The numerical index of Banach spaces and some related topics

🔖 The theory of numerical ranges has played a crucial role in the study of some algebraic structures and in the theory of Matrix operator. Of course there are many other Banach spaces whose numerical index is unknown such as Orlicz spaces. So we construct some new Banach sequence spaces which are subspaces of Orlicz spaces such as Nakano difference sequence space, Orlicz-Cesaro difference sequence space, Musielak-Orlicz and generalized Cesaro sequence space defined by weighted means, and give some geometric properties of these spaces. Finally we give a lower bound for the numerical index of the two-dimensional real Nakano sequence space. This gives a partial answer to Martin’s question [74] and generalizes the result obtained for the scalar case [71].