Nuclear operator between Banach spaces

Let $X$ and $Y$ be Banach spaces. A bounded linear operator $T:X\to Y$ is nuclear if there are bounded linear functionals $\varphi_n:X\to\mathbb K$ , vectors $y_n\in Y$ , and scalars $\lambda_n$ such that

T x=\sum_{n=1}^{\infty}\lambda_n\varphi_n(x)y_n \quad\text{for every }x\in X, \qquad \sum_{n=1}^{\infty}|\lambda_n|\,\lVert\varphi_n\rVert\,\lVert y_n\rVert<\infty.

The series converges absolutely in operator norm. Equivalently, $T$ is an absolutely summable sum of rank-one operators $x\mapsto\varphi_n(x)y_n$ .

Nuclear norm and consequences

The infimum of the displayed sums over all such representations is the nuclear norm. Every finite-rank operator is nuclear, and every nuclear operator is compact.

Nuclearity is stronger than compactness in general; the distinction matters when nuclearity is used for the linking maps between completed seminorm quotients of a nuclear space.

Reference

See Sections 2–3 of Kazhdan's notes on nuclear spaces for the Banach-space nuclear expansion and the locally convex nuclear-space criterion.