April 26, 2013 -
4:00 pm to 6:00 pm
Location: Cupples I, Room 199 |
Host: Prof. John McCarthy
Abstract: We examine two distinct problems about multivariate functions and their associated operators. First, we discuss the structure of Agler decompositions, which give a useful way to represent two-variable Schur functions on the bidisk. We will discuss an elementary proof of the existence of Agler decompositions that uses special shift-invariant subspaces of the Hardy space. These shift-invariant subspaces are specific cases of Hilbert spaces that can be defined from Agler decompositions and we will outline several properties of such Hilbert spaces. Time permitting, we will touch on the situation for rational inner functions and highlight how related analyses can be used to characterize stable polynomials on the polydisk. Second, we consider differentiation of matrix-valued functions. Specifically, multivariate, real-valued functions induce matrix-valued functions defined on the space of d-tuples of n-times-n pairwise-commuting self-adjoint matrices. We will briefly discuss the geometry of this space of matrix tuples and highlight reasons why a suitable notion of differentiation of these matrix-valued functions is differentiation along curves. We will then discuss our proof that real-valued Cm functions induce matrix-valued functions that can be m-times continuously differentiated along Cm curves.
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