Download Completeness of Root Functions of Regular Differential by Sasun Yakubov PDF

By Sasun Yakubov

ISBN-10: 0582236924

ISBN-13: 9780582236929

The ideal mathematical research of assorted average phenomena is an previous and tough challenge. This booklet is the 1st to deal systematically with the final non-selfadjoint difficulties in mechanics and physics. It bargains frequently with bounded domain names with soft limitations, but in addition considers elliptic boundary price difficulties in tube domain names, i.e. in non-smooth domain names. This quantity could be of specific worth to these operating in differential equations, sensible research, and equations of mathematical physics.

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27 (spJ9)_o. Hence, = {B-i{Ji^B)Y. 3 B-i{]X,B)H C ((sp5)fo,-B -i(7I,S)^) = 0. 20) On the other hand, uq G (spB)io and is an eigenvector of the operator B*, cor­ responding to the eigenvalue p. 20). ■ Let £? be a Banch space and { uk} ^ be a system of elements from E. The closure of the linear span of the system will be denoted by sp{ufc}. The system is called complete in the set F of the space B, if F C sp{ujb}. The system {ujb} is called complete in the space F , if sp{ujt} = E. If a functional w' G F ' orthogonal to spjufe} is orthogonal to the set F as well, then the system {ufc} is complete in the set F of the space F.

By the Liouville theorem, we have that under n > 0 F{X) = oio + Qfi A+ •••+ a,iX^ ^ (2*3) and under n = —1 F(X) = 0. Let us expand ii(A, A) in some neighborhood of Aq in the Taylor series: ii(A, A) = ii(Ao, A) + (A - Ao)ij2(Ao, A) + ... + (A - XoY^R^-^\Xo,A) + • • • . (A,A)u,u' > = < R{X o,A)u y > + ••• + (A - Ao)^‘ < ii’^+'(Ao, A)u,u' > + . * . 3) follows < > =0, ueE. So, a system of root vectors of the operator A is complete in D{{A - Ao/)"+2). ((j4 —Ao/)”+2) = oo. Prom this eind the completeness of the root vectors it follows that the set of different eigenvalues of the operator A is countable, which, in turn, implies the discreteness of the spectrum of the operator A.

J | A - A o |=€ |M -A o |= ffi dA A -^ ’ |/i- A o |= e i |A -A o |= e and observing that / d/z = —27TZ, A —/z |/i-A o |= € i / |A -A o |= e we obtain |A -A o |= e dA = 0, A —/1 2. UNBOUNDED POLYNOMIAL OPERATOR PENCILS 42 So, P is a projector. Commutativity of P with B follows from commutativity of B with P(A,P). Since a compact projector is finite-dimensional, then P is a finite­ dimensional projector. So, P = Pi + -E2, where P i = P E is a finite-dimensional invariant subspace of the operator B, and P 2 = {I —P) E is an infinite-dimensional invariant subspace of the operator P.

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