Operators on Hilbert Space

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Abstract : In broad terms, the purpose of this project is to increase our knowledge aboutlinear operators acting on a complex, separable, infinite dimensional Hilbert space.The proposer plans to continue work that was done under prior NSF grants, usingresults and techniques that were obtained in the recent past by the proposer, hiscoauthors, and his students, to accomplish this. In particular, the proposer willwork on the invariant subspace problem for various classes of operators suchas the Lomonosov-amenable operators, the sub-n-normal operators, the contractionswith spectral radius one, and the power bounded operators. These classes are moregeneral than some others in which the proposer has been instrumental in the solutionof the invariant subspace problem. Operators on Hilbert space may be thought of as the natural generalization offinite complex matrices, and such matrices play an important role presently inthe solution of many problems in the real world such as signal processing, populationstudies, various types of optimization procedures, etc. But the natural mathematicalsetting in which to cast certain problems in computing, quantum physics, nuclearfuel 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, obtainedfrom a purely mathematical investigation, very frequently is useful in resolvingreal-world problems of the sort mentioned above. This project aims to increasesuch basic knowledge about operators.