In algebra, I learned that if $\lambda$ is an eigenvalue of a linear operator $T$, I can have
\begin{equation}
Tx = \lambda x
\tag{1}
\end{equation}
for some $x\neq 0$, which is equivalent to $\lambda I-T$ not being invertible.
In functional analysis, it is said that if $\lambda$ is an element of a spectrum of the linear operator $T$, then $\lambda I – T$ is not invertible. However, my Professor never mentioned $(1)$.
Is the definition/concept in functional analysis the same as $(1)$ in linear algebra? Can I use $(1)$ in functional analysis too? Does it depend on which spaces we are in?
For example, suppose $\lambda$ is in the spectrum of $T$, where $T$ is a linear operator on $E$, a Banach space. I want to show $\lambda^n$ is in the spectrum of $T^n$. Would this problem is equivalent to showing if $\lambda$ is an eigenvalue of a linear operator $T$, then $\lambda^n$ is an eigenvalue of $T^n$?
Thank you.
Best Answer
Spectral theory in infinite-dimensional spaces is quite a bit more complicated than in the finite-dimensional case. In particular, we have to distinguish between the spectrum $\sigma(A)$ of an operator and its eigenvalues. Let $A$ be a linear operator on a Banach space $X$ over the scalar field $C$. We have $$ \sigma(A) = \{ \lambda \in C: (\lambda I - A) \text{ does not have a bounded inverse} \}. $$ An eigenvalue $\lambda$ of $A$ is a value such that there exists a nonzero eigenvector $x \in X$ such that $$ A x = \lambda x, $$ or equivalently, $\operatorname{ker}(\lambda I - A) \neq \emptyset$. We then call $\dim \operatorname{ker}(\lambda I - A)$ the geometric multiplicity of the eigenvalue $\lambda$.
An eigenvalue is always in the spectrum, as you can see from the definition, but not every element of the spectrum is an eigenvalue in general.
In increasing order of "complicatedness", we could say:
This list could of course be refined with more specific conditions. I'm still studying the theory myself and will go back and add details as I learn about them. (Suggestions are welcome.) If you want to learn more, I found some relatively digestible lecture notes by E. Kowalski, ETH Zürich.