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
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.
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.