Spectral theory studies a linear operator through the scalars at which it fails to be invertible in the required sense. For a bounded operator on a complex Banach space, its spectrum consists of for which has no bounded inverse defined on the whole space.
In finite dimensions this is the set of eigenvalues. In infinite dimensions there can be more: multiplication by on has spectrum but no eigenvectors. An eigenfunction at would have to be supported on the single point , a null set. Functions concentrated near that point nevertheless have arbitrarily small , ruling out a bounded inverse.
Inverse spectral theory reconstructs an operator or its coefficients from spectral data such as eigenvalues, spectral weights or boundary responses. It asks what the data determine and what additional information is needed for uniqueness.
Eigenvalues alone generally lose information. The matrices and have the same eigenvalues but different entries in the fixed coordinate system. Orthogonal conjugation changes eigenvectors while preserving the spectrum. An inverse problem must therefore specify whether recovery is sought in fixed coordinates, up to a change of basis, or within a more restricted class. Spectral weights can retain information about how selected measurements couple to eigenvectors.
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Spectral theory is a branch of mathematics that studies the spectrum of operators, particularly linear operators on function spaces or finite-dimensional vector spaces. It is closely related to linear algebra, functional analysis, and quantum mechanics, among other fields. The spectrum of an operator can be thought of as the set of values (often complex numbers) for which the operator does not behave like a regular linear transformation—in particular, where it does not have an inverse.