We characterize $k-$smoothness of bounded linear operators defined between infinite-dimensional Hilbert spaces. We study the problem in the setting of both finite and infinite-dimensional Banach spaces. We also characterize $k-$smoothness of operators on some particular spaces, namely $\mathbb{ L}(\mathbb{ X},\ell_{ \infty}^n),~\mathbb{ L}(\ell_{ \infty}^3,\mathbb{ Y}),$ where $\mathbb{ X}$ is a finite-dimensional Banach space and $\mathbb{ Y}$ is a two-dimensional Banach space. As an application, we characterize extreme contractions on $\mathbb{ L}(\ell_{ \infty}^3,\mathbb{ Y}),$ where $\mathbb{ Y}$ is a two-dimensional polygonal Banach space.
Categories: math.FA