---
tag: 81MZ023K1A
kind: concept
name: Subnormal operator
msc: [47B20]
specializes: [35XKRHEC3Q]
sources:
  - url: https://en.wikipedia.org/wiki/Subnormal_operator
same_as:
  wikidata: Q7631728
  wikipedia: Subnormal operator
---

## Summary

A subnormal operator is a bounded operator on a Hilbert space that is the restriction of a normal operator to an invariant subspace.
Every normal operator is subnormal, and every subnormal operator is [[hyponormal|35XKRHEC3Q]].
The unilateral shift is the basic example, and multiplication by $z$ on spaces of analytic functions gives many more.
Scott Brown showed that subnormal operators have nontrivial invariant subspaces.

## Explanations

### L1 human

A subnormal operator is a normal operator seen only on part of the space that it maps into itself.
Multiplication by $z$ on square-integrable functions on the unit circle is normal.
On the smaller space of functions that extend analytically into the disc it is no longer normal, but it is subnormal.
In finite dimensions a subnormal operator is already normal, so the notion only matters in infinite dimensions.