functional calculus

1 Basic Idea

Let $X$ be a normed vector space over a field $𝕂$. Let $T$ be a linear operator in $X$ and $I$ the identity operator in $X$.

The functional calculus refers to a specific process which enables the expression

$$f(T)$$

to make sense as a linear operator in $X$, for certain scalar functions $f:𝕂⟶𝕂$.

At first sight, and for most functions $f$, there is no reason why the above expression should be associated with a particular linear operator.

But, for example, when $f$ is a polynomial $f(x)=a_k{x}^k+\mathrm{\dots }+a_2{x}^2+a_{1}x+a_0$, the expression

$$f(T):=a_k{T}^k+\mathrm{\dots }+a_2{T}^2+a_{1}T+a_{0}I$$

does indeed refer to a linear operator in $X$.

As another example, when $T$ is a matrix in ${ℝ}^n$ or ${ℂ}^n$ one is sometimes led to the exponential (http://planetmath.org/MatrixExponential) of $T$

$$e^T=\sum _{k=0}^{\mathrm{\infty }}\frac{T^k}{k!}$$

Thus, we are applying the scalar exponential function to a matrix.

Note in this last example that $e^T$ is approximated by polynomials (the partial sums of the series). This provides an idea of how to make sense of $f(T)$ if $f$ can be approximated by polynomials:

If $f$ can be approximated by polynomials $p_n$ then one could try to define

$$f(T):=\underset{n\to \mathrm{\infty }}{lim}{p}_n(T)$$

But for that one needs to define what “approximated” means and to assure the above limit exists.

2 More abstractly

There is no reason why one should restrict to linear operators in a normed vector space. In this , we can consider instead a unital topological algebra $𝒜$ over a field $𝕂$.

There is no definition in mathematics of functional calculus, but the ideas above show that a functional calculus for an element $T\in 𝒜$ should be something like an homomorphism $(\cdot )(T):ℱ⟶𝒜$ from some topological algebra of scalar functions $ℱ$ to $𝒜$, that satisfied the following :

• $ℱ$ must contain the polynomial functions.

• $(\cdot )(T)$ is continuous.

• $p(T)=a_k{T}^k+\mathrm{\dots }+a_2{T}^2+a_{1}T+a_{0}I$ for each polynomial $p(x)=a_k{x}^k+\mathrm{\dots }+a_2{x}^2+a_{1}x+a_0$, where $I$ denotes the identity element of $𝒜$.

3 Functional Calculi

There are some functional calculi of . We give a very brief descprition of each one of them (please follows the links for entries with more detailed explanation).

• polynomial functional calculus -

  This is valid for any element $T$ in any algebra $𝒜$. It associates polynomials to elements in the algebra generated by $T$, as discussed above.

• functional calculus -

  This is valid for any element $T$ in a complex Banach algebra $𝒜$. It associates complex analytic functions defined on the spectrum (http://planetmath.org/Spectrum) of $T$ to elements in the algebra generated by $T$.

• continuous functional calculus -

  This is valid for normal elements in $C^{*}$-algebras (http://planetmath.org/CAlgebra). It associates continuous functions on the spectrum of $T$ to elements in the $C^{*}$-algebra generated by $T$.

• Borel functional calculus -

  This is valid for normal operators $T$ in a von Neumann algebra $𝒜$. It associates bounded Borel measurable functions on the spectrum of $T$ to elements in the von Neumann algebra generated by $T$.

• functional calculus for Hermitian matrices -

  A case. It is valid for Hermitian matrices $T$. It associates real valued functions on the spectrum of $T$ to elements in the algebra generated by $T$.

4 Applications

• Functional calculi provide an of constructing new linear operators having specified out of given ones.

• There are strong with spectral theory since one usually has $f(\sigma (T))=\sigma (f(T))$, where $\sigma (\cdot )$ denotes the spectrum of its . This is called the spectral mapping theorem.

• As the with spectral theory can possibly show, functional calculi are an tool for studying equations. For example, they can give sufficient conditions for the existence of a square root $\sqrt{T}$ of an $T$.