Borel functional calculus in the context of Operator (mathematics)

⭐ Core Definition: Borel functional calculus

In functional analysis, a branch of mathematics, the Borel functional calculus is a functional calculus (that is, an assignment of operators from commutative algebras to functions defined on their spectra), which has particularly broad scope.Thus for instance if T is an operator, applying the squaring function s → s to T yields the operator T. Using the functional calculus for larger classes of functions, we can for example define rigorously the "square root" of the (negative) Laplacian operator $-Δ$ or the exponential $e^{it\Delta }.$

The 'scope' here means the kind of function of an operator which is allowed. The Borel functional calculus is more general than the continuous functional calculus, and its focus is different than the holomorphic functional calculus.

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Borel functional calculus in the context of Spectral measure

In mathematics, particularly in functional analysis, a projection-valued measure, or spectral measure, is a function defined on certain subsets of a fixed set and whose values are self-adjoint projections on a fixed Hilbert space. A projection-valued measure (PVM) is formally similar to a real-valued measure, except that its values are self-adjoint projections rather than real numbers. As in the case of ordinary measures, it is possible to integrate complex-valued functions with respect to a PVM; the result of such an integration is a linear operator on the given Hilbert space.

Projection-valued measures are used to express results in spectral theory, such as the important spectral theorem for self-adjoint operators, in which case the PVM is sometimes referred to as the spectral measure. The Borel functional calculus for self-adjoint operators is constructed using integrals with respect to PVMs. In quantum mechanics, PVMs are the mathematical description of projective measurements. They are generalized by positive operator valued measures (POVMs) in the same sense that a mixed state or density matrix generalizes the notion of a pure state.

View the full Wikipedia page for Spectral measure

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Borel functional calculus in the context of Operator (mathematics)

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⭐ Core Definition: Borel functional calculus

Borel functional calculus in the context of Spectral measure