[proofplan] We work locally, because the definition of the class $I^m(X,Y;C)$ and its principal symbol is local modulo smoothing kernels. In a coordinate patch, the Schwartz kernel of $A$ is an oscillatory integral with phase $\phi(x,y,\theta)$ parametrizing $C$ and amplitude $a(x,y,\theta)$. The adjoint kernel is obtained by exchanging the two base variables and taking the fiberwise Hermitian adjoint of the amplitude together with complex conjugation of the exponential, giving the new phase $-\phi(x,y,\theta)$ written in the variables $(y,x)$. This new phase parametrizes $C^t$, the amplitude has the same symbolic order, and its leading homogeneous term is exactly the adjoint of the leading homogeneous term of $a$ under the induced half-density identification. [/proofplan]

[step:Write the operator locally by an oscillatory kernel associated to $C$] Fix coordinate neighbourhoods $U \subset X$ and $V \subset Y$ over which the bundles $F$ and $E$ are trivialised and over which the chosen positive densities are written as smooth positive multiples of [Lebesgue measure](/page/Lebesgue%20Measure). Let $N \in \mathbb{N}$ be the number of oscillation variables in this local parametrization, and let $\mathcal{L}^N$ denote $N$-dimensional Lebesgue measure on $\mathbb{R}^N$. Let $\ell \in \mathbb{R}$ denote the symbolic degree of the local amplitude corresponding to the FIO order $m$ under the adopted order convention. In these local data, the Schwartz kernel of $A$ is, modulo a smooth kernel, of the form \begin{align*} K_A(x,y) = \int_{\mathbb{R}^N} e^{i\phi(x,y,\theta)} a(x,y,\theta)\, d\mathcal{L}^N(\theta), \end{align*} where $\phi: U \times V \times (\mathbb{R}^N \setminus \{0\}) \to \mathbb{R}$ is a non-degenerate homogeneous phase function and \begin{align*} a: U \times V \times (\mathbb{R}^N \setminus \{0\}) \to \operatorname{Hom}(\mathbb{C}^{\operatorname{rank} E},\mathbb{C}^{\operatorname{rank} F}) \end{align*} is a classical symbol of the order corresponding to the stated FIO order $m$ under the adopted convention. The critical set of the phase in the oscillation variables is \begin{align*} \Sigma_\phi := \{(x,y,\theta) \in U \times V \times (\mathbb{R}^N \setminus \{0\}) : \partial_\theta \phi(x,y,\theta)=0\}. \end{align*} By the convention in the statement, $\phi$ parametrizes the local piece of $C$ by the map $\gamma_\phi: \Sigma_\phi \to T^*U \setminus 0 \times T^*V \setminus 0$ defined by \begin{align*} \gamma_\phi(x,y,\theta) = (x,\partial_x\phi(x,y,\theta); y,-\partial_y\phi(x,y,\theta)). \end{align*} [/step]

[step:Compute the adjoint kernel by exchanging variables and taking the Hermitian adjoint]Let $A: C_c^\infty(V;\mathbb{C}^{\operatorname{rank} E}) \to \mathcal{D}'(U;\mathbb{C}^{\operatorname{rank} F})$ denote the local form of the operator. The adjoint $A^*$ is defined by the identity \begin{align*} \int_U (Au)(x) \cdot \overline{v(x)}\, d\mu_X(x) = \int_V u(y) \cdot \overline{(A^*v)(y)}\, d\mu_Y(y). \end{align*} for test sections $u: V \to \mathbb{C}^{\operatorname{rank} E}$ and $v: U \to \mathbb{C}^{\operatorname{rank} F}$, where $d\mu_X$ and $d\mu_Y$ denote the fixed smooth positive densities in the chosen coordinates and the dot denotes the standard Hermitian pairing in the trivialised fibers. Thus the local Schwartz kernel of $A^*$ is \begin{align*} K_{A^*}(y,x) = K_A(x,y)^*, \end{align*} where $^*$ denotes the fiberwise Hermitian adjoint. Substituting the oscillatory representation of $K_A$ gives \begin{align*} K_{A^*}(y,x) = \int_{\mathbb{R}^N} e^{-i\phi(x,y,\theta)} a(x,y,\theta)^*\, d\mathcal{L}^N(\theta). \end{align*}[/step]

[guided]The adjoint is determined by the pairing of input and output sections. Locally, after trivialising the bundles and writing the fixed smooth densities as $d\mu_X$ and $d\mu_Y$, the defining identity is \begin{align*} \int_U (Au)(x) \cdot \overline{v(x)}\, d\mu_X(x) = \int_V u(y) \cdot \overline{(A^*v)(y)}\, d\mu_Y(y). \end{align*} If $K_A(x,y)$ is the kernel of $A$, then $Au$ is locally represented by pairing $K_A(x,y)$ with $u(y)$ and integrating over $Y$. Moving this pairing to the other side of the Hermitian [inner product](/page/Inner%20Product) exchanges the variables and takes the fiberwise Hermitian adjoint of the matrix acting between the fibers. Therefore the adjoint kernel is \begin{align*} K_{A^*}(y,x)=K_A(x,y)^*. \end{align*} Now insert the oscillatory formula for $K_A$. Since complex conjugation sends $e^{i\phi}$ to $e^{-i\phi}$ and the fiber map is replaced by its Hermitian adjoint, we obtain \begin{align*} K_{A^*}(y,x) = \int_{\mathbb{R}^N} e^{-i\phi(x,y,\theta)} a(x,y,\theta)^*\, d\mathcal{L}^N(\theta). \end{align*} No integration in the oscillation variable has been performed here; we have only exchanged the two base variables and conjugated the kernel. This point is what keeps the FIO order unchanged.[/guided]

