Adjoining is Linear

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Theorem

Let $\HH$ and $\KK$ be Hilbert spaces over $\Bbb F \in \set {\R, \C}$.

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

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


Then the operation of adjoining $^*$ satisfies, for all $\lambda \in \Bbb F$:

$(1): \quad \paren {\lambda A}^* = \overline \lambda A^*$
$(2): \quad \paren {A + B}^* = A^* + B^*$

where $\overline \lambda$ denotes the complex conjugate of $\lambda$.


That is,:

$^*: \map \BB {\HH, \KK} \to \map \BB {\KK, \HH}$

is a linear transformation.


Proof

Let $\innerprod \cdot \cdot_\HH$ and $\innerprod \cdot \cdot_\KK$ be inner products on $\HH$ and $\KK$ respectively.


Proof of $(1)$

Let $\lambda \in \Bbb F$, $h \in \HH, k \in \KK$.

Then:

\(\ds \innerprod {\paren {\lambda A} h} k_\KK\) \(=\) \(\ds \lambda \innerprod {A h} k_\KK\) Property $(2)$ of Inner Product
\(\ds \) \(=\) \(\ds \lambda \innerprod h {A^*k}_\HH\) Definition of Adjoint Linear Transformation
\(\ds \) \(=\) \(\ds \innerprod h {\paren {\overline \lambda A^*} k}_\HH\) Properties $(1)$ and $(2)$ of Inner Product


Thus, by Existence and Uniqueness of Adjoint:

$\paren {\lambda A}^* = \overline \lambda A^*$

$\Box$


Proof of $(2)$

Let $h \in \HH, k \in \KK$.

Then:

\(\ds \innerprod {\paren {A + B} h} k_\KK\) \(=\) \(\ds \innerprod {A h} k_\KK + \innerprod {B h} k_\KK\) Property $(3)$ of Inner Product
\(\ds \) \(=\) \(\ds \innerprod h {A^*k}_\HH + \innerprod h {B^*k}_\HH\) Definition of Adjoint Linear Transformation
\(\ds \) \(=\) \(\ds \innerprod h {\paren {A^* + B^*} k}_\HH\) Properties $(1), (3)$ of Inner Product


Thus, by Existence and Uniqueness of Adjoint:

$\paren {A + B}^* = A^* + B^*$

$\blacksquare$


Sources

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