---
tag: 86B7HF79PQ
kind: concept
name: Adjoint of an unbounded operator
msc: [47B25, 47A05]
sources:
  - url: https://en.wikipedia.org/wiki/Unbounded_operator
---

## Summary

The adjoint of a densely defined operator $T$ between Hilbert spaces is an operator $T^*$ in the opposite direction.
Its domain consists of the vectors $y$ for which $x \mapsto \ip{Tx}{y}$ is bounded on the domain of $T$.
It extends the [[Hilbert space adjoint|6HSJBJ8TTQ]] of bounded operators, and the domain of $T^*$ is part of the answer.
The adjoint is needed to define self-adjoint unbounded operators, the observables of quantum mechanics.

## Explanations

### L1 human

For bounded operators, the adjoint moves the operator from one side of the inner product to the other.
For an unbounded operator this can only be done for some vectors $y$, and the adjoint is defined exactly on those.
The domain of $T$ must be dense; otherwise the vector $T^* y$ would not be determined.
For the diagonal operator $(x_n) \mapsto (n x_n)$ on its natural domain in $\ell^2$, the adjoint is the same operator on the same domain.