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Concept
Borel functional calculus
Summary
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 its focus is different than the holomorphic functional calculus.
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.
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 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.
Official source
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