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**Comparison theorems for the spectra of linear operators and spectral asymptotics.**
*(Russian)*
Zbl 0532.47012

This work is organized on five sections. The first one contains auxiliary propositions on complex and operator holomorphic functions. The main result of the second section is the following comparison theorem. Let H be a selfadjoint operator with discrete spectrum and \(N(r,H)=\sum_{| \lambda_ j(H)| \leq r}1\) denote the distribution function of its eigenvalues. Let B denote a non-selfadjoint operator.

Theorem I. If the operator \(BH^{-p}\) is bounded for some \(p<1\) then there exists \(q>0\) such that the spectrum of the operator \(A=H+B\) lies in the domain \(| Im \lambda |<q| \lambda |^ p\) and \[ N(r,A)-N(r,H)=O(N(r+qr^ p,H)-N(r-qr^ p,H)). \] Theorem I is generalized to the case when H is a normal operator whose spectrum in some angle lies only on the bisectrice. In the third section a more general class of perturbations, those for which \(BH^{-1}\) is compact, is considered. The fourth section contains results on spectra and asymptotics for polynomial operator bundles. In particular, the complete proofs are given, under more general assumptions, of some results announced by M. V. Keldyš since 1951. The abstract results are applied to elliptic differential operators in the fifth section. The following theorem is obtained by applying Theorem I and some results of L. Hörmander.

Theorem II. Let \(\Omega\) be a compact \(C^\infty\)-manifold and \({\mathcal L}\) be an elliptic differential operator of order 2m on \(\Omega\) with \(C^\infty\)-coefficients, which is selfadjoint and semi-bounded in the space \(L^ 2(\Omega,dx),\) where dx is a positive \(C^\infty\)-density on \(\Omega\). If \({\mathcal L}_ 1\) is another differential operator of order \(<2m\) with bounded coefficients, then \[ N(r,{\mathcal L}+{\mathcal L}_ 1)=\omega r^{n/2m}+O(r^{n-1/2m}) \] where \[ \omega =(2\pi)^{- n}\int_{\Omega}mes\{\xi \in R^ n;\tilde {\mathcal L}(x,\xi)<1\}dx \] and \({\mathcal L}(x,\xi)\) is the principal symbol of \({\mathcal L}.\)

Another application of Theorem I is to boundary value problems.

Theorem I. If the operator \(BH^{-p}\) is bounded for some \(p<1\) then there exists \(q>0\) such that the spectrum of the operator \(A=H+B\) lies in the domain \(| Im \lambda |<q| \lambda |^ p\) and \[ N(r,A)-N(r,H)=O(N(r+qr^ p,H)-N(r-qr^ p,H)). \] Theorem I is generalized to the case when H is a normal operator whose spectrum in some angle lies only on the bisectrice. In the third section a more general class of perturbations, those for which \(BH^{-1}\) is compact, is considered. The fourth section contains results on spectra and asymptotics for polynomial operator bundles. In particular, the complete proofs are given, under more general assumptions, of some results announced by M. V. Keldyš since 1951. The abstract results are applied to elliptic differential operators in the fifth section. The following theorem is obtained by applying Theorem I and some results of L. Hörmander.

Theorem II. Let \(\Omega\) be a compact \(C^\infty\)-manifold and \({\mathcal L}\) be an elliptic differential operator of order 2m on \(\Omega\) with \(C^\infty\)-coefficients, which is selfadjoint and semi-bounded in the space \(L^ 2(\Omega,dx),\) where dx is a positive \(C^\infty\)-density on \(\Omega\). If \({\mathcal L}_ 1\) is another differential operator of order \(<2m\) with bounded coefficients, then \[ N(r,{\mathcal L}+{\mathcal L}_ 1)=\omega r^{n/2m}+O(r^{n-1/2m}) \] where \[ \omega =(2\pi)^{- n}\int_{\Omega}mes\{\xi \in R^ n;\tilde {\mathcal L}(x,\xi)<1\}dx \] and \({\mathcal L}(x,\xi)\) is the principal symbol of \({\mathcal L}.\)

Another application of Theorem I is to boundary value problems.

Reviewer: Şt.Frunză

### MSC:

47A56 | Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones) |

47A55 | Perturbation theory of linear operators |

47B25 | Linear symmetric and selfadjoint operators (unbounded) |

47A10 | Spectrum, resolvent |

35J40 | Boundary value problems for higher-order elliptic equations |