By Prof. Leiba Rodman (auth.)
This ebook presents an advent to the trendy idea of polynomials whose coefficients are linear bounded operators in a Banach house - operator polynomials. This thought has its roots and functions in partial differential equations, mechanics and linear structures, in addition to in glossy operator conception and linear algebra. during the last decade, new advances were made within the conception of operator polynomials in keeping with the spectral procedure. the writer, in addition to different mathematicians, participated during this improvement, and lots of of the new effects are mirrored during this monograph. it's a excitement to recognize aid given to me via many mathematicians. First i want to thank my instructor and colleague, I. Gohberg, whose information has been worthy. all through a long time, i've got labored wtih numerous mathematicians near to operator polynomials, and, accordingly, their principles have inspired 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 relating a variety of points of the e-book: I. Gohberg, M. A. Kaashoek, A. C. M. Ran, okay. Clancey, J. Rovnyak, H. Langer, P.
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Extra resources for An Introduction to Operator Polynomials
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).
So aCT) is a union of two disjoint compact sets aCT) n oCT) n ([\n o )' and consequently there is a direct sum decomposition no and Chap. 1 LINEARIZATIONS 22 where Yl and Y2 are T-invariant subspaces such that o(TIY 1 ) = oCT) n 00 and o(TIY 2 ) = oCT) n ([\00). 5) in the form [T11' :>ry, r:J C (. [:" I;Z l] [:(' l o ] y (V->r( [Tly,:>r , I;Z ~J This equali ty shows that TI'Y 1 is a linearization of L(X) with respect to 0. In case X is a Hilbert space or a separable Hilbert space, use L 2 (aO O'X) in the above arguments in place of c(aOo'X).
J=l (Sj(A»P<"". The class Sl which is of special importance is called the trace class. 2 in Gohberg-KreYn ». •. B ESp' then also A+B E S. Indeed, we have (see. 4 in p Gohberg-KreYn ) k ! (Sj(A+B»P ~ j=l k ! (Sj(A)+Sj(B»P, j=l k=1,2 ••.. , and an application of the Minkowski's inequality proves our claim. It follows from these two observations that Sp is an ideal in L(X) for every p ~ 1. The ideal Sp (for p 1) is a ~ Banach space with the norm It will be convenient to use the notation S"" to designate the ideal of all compact operators in L(X).