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Zeta-function of a transversely elliptic operator. (English. Russian original) Zbl 0569.58034

Sib. Math. J. 25, 959-966 (1984); translation from Sib. Mat. Zh. 25, No. 6(148), 158-166 (1984).
In the paper the zeta-function of a transversally elliptic operator is constructed. Let X be a closed, smooth G-manifold, where G is a compact Lie group. Let \(A: C^{\infty}(X,E)\to C^{\infty}(X,E)\) be a classical, pseudodifferential operator of order \(m>0\) and \(T^*_ GX\) the space of covectors conormal to the orbits of the action. It is assumed that for any \(\nu \in T^*_ GX\setminus 0\) and \(\Lambda\in (- \infty,0]\) there exists \((a_ m(\nu)-\Lambda)^{-1}\) where \(a_ m\) denotes the principal symbol of A. Using \(\theta\), an auxiliary symbol of 0-th order with support in a cone neighbourhood of \(T^*_ GX\) and equal to 1 on \(T_ GX\setminus 0\), we are able to construct \(A^ z_{\theta}\) complex power for such an operator, in a way analogous to that for the elliptic operator. Then we define \(\zeta (z)=tr_ GA^ z_{\theta}\) with values in D’(G), where \(<tr_ GB,\eta >=tr BT(\eta)\), \(\eta \in C^{\infty j}(G)\); \(T(\eta)=\int_{G}\eta (g)T(g)dg\) is a bounded operator in \(L^ 2(X,E)\) (T(g) denotes the action of g on sections). \(\zeta\) (z) is holomorphic for \(Re(z)<-\dim X/m\). It is shown in the paper that it extends to the meromorphic function in C with poles of the order \(<\dim X+\dim G\), which lie on the points of arithmetic progression (in R). Moreover the singular part does not depend on the choice of an auxiliary symbol. Namely the following fact is proved: Let \(\theta_ 1\), \(\theta_ 2\) be two auxiliary symbols, then \(\zeta_ 1(z)-\zeta_ 2(z)=tr_ GA^ z_{\theta_ 1}-tr_ GA^ z_{\theta_ 2}\) extends to the holomorphic function in the whole complex plane.
Reviewer: K.Wojciechowski

MSC:

58J50 Spectral problems; spectral geometry; scattering theory on manifolds
35P20 Asymptotic distributions of eigenvalues in context of PDEs
57S15 Compact Lie groups of differentiable transformations
35S05 Pseudodifferential operators as generalizations of partial differential operators
43A99 Abstract harmonic analysis
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References:

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