A Szegö limit formula for conjugates of ψ DO's by Hecke operators

Steven Zelditch

doi:10.1016/0022-1236(88)90084-5

A Szegö limit formula for conjugates of ψ DO's by Hecke operators

Steven Zelditch^*

^*Corresponding author for this work

Mathematics

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

Abstract

Let T_g be a Hecke operator and let Op(a) be a ψ DO on L²(Γ{minus 45 degree rule}b). Further, let π_λ be the projection onto span {u_j: -Δu_j = λ_ju_j, T_gu_j = ρ{variant}_j(g) u_j, √λ_j ≤ λ}. We will determine the Szegö limit measure μ_g,a for the operators T_g^*Op(a) T_g: ∝_-∞^∞ x^kdμ_g,a(x) = lim_{λ → ∞} (1/rank π_λ tr(π_λT _g^* Op(a) T_gπ_λ)^k. Comparison of μ_g,a with the Szegö limit measure of Op(T_ga) shows that T_g^* Op(a) T_g is quite remote spectrally from Op(T_ga).

Original language	English (US)
Pages (from-to)	198-210
Number of pages	13
Journal	Journal of Functional Analysis
Volume	77
Issue number	1
DOIs	https://doi.org/10.1016/0022-1236(88)90084-5
State	Published - Mar 1988

ASJC Scopus subject areas

Analysis

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title = "A Szeg{\"o} limit formula for conjugates of ψ DO's by Hecke operators",

abstract = "Let Tg be a Hecke operator and let Op(a) be a ψ DO on L2(Γ{minus 45 degree rule}b). Further, let πλ be the projection onto span {uj: -Δuj = λjuj, Tguj = ρ{variant}j(g) uj, √λj ≤ λ}. We will determine the Szeg{\"o} limit measure μg,a for the operators Tg*Op(a) Tg: ∝-∞∞ xkdμg,a(x) = limλ → ∞ (1/rank πλ tr(πλT g* Op(a) Tgπλ)k. Comparison of μg,a with the Szeg{\"o} limit measure of Op(Tga) shows that Tg* Op(a) Tg is quite remote spectrally from Op(Tga).",

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N2 - Let Tg be a Hecke operator and let Op(a) be a ψ DO on L2(Γ{minus 45 degree rule}b). Further, let πλ be the projection onto span {uj: -Δuj = λjuj, Tguj = ρ{variant}j(g) uj, √λj ≤ λ}. We will determine the Szegö limit measure μg,a for the operators Tg*Op(a) Tg: ∝-∞∞ xkdμg,a(x) = limλ → ∞ (1/rank πλ tr(πλT g* Op(a) Tgπλ)k. Comparison of μg,a with the Szegö limit measure of Op(Tga) shows that Tg* Op(a) Tg is quite remote spectrally from Op(Tga).

AB - Let Tg be a Hecke operator and let Op(a) be a ψ DO on L2(Γ{minus 45 degree rule}b). Further, let πλ be the projection onto span {uj: -Δuj = λjuj, Tguj = ρ{variant}j(g) uj, √λj ≤ λ}. We will determine the Szegö limit measure μg,a for the operators Tg*Op(a) Tg: ∝-∞∞ xkdμg,a(x) = limλ → ∞ (1/rank πλ tr(πλT g* Op(a) Tgπλ)k. Comparison of μg,a with the Szegö limit measure of Op(Tga) shows that Tg* Op(a) Tg is quite remote spectrally from Op(Tga).

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A Szegö limit formula for conjugates of ψ DO's by Hecke operators

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