All Questions
Tagged with operator-theory spectral-theory
1,079
questions
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Is the resolvent of a local operator local?
Let $A$ denote a bounded linear operator on the Hilbert space $l^2(\mathbb{Z})$. We call $A$ a local operator if and only if there exists a $C \geq 0$ such that $\langle e_x | A | e_y \rangle = 0 $ if ...
- 133
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Basis of eigenfunctions and spectrum
I have a linear, closed and densely defined operator $A$ defined on a Hilbert space $H$. I prove that the point spectrum consists of discrete eigenvalues, and that the eigenfunctions of $A$ and of $A^*...
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On self-adjoint extensions and multiplicity of eigenvalues
I hope you can help me with the following question.
Let $B$ be a densely defined closed symmetric operator on a infinite-dimensional separable complex Hilbert space $\mathcal{H}$ with deficiency ...
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spectral theory of pseudo-differential operators of class $S^m$
I would be very grateful if you could give me the titles of books that deal with the spectral theory of pseudo-differential operators of class $S^m$
Thank you very much.
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Thus $MV - VM = V^2$. So the spectrum of $V^2$ is $\sigma(V^2) = (\sigma(V))^2=0$. Why??
I am curious if this statement holds (it doesn't make much sense to me, but it was written in solutions in this form):
$\sigma(A)=0\implies\sigma(A^2)=(\sigma(A))^2=0.$
Can anybody explain to me why ...
user1315539
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Spectrum of $T(\cdots , x_{-2}, x_{-1}, (x_{0}), x_{1}, x_{2},\cdots )=(\cdots ,x_{-2}, x_{-1}, 4x_{0}, (3x_{-1}), x_{0}, x_{1}, x_{2},\cdots )$
Find spectrum, eigenvalues and eigenvectors of operator $T:\ell^2(\mathbb{Z})\to \ell^2(\mathbb{Z})$, defined by
$$T(\cdots , x_{-2}, x_{-1}, (x_{0}), x_{1}, x_{2},\cdots )=(\cdots ,x_{-2}, x_{-1}, ...
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Proof that the hausdorff distance $d_H( \sigma (A), \sigma(B)) \le r(A-B)$ for commuting operators $A,B$
I am supposed to prove that for two bounded commuting operators $A, B$ on a banach space, the hausdorff distance of the spectra $d_H(\sigma(A),\sigma(B))$ is less than or equal to the spectral radius ...
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spectrum of translation operator
Let $\displaystyle A: c \rightarrow c, (Ax)(n) = \frac{x(n+2)}{n^2}$. The task is to find point spectrum, continuous spectrum, and residual spectrum of $A$ and of adjoint operator $A^*$.
$c$ is Banach ...
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Characterization of normal Fredholm Operators
I'm working on the following problem, but I'm getting stuck on the last part. Here is the problem:
Let $T \in \mathcal{B}(H)$ be normal. Prove $T$ is Fredholm if and only if $0$ isn't a limit point ...
- 1,399
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About the spectrum family of the multiplication operator
Let $<,>$ be the inner product of $L^2(\mathbb{R})$.
For a measurable function $F:\mathbb{R}^d\to\mathbb{R}$, we define
a multiplication operator $M_F:L^2(\mathbb{R}^d)\to L^2(\mathbb{R}^d)$, ...
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Does this integral operator between Banach spaces have a non-trivial kernel?
In a question which was recently asked, the goal was to (dis)prove that the set of functions which satisfy a certain equation is trivial. There is already a very neat solution there, but I came up ...
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Definition of closure of operator
Let $\mathcal{H}$ be a Hilbert space and $A:\mathrm{dom}(A)\to\mathcal{H}$ be an unbounded operator. I have seen the following definition of the closure in a lecture:
$$\mathrm{dom}(\overline{A})=\{x\...
- 2,876
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Applying Spectral Mapping Theorem to determine if $f(T)$ is compact
Given a self-adjoint, compact operator $T$ and a continuous $f:\sigma(T)\to\mathbb{R}$, I'm trying to determine the conditions under which $f(T)$ is also compact. I know of the Spectral Mapping ...
- 529
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Spectral Permanence Remark in Murphy's C*-algebras
In Murphy's $C^{*}$-algebras book, he states theorem $2.1.11$ which is that if $\mathfrak{B} \subset \mathfrak{A}$ are $C^{*}$-algebras with $\mathfrak{A}$ unital such that $1_{\mathfrak{A}} \in \...
- 1,399
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uniqueness of the inversion to Riemann-Stieltjes integral equation
I believe that if, for a Riemann-Stieltjes integral with $h(s)$ of bounded variation,
$$ \int_0^1 s^\alpha dh(s) = 0 \qquad\text{for any }\alpha\in(\alpha_0,\alpha_1) ,
\tag{1}\label{eq1}$$
then $h$ ...
