From Prof. J. Nathan Kutz’ Lecture 003
Linear Operators And Thier Adjoints
Fredholm Alternative Theorem
Given a matrix \(\mathbf{A} \in \mathbb{C}^{m\times n}\), then the vector \(\mathbf{Ax}\), where \(x \in \mathbb{C}^n\) must be orthogonal to the null space of \(\mathbf{A}^{\mathsf{H}}\). First recall the definition of inner product is
\[\braket{\mathbf{a},\mathbf{b}} = \mathbf{b}^{\mathsf{H}} \mathbf{a}\]thus
\[\begin{align} \braket{\mathbf{Ax}, \mathbf{y}} &= \braket{\mathbf{x},\mathbf{A}^{\mathsf{H}}\mathbf{y}}\\ \braket{\mathbf{x},\mathbf{Ay}} &= \braket{\mathbf{A}^{\mathsf{H}}\mathbf{x}, \mathbf{y}} \end{align}\]Assume \(\mathbf{A}^{\mathsf{H}} \mathbf{y}=0\), then
\[\braket{\mathbf{Ax},\mathbf{y}} = \braket{\mathbf{x},\mathbf{A}^{\mathsf{H}}\mathbf{y}} = 0\]Linear Operators
\[Lu=f\]where \(L\) will be a linear, differential operator on the domain \(x\in [0,l]\) and with boundary conditions specified at \(x=0\) and \(x=l\) which will be specified shortly. The definition of inner product is given by
\[\braket{u,v} = \int_{0}^{l} uv^{*}dx\]we want to find the adjoint operator such that
\[\braket{v,Lu}=\braket{L^{\dagger}v,u}\]For example
\[L = a(x)\dfrac{d^2}{dx^2} + b(x)\dfrac{d}{dx} + c(x)\]on the domain \(x\in[a,b]\) with boundary conditions
\[\begin{align} \alpha_1 u(a) + \beta_1 \dfrac{d u(a)}{dx} &= 0\\ \alpha_2 u(b) + \beta_2 \dfrac{d u(b)}{dx} &= 0 \end{align}\]assume we are working on real functions \(u,v\) we can calculate
\[\begin{align} \braket{v,Lu} &=\int_{a}^b v \bigg( a(x)\dfrac{d^2 u}{dx^2} + b(x)\dfrac{du}{dx} + c(x)u \bigg)dx\\ &=\int_a^b \bigg( a(x)v\dfrac{d^2 u}{dx^2} + b(x)v\dfrac{du}{dx} + c(x)vu \bigg)dx \end{align}\]using integration by part
\[\int udv = uv - \int vdu\] \[u= av, \quad v =u_x\] \[du = (av)_x dx, \quad dv = u_{xx}dx\] \[\begin{align} \int_{a}^b avu_{xx}dx &= avu_x\vert_{a}^b - \int_a^b u_x (av)_x dx\\ &= avu_x\vert_{a}^b - (av)_x u\vert_{a}^b + \int_a^b u(av)_{xx} dx \end{align}\] \[\int_a^b bvu_x dx = buv\vert_a^b - \int_a^b u(bv)_x dx\] \[\braket{v,Lu} = (avu_x - (av)_x u + uvb)\vert_a^b + \int_a^b [(av)_{xx} u - (bv)_x u + cvu] dx\]Thus the formal adjoint is given by
\[L^{\dagger}v = \dfrac{d^2}{dx^2}(a(x)v) - \dfrac{d}{dx}(b(x)v) + c(x)v\]also the boundary condition for operator \(L^{\dagger}\) can be given by
\[(avu_x - (av)_x u + uvb)\vert_a^b = 0\]