---
tag: N44162Y9XS
kind: concept
name: Approximate point spectrum
msc: [47A10]
sources:
  - url: https://en.wikipedia.org/wiki/Spectrum_(functional_analysis)
---

## Summary

The approximate point spectrum of an operator $T$ is the set of numbers $\lambda$ that are eigenvalues up to an arbitrarily small error.
This means that there are unit vectors $x$ with $Tx - \lambda x$ as small as one likes.
It contains every eigenvalue and the whole continuous spectrum, and also the boundary of the spectrum.
For self-adjoint and normal operators on Hilbert spaces it is the whole spectrum.

## Explanations

### L1 human

An eigenvector is a vector $x$ with $Tx = \lambda x$ exactly.
An approximate eigenvector only needs $Tx$ to be very close to $\lambda x$, compared with the size of $x$.
For the operator that divides the $n$-th entry of a sequence by $n$, the $n$-th unit vector is sent to a vector of length $1/n$.
So $0$ is an approximate eigenvalue, although it is not an eigenvalue.