Operator Theory, Free Harmonic Analysis, and Related Problems

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ABSTRACTDMS-0070459The proposed research involves the study of operators and their invariantsubspaces, and some problems arising in the theory of free random variables. The problems on invariant subspaces will focus on the structure andclassification results which can in special cases be inferred from the combinatorics ofYoung tableaux. Problems in free probability will concern the weak lawsfor arbitrary arrays, and the study of multiplicative stability. Thefiner harmonic analysis of free convolutions will also be studied.The work proposed here is mostly of a theoretical nature, and it isintended to clarify some (sometimes unexpected) connections between objects studiedin different areas (such as operator theory, complex analysis, andcombinatorial analysis). This kind of work does often have connectionswith more applied areas. The one specific applied area closely related to the operator theoryis control theory and optimization where operators have indeed found rather strikingapplications.