Consider the following fragment from Takesaki's book "Theory of operator algebra I":
In this proof, we encounter expressions like
$$x = \int_{-\|x\|}^{\|x\|}\lambda de(\lambda).$$
Can someone explain me how to understand this integral or give me an 'accessible' reference where I can read more about these kinds of integrals? Is this related to the "Borel functional calculus"?
Best Answer
This is a result of the spectral theorem for self adjoint, bounded operators on a Hilbert space. The theorem which can be found in Reed and Simons for example provides a way of constructing an operator out of an integral over the spectrum of the operator with respect to a special kind of measure called a spectral measure.
In the case of self adjoint $x$ we know $\sigma(x)\subset \mathbb{R}$ and moreover, the spectral radius is equal to $\|x\|$.
The $de(\lambda)$ is the spectral measure defined on $\sigma(x)$ and it allows you to integrate continuous functions or even measurable functions with respect to this measure to define the analogue of the given function on the operator $x$. For example, if $f(\lambda)=e^\lambda$ defined on the spectrum of $x$, then the spectral theorem says that $$f(x)=\int_{\sigma(x)}f(\lambda)de(\lambda)$$ So you can give meaning to the exponential of an operator (in the case of bounded operators this machinery is not necessary since you can represent it as a power series in the operator norm). In the case where $f(\lambda)=\lambda$ i.e the $id_{\sigma(x)}$ then the theorem yields $$x=\int_{\sigma(x)}\lambda de(\lambda)$$ But remember if $x$ is self adjoint then the spectral radius is equal to $\|x\|$ and $\sigma(x)\subset \mathbb{R}$ we have $\sigma(x)\subset [-\|x\|,\|x\|]$ (if $\sigma(x)\subsetneq [-\|x\|,\|x\|]$ you can extend the integration to the endpoints by declaring that the spectral measures are exactly zero in the compliment) and hence $$x=\int_{-\|x\|}^{\|x\|}\lambda de(\lambda)$$