Abstract
We present sucient conditions in order (the space of) a Riesz op- erator T in a Hilbert space H have a Jordan-Schur basis with respect to a scalar product equivalent to the original one. This is related to Schur's lemma for a compact operator, which is an extension of Schur's classical theorem on unitary triangularization in a finite dimensional space. The finite dimensional case is also studied.
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