On Tensor Products of Bounded Linear Operators

This paper studies tensor products of bounded linear operators on Hilbert spaces. It establishes bilinearity of the tensor product mapping and shows that key operator properties, including rank-one, finite-rank, compactness, and injectivity, are preserved under tensorization. Spectral behavior is analyzed at the level of eigenvalues, where multiplicativity is obtained. In addition, a structural decomposition for tensor products involving bounded and diagonal operators is derived. The paper also shows that compactness is preserved under tensor products when both factor operators are compact, and in particular when one factor is compact and the other acts on a finite-dimensional space. These results provide a coherent and systematic framework for understanding tensor products of bounded linear operators and lay a foundation for further developments in spectral theory.Subject AreasAlgebra, Applied Physics