By Prof. Leiba Rodman (auth.)
This booklet presents an advent to the trendy conception of polynomials whose coefficients are linear bounded operators in a Banach area - operator polynomials. This thought has its roots and functions in partial differential equations, mechanics and linear structures, in addition to in smooth operator conception and linear algebra. over the past decade, new advances were made within the conception of operator polynomials in response to the spectral technique. the writer, besides different mathematicians, participated during this improvement, and plenty of of the new effects are mirrored during this monograph. it's a excitement to recognize support given to me by means of many mathematicians. First i need to thank my instructor and colleague, I. Gohberg, whose tips has been worthy. all through decades, i've got labored wtih a number of mathematicians with reference to operator polynomials, and, for this reason, their rules have stimulated my view of the topic; those are I. Gohberg, M. A. Kaashoek, L. Lerer, C. V. M. van der Mee, P. Lancaster, ok. Clancey, M. Tismenetsky, D. A. Herrero, and A. C. M. Ran. the next mathematicians gave me suggestion bearing on numerous facets of the e-book: I. Gohberg, M. A. Kaashoek, A. C. M. Ran, ok. Clancey, J. Rovnyak, H. Langer, P.
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Additional info for An Introduction to Operator Polynomials
Then clearly dim Ker L(X) = n-p for all X in this punctured disc. and the auxilliary statement is proved. 3. Let Xo E C\O be the spectrum of L(X). and let Xl E C\O be such that L(X 1 ) is invertible. Pick a connected compact set K c C\O that contains both points Xo and Xl. For every X' E K let E(X') > 0 be such that dim Ker L(X) is constant for o < IX-X'I < E(X'). Using compactness of K. • X(m)}. Since dim Ker L(X) = 0 in a neighborhood of Xl' we have dim Ker L(~) = 0 in a punctured neighborhood of XO.
These properties define Y correctly and. uniquely in view of the i £-1 invertibility of col[XT ]i=O' The triple of operators (X,T,y) will be called spectral triple of L(A). Basic properties of spectral triples are summarized in the following proposition. SPECTRAL TRIPLES Sec. 1. Y 2 ) are two spectral triples of L(~). then there is a unique invertible oper~tor S such that X2 = X1 S. T2 = S L(~). -1 -1 T1 S. TY ••••• T Y] = [0···01]. Y) is a spectral triple of L(~). PROOF. 1. X), T and the corresponding Y turns out to be Y I l = col[OliI1i=1 E L(X,X).
1 represents a very familiar situation. Here one is often interested not only in the Jordan form of L(A) = AI+AO (which is fully described by the class of operators similar to CL -Ao )' but also in the eigenvectors and generalized eigenvectors for L(A). More generally, in the case 0: of finite dimensional X which we identify with c n , the following construction of eigenvectors and generalized eigenvectors for a monic operator polynomial L(A) is used. Chap. 2 REPRESENTATIONS AND DIVISORS 40 L(~) = ~tI t-l +!
An Introduction to Operator Polynomials by Prof. Leiba Rodman (auth.)