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Riesz bases and positive operators on Hilbert space. (English) Zbl 1027.46023

Summary: It is shown that a normalized Riesz basis for a Hilbert space \(H\) (i.e., the isomorphic image of an orthonormal basis in \(H\)) induces in a natural way a new, but equivalent, inner product on \(H\) in which it is an orthonormal basis, thereby extending the sense in which Riesz bases and orthonormal bases are thought of as being the same. A consequence of the method of proof of this result yields a series representation for all positive isomorphisms on a Hilbert space.

MSC:

46C05 Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product)
46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
47B65 Positive linear operators and order-bounded operators
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