ORIGINS IN SINGLE OPERATOR THEORY 3 of finite-dimensional projections such that Pn - 1 strongly, and (1 - P„)TPn = 0, « = 1,2,.... The dimensions of the projections Pn are allowed to increase arbitrarily fast. However, since the restriction of Ô to the ränge of each Pn is a finite-dimensional Operator, one may use conventional linear algebra to find a new sequence {P„'} of invariant projections, which refines the original sequence {Pn} in the sense that {Pn} c {/*„'}, which satisfies P'n Pn'+i, and is such that P'n isrt-dimensionalfor every n. CONCLUSION. An Operator Ô is triangulär iff there is an orthonormal basis {el9 e2,...} with respect to which the matrix of Ô is upper triangulär: * * * 0 * * ô~ é ï ï * Let us write F for the dass of all triangulär Operators (I am going to ignore the obvious set-theoretic difficulties associated with such a definition, leaving it for the reader to reformulate the definition of ^"so as to obtain a bonafide set). An Operator à ^ ï ? ( / ) is called quasitriangular if there is an increasing sequence {Pn} of finite-dimensional projections in ££(34?) such that Pn -* 1 strongly and ||(i-pjrpJ-»o, as ç -* oo. Let us write ¢^ for the dass of all quasitriangular Operators. Now if {Pn} is any sequence of projections which tends strongly to the identity, then one can show easily that lim \\(l-Pn)KPn\\ = 0 n-* oo for every compact Operator K. It follows that every compact perturbation of a triangulär Operator is quasitriangular. Significantly, these two classes of Operators actually coincide [37]. THEOREM 1.3. £^= y+ X. SKETCH OF PROOF. Let Á be a quasitriangular Operator and let {Pn} be a sequence of finite rank projections which increases to 1 and satisfies (1.4) lim ||(1 - Pn)APn\\ = 0. «-•o o We may find a subsequence Ñ , / , . . . of {Pn} so that the numbers ||(1 - Pn )AP„ \\ tend to zero as fast as we like, and in particular there is a subsequence Q1 = Ñ , Q2 = P„2,... such that 00 (i.5) Ó ||(é-â*ìâ*||ïï . k = \
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