In this project, I plan to study the interaction between operator theory, function theory and algebraic geometry in some reproducing kernel Hilbert spaces (and their multiplier algebras), which live on subvarieties of the unit ball. The reproducing kernel Hilbert spaces (RKHSs) that I shall consider in this project are the quotients of Drury-Arveson space by a radical, homogeneous ideal.
I plan to address four main problems. First, I plan to compute the essential norm of the continuous multipliers on these RKHSs. Second, I plan to compute the C*-envelope of the operator algebra given by the image of the continuous multipliers on a RKHS in the Calkin algebra of that RKHS. Third, I plan to prove an effective Hilbert's Basis Theorem by showing that every radical homogeneous ideal has the stable division property. Finally, I plan to use the above results to prove some versions of Arveson's conjecture, which states that every quotient of the Drury-Arveson space by a graded submodule is essentially normal.
Fields of science
Call for proposal
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