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 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 its focus is different than the holomorphic functional calculus one. More precisely, the Borel functional calculus allows for applying an arbitrary Borel function to a self-adjoint operator, in a way that generalizes applying a polynomial function. Motivation If T is a self-adjoint operator on a finite-dimensional inner product space H, then H has an orthonormal basis consisting of eigenvectors of T, that is Thus, for any positive integer n, If only polynomials in T are considered, then one gets the holomorphic functional calculus. Is it possible to get more general functions of T? Yes it is. Given a Borel function h, one can define an operator h(T) by specifying its behavior on the basis: Generally, any self-adjoint operator T is unitarily equivalent to a multiplication operator; this means that for many purposes, T can be considered as an operator acting on L2 of some measure space. The domain of T consists of those functions whose above expression is in L2. In such a case, one can define analogously For many technical purposes, the previous formulation is good enough. However, it is desirable to formulate the functional calculus in a way that does not depend on the particular representation of T as a multiplication operator. That's what we do in the next section. The bounded functional calculus Formally, the bounded Borel functional calculus of a self adjoint operator T on Hilbert space H is a mapping defined on the space of bounded complex-valued Borel functions f on the real line, such that the following conditions hold is an involution-preserving and unit-preserving homomorphism from the ring of complex-valued bounded measurable functions on R. If ξ is an element of H, then is a countably additive measure on the Borel sets E of R.