Hello and welcome to my complete video course about Spectral Theory consisting of 12 videos. Alongside the videos, I provide helpful text explanations. To test your knowledge, take the quizzes, work through the included exercises, and refer to the PDF versions of the lessons if needed. If you have any questions, feel free to ask in the community forum. Now, without further ado, let’s get started!
Part 1 - Complex Measures
In this video series, we will consider different integrals to represent a self-adjoint operator on a Hilbert space. In order to understand these integrals, we first have to talk about complex measures. They are essentially ordinary measures where we also allow complex numbers as values. So we extend the concept of a volume into the complex realm as well. This is needed since we will consider complex Hilbert spaces later on. But first, let’s show that a complex measure is also continuous at the empty set:
Part 2 - Integration of Complex Measures
We already know what complex measures are, but in order to make the useful we definitely need an integral with respect to a complex measure. We could do a whole construction of such an integral as we have done in Measure Theory for ordinary measures. However, this is not necessary because we can just use this theory for a more general definition as well. First, we need to define the total variation measure to connect the complex measure $\mu$ to an ordinary measure $| \mu |$. From now on, ordinary measures are just called positive measures to emphasize the codomain for them. Since $| \mu |$ is such a positive measure, we can just integrate with respect to it and use a polar decomposition to extend this integral to complex integral with respect to $\mu$.
Part 3 - Operator-Valued Measures
We will see soon that the spectral theorem requires integrals that produce an operator on a Hilbert space. Therefore, it makes sense to define measures with values in the space of bounded linear operators on a Hilbert space. We can simply do that by using the complex measures that such maps induce.
Part 4 - Integral is Well-Defined
Let’s finish the discussion from the last video and show that the integral with respect to an operator-valued measure is well-defined. For this, we will prove a general result about Hilbert spaces and so-called sesquilinear forms. These are just maps on $X \times X$ that are almost bilinear, just that linearity in one component is just a conjugate-linear. The inner product itself is the best example of such a sesquilinear form. Now it turns out the bounded sesquilinear form are in 1:1 correspondence to bounded linear operators on $X$. This fact is enough to show the well-definedness of the integral from the last video.
Part 5 - Spectral Measures
Although the concept of operator-valued measures works for the general class of bounded linear operators on Hilbert Spaces, we are actually only interested in particular class: the orthogonal projections. Recall that self-adjoint idempotent operators are exactly the orthogonal projections. Then we speak of a spectral measure $E$ if we map the Borel sets of $\mathbb{R}$ into these orthogonal projections with the additional condition $E(\mathbb{R}) = I$. We can also prove that the $\sigma$-additivity is even given with respect to the strong operator topology, which is more than we have just for a general operator-valued measure where only have the $\sigma$-additivity with respect to the weak operator topology
Other videos related to this topic:
Part 6 - Property of Spectral Measure
Spectral measures are special operator-valued measure but they have much more useful properties. Since all outcomes are orthogonal projections, they have to be compatible in some sense. In particular, an intersection of Borel sets translates into the product of projections.
Part 7 - Spectral Integral
Now, we are ready to define the spectral integral by looking at the total variation measures of the complex spectral measures. It turns out that we can always integrate bounded functions.
Part 8 - Theorem on Spectral Integrals
From the last video, we know that the spectral integral for bounded functions exists. We can even extend that since they only need to be bounded on a support of the spectral measure. This makes the notation of the spectral integral more versatile but does not change much overall. More importantly, we can prove a lot of properties of the spectral integral. For example, how it changes under building adjoints and products.
Part 9 - Spectral Theorem for Bounded Self-Adjoint Operators
Now we are finally ready to formulate the famouse spectral theorem. We focus on the one for self-adjoint bounded operators where the corresponding spectral measure has a compact support. If you want to describe unbounded operators, you have to drop this additional requirement but it also gets more technical. Indeed, we already have the first part of the spectral theorem, namely that for every such spectral measure we can construct a particular self-adjoint operator by using the identity map $ t \mapsto t $. It also works the other way around: for every self-adjoint operator, there is such a representation with special spectral measure. In fact, the spectral measure can be constructed by Stone’s formula, which uses the resolvent of the operator.
Part 10 - Spectrum and Numerical Range
Let’s go back to basics and explain why even speak of spectral measures and spectral theorem. It all comes down to the spectrum of a bounded linear operator $T$. It just consists of the complex numbers where $T-\lambda I$ is not invertible. It turns out that numerical range can be used to describe the spectrum roughly.
Other videos related to this topic:
- Hilbert Spaces 19 | Properties of the Adjoint
- Functional Analysis 27 | Bounded Inverse Theorem and Example
- Functional Analysis 30 | Properties of the Spectrum
- Functional Analysis 31 | Spectral Radius
Part 11 - Spectrum of Self-Adjoint Operators
The result from the last video can directly be applied to self-adjoint operators. We get that the spectrum is always a subset of the real numbers. This is quite important because we want to define a spectral measure for self-adjoint operators later. Moreover, we can also give an estimate of the operator norm of the resolvent.
Other videos related to this topic:
Part 12 - Herglotz Theorem
Connections to other courses
Summary of the course Spectral Theory
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