OperatorproduktExpansion

Operatorprodukt, or operator product, refers to the composition of two linear operators acting on a vector space or Hilbert space. The product AB is defined on the subset of vectors x for which x lies in the domain of B and Bx lies in the domain of A. In finite-dimensional settings or when the operators are defined on the whole space, AB corresponds to the standard composition of linear maps and, equivalently, to matrix multiplication.

Key properties include associativity and, in normed spaces, boundedness under suitable conditions. If A and B are linear maps with B defined everywhere and A bounded, then AB is bounded and its norm satisfies ||AB|| ≤ ||A|| ||B||. For bounded operators on a Banach space, the operation is associative: (AB)C = A(BC). When dealing with unbounded operators, domains must be specified carefully, as AB may be densely defined only under particular assumptions and may fail to be defined on all vectors.

Common examples include differential operators and multiplication operators. For instance, D denotes differentiation and M_f denotes multiplication by a function f; their product AB represents applying B first and then A, subject to domain compatibility. In quantum mechanics, operator products often fail to commute, reflecting fundamental physical principles.

Related topics encompass operator algebras, tensor products of operators, and spectral theory. In theoretical physics, the operator product expansion describes the short-distance behavior of the product of local operators, providing a different but conceptually connected use of the term “operator product.”