Mathematical Sciences: Operator Algebras and Operator Theory

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9401544 Larson One area of the research plan concerns prior work on quasitriangularity considerations within von Neumann algebras. Work by this investigator and co-workers on quasitriangularity led to the development of a sub-optimization formula which has impacted recent work by others in control theory with applications to electrical and aeronautical engineering. The plan is to continue work in this direction. A second area of the plan concerns problems related to triangular and semitriangular operators, and to reflexivity, that arose out of prior work by this investigator and co-workers which yielded counterexamples to open questions on these topics. The third area of the plan concerns an operator-theoretic approach to certain aspects of orthogonal wavelet theory obtained by interpreting orthogonal wavelets as wandering vectors for a special set of unitary operators on Hilbert space. The new idea used is that the set of all orthogonal wavelets for the usual dilation-translation wavelet system (D,T) on the Hilbert space of square- integrable complex functions on the real line can be parameterized in a natural way by a single wavelet v together with the set of all unitary operators in the local commutant of the system at x. Analysis of this unitary set yields information concerning the set W(D,T) of all orthogonal wavelets. The research plan involves three separate but related projects. The first builds onto previous work by this investigator in the area of operator algebras. A formula which was derived three years ago to solve an open problem in pure mathematics has recently impacted applications-oriented research projects of several mathematicians and engineers working incontrol theory. Further work in this connection is being pursued. The second project continues previous work on properties of single operators and dual operator algebras. Recent successes here include counter examples which have impacted work of several other mathemati cians. The third project concerns wavelets. Wavelet analysis has been the scene of a sizable research drive in mathematics and engineering during the past few years. The new concept utilized in this project is the idea of determining new wavelets and functional properties of wavelets by analyzing structural properties of a certain naturally related set of operators, using established techniques of single operator theory and operator algebras. This overall plan accomodates the dissertation research of four doctoral students. ***