This questions arises from a proof of the existence of a generalized singular value expansion (an extension of the generalized singular value decomposition developed by Van Loan in the 70's).
The theorem is starts as follows: Suppose that there are two compact operators $A$ and $B$ on a separable Hilbert space $\mathcal{H}$, such that $A:\mathcal{H} \rightarrow \mathcal{H_{A}}$ and $B:\mathcal{H} \rightarrow \mathcal{H_{B}}$ for separable Hilbert spaces $\mathcal{H_{A}}$ and $\mathcal{H_{B}}$.
- (i) Let $\{d_{jA}, f_{jA},g_{jA}\}$ and $\{d_{jB}, f_{jB},g_{jB}\}$ be the singular systems of $A$ and $B$ respectively, and let $\{d_{j}, f_{j},g_{j}\}$ be the singular system of $C=[A \quad B]^{T}$.
My question is how do we find the singular value system for $C=[A \quad B]^{T}$? We have a compact operator $C:\mathcal{H} \rightarrow \mathcal{H_{A}} \times \mathcal{H_{B}}$ that admits a singular value expansion. But to find this expansion we first need the adjoint operator $C^{*}:\mathcal{H_{A}} \times \mathcal{H_{B}} \rightarrow \mathcal{H}$. I am unsure of how to find the adjoint operator in this case. I am unsure of how to proceed after finding the adjoint as well. Any references would be appreciated.
Note: this theorem comes from chapter 2 of a dissertation titled "Some Topics Concerning the Singular Value Decomposition and Generalized Singular Value Decomposition" by Qing Huang.
You cannot expect a simple expression for the singular values of $C$ in terms of the singular values of $A$ and $B$. Position is important, which means to account for the orthonormal bases.
Take $$A=\begin{bmatrix}2&0\\0&0\end{bmatrix},\ \ \ \ B=\begin{bmatrix}3&0\\0&0\end{bmatrix}. $$ Then $$ C^*C=\begin{bmatrix}A^* & B^*\end{bmatrix}\,\begin{bmatrix} A\\ B\end{bmatrix}=A^*A+B^*B=\begin{bmatrix}13&0\\0&0\end{bmatrix}, $$ so the singular values of $C$ are $\sqrt{13}$ and $0$. But if now $$A=\begin{bmatrix}0&0\\0&2\end{bmatrix},\ \ \ \ B=\begin{bmatrix}3&0\\0&0\end{bmatrix}. $$ Then $$ C^*C=\begin{bmatrix}A^* & B^*\end{bmatrix}\,\begin{bmatrix} A\\ B\end{bmatrix}=A^*A+B^*B=\begin{bmatrix}9&0\\0&4\end{bmatrix}, $$ and the singular values of $C$ are $3$ and $2$.