Spectral Theory 1 | Complex Measures

Indexé : 2026-05-22

[00:00:00]Hello and welcome to spectral theory, the new video series where I want to talk about the spectral theorem for self-adjoint operators on Hilbert spaces.

[00:00:10]This is a really important result with a lot of applications such that I want to put this into a separate video series.

[00:00:18]So, the goal here is to prove the spectral theorem and to show what we can do with it.

[00:00:24]However, as always, first I want to thank all the nice people who support the channel on Study, here on YouTube, or via other means.

[00:00:31]Indeed, only because of your support am I able to make these in-depth videos.

[00:00:37]And now, without further ado, I can already tell you how this new video course is embedded in my network of video courses.

[00:00:46]In fact, you can see it as a continuation of my functional analysis course and my course about Hilbert spaces.

[00:00:53]This makes sense because we will talk about self-adjoint operators defined on Hilbert spaces.

[00:00:59]Therefore, you should already have a basic knowledge of general Hilbert spaces and operators on them.

[00:01:06]And then we will see how we can generalize stuff from self-adjoint matrices to self-adjoint operators on infinite-dimensional spaces.

[00:01:16]So, if you want to see the finite-dimensional version of the spectral theorem, you can already check that out in my abstract linear algebra course.

[00:01:24]However, now it turns out that the infinite-dimensional version needs some measure theory in the description.

[00:01:31]Therefore, it's definitely good to have some basic knowledge of measure theory because these will be the tools we will use.

[00:01:39]In fact, in today's part one, we will start exactly with that. We will define so-called complex measures.

[00:01:46]But before we do that, perhaps we should write down the finite-dimensional version of our spectral theorem from linear algebra.

[00:01:54]Here I will formulate it for self-adjoint matrices, but if you're interested in the general formulation for normal matrices, you can check out part 47.

[00:02:04]In any case, the assumption is always that we have a square matrix A with complex entries.

[00:02:10]And we know that this induces a linear map from CN into CN.

[00:02:16]And now this linear map or the matrix representation A is called self-adjoint if A star is equal to A.

[00:02:23]And here please recall for matrices, this star is the transpose together with the complex conjugation.

[00:02:31]And now it turns out that these matrices are quite nice because we can unitarily diagonalize them.

[00:02:38]And this simply means that we find a unitary matrix U such that U star A U is a diagonal matrix.

[00:02:47]And here please don't forget, unitary means that U star is equal to the inverse of U.

[00:02:53]Hence, A is unitarily similar to a diagonal matrix.

[00:02:58]And moreover, on this diagonal, we find exactly the eigenvalues of the matrix A.

[00:03:05]And this fits because if we count the eigenvalues with multiplicities, we have exactly n of them.

[00:03:12]And there you might already see the big problem in infinite dimensions because there the spectrum consists of more than just eigenvalues.

[00:03:21]Hence, if we want to generalize this diagonalization here, we need to talk about the whole set of spectral values.

[00:03:28]And at that point, the tools of measure theory will help us a lot.

[00:03:33]But still, we have to extend the common treatment of measure theory because we have to deal with so-called complex measures.

[00:03:41]This simply means that the measures we consider here can have values in the complex numbers.

[00:03:48]So, let's immediately start with the definition. We consider a measure on an arbitrary set X.

[00:03:54]And this means we will take a sigma algebra A that consists of subsets of X.

[00:04:00]So, formally, A is a subset of the power set of X.

[00:04:06]And now the concept of a sigma algebra tells us that the collection A at least contains the empty set and X itself.

[00:04:15]And moreover, the collection of subsets is stable under forming complements and countable unions.

[00:04:23]And then we can rightfully speak of a sigma algebra on the set X.

[00:04:28]And this is important because any measure mu is always defined on a sigma algebra. So, A is the domain of mu and the codomain is given by the complex numbers.

[00:04:40]And now you already know, this is the reason why we speak of a complex measure.

[00:04:46]Of course, we still require the same two properties as in measure theory, but now we change the codomain from the positive real number line to C.

[00:04:56]Hence, we still want that the empty set has measure zero.

[00:05:00]So, this is the first property, and the second one is the sigma additivity.

[00:05:06]And this one tells us that we can take any countable collection of subsets, and let's call them AI, and we want that this collection is pairwise disjoint.

[00:05:16]This means AI intersected with AJ is the empty set.

[00:05:22]But obviously, only in the case that the two indices I and J don't coincide.

[00:05:29]And then we just want to measure the whole union of all these sets.

[00:05:33]And in order to keep it simple, let's say that our index I goes from one to infinity.

[00:05:40]And now the idea of a measure is that we have a partition of the set, and we can just measure the single parts and sum them up.

[00:05:48]In other words, we get an infinite sum of the measures of AI.

[00:05:54]And there you know, this is what we call sigma additivity.

[00:05:58]But now, in contrast to the original positive measure, the symbol infinity is not allowed as an outcome of the measure.

[00:06:06]Therefore, we have to add an additional requirement here, namely that this sum is always absolutely convergent. And this guarantees that the value of the sum does not change when we reorder the entries.

[00:06:21]And this is definitely needed because the left-hand side here does not care about any order of the sets AI.

[00:06:28]So, this is the sigma additivity as we needed for a complex measure.

[00:06:33]Therefore, the first thing you can already remember here is that a complex measure is always a finite measure.

