Operator Theory & Functional Analysis

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Abstractions about linear operators on normed and Hilbert spaces and the function spaces they act on, covering operator classes (normal, unitary, hyponormal, Toeplitz), spectral notions (spectrum, pseudospectrum, spectral abscissa), norms and inequalities (Schatten norm, Cauchy-Schwarz), and structures like seminorms and operator topologies.

28 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.

• Besov space — A scale of function spaces measuring smoothness through integrability and multiscale difference or frequency-decay parameters, generalizing Sobolev and Hölder spaces.
• Bounded operator — A linear operator between normed spaces whose output norm is at most a fixed constant times the input norm.
• Cauchy–Schwarz inequality — The inequality bounding the absolute inner product of two vectors by the product of their norms, with equality exactly when the vectors are linearly dependent under the usual hypotheses.
• Complementarity theory — The theory of optimization and equilibrium problems seeking nonnegative vectors whose paired components have zero product, so each constraint and associated slack cannot both be positive.
• Dedekind eta function — A holomorphic function on the upper half-plane defined by q^(1/24) times the infinite product of (1−q^n), with a weight-one-half modular transformation law.
• Fourier algebra — The commutative Banach algebra of coefficient functions of the left regular representation of a locally compact group, under pointwise multiplication.
• Frobenius–Schur indicator — An invariant distinguishing whether an irreducible complex representation is real, complex, or quaternionic in type.
• 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.
• Indefinite inner product space — A vector space with a Hermitian sesquilinear form that can assign positive, negative or zero squared length to nonzero vectors.
• 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.
• Jacobi operator — A self-adjoint or symmetric tridiagonal operator on a sequence space determined by positive off-diagonal and real diagonal coefficient sequences.
• Lomonosov's invariant subspace theorem — A theorem stating that every bounded operator on an infinite-dimensional complex Banach space that commutes with a nonzero compact operator has a nontrivial closed invariant subspace.
• 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.
• Quaternionic representation — A complex group representation carrying an invariant antilinear equivariant operator whose square is minus the identity, equivalently a representation of quaternionic type.
• Schatten norm — The p-norm of a compact operator's singular-value sequence, generalizing matrix Frobenius and nuclear norms to operators on Hilbert spaces.
• Seminorm — A nonnegative subadditive absolutely homogeneous function on a vector space that may vanish on nonzero vectors.
• 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.
• Unbounded operator — A linear operator whose domain is typically a proper dense subspace of a normed space and which is not required to satisfy a global boundedness estimate.
• Unitary operator — A surjective linear operator on a Hilbert space that preserves inner products, equivalently one whose adjoint is its inverse.