Definition:Adjoint Linear Transformation

Definition

This article is complete as far as it goes, but it could do with expansion. In particular: This definition can also be applied to a general linear transformation on a general inner product space, according to 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.). You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Expand}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Let $\HH$ and $\KK$ be Hilbert spaces.

Let $\map \BB {\HH, \KK}$ be the set of bounded linear transformations from $\HH$ to $\KK$.

Let $A \in \map \BB {\HH, \KK}$ be a bounded linear transformation.

By Existence and Uniqueness of Adjoint, there exists a unique bounded linear transformation $A^* \in \map \BB {\KK, \HH}$ such that:

$\forall h \in \HH, k \in \KK: {\innerprod {\map A h} k}_\KK = {\innerprod h {\map {A^*} k} }_\HH$

where $\innerprod \cdot \cdot_\HH$ and $\innerprod \cdot \cdot_\KK$ are inner products on $\HH$ and $\KK$ respectively.

$A^*$ is called the adjoint of $A$.

The operation of assigning $A^*$ to $A$ may be referred to as adjoining.

Also known as

An adjoint linear transformation is also known as an adjoint operator.

Also see

• Existence and Uniqueness of Adjoint, which ensures this concept is well-defined.

• Definition:Hermitian Operator
• Definition:Unitary Operator

• Results about adjoint linear transformations can be found here.

Sources

• 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next) $\text {II}.2.4$
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): adjoint (of a linear map)