#
nLab

adjoint operator

### Context

#### Functional analysis

## Overview diagrams

## Basic concepts

## Theorems

## Topics in Functional Analysis

# Contents

## Definition

Let $A: H\to H$ be an unbounded operator on a Hilbert space $H$. An unbounded operator $A^*$ is its **adjoint** if

- $(A x|y) = (x|A^*y)$ for all $x\in dom(A)$ and $y\in dom(A^*)$; and
- every $B$ satisfying the above property for $A^*$ is a restriction of $A$.

An adjoint does not need to exist in general.