[00:06:40]Of course, this is kind of clear because only complex numbers are allowed as the outcome anyway.

[00:06:47]Hence, if we put in the whole set X, we also get out a complex number.

[00:06:53]In particular, the absolute value of this complex number is not equal to the symbol infinity.

[00:06:59]I point this out because the symbol infinity was actually allowed in measure theory before.

[00:07:06]So, in conclusion, the notion complex measure is on the one hand a generalization of an ordinary measure from before, but on the other hand also a restriction.

[00:07:17]So, for the moment, we can only deal with finite measures.

[00:07:21]And therefore, for the first example, I want to look at a bounded set in the real number line.

[00:07:27]And to keep it simple, let's take the unit interval.

[00:07:31]And there we can consider the ordinary Lebesgue measure defined on the Borel sigma algebra.

[00:07:37]So, as an ordinary measure, this is also a finite measure because the measure of the whole interval here is just one.

[00:07:45]However, usually in measure theory, we would say that the codomain is given by zero to infinity, where infinity is allowed. And now, in the same way, we can also look at the Dirac measure at the origin.

[00:07:58]This one is also a well-defined ordinary measure on the Borel sigma algebra.

[00:08:04]And here, what I want to do is to combine both measures into a complex measure.

[00:08:10]And indeed, this is quite simple because both measures are already complex measures by this definition.

[00:08:16]They satisfy both conditions because they don't need the symbol infinity.

[00:08:21]But moreover, I can also say that we consider the measure lambda plus i delta.

[00:08:28]This is not an ordinary measure again, but a proper complex measure.

[00:08:33]Indeed, let's put in the unit interval, and then you see what we get.

[00:08:38]Namely, we first have one by the Lebesgue measure, and also one by the Dirac measure because zero is included here.

[00:08:47]In other words, the result is the complex number 1 + i.

[00:08:52]So, maybe this seems a little bit strange, but I can already tell you that this will help us because our operators are defined on a complex Hilbert space.

[00:09:01]So, complex numbers are intrinsically needed in the spectral theorem.

[00:09:06]The spectrum of any operator is always a subset of the complex numbers.

[00:09:12]But how we can use the complex measures in Hilbert spaces, we will see soon.

[00:09:17]But first, let's discuss some important properties of complex measures.

[00:09:22]The first thing is that any complex measure is always continuous at the empty set.

[00:09:28]Which simply means if we make the set smaller and smaller that we put into the measure, then we also make the measure smaller and smaller.

[00:09:38]And moreover, if the limit of the sets is given by the empty set, which simply means that the intersection is the empty set, then the measure of AJ should also converge to the measure of the empty set.

[00:09:52]And as we already know, the measure of the empty set should be equal to zero.

[00:09:58]In other words, we get this convergence for any sets from the sigma algebra that are decreasing in that sense.

[00:10:06]And this is an important property that we will definitely use later, so let's prove it.

[00:10:12]And in fact, it immediately follows from the sigma additivity of the measure.

[00:10:17]So maybe let's say this one is A1, and then we have A2 in it, and so on.

[00:10:23]And now, in order to use the sigma additivity, you know that we need disjoint sets.

[00:10:29]And for example, we can construct disjoint sets by forming set differences.

[00:10:35]So in the picture, we have A1 without A2, and we can continue that for every index K.

[00:10:41]So let's say we have the set CK, which is AK without AK + 1.

[00:10:48]So in the picture, the sets CK have this ring structure here.

[00:10:52]And then immediately by definition, we have disjointness of the CK sets.

[00:10:58]And in addition, it's also not hard to see that the whole union is just our original set A1.

[00:11:05]And with that, we have a nice collection of subsets where we can apply our sigma additivity of mu.

[00:11:12]And since there is an infinite sum included in the sigma additivity, we can first have a look at the finite sum.

[00:11:20]So we have mu of CK, and we sum from 1 to N.

[00:11:25]And here, since CK is given by a set difference, we also know what the result of the measure is by the sigma additivity.

[00:11:34]Namely, it has to be the difference of the two measures.

[00:11:38]And there you recognize that we have an alternating sign in our sum, so we actually have a telescoping sum.

[00:11:46]This means everything will cancel out except the first entry and the last one.

[00:11:51]So, the finite sum actually just consists of two parts.

[00:11:56]And obviously, this also helps us for the infinite sum.

[00:12:00]First of all, the measure of A1 can be written as the measure of this union.

[00:12:06]And since the entries in the union are pairwise disjoint, we can use the sigma additivity. So, we get an infinite sum of complex numbers that is absolutely convergent.

[00:12:17]Which means the value of the sum can be expressed as a limit of a finite sum.

[00:12:22]And now, the good thing is that we already calculated this finite sum.

[00:12:27]It's just a constant minus the measure of AN.

[00:12:32]And now, since this constant is exactly the same number as we have it on the left-hand side, this limit has to be equal to zero.

[00:12:41]And there we have it. This is exactly what we wanted to show.

[00:12:45]So, please remember this nice property of complex measures.

[00:12:50]Indeed, we will use that in the next videos where we will generalize the measures even more.

[00:12:56]For example, we will talk about measures that have values in a space of operators.

[00:13:02]So, I really hope I meet you there again, and I wish you a nice day.

[00:13:05]Bye-bye.

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