Zelfadjooint

Zelfadjooint is a term found in some Dutch-language mathematical writings to describe a self-adjoint joint operator concept. The usage is not standardized, and in many contexts the standard terminology self-adjoint (or Hermitian) together with commutativity is used instead. When people speak of zelfadjooint, they typically mean a collection of operators that are individually self-adjoint and possess a joint spectral structure allowing simultaneous analysis.

In formal terms, on a complex Hilbert space H, a set of operators A1, A2, … is called zelfadjooint if each Ai is self-adjoint and the operators commute pairwise (Ai Aj = Aj Ai for all i, j). This commutativity guarantees the existence of a joint spectral measure E on Rn, enabling a functional calculus for functions f of several variables: f(A1, A2, …) can be defined using E. In finite dimensions this reduces to the operators sharing a common eigenbasis, so their eigenvalues can be read off simultaneously.

Key properties include real spectra for each Ai, the possibility of defining joint functions through the joint spectral theorem, and the ability to analyze measurements that correspond to multiple commuting observables in quantum-mechanical or signal-processing contexts.

Examples are straightforward in finite-dimensional spaces: two commuting Hermitian matrices, such as diagonal matrices, admit a common eigenbasis and a clear joint spectrum. The term zelfadjooint remains largely a descriptive or nonstandard label; standard references typically speak of commuting self-adjoint operators and their joint spectral theory.

See also: self-adjoint, Hermitian, adjoint, commuting operators, spectral theorem.