A bounded linear operator is strictly singular if it is bounded below on no infinite-dimensional subspace. Equivalently, every infinite-dimensional subspace contains unit vectors with arbitrarily small images. On a hereditarily indecomposable Banach space, an operator which is arbitrarily small on unit vectors in every finite-codimensional subspace is strictly singular: construct an infinite basic span on which its operator norm is small, then use the zero-angle property of hereditary indecomposability.
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In functional analysis, a strictly singular operator is a type of linear operator that exhibits particularly strong properties of compactness. Specifically, an operator \( T: X \to Y \) between two Banach spaces \( X \) and \( Y \) is defined as strictly singular if it is not an isomorphism on any infinite-dimensional subspace of \( X \).