By Prof. Leiba Rodman (auth.)

ISBN-10: 3034891520

ISBN-13: 9783034891523

ISBN-10: 3034899289

ISBN-13: 9783034899284

This ebook offers an advent to the fashionable concept of polynomials whose coefficients are linear bounded operators in a Banach area - operator polynomials. This idea has its roots and functions in partial differential equations, mechanics and linear structures, in addition to in smooth operator concept and linear algebra. during the last decade, new advances were made within the thought of operator polynomials in keeping with the spectral procedure. the writer, in addition to different mathematicians, participated during this improvement, and plenty of of the new effects are mirrored during this monograph. it's a excitement to recognize aid given to me by way of many mathematicians. First i need to thank my instructor and colleague, I. Gohberg, whose assistance has been helpful. all through a long time, i've got labored wtih numerous mathematicians almost about operator polynomials, and, therefore, their principles have stimulated my view of the topic; those are I. Gohberg, M. A. Kaashoek, L. Lerer, C. V. M. van der Mee, P. Lancaster, okay. Clancey, M. Tismenetsky, D. A. Herrero, and A. C. M. Ran. the next mathematicians gave me suggestion bearing on numerous elements of the publication: I. Gohberg, M. A. Kaashoek, A. C. M. Ran, okay. Clancey, J. Rovnyak, H. Langer, P.

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2) are called right and left canonical forms of PROOF. L(~), respectively. 2) is analogous). 1) for Sec. 2 REPRESENTATIONS BY PAIRS 45 the companion operator of L(X), (cf. 1). As in this case, th t where Aj is the j coefficient of L(X). and Vj = col[OjiIX1i=1 (here and elsewhere in this book. 0ji stands for the Kronecker symbol: 0ji = 1 if j = i and 0ji = 0 if j _ i). 1) follows immediately. • In particular, a right (or left) spectral pair defines uniquely the monic operator polynomial. 1. 2. X), T E L(Y) Is a right spectral pair for a monic operator polynomial L(A) of degree t if and only if is invertible.

4) is an analytic function on X for IX-Xol < El . Let p be the largest size of a not identically zero minor of D(X). and let M(X) be some minor of D(X) of size p with M(X) • O. As M(X) is an analytic function of X. for some E. 0 < E ~ E1 • the punctured disc does not contain zeros of M(X). 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.

Let e 1 ,e 2 , ... ,ek be a bas1s for l and assume that the representation of Til in this basis is a Jordan normal form. If ei,e i + 1 , ... ,e i +r is a basis for a Jordan block of Til corresponding to eigenvalue AO' then {xei,Xe i + 1 , ••• ,xe i + r } is a Jordan chain for L corresponding to eigenvalue Ao. LINEARIZATIONS Sec. 3 For the proof. we refer the reader to Gohberg-LancasterRodman [4]. 3 Linearizations We study here global linearizations of monic operator polynomials. Given a Banach space X.

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An Introduction to Operator Polynomials by Prof. Leiba Rodman (auth.)

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