---
tag: V2H578D4RT
kind: definition
name: Singular values of a compact operator
msc: [47B10, 47B07, 15A18]
defines: QSJKPKPM7H
uses: [HM2HHE3J51, M2NEP1K5AE, 6HSJBJ8TTQ, 77020CY5ZW, 3JY5HNFRMF, A1G25R99PY, 0BQAA2BK3Q, 09VMDSRXWJ, Z8FABSYQTS]
sources:
  - url: https://en.wikipedia.org/wiki/Singular_value
---

## Definition

Let $H$ and $K$ be complex [[Hilbert spaces|HM2HHE3J51]], and let $T \colon H \to K$ be a [[compact operator|M2NEP1K5AE]] with [[adjoint|6HSJBJ8TTQ]] $T^*$.
Let $\abs{T} = (T^* T)^{1/2}$ be the [[square root|3JY5HNFRMF]] of the [[positive operator|77020CY5ZW]] $T^* T$; it is compact and self-adjoint.
List the nonzero [[eigenvalues|0BQAA2BK3Q]] of $\abs{T}$ in decreasing order, each repeated as many times as the dimension of its eigenspace.
If there are only finitely many, append zeros.
The resulting sequence $s_1(T) \ge s_2(T) \ge \cdots \ge 0$ is the sequence of *singular values* of $T$.

## Remarks

The list is well defined by the [[spectral theorem for compact self-adjoint operators|A1G25R99PY]], and $s_n(T) \to 0$.
One has $s_1(T) = \norm{T}$, and more generally
$$s_n(T) = \min \set{\norm{T - F} : F \text{ of rank less than } n},$$
the distance from $T$ to the [[finite-rank operators|Z8FABSYQTS]] of rank less than $n$.
There are orthonormal families $(u_n)$ in $H$ and $(v_n)$ in $K$ such that $T x = \sum_n s_n(T) \ip{x}{u_n} v_n$ for all $x \in H$, the Schmidt decomposition of $T$.
Example: if $T$ is diagonal in an orthonormal basis $(e_n)$ of $\ell^2$, with $T e_n = d_n e_n$ and $d_n \to 0$, then $\abs{T} e_n = \abs{d_n} e_n$, so the singular values are the numbers $\abs{d_n}$ arranged in decreasing order.
Non-example: the identity of an infinite-dimensional Hilbert space is not compact, so it has no singular values in this sense; the eigenvalue $1$ of $\abs{I} = I$ has infinite multiplicity.
For matrices, this is the usual notion of [[singular value|09VMDSRXWJ]].