skewadjoint

Skewadjoint (or skew-adjoint) refers to a linear operator T on a complex inner product space, typically a Hilbert space, for which the adjoint equals the negative of the operator: T* = -T. Equivalently, for all vectors x and y, ⟨Tx, y⟩ = -⟨x, Ty⟩. This identity implies that ⟨Tx, x⟩ is purely imaginary for every x, and multiplying T by i yields a self-adjoint operator.

In the finite-dimensional setting over the complex field, skewadjoint means the matrix A satisfies A* = -A. Over real inner product spaces, this reduces to the condition A^T = -A, i.e., A is skew-symmetric.

Spectral and dynamical properties follow from the definition. If T is bounded on a complex Hilbert space, its spectrum lies on the imaginary axis, and eigenvalues are purely imaginary. Eigenvectors corresponding to distinct eigenvalues are orthogonal. The operator iT is self-adjoint, and the one-parameter family e^{tT} is unitary for real t. By Stone’s theorem, every densely defined skewadjoint operator generates a strongly continuous one-parameter unitary group U(t) = e^{tT}.

Examples illustrate the concept: the 2×2 real matrix J = [[0, -1], [1, 0]] is skew-adjoint (J^T = -J) and generates rotations in the plane. More generally, any real skew-symmetric matrix is skew-adjoint, and on C^n the operator iI is skew-adjoint.

Skewadjoint operators form a real Lie algebra under the commutator and are closely related to the Lie algebra of the unitary group. They are also called skew-Hermitian in many texts.