# Linear Operators and Operator Equations by V. I. Derguzov, V. A. Yakubovich (auth.), Academician V. I.

By V. I. Derguzov, V. A. Yakubovich (auth.), Academician V. I. Smirnov (eds.)

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**Example text**

We are now in a position to give the exact definition of the notion of degeneracy mentioned above. Definition 4. 15) by the elimination of numbers Ai with identical indices. for any p such that n~~=O, s=1, 2, ... , Nm; m=I, 2, .... N. 18) * It should be emphasized that we are interested in identical indices. FOMIN 52 then we will say that the frequency vector Wo is nondegenerate. 17), t h e n t h e v e c tor W 0 will b e s ai d tob e d e gen e rat e an d. i n t his c ase, t h e I number of nonzero vectors generacy of the vector wo.

15) there may be numbers with identical indices. * Using the elimination method. 15) into a form in which the right-hand sides do not contain numbers Ai with identical indices. The transformed system can be written as p = I, 2, ... 16) and Mare integers. N The quantity ~ N, - M characterizes the number of eliminations performed and. con1-1 N sequently. 16). numbers Ai with identical indices cannot have coefficients n~~ that are simultaneously different from zero. We are now in a position to give the exact definition of the notion of degeneracy mentioned above.

N. FOMIN 35 in such a manner that the eigenvalues of Bo with the largest real parts will be situated in a given neighborhood of the line and will be separated by this line from the remaining spectrum. Performing the replacement y = ejrt in Eqs. I = B o- /lI. I coincides with the imaginary axis. 6) Sj= (~~~j ~d6 ~d~ (SI-B~)~ll~- B~rl r; r , j= 1,2. 1 r 2* are positively oriented con- LEMMA 7. 2 < ß2) which satisfy the following inequalities: aj IlP JxII2 =e:;;;(SJx, x)=e:;;;~JIIPJxII2, for any element x belonging to H.