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Section 8.2 Adjoint Operators

In this section we prove the spectral theorem for symmetric operators. For this we need the concept of an adjoint operator.
Definition 8.2.1.
Let \(A:X\to Y\) be linear with dense domain \(D(A)\text{.}\) We define the adjoint operator on the dual spaces \(A^*: Y^*\to X^*\) by
\begin{equation*} A^*(y)=w \qquad\mbox{if and only if} \qquad y(Au) = w(u), \qquad y\in Y^*, w\in X^*, u\in X, \end{equation*}
i.e., \(A^* y\) has the action of \(y\circ A\) on \(u\in X\text{.}\) We call \((A^*, D(A^*))\) the adjoint operator, where \(D(A^*)\) denotes the natural domain of definition of \(A^*\text{.}\)
Adjoint operators for bounded operators \(A\) are easily controlled.
This is a direct consequence of the Corollay [cross-reference to target(s) "HHH" missing or not unique] of the Hahn-Banach Theorem.
In Hilbert spaces the adjoint can be expressed through the inner product. In that case we use the more common notation of \(A^*\) for the adjoint.
Definition 8.2.3.
Let \(A:H_1\to H_2\) be a linear map between Hilbert spaces. Then \(A^*:H_2\to H_1\) such that
\begin{equation*} (v, Au)_{H_2} = (A^* v, u)_{H_1} . \end{equation*}
\((A^*, D(A^*))\) is called the Hilbert adjoint, or simply the adjoint of \(A\text{.}\)
Notice that, in principle, \(A^*\) is defined on \(H_2^*\) rather than on \(H_2\) and takes values in \(H_1^*\) rather than on \(H_1\text{.}\) Since, though, there is a canonical identification between \(H_2^*\) and \(H_2\) and between \(H_1^*\) and \(H_1\text{,}\) it won't be ambiguous considering \(A^*\) as an operator from \(H_2\) to \(H_1\text{.}\)

Let us consider range and null space of the adjoint operator.

If \(z\in (\mbox{Range}(A))^\perp\) then \((z,Au)=0\) for all \(u\in H\text{.}\) This implies \((A^* z, u) =0\text{,}\) i.e. \(A^* z= 0\) and \(z\in N(A^*)\text{.}\) On the other hand, if \(z\in N(A^*)\) then \((z, Au)=(A^*z, u) =0\) and \(z\in (\mbox{Range}(A))^\perp\text{.}\) Finally, if \(\mbox{Range}(A)\) is closed, then
\begin{equation*} \mbox{Range}(A) = \mbox{Range(A)}^{\perp\perp} = N(A^*)^\perp. \end{equation*}
Definition 8.2.5.
A is symmetric if \((Au, v) = (u, Av)\) for all \(u,v\in D(A)\text{.}\) \(A\) is self-adjoint if it is symmetric and \(D(A) = D(A^*)\text{.}\)
Consider \(A=-\Delta\) on \(L^2(\Omega)\text{,}\) where \(\Omega\) is a smooth domain. In the case of Dirichlet boundary conditions, we define \(A\) on
\begin{equation*} D(A) = \{ u\in H_0^1(\Omega)\cap H^2(\Omega): u\Big|_{\partial \Omega}=0\}. \end{equation*}
Then we have
\begin{equation} (u, Av) = -\int_\Omega u \Delta v \, dx = \int_\Omega \nabla u \nabla v dx = -\int_\Omega \Delta u \; v\, dx = (Au, v). \tag{8.2.1} \end{equation}
Hence \(A^*=A\) and we can chose \(D(A^*)=D(A)\) to obtain a self-adjoint operator.
Let \(A=\frac{\partial}{\partial x}\) in \(L^2(\RR)\) and \(D(A) = H_0^1(\RR)\text{.}\) Then
\begin{equation*} (u, Av) = \int u v' dx = -\int u' v dx = - (Au, v). \end{equation*}
Hence \(A^* = -A\) and \(D(A^*)=D(A)\) and we call \(A\) to be skew-adjoint.

Recall the definition of symmetric operator in Definition 4.6.3. Before we prove the important spectral theorem for symmetric operators, we need a small technical result on bounds for the resolvent:
Since \(A\) is surjective, for each \(y\in Y\) there exists an \(x\in X\) such that \(A x = y\text{.}\) We call this \(x=A^{-1} y\text{.}\) Then
\begin{equation} \|A^{-1}\| = \sup_{y\in Y} \frac{\|A^{-1} y\|}{\|y \|} = \sup_{x\in X} \frac{\|A^{-1}(Ax)\|}{\|Ax\|} = \sup_{x\in X}\frac{\|x\|}{\|Ax\|}\leq \frac{1}{\inf_{x} \frac{\|Ax\|}{\|x\|}} \leq \frac{1}{\delta}. \tag{8.2.2} \end{equation}
The inverse is well defined, since when there are two \(x_1, x_2\) with \(Ax_1 = Ax_2 =y\text{,}\) then
\begin{equation*} \|x_1-x_2\| = \|A^{-1}y - A^{-1} y\| = 0, \end{equation*}
since \(A^{-1}\) is bounded.
  1. On a complex Hilbert space we have \((a,b)=\overline{(b,a)}\text{.}\) Then
    \begin{equation*} \overline{(Ax, x)} = (x, Ax) = (Ax, x) \end{equation*}
    and \((Ax, x)\in \RR\text{.}\)
  2. Let \((\lambda, \varphi)\) be an eigenpair of \(A\) (i.e. \(\lambda\) is an eigenvalue and \(\varphi\) is a corresponding eigenvector). Then
    \begin{equation*} (A\varphi, \varphi) = \lambda \|\varphi\|^2 \end{equation*}
    and \(\lambda\) must be real.
  3. Let \((\lambda_1,\varphi_1), (\lambda_2,\varphi_2)\) be two eigenpairs. Then
    \begin{equation*} \lambda_1(\varphi_1, \varphi_2) = (A\varphi_1, \varphi_2) = (\varphi_1, A\varphi_2) = \lambda_2(\varphi_1, \varphi_2) \end{equation*}
    and either \(\lambda_1=\lambda_2\) or \(\varphi_1 \perp\varphi_2\text{.}\)
  4. Let \(\lambda\in \sigma_c(A)\) and \(\lambda = \gamma+i\mu\text{,}\) \(\gamma, \mu \in \RR\text{.}\) We want to show that \(\mu=0\text{.}\) We compute
    \begin{align*} \|(A-\lambda I) x\|^2 \amp=\amp (Ax-\gamma x-i\mu x, Ax-\gamma x-i\mu x)\\ \amp=\amp (Ax -\gamma x, Ax-\gamma x) + (i\mu x, \gamma x) + (i\mu x, i\mu x)\\ \amp\amp - (i\mu x, Ax) - (Ax, i\mu x) + (\gamma x, i \mu x)\\ \amp=\amp \|Ax-\gamma x\|^2 -i(\mu x, \gamma x) +i(\gamma x,\mu x)\\ \amp\amp + i \mu(x, Ax) - i \mu (Ax, x) + \mu^2 \|x\|^2\\ \amp=\amp\|Ax-\gamma x\|^2 +\mu^2\|x\|^2. \end{align*}
    If \(\mu>0\text{,}\) then \(\|(A-\lambda I)x\|\) is bounded below, away from \(0\text{.}\) Then, by Corollary Corollary 8.2.9, the inverse \(R_\lambda(A)\) exists and is bounded. But in this case \(\lambda\not\in \sigma_c(A)\text{,}\) which is a contradiction. Hence we must have \(\mu=0\) and \(\lambda\) is real.