Questions tagged [functional-calculus]

Question 1

I am trying to fill in the details of the following (already discussed in these posts for example and also in Rudin, Reed and Simon, etc) fact: If $\lambda \in \sigma(A)$ is isolated for $A$ self-...

Question 2

I'm currently trying to understand the action of a unitary operator on $L^2(\mathbb{R})$ (as a complex Hilbert space). For instance, let's assume we have an operator $P\colon D(P) \to L^2(\mathbb{R}), ...

Question 3

Let $A$ be an unitial $C^*$-algebra, $X \subseteq \mathbb{C}$ a compact set and $f : X \to \mathbb{C}$ a continuous function. Prove that for any $\epsilon > 0$ there is a $\delta > 0$ such that ...

Question 4

Let $H$ be a Hilbert space. Let $A:D(A)\to H$ be a self-adjoint operator. When can we say $A^2:D(A^2) \to H$ is self-adjoint? Is it enough to define $D(A^2)=\{u\in D(A) | Au\in D(A)\}$? And can you ...

Question 5

The following is from Hsing/Eubank's Theoretical Foundations of Functional Data Analysis, in the proof of Theorem 11.1.1. I have a compact operator $\mathcal{T}$ from the Sobolev space $\mathbb{W}_q[0,...

Question 6

Let $T \in B(H)$ be positive and injective, so that $\log(T) \in B(H)$ makes sense. One can define $T^{iz} := \exp(iz\log(T))$. I am trying to verify that $z \mapsto T^{iz}$ is analytic for $z \in \{...

Question 7

The question is, as the title already says: Why do we introduce the continuous functional calculus for self-adjoint operators? To be more specific, why self-adjoint operators? It is introduced (in ...

Question 8

Recently I learned about the Borel measurable functional calculus. Now I am learning about spectral measures. The motivation behind spectral measure was that for characteristic functions, $\Phi(\chi_A)...

Question 9

Context: For some function $f\in\mathcal{M}(\mathbb C)$ (meromorphic function on $\mathbb C$), I am interested in linear operators $T_f$ that act on functions of the form $g_a:x\mapsto \exp(ax)$ in ...

Question 10

I am trying to understand the proof of the Borel measurable functional calculus, especially the construction of the map $\tilde{\Phi}:B_b(\sigma(T)) \rightarrow B(H)$. For that reason I'll try to ...

Question 11

I am learning about measurable functional calculus at the moment. I have learned that given a operator $T \in B(H)$ that is self adjoint, there exists a measurable functional calculus $\overline{\Phi}:...

Question 12

Title: When does $\{1, a, a^*\}$ being linearly dependent imply that $a$ is normal and $\sigma(a)$ lies on a line? I'm working through a problem in spectral theory involving elements of a unital $C^*$...

Question 13

I want to make sense to functions of diagonal operators $\mathrm{diag}[a_1, \dots, a_n, \dots]$ in $\ell_2$. Such operators are normal, not self-adjoint. So, generally, functional calculi may not ...

Question 14

I have to proof that if $X$ is a C$^*$-algebra and $x\in X$ is invertible, there exist an hermitean $y\in X$ and an $u\in X^+$ such that $x=yu$. If $x$ is normal, I found this decompositian using ...

Question 15

Our teaching assistant claimed the following. If you are in a let's say commutative $C^*$-algebras $X,Y$ and a linear $*$-morphism $\varphi:X\to Y$, let $x\in X$ and let $f\in C(\text{sp} (x),\mathbb{...

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Questions tagged [functional-calculus]

Isolated point of spectrum of a self-adjoint operator and spectral projectors

How to derive the action of the unitary operator $e^{iP}$ on $L^2(\mathbb{R})$?

Almost commuting elements and functional calculus

$A$ self-adjoint then $A^2$ self-adjoint

Showing equality of norms for two self-adjoint operators

Verifying Holomorphicity of an Operator Valued Map

Why do we introduce the continuous functional calculus for self-adjoint operators?

Connection between Measurable Functional Calculus and spectral measures

Meromorphic symbol pseudodifferential operator

Extending the continuous functional calculus to Borel functional calculus

Spectral measure and measurable functional calculus

When is $\{1, a, a\}$ linearly dependent in a C-algebra, and what does it imply about a?

Which functional calculi may be extended to normal operators

How to decompose invertible elements into positive and unitary elements in C$^*$-algebras?

Does functional calculus commute with -morphisms in $C^$-algebras?

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