95-337 Brenner A.

MULTIPARAMETER PSEUDODIFFERENTIAL OPERATORS AND RELATED SPECTRAL ASYMPTOTICS (172K, LaTeX) Jun 28, 95

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Abstract. We introduce the notion of ellipticity with parameters and study elliptic boundary value problems with parameters in a bounded domain. Furthermore, the notion of the ellipticity with parameters for differential and pseudodifferential operator pencils is applied to the resolvent construction which leads to the definition of the corresponding complex powers and $\ \zeta -$function, i.e. to a new functional calculus of operators with promising applications to the multiparameter spectral theory. More precisely, we have proved the extension theorem for traces of the above defined kernels of the complex powers and the corresponding $\ \zeta -$function. The last appears to be a meromorphic function in $\ C^m\ $ ($m$ is the number of parameters) with polar sets of the first order and the corresponding holomorphic residue-forms. In particular, for $m=2$ this investigation gives the opportunity to write the two - parameter spectral asymptotics in terms of these forms using the two - dimensional refinement of the Ikehara Tauberian theorem.

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