Some context tangential to the question: I'm currently preparing for a final on an introductory course in functional analysis, which consists of a short presentation. Looking through these notes, I have found an elementary consequence of the spectral theorem for compact and self adjoint operators which seems interesting to investigate.
The author gives a version of functional calculus for these type of operators (namely Corollary 3.10 and the subsequent observation, which includes equation 3.11). Paraphrasing (and focusing only on the case over $\mathbb{R}$ in which the Hilbert space is separable, which is the one I'm interested in),
Theorem. Let $A \in \mathscr{L}(H)$ be a compact self adjoint operator on a separable Hilbert space. Given an orthonormal basis of eigenvectors $\{e_n\}_{n \geq 1}$ with corresponding eigenvalues $\{\alpha_n\}_{n \geq 1} \subset \mathbb{R}$. There exists a Banach algebra continuous homomorphism
$$
f \in B(\{\alpha_n\}_{n \geq 1},\mathbb{R}) \mapsto f(A) \in \mathscr{L}(H)
$$
via
$$
f(A)(x) := \sum_{n \geq 1}f(\alpha_n)\langle e_n,x \rangle e_n,
$$
which sends $\mathsf{1}$ to $id_{H}$ and $id_{\{\alpha_n\}_{n \geq 1}}$ to $A$.
In particular, we have
Remark: given $A$ as above, and $ z \not \in \{0\} \cup \{\alpha_n\}_{n \geq 1}$, then
$$
(A-zI)^{-1}(x) = \sum_{n \geq 1}\frac{1}{\alpha_n – z}\langle e_n,x \rangle e_n.
$$
I have a couple of questions motivated by the former results:
-
It is my impression that the theorem should be generalizable to continuous real valued functions by precomposing with the restriction $f \mapsto f|_{\{\alpha_n\}_n}$, which if I am not mistaken, is a continuous Banach algebra homomorphism. Is this the case?
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What are the relations between the spectrum of $A$ and $f(A)$? By a direct verification it seems that we have $f(\sigma(A)) \subset \sigma(f(A))$.
-
What are (if any) some sufficient conditions to guarantee that $f(A)$ is self adjoint and/or compact?
-
Are there any available references which treat functional calculus in the specific case of compact self adjoint operators?
Best Answer