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 of two research concepts, a classical concept whose aim is to solve difficult problems, and a modern concept whose aim is to introduce innovative techniques. The test of success is measured by the successful application of new methods to old problems. The Riemann hypothesis has been a consistent aim of the present work since the doctoral thesis. A successful conclusion of this longstanding project would be a major contribution to mathematics.
