Spårmängder

Spårmängder, literally “trace sets” in Swedish, are a mathematical concept used to describe the collection of trace values associated with a family of linear operators on a Hilbert space. They arise when one fixes a class of operators, such as trace‑class or finite‑rank operators, and looks at all possible traces produced by members of that class or by certain substructures derived from them.

A common construction defines a spårmängd as the set of traces obtained from a given operator family. If F is a family of trace‑class operators on a Hilbert space H, the spårmängd is S = { tr(A) : A ∈ F }. Another variant considers traces of restricted operators, for example S = { tr(PAP) : P ranges over a chosen collection of projections on H }. The trace is a linear functional on the trace‑class operators, and its value is independent of the chosen orthonormal basis.

Key properties follow from standard trace theory. If A is trace‑class, tr(A) is finite and tr(A*) = overline{tr(A)}. For positive trace‑class operators A ≥ 0, tr(A) ≥ 0. In finite dimensions, the trace equals the sum of eigenvalues (counting multiplicities), and for suitable A, B one has tr(AB) = tr(BA). These properties underpin the way spårmängder reflect the spectral data of the underlying operators.

Examples appear in quantum mechanics, where density operators have trace 1, producing trace sets contained in a bounded interval. In operator algebras and functional analysis, spårmängder help study linear functionals, moments, and the structure of state spaces.

See also: trace class, trace functional, Hilbert space, operator theory, spectral theory.