[step:Show that the adjoint phase parametrizes the transposed canonical relation] Define the adjoint phase $\psi: V \times U \times (\mathbb{R}^N \setminus \{0\}) \to \mathbb{R}$ by \begin{align*} \psi(y,x,\theta) = -\phi(x,y,\theta). \end{align*} Its critical set in the oscillation variables is \begin{align*} \Sigma_\psi = \{(y,x,\theta) : \partial_\theta \psi(y,x,\theta)=0\} = \{(y,x,\theta) : \partial_\theta \phi(x,y,\theta)=0\}. \end{align*} Thus the exchange map $\tau_\Sigma: \Sigma_\phi \to \Sigma_\psi$ defined by \begin{align*} \tau_\Sigma(x,y,\theta) = (y,x,\theta) \end{align*} is a diffeomorphism. For $(x,y,\theta) \in \Sigma_\phi$, the canonical relation parametrized by $\psi$ is \begin{align*} (y,\partial_y\psi(y,x,\theta); x,-\partial_x\psi(y,x,\theta)) = (y,-\partial_y\phi(x,y,\theta); x,\partial_x\phi(x,y,\theta)). \end{align*} Since $\phi$ parametrizes \begin{align*} (x,\partial_x\phi(x,y,\theta); y,-\partial_y\phi(x,y,\theta)) \in C, \end{align*} the phase $\psi$ parametrizes \begin{align*} (y,-\partial_y\phi(x,y,\theta); x,\partial_x\phi(x,y,\theta)) \in C^t. \end{align*} Non-degeneracy is preserved because replacing $\phi$ by $-\phi$ and exchanging the base variables does not change the rank of the differentials defining the phase critical set. Hence the adjoint kernel is locally associated to $C^t$. [/step]

[step:Check that the symbolic order is unchanged] Define the adjoint amplitude $b: V \times U \times (\mathbb{R}^N \setminus \{0\}) \to \operatorname{Hom}(\mathbb{C}^{\operatorname{rank} F},\mathbb{C}^{\operatorname{rank} E})$ by \begin{align*} b(y,x,\theta) = a(x,y,\theta)^*. \end{align*} Since taking the Hermitian adjoint is a linear operation on the finite-dimensional matrix fibers and the variable exchange $(x,y,\theta) \mapsto (y,x,\theta)$ is smooth, the symbol estimates for $a$ imply the same symbol estimates for $b$ with the same symbolic order. Moreover, if \begin{align*} a(x,y,\theta) \sim \sum_{j=0}^{\infty} a_{\ell-j}(x,y,\theta) \end{align*} is the classical homogeneous expansion of $a$, then \begin{align*} b(y,x,\theta) \sim \sum_{j=0}^{\infty} a_{\ell-j}(x,y,\theta)^*. \end{align*} Thus the leading homogeneous term has the same degree. Since the number of oscillation variables and the excess convention are unchanged, the FIO order remains $m$. [/step]

[step:Identify the principal symbol under the half-density transport] Let $\tau: C \to C^t$ be the transposition diffeomorphism defined by $\tau(x,\xi;y,\eta)=(y,\eta;x,\xi)$. The principal symbol of $A$ is represented locally by the leading homogeneous amplitude, together with the phase-induced half-density and Maslov factor fixed in the theorem statement. The phase $\psi=-\phi$ has the same critical set after the exchange map $\tau_\Sigma$. The standard principal-symbol transformation law for Fourier integral operators applies because $\phi$ and $\psi$ are non-degenerate homogeneous phase functions parametrizing the corresponding local pieces of $C$ and $C^t$, and because the statement fixes the phase-induced half-density and Maslov convention. Under this law, the transposition diffeomorphism $\tau$ identifies the symbol line over $q \in C$ with the symbol line over $\tau(q) \in C^t$ by transporting the phase-induced half-density and Maslov factor along $\tau_\Sigma$. The replacement of $\phi$ by $-\phi$ conjugates the oscillatory factor and is exactly matched by taking the Hermitian adjoint of the leading amplitude, while the underlying positive half-density is unchanged under this transposition. Therefore the principal symbol of $A^*$ at $\tau(q) \in C^t$, for $q \in C$, is \begin{align*} \sigma_m(A^*)(\tau(q)) = \sigma_m(A)(q)^*, \end{align*} where $^*$ is the fiberwise Hermitian adjoint \begin{align*} \operatorname{Hom}(E_y,F_x) \to \operatorname{Hom}(F_x,E_y). \end{align*} For scalar operators, this operation is ordinary complex conjugation. The membership assertion is local modulo smooth kernels: on every pair of coordinate patches and bundle trivialisations the adjoint kernel has the oscillatory representation just constructed, while changes of phase and trivialisation are handled by the same principal-symbol transformation law. Proper support ensures that $A^*$ is a globally defined continuous operator on test sections and that these local adjoint kernels patch modulo smooth kernels. Therefore \begin{align*} A^* \in I^m(Y,X;C^t;\operatorname{Hom}(F,E)). \end{align*} This is exactly the asserted adjoint theorem and the stated principal-symbol formula. [/step]