arXiv subjects
Publications and source records attributed to Deepesh K P.
We establish a spectral representation theorem for compact operators that are self-adjoint with respect to the duality mapping on smooth Banach spaces. The result is a natural analogue of the classical spectral theorem for compact self-adjoint operators on Hilbert spaces.
This article introduces classes of normal and unitary operators on smooth Banach spaces, providing extensions of the classical notions of normal and unitary operators from Hilbert spaces to the smooth Banach space setting. The proposed class of normal operators contains, in particular, the class of self-adjoint operators on Banach spaces known in the literature. In addition, 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. Further, we obtain characterisations and spectral properties of the newly introduced classes of normal and unitary operators. Our results demonstrate close analogies with the corresponding theory of self-adjoint, normal, and unitary operators on Hilbert spaces.
This article investigates the convergence properties of s-numbers of certain truncations of bounded linear operators between Banach spaces. We prove a generalized version of a known convergence result for the approximation numbers of truncations of operators, removing the restrictive assumption of separability from the underlying spaces. This generalization extends several existing results in the literature and establishes a close connection between two significant problems concerning approximation numbers. By exploring the relationships between approximation numbers and other prominent s-numbers, we also derive results on the convergence of s-numbers of truncations to those of the original operator, as applications of the generalized convergence result.