Nuclear map

Nuclear operators between Hilbert spaces

Let X and Y be Hilbert spaces and let N : X → Y be a continuous linear map. Suppose that ${\displaystyle N=UR}$ where R : X → X is the square-root of ${\displaystyle N^{*}N}$ and U : X → Y is such that ${\displaystyle U{\big \vert }_{\operatorname {Im} R}:\operatorname {Im} R\to \operatorname {Im} N}$ is a surjective isometry. Then N is a nuclear map if and only if R is a nuclear map; hence, to study nuclear maps between Hilbert spaces it suffices to restrict one's attention to positive self-adjoint operators R.