Self-adjoint polynomial operator pencils, spectrally equivalent to self-adjoint operators

Abstract

The solving of various problems of mathematical physics by the Fourier method leads naturally to appropriate polynomial operator pencils of the form L(~) = ~Ao + ~-~A~+. . . + ~A~-~ + A~ (1) with operator coefficients A~ = A~, i = 0, I ..... n, A~IE~(H), H being a Hilbert space. One of the possible approaches to the spectral investigation of such pencils consists in their linearization. The latter is especially effective in those cases when the equivalent linear problem can be symmetrized by some self-adjoint operator, possessing the definiteness property. This takes place in the case of the so-called strongly Hilbert operator pencils [I, 2]. In the space H of the direct orthogonal sum of n copies of the initial space H ((y.z) = k ) ((y~y~ . . . . . y~)~,(zt, z~ . . . . . z~/) = ~_~(y~,%) , to the penc i l (~) one a s s o c i a t e s the l i n e a r penc i l (2) where L (~) = ~ -~'