, compact symmetric spaces
Article copyright: © Copyright 1982 American Mathematical Society