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By I. M Glazman

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Example text

13. On differential operations on vector-functions. Let a differential operation on vector-functions be defined by a da-r da-r (44) r-0 where the coefficients P,(x)=lpr,, k(x)C k-1 (m 0. Let us denote by V2(0, oo) the Hilbert space of the m-component vectorfunctions , (x) = )y,, (x))R,1 having the scalar product - m ( 7) - f I (pk (x) Tk (x) dx. 0 k-1 The operation 1 defines an operator L(symmetrical in X2(0, oo)) on the manifold q.

Furthermore the same lemma shows that the A- and B-metrics are equivalent. Thus the real quadratic functional (Bf. f)-(Af, f)is bounded in HA; therefore there exists in HA a bounded self-adjoint operator T such that [f' g1B-If. g1A=[f Tg1A 2146 22 (28) for all f E HA. For f E ZA we have if, g]B=(Bf g)=(AA-'Bf. g)=[A-'Bf. g1A and relation (28) gives A 'Bf-f=Tf, or A _1(B - A)f = TI. so that for f E I)A Tf=A-'Kf (29) therefore the operator T is completely continuous in HA. Setting for any h E H, we obtain for all f E HA If, T1.

Weyl, since under the given condition the operator L2-L1 is completely continuous. From the decomposition principle it also follows that the type of each point of the continuous part of the spectrum C(L), as well as the non-existence of a lower and upper bound to the spectrum S(L) do not depend on the values of a finite number of elements of matrix (47). Introducing the integer-variable functions N(m; A0, 8), N, (m; µ), and N2(m; µ) for the number of eigenvalues of the square matrix of m-th order / Y6 I a1 aa1 Y1 0 .