Operator Theory & Spectral Analysis

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Abstractions about linear operators on functional spaces, their spectra, norms, topologies, invariant subspaces, and projection structure. They include bounded, normal, unitary and nuclear operators, functional calculus, pseudospectra, Toeplitz forms, and covariance operators.

22 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Bounded operator — A linear operator between normed spaces whose output norm is at most a fixed constant times the input norm.
• Covariance operator — The linear operator encoding second-order variation of a random element by mapping a direction to its expected covariance-weighted displacement.
• Fourier algebra — The commutative Banach algebra of coefficient functions of the left regular representation of a locally compact group, under pointwise multiplication.
• Functional principal component analysis — A dimension-reduction method representing random curves or functions in the eigenbasis of their covariance operator.
• Hilbert–Schmidt integral operator — An integral operator whose square-integrable kernel makes it a compact Hilbert–Schmidt operator.
• Holomorphic functional calculus — A calculus assigning f(T) to a bounded operator T for every function holomorphic near its spectrum through a contour integral of the resolvent.
• Hyponormal operator — A bounded Hilbert-space operator whose self-commutator TT−TT is positive semidefinite.
• Integration by parts operator — Represent the adjoint of directional differentiation relative to a measure by mapping admissible vector fields to scalar divergences that satisfy a measure-specific integration-by-parts identity.
• Invariant subspace problem — The family of questions asking whether every operator in a specified class on a topological vector space has a nonzero proper closed subspace mapped into itself.
• Jacobi operator — A self-adjoint or symmetric tridiagonal operator on a sequence space determined by positive off-diagonal and real diagonal coefficient sequences.
• Normal operator — A bounded linear operator on a complex Hilbert space that commutes with its adjoint.
• Nuclear operators between Banach spaces — Linear operators admitting a summable rank-one representation through dual functionals and target vectors.
• Operator topologies — Standard topologies on spaces of bounded linear operators that distinguish norm, strong, weak and weak-star modes of operator convergence.
• Partial isometry — A Hilbert-space operator that acts isometrically on the orthogonal complement of its kernel and vanishes on the kernel, mapping an initial subspace onto a final subspace.
• Pseudospectrum — For an operator and tolerance epsilon, the set of spectral values attainable under perturbations of size epsilon, equivalently points where the resolvent is large.
• Riesz projector — A contour-integral projection onto the invariant spectral subspace associated with an isolated portion of an operator’s spectrum.
• Schatten norm — The p-norm of a compact operator's singular-value sequence, generalizing matrix Frobenius and nuclear norms to operators on Hilbert spaces.
• Spectral abscissa — The greatest real part among the eigenvalues or spectral values of a matrix or bounded linear operator.
• Spectrum (functional analysis) — The set of scalars for which an operator minus that scalar times the identity fails to possess an everywhere-defined bounded inverse.
• Subnormal operator — A bounded operator on a Hilbert space that is the restriction of a normal operator to an invariant subspace of a larger Hilbert space.
• Toeplitz operator — The compression to Hardy space of multiplication by a bounded function on the unit circle, yielding an operator with constant matrix diagonals.
• Unitary operator — A surjective linear operator on a Hilbert space that preserves inner products, equivalently one whose adjoint is its inverse.