In functional analysis, a branch of mathematics, a strictly singular operator is a bounded linear operator between normed spaces which is not bounded below on any infinite-dimensional subspace.

Let X and Y be normed linear spaces, and denote by B(X,Y) the space of bounded operators of the form ${\displaystyle T:X\to Y}$. Let ${\displaystyle A\subseteq X}$ be any subset. We say that T is bounded below on ${\displaystyle A}$ whenever there is a constant ${\displaystyle c\in (0,\infty )}$ such that for all ${\displaystyle x\in A}$, the inequality ${\displaystyle \|Tx\|\geq c\|x\|}$ holds. If A=X, we say simply that T is bounded below.

Now suppose X and Y are Banach spaces, and let ${\displaystyle Id_{X}\in B(X)}$ and ${\displaystyle Id_{Y}\in B(Y)}$ denote the respective identity operators. An operator ${\displaystyle T\in B(X,Y)}$ is called inessential whenever ${\displaystyle Id_{X}-ST}$ is a Fredholm operator for every ${\displaystyle S\in B(Y,X)}$. Equivalently, T is inessential if and only if ${\displaystyle Id_{Y}-TS}$ is Fredholm for every ${\displaystyle S\in B(Y,X)}$. Denote by ${\displaystyle {\mathcal {E}}(X,Y)}$ the set of all inessential operators in ${\displaystyle B(X,Y)}$.

An operator ${\displaystyle T\in B(X,Y)}$ is called strictly singular whenever it fails to be bounded below on any infinite-dimensional subspace of X. Denote by ${\displaystyle {\mathcal {SS}}(X,Y)}$ the set of all strictly singular operators in ${\displaystyle B(X,Y)}$. We say that ${\displaystyle T\in B(X,Y)}$ is finitely strictly singular whenever for each ${\displaystyle \epsilon >0}$ there exists ${\displaystyle n\in \mathbb {N} }$ such that for every subspace E of X satisfying ${\displaystyle {\text{dim}}(E)\geq n}$, there is ${\displaystyle x\in E}$ such that ${\displaystyle \|Tx\|<\epsilon \|x\|}$. Denote by ${\displaystyle {\mathcal {FSS}}(X,Y)}$ the set of all finitely strictly singular operators in ${\displaystyle B(X,Y)}$.

Let ${\displaystyle B_{X}=\{x\in X:\|x\|\leq 1\}}$ denote the closed unit ball in X. An operator ${\displaystyle T\in B(X,Y)}$ is compact whenever ${\displaystyle TB_{X}=\{Tx:x\in B_{X}\}}$ is a relatively norm-compact subset of Y, and denote by ${\displaystyle {\mathcal {K}}(X,Y)}$ the set of all such compact operators.

Strictly singular operators can be viewed as a generalization of compact operators, as every compact operator is strictly singular. These two classes share some important properties. For example, if X is a Banach space and T is a strictly singular operator in B(X) then its spectrum ${\displaystyle \sigma (T)}$ satisfies the following properties: (i) the cardinality of ${\displaystyle \sigma (T)}$ is at most countable; (ii) ${\displaystyle 0\in \sigma (T)}$ (except possibly in the trivial case where X is finite-dimensional); (iii) zero is the only possible limit point of ${\displaystyle \sigma (T)}$; and (iv) every nonzero ${\displaystyle \lambda \in \sigma (T)}$ is an eigenvalue. This same "spectral theorem" consisting of (i)-(iv) is satisfied for inessential operators in B(X).

Classes ${\displaystyle {\mathcal {K}}}$, ${\displaystyle {\mathcal {FSS}}}$, ${\displaystyle {\mathcal {SS}}}$, and ${\displaystyle {\mathcal {E}}}$ all form norm-closed operator ideals. This means, whenever X and Y are Banach spaces, the component spaces ${\displaystyle {\mathcal {K}}(X,Y)}$, ${\displaystyle {\mathcal {FSS}}(X,Y)}$, ${\displaystyle {\mathcal {SS}}(X,Y)}$, and ${\displaystyle {\mathcal {E}}(X,Y)}$ are each closed subspaces (in the operator norm) of B(X,Y), such that the classes are invariant under composition with arbitrary bounded linear operators.

In general, we have ${\displaystyle {\mathcal {K}}(X,Y)\subset {\mathcal {FSS}}(X,Y)\subset {\mathcal {SS}}(X,Y)\subset {\mathcal {E}}(X,Y)}$, and each of the inclusions may or may not be strict, depending on the choices of X and Y.