Structure of closed subideals of $\mathcal L(X)$

The closed subalgebra $\mathcal J$ of the Banach algebra $\mathcal L(X)$ of bounded linear operators on the Banach space $X$ is a non-trivial closed $\mathcal I$-subideal of $\mathcal L(X)$ if $\mathcal I$ is a closed ideal of $\mathcal L(X)$ and $\mathcal J$ is an ideal of $\mathcal I$, but $\mathcal J$ is not an ideal of $\mathcal L(X)$. We obtain a variety of examples of non-trivial closed subideals of $\mathcal L(X)$ for different spaces $X$, which highlight further significant differences compared to the class of closed ideals. We study the concept of a closed $n$-subideal of $\mathcal L(X)$ for $n \ge 3$, which is a natural generalization of that of a closed subideal. In particular, we find explicit spaces $X$ for which $\mathcal L(X)$ contains a decreasing sequence $(\mathcal M_n)_{n\in \mathbb N}$ of closed subalgebras, where for all $n\in\mathbb N$ the subalgebra $\mathcal M_n$ is an $(n+1)$-subideal of $\mathcal L(X)$ but not an $n$-subideal. Moreover, we construct closed $n$-subideals contained in the compact operators $\mathcal K(X)$ for certain Banach spaces $X$ which fail the approximation property.