The project aims at developing a Fredholm theory encompassing both a single operator and a tuple of commuting operators acting on a common Hilbert space. The Koszul complex approach will be emphasized in order to introduce ideas and methods in commutative algebra to operator theory. The strategy is to introduce stabilized invariants such as the Samuel multiplicities into operator theory to calculate the multivariable Fredholm index. This has many connections with existing topics such as the codimension of invariant subspaces of the Bergman space and the Dirichlet space, Toeplitz operators on the Hardy space, Arveson's curvature invariant on the symmetric Fock space, function theory on Hardy spaces over higher dimensional domains, Apostol's triangular representation.
The Fredholm index of a linear operator is one of the most intensely studied numerical invariants in mathematics. The content of the celebrated Atiyah-Singer Index Theorem, one of the deepest results in mathematics, is on how to calculate the Fredholm index of a special class of operators. This project will focus on developing a multivariable version of the traditional one variable Fredholm theory. This allows one to introduce ideas from commutative algebra, a seemingly distant area in mathematics, into multivariable operator theory, which in turn shed new light on traditional problems in one variable operator theory.
