Functional calculus is a mathematical framework that extends the notion of functions applied to real or complex numbers to functions applied to linear operators, particularly in the context of functional analysis and operator theory. It allows mathematicians and physicists to manipulate operators (usually bounded or unbounded linear operators on a Hilbert space) using functions. This methodology is particularly useful in quantum mechanics and other fields involving differential operators.
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A functional calculus assigns an linear operator to suitable scalar functions and an operator , extending polynomial substitutionA useful calculus preserves algebraic identities, such as , on its allowed function class. The admissible functions and continuity properties depend on the operator: polynomial, holomorphic functional calculus, continuous and measurable spectral calculi are different constructions.
For a diagonal matrix , substitution gives . A nondiagonal Jordan block shows why values alone need not suffice: if with , then . This follows by expanding each power and discarding all terms containing .
For bounded self-adjoint operators, the spectral theorem supplies a calculus for continuous functions on the spectrum, and more generally bounded measurable functions through spectral integration. It defines objects such as positive square roots without choosing a basis of eigenvectors.