M-ideals of compact operators and Norm attaining operators

We investigate M-ideals of compact operators and two distinct properties in norm-attaining operator theory related with M-ideals of compact operators called the weak maximizing property and the compact perturbation property. For Banach spaces X and Y, it is previously known that if K (X, Y) is an M-ideal or (X, Y) has the weak maximizing property, then (X, Y) has the adjoint compact perturbation property. We see that their converses are not true, and the condition that K (X, Y) is an M-ideal does not imply the weak maximizing property, nor vice versa. Nevertheless, we see that all of these are closely related to property (M), and as a consequence, we show that if K (ₚ, Y) (1<p<) is an M-ideal, then (ₚ, Y) has the weak maximizing property. We also prove that (₁, ₁) does not have the adjoint compact perturbation property, and neither does (₁, Y) for an infinite dimensional Banach space Y without an isomorphic copy of ₁ if Y does not have the local diameter 2 property. As a consequence, we show that if Y is an infinite dimensional Banach space such that L (₁, Y) is an M-ideal, then it has the local diameter 2 property. Furthermore, we also studied various geometric properties of Banach spaces such as the Opial property with moduli of asymptotic uniform smoothness and uniform convexity.