International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 18, Pages 1173-1174

Riesz bases and positive operators on Hilbert space

James R. Holub

Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg 24061-0123, VA, USA

Received 2 February 2002

Copyright © 2003 James R. Holub. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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.