A closed subalgebra $J\subset L(X)$ of the bounded operators on a Banach space $X$ is called a closed subideal of $L(X)$ if there is a closed ideal $I$ of $L(X)$ such that $J$ is a closed ideal of $I$. The subideal $J$ is called non-trivial if it is not an ideal of $L(X)$.
More generally, we call $J$ a closed $n$-subideal of $L(X)$ if there are closed subalgebras $J_0,\ldots,J_n$ of $L(X)$ such that $J=J_n\subset\ldots\subset J_1\subset J_0=L(X)$ and each $J_k$ is a closed ideal of $J_{k-1}$.
In this talk I will describe examples and properties of non-trivial closed subideals and closed $n$-subideals of $L(X)$ for various Banach spaces $X$. The talk is based on a joint ongoing work with Hans-Olav Tylli (Helsinki).