摘要：In this talk I will present my recent contribution to solve the Banach spectral problem from the Scottich book. I will further give some ingredients of the proof. One of the main ingredients is the fact that the following three problem are equivalent if we restrict our self to the rank one maps:

1.the spectral Banach problem,

2 the Mahler problem in the class of Newman-Bourgain polynomials and

3.the flatness polynomial problem.

The previous fact was established by Nadkarni and the speaker. The second ingredient is based on the nice combinatorial properties of the Singer and Sidon sets and the third ingredient is the Marcinkiewicz-Zygmund inequalities in the $H^p$ interpolation theory, and its recent refinements by Chui-Shen and Zhong.\In this talk I will further present a proof that the $L^4$ classical strategy fails to solve the problem of $L^1$-flatness in the class of Newman-Bourgain polynomials. Nevertheless, the answer to Erd"{o}s-Newman-Littlewood problem is positive. From this we deduce that there exist a rank one map acting on a space of infinite measure with simple Lebesgue spectrum which gives an affirmative answer to the Banach problem. As a consequence,

I will further present my answer to the Bourgain question on supremum on the $L^1$ over all Newman-Bourgain polynomials.

欢迎广大师生参加！