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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 [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 ••.. , 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).

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