Nuclear operator

Builds on Banach space Continuous linear map Continuous dual space Compact operator Finite-rank operator Trace class operator

Definition

Let $X$ and $Y$ be Banach spaces over $\R$ or $\C$ , and let $X'$ be the continuous dual of $X$ . A bounded linear map $T \colon X \to Y$ is nuclear if there are sequences $(f_n)$ in $X'$ and $(y_n)$ in $Y$ such that $\sum_{n=1}^{\infty} \norm{f_n} \norm{y_n} < \infty \quad \text{and} \quad T x = \sum_{n=1}^{\infty} f_n(x) \, y_n \quad \text{for all } x \in X.$ The nuclear norm $\nu(T)$ is the infimum of $\sum_n \norm{f_n} \norm{y_n}$ over all such representations.

Remarks

Every finite-rank operator is nuclear, with a finite sum. Example: on $\ell^2$ , let $T x = \sum_n d_n x_n e_n$ with $\sum_n \abs{d_n} < \infty$ ; take $f_n(x) = d_n x_n$ and $y_n = e_n$ , so $\norm{f_n} \norm{y_n} = \abs{d_n}$ is summable and $T$ is nuclear. Non-example: the identity of an infinite-dimensional Banach space is not nuclear, because it is not compact. Every nuclear operator is a norm limit of the partial sums, which are of finite rank, so it is compact. When $X$ and $Y$ are Hilbert spaces, the nuclear operators are exactly the trace class operators, and $\nu(T)$ is the trace norm.

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Tag	`3F94J43P2C`
URI	`m2://3F94J43P2C`
Kind	definition
Writing	W0 stub
MSC	47B10, 46B28
Defines	Nuclear operator
Uses	Banach space, Continuous linear map, Continuous dual space, Compact operator, Finite-rank operator, Trace class operator
Changed	2 November 2026

Nuclear operator

Definition

Remarks

Reading path

Further reading

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