By Prof. Leiba Rodman (auth.)
This publication offers an advent to the fashionable conception of polynomials whose coefficients are linear bounded operators in a Banach house - operator polynomials. This conception has its roots and purposes in partial differential equations, mechanics and linear platforms, in addition to in smooth operator conception and linear algebra. during the last decade, new advances were made within the conception of operator polynomials in line with the spectral method. the writer, in addition to different mathematicians, participated during this improvement, and lots of of the hot 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 tips has been useful. all through decades, i've got labored wtih numerous mathematicians as regards to operator polynomials, and, therefore, their rules have motivated 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 quite a few points of the ebook: I. Gohberg, M. A. Kaashoek, A. C. M. Ran, ok. Clancey, J. Rovnyak, H. Langer, P.
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Extra info for An Introduction to Operator Polynomials
Example text
4) Al A2 ... At_II A2 At _ 1 I 0 I A2 ... At_II I 0 I 0 0 0 -AO I 0 0 -AI 0 I I 0 0 I -Ae _2 -Ae _1 0 0 ... This formula can be verified by a straightforward multiplication. 3. Let (X,T), where X e L(Y,X), T e L(Y), be a right spectral pair for the monic operator polynomial and suppose that l in L(~), is a fin1te dimensional T-invariant subspace Y. 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.
Let A E L(X) be a compact operator. Then the positive semidefinite operator (A * A) 1/2 is compact as well. 6) the number of times equal to its algebraic INVERSE LINEARIZATIONS Sec. 6) is augmented by infinite number of zeros. 6) is infinite. 6) has the property that ""! j=l (Sj(A»P<"". The class Sl which is of special importance is called the trace class. 2 in Gohberg-KreYn [1]». •. B ESp' then also A+B E S. Indeed, we have (see. 4 in p Gohberg-KreYn [1]) k ! (Sj(A+B»P ~ j=l k ! (Sj(A)+Sj(B»P, j=l k=1,2 ••..
Put t i-1 M(A) = A I-P 1B(V1 +V 2 A+ ... +Vt A ), where [V 1 V 2 • .. V t ] t Vi: X ~ X. Then one easily verifies the equality Q(B)B = CMQ(B). Further, ~ X 1 Q(B)-, IIQ(B)-1_ I11 S K1 I1B-C LII. IIBi_C~1I S K2 I1B-C LII. t for some positive constants K1 and K2 (where B E L(X ) is such that IIB-C LI! < E) • . 1). 2). 1. , in terms of the operator itself) the global linearizations of monic operator polynomials. In the finite-dimensional case (dim X < problem is solved. 3. ~), this Let X = cn, and let X e L(Xi ).