On numerical range properties of self-adjoint operators on smooth Banach spaces

In this article, we study several properties of self-adjoint operators on smooth Banach spaces, with emphasis on the norm, minimum modulus, numerical radius, and Crawford number, as well as the corresponding attainment properties and the relations among these quantities. For a special class of self-adjoint operators, we characterise norm, numerical radius, minimum modulus and Crawford number attainment in terms of eigenvalues, extending several classical Hilbert space results. We further establish results on the minimum modulus and Crawford number of powers of operators. These results reveal connections between spectral theory, numerical range theory, and the geometry of smooth Banach spaces, while also extending several classical matrix-theoretic phenomena to a broader infinite-dimensional setting.