Adjoining is Linear

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Theorem

Let $H, K$ be Hilbert spaces over $\Bbb F \in \left\{{\R, \C}\right\}$.

Let $A, B \in B \left({H, K}\right)$ be bounded linear transformations.

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

$(1): \qquad \left({\lambda A}\right)^* = \overline \lambda A^*$

$(2): \qquad \left({A + B}\right)^* = A^* + B^*$

That is, $^*: B \left({H, K}\right) \to B \left({K, H}\right)$ is a linear transformation.

Let $\lambda \in \Bbb F$, $h \in H, k \in K$. Then:

$\displaystyle $	$\displaystyle \left\langle{\left({\lambda A}\right)h, k}\right\rangle_K$	$=$	$\displaystyle \lambda \left\langle{Ah, k}\right\rangle_K$	$\displaystyle $	Property $(2)$ of inner products
$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \lambda \left\langle{h, A^*k}\right\rangle_H$	$\displaystyle $	Definition of adjoint
$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \left\langle{h, \left({\overline \lambda A^*}\right) k}\right\rangle_H$	$\displaystyle $	Properties $(1), (2)$ of inner products

Thus, by Existence and Uniqueness of Adjoint, $\left({\lambda A}\right)^* = \overline \lambda A^*$.

$\Box$

Let $h \in H, k \in K$. Then:

$\displaystyle $	$\displaystyle \left\langle{\left({A + B}\right)h, k}\right\rangle_K$	$=$	$\displaystyle \left\langle{Ah, k}\right\rangle_K + \left\langle{Bh, k}\right\rangle_K$	$\displaystyle $	Property $(3)$ of inner products
$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \left\langle{h, A^k}\right\rangle_H + \left\langle{h, B^k}\right\rangle_H$	$\displaystyle $	Definition of adjoint
$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \left\langle{h, \left({A^* + B^*}\right) k}\right\rangle_H$	$\displaystyle $	Properties $(1), (3)$ of inner products

Thus, by Existence and Uniqueness of Adjoint, $\left({A + B}\right)^* = A^* + B^*$.

$\blacksquare$

\(\displaystyle \)	\(\displaystyle \left\langle{\left({\lambda A}\right)h, k}\right\rangle_K\)	\(=\)	\(\displaystyle \lambda \left\langle{Ah, k}\right\rangle_K\)	\(\displaystyle \)	Property $(2)$ of inner products
\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \lambda \left\langle{h, A^*k}\right\rangle_H\)	\(\displaystyle \)	Definition of adjoint
\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left\langle{h, \left({\overline \lambda A^*}\right) k}\right\rangle_H\)	\(\displaystyle \)	Properties $(1), (2)$ of inner products

\(\displaystyle \)	\(\displaystyle \left\langle{\left({A + B}\right)h, k}\right\rangle_K\)	\(=\)	\(\displaystyle \left\langle{Ah, k}\right\rangle_K + \left\langle{Bh, k}\right\rangle_K\)	\(\displaystyle \)	Property $(3)$ of inner products
\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left\langle{h, A^k}\right\rangle_H + \left\langle{h, B^k}\right\rangle_H\)	\(\displaystyle \)	Definition of adjoint
\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left\langle{h, \left({A^* + B^*}\right) k}\right\rangle_H\)	\(\displaystyle \)	Properties $(1), (3)$ of inner products