Operators on Hilbert Space

Texas A&M Research Foundation, College Station TX

Investigators

Abstract

Abstract : In broad terms, the purpose of this project is to increase our knowledge about linear operators acting on a complex, separable, infinite dimensional Hilbert space. The proposer plans to continue work that was done under prior NSF grants, using results and techniques that were obtained in the recent past by the proposer, his coauthors, and his students, to accomplish this. In particular, the proposer will work on the invariant subspace problem for various classes of operators such as the Lomonosov-amenable operators, the sub-n-normal operators, the contractions with spectral radius one, and the power bounded operators. These classes are more general than some others in which the proposer has been instrumental in the solution of the invariant subspace problem. Operators on Hilbert space may be thought of as the natural generalization of finite complex matrices, and such matrices play an important role presently in the solution of many problems in the real world such as signal processing, populationstudies, various types of optimization procedures, etc. But the natural mathematical setting in which to cast certain problems in computing, quantum physics, nuclear fuel processing, etc. is in the more general setting of operators on Hilbert space. And there is ample evidence to show that knowledge about such operators, obtained from a purely mathematical investigation, very frequently is useful in resolving real-world problems of the sort mentioned above. This project aims to increase such basic knowledge about operators.