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 → s2 to T yields the operator T2. 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
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 has a different focus from the holomorphic functional calculus.
More precisely, the Borel functional calculus allows us to apply an arbitrary Borel function to a self-adjoint operator, in a way which generalizes applying a polynomial function.
If T is a self-adjoint operator on a finite-dimensional inner product space H, then H has an orthonormal basis {e1, ..., eℓ} consisting of eigenvectors of T, that is
Thus, for any positive integer n,
If only polynomials in T are considered, then one arrives at the holomorphic functional calculus. Are more general functions of T possible? Yes. Given a Borel function h, one can define an operator h(T) by specifying its behavior on the basis:
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