2
votes
2answers
38 views

Definition of continuous spectrum of a bounded operator

Let $T$ be a bounded operator acting on a Banach space $X$. The point spectrum $\sigma_p(T)$ is of $T$ is defined to be $$\sigma_p(T):=\{\lambda\in\mathbb C~|~T-\lambda\text{ has nonempty kernel}\}$$ ...
0
votes
1answer
20 views

Banach Algebras: Continuity of Inversion?

Context: This question is related to this thread: Spaces of Functions Given a topological space $X$ and a Banach algebra with unit $B$. Consider a continuous map $F:X\to B$ that is invertible ...
0
votes
2answers
52 views

Space of Functions: Characterizations of Positivity

Context The problem here is about the characterization of positivity for real or complex valued functions: $$\sigma(f)\geq 0\iff\sigma(f(x))\geq 0\text{ for all }x\in X\iff f(x)\geq 0\text{ for all ...
1
vote
1answer
48 views

Spectrum of a finite rank operator

If $ T\in B(H)$ is a finite rank operator, then there are orthonormal vectors $e_1,...,e_n$ and vectors $g_1,...,g_n$ such that $Th=\sum_{i=1}^n (h,e_i )g_i$, then we can easily see that $T$ is ...
1
vote
0answers
24 views

Helffer-Sjöstrand-Formula: Idea behind?

I have to present the Helffer-Sjöstrand-Formula. Now I'm wondering: Why does it include a factor $\chi(y\langle x\rangle^{-1})$ for some bump function $\chi$ and the chinese symbol ...
0
votes
0answers
38 views

Riemann Sphere: Holomorphic Functional Calculus

Why do we consider the holomorphic functional calculus on the Riemann sphere rather than the complex plane only? Is there a serious problem? Moreover isn't any curve encircling the spectrum ones ...
1
vote
1answer
66 views

“right shift” il $L^1$

Let $X=L^1(\mathbb{R})$ be the space of Lebesgue integrable functions $f:\mathbb{R}\rightarrow \mathbb{C}$ with the usual norm. Let $T\in B(X)$ be defined by $$(Tf)(t)= f(t+1)$$ I need to find the ...
1
vote
1answer
57 views

Operator: not closable!

Is there an operator between Banach spaces with the following properties: $$T:\mathcal{D}(T)\subseteq X\to Y:\text{ injective, dense range, continuously invertible, not closable!}$$ (Note that the ...
4
votes
1answer
84 views

What makes compact operators special?

I would like to understand why compact operators are considered so special to consider them as an extra class of operators. Over Hilbert spaces these (as far as I know) these are the ones with ...
1
vote
1answer
27 views

Inequality norms

Let $A$ be a bounded linear operator on a Banach space $X$. Can we show that for an arbitrary $n \in \mathbb{N}$ and $x \in X$ such that $\|x\|_X \geq 1$ we have that $$\|A^n x \| \leq \|Ax\|^n.$$ ...
2
votes
2answers
120 views

Resolvent Set: Definition

Given Banach spaces: $X,Y$ Consider a linear operator: $T:\mathcal{D}(T)\to Y$ (not necessarily bounded nor closed nor closable nor densely defined) Define for the shorthand the shifted operator: ...
0
votes
1answer
47 views

Positive Operator: Norm Estimate

In class we encountered the statement: $$H\geq C\mathrm{Id},C>0\Rightarrow\|\mathrm{e}^{-\beta H}\|<1,\beta>0$$ How does one prove this? Moreover what about the weakened version: $$H\geq ...
5
votes
0answers
113 views

renorm a Banach space to make an operator have spectral radius equal to norm

Let $X$ be an infinite-dimensional complex Banach space equipped with the norm $\lVert\cdot\rVert$, and let $T\in\mathcal{L}(X)$ a bounded linear operator on $X$. Let $r(T)$ denote the spectral ...
3
votes
0answers
46 views

operator on separable banach space whose spectrum and point spectrum is prescribed compact set

I am interested in obtaining the following paper: G. K. Kalisch, "On operators with large point spectrum," Scripta Math. 29 No. 3-4, (1973), 371-378. According to Ben Mathes, "Strictly Cyclic ...
1
vote
1answer
69 views

Resolvent lemma

I would like to proof a lemma that I am quite sure should be correct as I found it somewhere, I am writing a thesis about quantum walks and need this to get through an article. Let $X$ be Banach ...
0
votes
1answer
125 views

Image of a set under a mapping

I need to show that the image of the closed unit ball in $\mathbb{C}$, under the polynomial mapping $p(x) = (1-x)^2$ is the cardioid: ${re^{i\theta} : 0 \leq \theta < 2π, 0 ≤ r ≤ 2 + 2 ...
3
votes
1answer
56 views

Evaluating difficult spectrum

Can anyone see how to show the spectrum of the bounded linear operator $T$ on $l^1$ defined by $$T((\alpha_j)) = (\alpha_j - 2\alpha_{j+1} + \alpha_{j+2})$$ is the cardioid $$\{(r, θ) : 0 ≤ θ < 2π, ...
9
votes
1answer
293 views

An approximate eigenvalue for $ T \in B(X) $.

This is a problem from Conway’s Functional Analysis: Definition An approximate eigenvalue for $ T \in B(X) $ is a scalar $ \lambda $ such that there is a sequence of unit vectors $ x_{n} \in X $ ...
4
votes
2answers
98 views

$(\lambda-a)^{-1}$ as limits of 'polynomials'

For a unital $C^*$-algebra $\mathcal{A}$ the spectral permanence gives \begin{equation} \sigma_{\mathcal{B}}(a)=\sigma_{\mathcal{A}}(a) \end{equation} for any unital $C^*$-subalgebra $\mathcal{B}$. ...
2
votes
2answers
101 views

If $Lat(\mathcal{A})$ is trivial then $\mathcal{A}'$ consists of scalars.

This is related to Exercise 3 of Section 2.5 of Arveson's book on spectral theory. For those who are interested, we were asked to show the following $\mathcal{A}$ is a Banach *-algebra. ...
4
votes
1answer
192 views

$\sigma(x)$ has no hole in the algebra of polynomials

Let $A$ be a unital banach algebra generated by two elements $1$ and $x$. Then it seems $\sigma(x)$ cannot have holes. At least this is true in the case for disk algebras. This amounts to prove that ...
2
votes
2answers
172 views

Spectrum of an element in sub-algebra: $\sigma_A(b)\setminus \{0\}\subseteq \sigma_B(b) \setminus \{0\}$

Please help me to prove this:(or give me some references for this.) Thanks very much! Let $A$ be a (unital) algebra and $B\subset A$ a (unital) sub-algebra. Then for all $b\in B$: ...
42
votes
1answer
2k views

Example of a compact set that isn't the spectrum of an operator

This question is a follow-up to this recent question and related to that one. Is there an easy example of an (infinite-dimensional) Banach space $X$ and a non-empty compact set $K \subset ...