NUMERICAL RANGES AND NUMERICAL INDICES

Three lectures are going to be given on numerical ranges of operators, Banach spaces with numerical index one, and polynomial numerical indices of Banach spaces. We give now a briefly account of each of the topic. 1. Numerical ranges of operators We give an exposition about the numerical ranges of operators, start-ing with the emergence of this concept in 1918 for matrices and opera-tors on Hilbert space, presenting the mayor results here. Then, we present the definitions, preliminary properties and examples of numerical ranges of operators on Banach spaces. The relation between numerical ranges and one-parameter semigroups of operators is then presented to finally get a description of the Lie-algebra of a Banach space. The basic references for this part are the books by Bonsall and Duncan about numerical ranges in normed algebras from the 1970s [2, 3], and a paper by Rosenthal [22] which contains proofs of some of the results written just for operators. The rest of the talk is devoted to present two recent results on surjectiv