Hello and welcome to my ongoing video course about Unbounded Operators, already consisting of 14 videos. This series explores important concepts and is presented in a logical order, with more content to come. Along with the videos, you’ll find helpful text explanations. PDF versions of the lessons are available for reference. If you have any questions, feel free to ask in the community forum. Let’s dive in!
Part 1 - Introduction and Definitions
Let’s start with the definition of operators.
With this you now know the definition of the imporant operators in this series. Let’s also state which topics we will cover. Some important bullet points are closed operators, adjoints, spectrum and selfadjoint operators. So in order to understand these things, let’s go more into the general theory of unbounded operators.
Part 2 - Examples
In this video, will first talk about the inverse of an operator, which is also defined for injective operators. However, in this case, the resulting inverse could be an unbounded operator even in the case the original operator was bounded. Moreover, we will look at examples for unbounded operators $ C([0,1]) \rightarrow C([0,1]) $.
Part 3 - Closed Operators
Now, we define a subset of operators. We are mostly interested in so-called closed operators because they carry some nice properties. Note that our bounded operators from the functional analysis series are always closed operators.
Part 4 - Closable Operators
Sometimes the reason that an operator is not closed is just given in the domain. So by increasing the domain, one could make the operator into a closed one. Such operators are then called closable.
Part 5 - Example
Now, we are ready for an explicit example for an operator on an infinite-dimensional space we already know, namely $ \ell^2(\mathbb{N}, \mathbb{C}) $.
Part 6 - Closed Graph Theorem
In this video, we will formulate and prove the so-called closed graph theorem. It roughly says that between Banach Spaces closed operators that are everywhere defined are necessarily bounded. It turns out that this theorem is heavily related to the open mapping theorem* from our functional analysis course.
Part 7 - Graph Norm and Closed Operators
In addition to the different possibilities to describe closed operators from the last videos, we can also use another one. This one needs the so-called graph norm and formulates the closedness of the operator with a completeness of the corresponding graph space.
Part 8 - Adjoint Operators
As one usually does it for bounded operators, one can also define so-called adjoints for unbounded operators. This happens in the Banach space setting and similarly in the Hilbert space setting. However, we really have to put the domains of the operators in the focus.
Part 9 - Extension and Restriction for Adjoints
Let’s talk about how the domain changes for the domain of the Banach space adjoint and the Hilbert space adjoint if we change the domain of the orignal operator.
Part 10 - Adjoint of Multiplication Operator
Let’s look at a special case of operators to see how we can calculate the adjoint operator.
Part 11 - Double Adjoint
We already know that the Hilbert space adjoint $T^\ast$ is well-defined if $T$ is densely defined. This also means that it is not clear if the adjoint of the adjoint $T^{\ast \ast}$ actually exists. We will discuss that and the properties we have for the double adjoint.
Part 12 - Kernel and Range for Adjoints
The operators $T$ and $T^\ast$ are connected via an orthogonal complement of their corresponing graphs. This results into a relation between kernel and range which one can easily remember. Moreover, we can also write the Hilbert spaces as a direct sum or, more precisely, as an orthogonal sum.
Part 13 - Self-adjoint and Symmetric Operators
In contrast to bounded linear operators, the term self-adjoint cannot just be defined by changing the action of the operator inside the inner product, meaning $ \langle y, Tx \rangle = \langle Ty, x \rangle $. This is simply because the domain of the adjoint can still be larger than the domain of the original operator. Therefore, we usually speak of symmetric operators and only use the term self-adjoint if the two domains actually coincide. Moreover, we will also define the notion essentially self-adjoint for closable operators $T$.
Part 14 - Properties of (Essentially) Self-Adjoint Operators
It turns out that self-adjoint operators are necessarily closed. So whenever we search for a self-adjoint extension of an operator, the closure should be the first thing to check. This explains why the term essentially self-adjoint is so important and, in fact, we are able to show some nice equivalences there.
Summary of the course Unbounded Operators
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