unitarylike

Unitarylike is an informal term used in mathematics and related fields to describe objects that resemble unitary operators or matrices in some respects, but do not satisfy the defining condition exactly. Because it is not a standard technical term, its precise meaning depends on context.

In operator theory and functional analysis, unitary-like may refer to operators that are isometric on a subspace or that are close to being unitary in the sense that T* T is close to the identity, often in a norm sense. A related concept is a partial isometry, which acts unitarily on its initial space and is zero on the orthogonal complement, and can be viewed as unitary-like on the parts where it is defined.

In numerical linear algebra, a unitary-like matrix is typically one that approximately preserves norms, meaning A* A is close to the identity matrix within a specified tolerance. Such matrices retain many favorable properties of unitary matrices, including stable behavior under multiplication and favorable spectral properties, with eigenvalues near the unit circle.

Examples include a unitary matrix perturbed by a small noise term, or a matrix A for which ||A* A - I|| is small. The exact interpretation depends on the chosen tolerance or property of interest.

Relation to standard concepts: a unitary operator U satisfies U* U = I; unitary-like objects are not required to satisfy this exactly but share one or more analogous features, such as norm preservation on a subspace, near-normality, or approximate isometry.

See also: unitary operator, isometry, partial isometry, normal operator, numerical range, perturbation theory.