Topics in Dilation Theory

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PI: Scott McCulloughProposal Number: 0140112ABSTRACTThe Sz. Nagy Dilation Theorem, which models a contractionoperator on Hilbert space as the adjoint of an isometry restricted to an invariant subspace has had a profound influence on operator theory and its applications. The existence of the Agler boundary for families of representations and of boundary representations (not necessarily irreducible) and the C-star envelope for operator algebras are abstract generalizations of the Sz.-Nagy dilation theorem which naturally encode Pick interpolation and in many cases commutant lifting. They also provide a natural setting for versions of Beurling's theorem. The Agler boundary and C-star envelope have been explicitly computed for the families of completely contractive representations of the algebra of multipliers of symmetric Fock space, the algebra of multipliers of the Dirichlet space, as well as quotients of the disc and annulus algebra. The PI will investigate situations in which strong versions of Pick interpolation, Beurling's theorem, and commutant liftinghold as a function of the complexity of the correspondingAgler boundary and C-star envelope. The investigation will make contact with function theory, special functions, differential equations, and the general theory of operator algebras. Thre are also plans to continue studyingfactorization of polynomials in several non-commuting variables pursuing the theme that one-variable resultsmost naturally generalize to the non-commutative setting.Operator theory has a history of rich interplay with engineering and physics as well as other vital areas of mathematics including complex function theory and algebraic geometry. Originally developed as a tool to study integral and differential equations arising in physics, operator theory, operator algebras, and operator systems play an important role in modern quantum physics. Four fundamental themes in operator theory, Sz.-Nagy dilation, Pick interpolation, commutant lifting, and Beurling's theorem are now basic technology in systems theory which in turn has important applications in image processing and control theory - the mathematics behind automatic controllers such as autopilots. A major emphasis in the proposed work is generalizations of these themes with a view toward applications to systems theory, control theory, operator algebras, and function theory. McCullough will also study factorization problems for polynomials in several non-commuting variables. (Here xy may not equal yx.) This investigation isexpected to make connection with non-commutative algebraic geometry and Linear Matrix Inequalities or LMI's which frequently arise in engineering applications.