Spectral Theory

Hello and welcome to my complete video course about Spectral Theory consisting of 4 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:

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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 $\mu$.