Linear Analysis, Complex Analysis, and Number Theory

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ABSTRACT:The principal research initiative results from a theory of Hilbert spaces of entire functions produced in the postdoctoral years 1957--1962 and from a related theory of Hilbert spaces of analytic functions then produced in joint work with James Rovnyak. A striking application was made in 1984 to a proof of the Bieberbach conjecture. Another research initiative results from a proof obtained in 1959 of the Stone--Weierstrass theorem. An application to the invariant subspace problem was made in 1991 by Victor Lomonosov. Existence theorems for invariant subspaces are currently pursued by methods of the Stone--Weierstrass theorem and of Hilbert--spaces of analytic functions.Hilbert spaces of entire functions are being applied to a proof of the Riemann hypothesis.The significance of the present work lies in its effective combination oftwo research concepts, a classical concept whose aim is to solve difficult problems, and a modern concept whose aim is to introduce innovativetechniques. The test of success is measured by the successful application ofnew methods to old problems. The Riemann hypothesis has been a consistent aim of the present work since the doctoral thesis. A successful conclusion ofthis longstanding project would be a major contribution to mathematics.