International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 18, Pages 1173-1174
doi:10.1155/S0161171203202349

Riesz bases and positive operators on Hilbert space

Abstract

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.