## 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

### Keywords:

Riesz basis; positive isomorphisms on a Hilbert space
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