Mathematical Sciences: Singly Generated Dual Algebras

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The purpose of this project is to accumulate more knowledge leading to a better understanding of the structure of bounded linear operators acting on a complex Hilbert space of infinite dimension. In the past ten years much progress has been made in this direction by employing the concept of the dual algebra generated by a single operator. Professor Pearcy and others have made an extensive analysis of the predual of such a dual algebra, and this has led to considerable progress on the dilation theory of contraction operators, and on the invariant subspace problem. Professor Pearcy plans to continue this program of mathematical research, focusing now on the the class of contraction operators having spectral radius one. The kind of mathematics to be pursued here is operator theory. Operators may be thought of, very roughly, as enriched (real or complex) numbers. One can do arithmetic with operators just as with numbers, except that multiplication of operators depends in general on the order in which the factors are taken, and not every non-zero operator has an inverse. Furthermore, numbers are inherently one-dimensional, whereas operators generally act on an infinite-dimensional space. The notion that physical quantities are more realistically represented by operators than by numbers lies at the heart of quantum mechanics, a subject whose development provided much of the original impetus to operator theory.