Several Problems Concerning Multivariate Functions and Associated Operators

This dissertation examines two distinct problems about multivariate functions and their associated operators. It first discusses the structure of Agler decompositions, which give useful ways to represent two-variable Schur functions on the bidisk using positive kernels. An elementary proof of the existence of Agler decompositions is provided, which 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 their properties are analyzed. More specific results are obtained for certain classes of polynomials and rational inner functions. Secondly, this dissertation examines differentiation of matrix-valued functions. Specifically, multivariate, real-valued functions induce matrix-valued functions defined on the space of d-tuples of n x n pairwise-commuting self-adjoint matrices. The geometry of this space of matrix tuples is characterized, and it is shown that the best notion of differentiation of these matrix-valued functions is differentiation along curves. The main result states that real-valued m-times continuously differentiable functions induce matrix-valued functions that can be m-times continuously differentiated along m-times continuously differentiable curves