VICTOR J. MIZEL
Carnegie-Mellon University, University of California, Pittsburgh, PA 15213 Riverside, CA 92521, USA
M. M. RAO
Carnegie-Mellon University, University of California, Pittsburgh, PA 15213 Riverside, CA 92521, USA

Received 7 August 2008
Revised 14 October 2008

In this paper bounded linear operators in Hilbert space satisfying general quadratic equations are characterized. Necessary and sufficient conditions for sets of operators satisfying two such equations to compare relative to a weak ordering are presented. In addition, averaging operators in finite dimensional spaces are determined, and in this case it is shown that they are unitary models for all projections. It is pointed out, by an example, that the latter result does not hold in infinite dimensions. A key application to certain second order random fields of Karhunen type is given. The main purpose is to present the structure of bounded non-self adjoint operators solving quadratic equations, and indicate their use.

Nonsymmetric operators; integral representations; averaging operators; weak ordering; Karhunen type random fields; conditional expectations
AMSC numbers: Primary: 47A65, Primary: 47D05, Primary: 60G57, Secondary: 60G05, Secondary: 46G10