# existence of adjoints of bounded operators

Let $\mathscr{H}$ be a Hilbert space^{} and let $T:\mathcal{D}(T)\subset \mathscr{H}\u27f6\mathscr{H}$ be a densely defined linear operator^{}.

Theorem - If $T$ is bounded (http://planetmath.org/ContinuousLinearMapping) then its adjoint^{} ${T}^{*}$ is everywhere defined and is also bounded.

Proof : Since $T$ is densely defined and bounded, it extends uniquely to a bounded (everywhere defined) linear operator on $\mathscr{H}$, which we denote by $\stackrel{~}{T}$.

For each $z\in \mathscr{H}$, the function $f:\mathscr{H}\u27f6\u2102$ defined by
$f(x)=\u27e8\stackrel{~}{T}x,z\u27e9$ defines a bounded linear functional^{} on $\mathscr{H}$. By the Riesz representation theorem^{} there exists $u\in \mathscr{H}$ such that

$$f(x)=\u27e8x,u\u27e9$$ |

i.e.

$$\u27e8\stackrel{~}{T}x,z\u27e9=\u27e8x,u\u27e9.$$ |

Since $\stackrel{~}{T}$ extends $T$, we also have that for every $z\in \mathscr{H}$ there exists $u\in \mathscr{H}$ such that

$$\u27e8Tx,z\u27e9=\u27e8x,u\u27e9\text{for every}x\in \mathcal{D}(T).$$ |

We conclude that ${T}^{*}$ is everywhere defined. To see that it is bounded one just needs to check that

$$\underset{z\ne 0}{sup}\frac{\parallel {T}^{*}z\parallel}{\parallel z\parallel}=\underset{\begin{array}{c}z\ne 0\\ {T}^{*}z\ne 0\end{array}}{sup}\frac{|\u27e8{T}^{*}z,{T}^{*}z\u27e9|}{\parallel {T}^{*}z\parallel \parallel z\parallel}\le \underset{\begin{array}{c}z\ne 0\\ x\ne 0\end{array}}{sup}\frac{|\u27e8x,{T}^{*}z\u27e9|}{\parallel x\parallel \parallel z\parallel}=\underset{\begin{array}{c}z\ne 0\\ x\ne 0\end{array}}{sup}\frac{|\u27e8Tx,z\u27e9|}{\parallel x\parallel \parallel z\parallel}\le \parallel T\parallel $$ |

where the last inequality comes from the Cauchy-Schwarz inequality and the fact that $T$ is bounded. $\mathrm{\square}$

Remark - This theorem shows in particular that bounded linear operators $T:\mathscr{H}\u27f6\mathscr{H}$ have bounded adjoints ${T}^{*}:\mathscr{H}\u27f6\mathscr{H}$.

Title | existence of adjoints of bounded operators |
---|---|

Canonical name | ExistenceOfAdjointsOfBoundedOperators |

Date of creation | 2013-03-22 17:33:44 |

Last modified on | 2013-03-22 17:33:44 |

Owner | asteroid (17536) |

Last modified by | asteroid (17536) |

Numerical id | 4 |

Author | asteroid (17536) |

Entry type | Theorem |

Classification | msc 47A05 |

Synonym | bounded operators^{} have (bounded) adjoints |