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So, welcome to lecture number 11. Today we'll finally prove what is left to be proven to have the spectral theorem

and I'll present an application to that and there will be many more applications to follow.

In fact, some of the definitions already carry some applications in them, although today we won't emphasize on that.

So, please recall that from the previous lecture we proved the inverse spectral theorem,

which goes like this. Given a PVM, given a projection valued measure, for which we wrote PVM for short,

given a projection valued measure P, we have a self-adjoint operator, so we can construct from it an operator A,

by simply taking the integral over the measurable function identity on R, integrated over the projection valued measure P.

And if you recall, there is another notation for that, so if I don't want to specify this function abstractly,

but rather point-wise, the identity on R of course sends a lambda in R to a lambda, so that's lambda,

and then you know that the notation is that when we write P, the lambda, but this is really just notation

that provides us with the freedom to define the F point-wise in there, in order to avoid having to be overly formal,

but in fact this is the formally more useful definition.

Okay, so today the idea is to invert this, solve for this P, so not given P construct an A,

but given an A, find the P such that A can be written like that. And in order to get an idea, or the idea of how to do that,

we will start with the assumption that the A can be decomposed like this, and we will check first for a case where it can be decomposed,

how we could get this P once we constructed the A in this fashion.

And then we will find all the necessary ingredients in order to more abstractly require that for any A,

even if we don't know A priori it's given in this fashion, how to reconstruct the P.

Okay, that's the strategy, and the first section however is already an application of the spectral theorem,

first of all restricted to self-adjoint operators that can be decomposed this way.

Well, the spectral theorem will tell you every self-adjoint operator can finally be written like this,

but at the moment we don't have that result yet, at the moment we say let's have a spectral and self-adjoint operator that can be written like this.

Of course, later on the spectral theorem allows us to drop that. So the first section is measurable functions,

function applied to a spectrally decomposable self-adjoint operator.

And as I just said, this assumption that the self-adjoint operator is spectrally decomposable means it has this form,

later on there will be a triviality, that's always the case, but at the moment we've got to assume this.

So definition, let A be spectrally decomposable, that is, there exists a projection valued measure P such that A can be written in above fashion.

That's the definition in the definition. Then for any measurable function f,

and we take f only from R to R, and here the A should also be self-adjoint.

Then for any measurable function f from R to R, we define the operator f of A.

So we take a decomposable operator, decomposable self-adjoint operator,

and we now want to define the application of a real function to it.

And this is supposed to be an operator. Well, formally speaking, we have to have a definition here,

because I cannot plug in an operator where only an R can be eaten, right? That's not possible.

But so therefore, if I write such a symbol, I need to define it. Then the definition is as follows.

You apply the f to the lambda, and then you use the P that decomposes A, P, the lambda.

So now I use the pointwise notation here, but I can also translate this into the not pointwise.

So this is again notation. This would be just f after the identity on R, right? dP.

So this is again just notation.

However, at this point, this is only defined for A that we know from the beginning to be decomposable.

And I should also say from where to where this operator maps. Well, it maps obviously from f

after the identity is just f dP, and we know this domain. We had it last time.

You remember this domain is all those psi in H for which the integral over f absolute value squared d mu psi.

Here's the psi that we're conditioning reappearing again, where this is this.

And you should remember that this Borel measure in turn is given by mu psi of a Borel set omega.

So this is in sigma or is defined as psi P of omega psi. OK? So those were the definitions.

We define this operator and we call it the application of the measurable function f to the so far spectral decomposable.

The condition will drop self adjoint operator A. OK? So remark.

The spectral theorem, which I mentioned at the beginning of last lecture and which we will show today,