Triangularizability of operators with increasing spectrum
Let \((X,\mu)\) be a measure space, let \(H=L^2(X,\mu)\) and let \(M^\infty(X,\mu)\) be the algebra of all multiplication operators \(M_f\) with \(f\in L^\infty(X,\mu)\). An operator \(T\in B(H)\) is said to have a standard triangularization provided that there exists a maximal totally ordered family \(\{P_\lambda\}_{\lambda\in\Lambda}\) of \(T\)-invariant projections in \(M^\infty(X,\mu)\). A triangularization \(\{P_\lambda\}_{\lambda\in\Lambda}\) is called multiplicity-free if its linear span is weakly dense in \(M^\infty(X,\mu)\). An operator \(T\) is said to have increasing spectrum if, whenever \(P,Q\in M^\infty(X,\mu)\) are projections with \(P\leq Q\), we have \(\sigma(PTP)\subset\sigma(QTQ)\). In this paper, conditions are given under which a given operator \(T\in B(H)\) which has increasing spectrum admits a multiplicity-free standard triangularization. For instance, in the case when \(\mu\) is the counting measure on \(X\), it is shown that every operator \(T\in B(H)\) with increasing spectrum relative to the set of all finite rank projections in \(M^\infty(X,\mu)\) has a multiplicity-free standard triangularization. A similar result is proved for the case when \(T\) has finite rank and \(\mu\) is a \(\sigma\)-finite Borel measure on a locally compact Hausdorff-Lindelöf space \(X\), but the proof is much more difficult. The authors construct an example of a trace class operator with increasing spectrum relative to some measure space and which does not admit any multiplicity-free standard triangularization. It seems to be still open whether every compact operator with increasing spectrum admits a (not necessarily multiplicity-free) triangularization. The authors obtain positive results in this direction for non-negative compact operators on the space \(L^2[0,1]\).
scientific article; zbMATH DE number 5640321