Linear Analysis, Complex Analysis, and Number Theory

摘要：本文的主要研究成果来源于1957- 1962年博士后期间提出的全函数的希尔伯特空间理论，以及随后与詹姆斯·罗夫尼亚克共同提出的解析函数的希尔伯特空间理论。1984年，对比伯巴赫猜想的证明提出了一个引人注目的应用。另一项研究源于1959年对斯通-魏尔斯特拉斯定理的证明。维克多·罗蒙诺索夫（Victor Lomonosov）在1991年提出了一个应用于不变子空间问题的方法。不变子空间的存在性定理目前是用解析函数的Stone—Weierstrass定理和Hilbert—空间的方法来研究的。整个函数的希尔伯特空间被应用于黎曼假设的证明。本文的意义在于将以解决难题为目的的古典概念和以引进创新技术为目的的现代概念这两种研究理念有效地结合起来。衡量成功的标准是把新方法成功地应用于老问题。自博士论文发表以来，黎曼假设一直是当前工作的一贯目标。这个长期项目的成功结束将是对数学的重大贡献。

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.