What is a "singular system" mean? Currenly I'm working on a problem as an apprentice. I'm currently reading the book "An introduction to the Mathematical Theory of Inverse Problems" by Andreas Kirshch.
In that book, on page 32, in Theorem 2.6, there is a sentence like this

Let $K: X \to Y$ be compact with singular system $(\mu_j, x_j, y_j )$ ...

I am not sure what was the author meant when he mentioned "singular system" here. I have the most basic knowledge about functional analysis, and I also search it up but didn't come up with a proper answer. Please help me with this.
Sorry if I missed something crucial. Thank you.
 A: In the Appendix, theorem $A.53$ clarifies your doubt!

Theorem $A.53$ (Singular Value Decomposition).
Let $K : X \rightarrow Y$ be a linear
compact operator, $K^∗ : Y \rightarrow X$ its adjoint operator, and $μ_1 ≥ μ_2 ≥ μ_3 . . . > 0$
the ordered sequence of the positive singular values of $K$, counted relative to its
multiplicity. Then there exist orthonormal systems $(x_j) ⊂ X$ and $(y_j) ⊂ Y$ with the following properties:
$$Kx_j = μ_jy_j$$ and $$K^∗y_j =μ_jx_j$$ for all $j ∈ J$.
The system $(μ_j ,x_j ,y_j)$ is called a singular system for $K$.

A: You shall looking for Singular Value Decomposition for compact operators (SVD). It's, in somehow, a generalisation of the spectral theorem for self-adjoint compact operators to ones not necessarily self-adjoint just compact. In fact, if $A$ is just compact, it's just the spectral theorem applied to the self-adjoint compact operator $A^* A$.
The SVD plays an important role to measure the degree of ill-posedness in inverse problems.
For more details see Chapter 2 : Functional analysis background of ill-posed problems. Section 2.4 : Singular Value Decomposition, from the book

A. Hasanov, V. Romanov. Introduction to inverse problems for differential equations (2017).

