Subnormal operator

Builds on Hilbert space Invariant subspace

Definition

Let $H$ be a complex Hilbert space and let $S \colon H \to H$ be a bounded linear operator. The operator $S$ is subnormal when there are a Hilbert space $K \supseteq H$ , with $H$ closed in $K$ , and a bounded normal operator $N \colon K \to K$ such that

$N(H) \subseteq H$ , that is, $H$ is an invariant subspace of $N$ ;
$Nx = Sx$ for all $x \in H$ .

Such an $N$ is a normal extension of $S$ .

Remarks

Example: the unilateral shift $S$ on $\ell^2(\N)$ is subnormal. View $\ell^2(\N)$ as the sequences in $\ell^2(\Z)$ that vanish at negative indices, and let $N$ be the bilateral shift $(Nx)_n = x_{n-1}$ , which is unitary, hence normal. $N$ maps sequences vanishing at negative indices to sequences of the same kind, and it agrees with $S$ there.

Non-example: $S = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ on $\C^2$ is not subnormal. A subnormal operator is hyponormal, that is $S^*S - SS^* \ge 0$ , but here $S^*S - SS^* = \mathrm{diag}(-1, 1)$ , which is not positive.

A normal extension $N$ can be chosen minimal: $K$ is the smallest closed subspace containing $H$ and invariant under $N$ and $N^*$ . The minimal normal extension is unique up to unitary equivalence.

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Tag	`273NZSYWZA`
URI	`m2://273NZSYWZA`
Kind	definition
Writing	W0 stub
MSC	47B20
Defines	Subnormal operator
Uses	Hilbert space, Invariant subspace
Changed	2 November 2026

Subnormal operator

Definition

Remarks

Reading path

Further reading

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