Spectral theory is the study of generalized notions of eigenvalues and eigenvectors for linear operators in Banach spaces.

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Compactness of $(x_1,x_2,…)\mapsto(0,x_1,x_2/2…)$

I read that the linear operator in the Hilbert space $\ell_2$ defined by $(x_1,x_2,...,x_n,...)\mapsto(0,x_1,x_2/2,...,x_n/n,...)$ is compact. I wanted to prove it by proving that the image of the ...
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42 views

Finite number of eigenvalues outside circle $\|\lambda\|>\delta>0$

I know that, for any $\delta>0$, a compact operator $A$ defined on a linear variety of a Banach space has only a finite number of linearly independent eigenvectors corresponding to the eigenvalues ...
2
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1answer
64 views

Spectrum of integral operator

Given $g\in C^1([0,1]\times[0,1])$, consider the operator $$Tu(x) = \int_0^1 g(x,t) u(t) dt$$ defined on $u\in C([0,1])$. Discuss the spectrum of T. My attempt: First I can show that $T$ is ...
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63 views

Spectral Decomposition of Function-of-Normal-Operator

In Arveson's book A Short Course on Spectral Theory, on page 64 (section on spectral measures) the author mentions the usual spectral decomposition of a normal operator $N$ as $$N=\sum_{\lambda \in ...
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98 views

Exercise about an operator (adjoint and spectrum)

Let $y\in c_0$ and define the operator from $l^2 \rightarrow l^2$ as the following $$T\bigg(\sum x_n e_n\bigg) \mapsto \sum y_n x_n e_n.$$ I have shown that the operator is continuous, compact and ...
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244 views

Spectrum of self-adjoint operator on Hilbert space real

My book says that a self-adjoint bounded linear operator $A:H\to H$ on a complex Hilbert (not sure if separability is needed) space has a real spectrum. I guess that the key is in the fact that any ...
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59 views

Spectrum of Verschiebung

I read that the shift operator $A:\ell_2\to\ell_2$, $(x_1,x_2,x_3,...)\mapsto(0,x_1,x_2,...)$ contains $0$ in its spectrum, and that's clear to me. It is also clear to me that it has no eigenvalue. ...
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61 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 ...
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59 views

Eigenfunctions and spectrum of $T:H \to H^*$ where $H$ is a Hilbert space

Let $H$ be a Hilbert space with dual $H^*$. Suppose $T:H \to H^*$ is a linear bounded symmetric operator. (We probably don't want to identify $H$ with $H^*$). Can we talk about the ...
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spectral decomposition of a bivariate function

Now I have a function $f=f(x,y)$, smooth and symmetric(i.e. $f(x,y)=f(y,x)$ everywhere), with arguments defined on a compact set: $(x,y)\in[0,1]\times[0,1]$. I'd wish to know if $f$ can be expanded ...
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Is a bounded Borel function of a normal operator normal

I am playing around with Borel functional calculus to try to understand it, and made the following argument: Let $T\in B(H)$ (bounded operator on Hilbert space) be normal. Let $f\in C(\sigma(T)) $ ...
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149 views

Spectrum of an integral operator.

For any $f\in C([0,1],\mathbb{R})$ set $$ Tf(x) = \int_0^1 [\min\{x,y\}\cdot f(y)]dy. $$ I have just proved that $T$ is a compact operator from $C([0,1],\mathbb{R})$ into itself. I would like to know ...
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43 views

Invariant functions on product of ergodic systems is determined by eigenfunctions?

Given an ergodic measure-preserving system $(X,\mathcal{B},\mu,T)$, the product system $(X\times X, T \times T, \mu \times \mu )$ need not be ergodic, in other words: It may have non-trivial invariant ...
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1answer
54 views

Positive Linear Transformations: What good for?

Positivity is a concept appearing quite frequently in the study of algebras and its related spectral theory. Positive elements naturally give rise to an ordering and therefore allows to construct ...
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225 views

Riesz Functional Calculus vs. Holomorphic Functional Calculus

"Functional calculus" is a word used to describe the practice of taking some functions or formulas defined on complex numbers, and apply them in some way to certain kinds of operators, despite that ...
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Function Spaces: 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 ...
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151 views

Normal Operator: Empty Spectrum

Given a Hilbert space $\mathcal{H}$. For normal operators: $$N^*N=NN^*:\quad\sigma(N)\neq\varnothing$$ How can I check this?
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29 views

How to derive the spectral projection operator in finite element method?

For a compact self-adjoint operator $T$, the spectral operator can be defined as follows: $$E(\lambda) = \frac{1}{2\pi i}\int_{\Gamma}(z-T)^{-1}dz$$. For a finite dimensional approximate operator ...
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39 views

Question about the Averson's proof of the bicommutant theorem.

In the Averson's proof of the bicommutant theorem is proved that, if $A$ is a self-adjoint algebra of operators with trivial null space and $T \in A''$, for every $\epsilon>0$, $n=1,2..$ and every ...
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Question about a passage in the Bicommutant Theorem's proof.

In the Averson's book, in the proof of the Von Neumann's Bicommutant theorem there is this passage: ($A $ is a self-adjoint algebra of operators in $L(H)$) "Let $\xi_1$ be an element of the Hilbert ...
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A characterization of a certain family of matrices in terms of another matrix.

Consider a real matrix $A$ of dimension $n \times n$. Assume $k \leq n$ is given. I am looking for ways to describe the following set of matrices in terms of properties of $A$. $\mathcal{S}(A) = \{B ...
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30 views

Commutant of a set of operators and norm topology.

In the references I have it's remarked that the commutant $S'$ of a set $S$ in $B(H)$, where $H$ is a Hilbert space, is closed in the weak operator topology. And this is true because if ...
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Why is the Gelfand transform injective?

There is a theorem that proves that if $A$ is a commutative C*-Algebra, the Gelfand map is an isometric *-isomorphism of $A$ onto $\hat{A}$ i.e. the spectrum of $A$. (Theorem 1.1 in Averson's "An ...
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118 views

Continuity of the spectral radius

Let $M \in \mathbb{R}^{n\times n}$ be a nonnegative irreducible matrix with strictly positive entries on its main diagonal. Then $M$ is primitive and by the Perron-Frobenius Theorem we know that the ...
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Resolvent and spectrum of a self-adjoint extension

In this paper, they give the resolvent, spectrum, and eigenfunctions of the self-adjoint extension of the Laplacian on a rectangle that corresponds to a delta potential at an arbitrary point (items ...
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82 views

Finding the smallest max eigenvalues for related matrices?

While messing around with a spectral approach to a graph coloring question, I happened upon a type of problem I hadn't seen before. Suppose you have two symmetric $n$ x $n$ matrices in the form ...
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92 views

Spectrum of $Tu=\int^1_{-1} (1-|x-y|)u(y)dy $

Consider the operator $$ Tu(x)=\int^1_{-1} (1-|x-y|)u(y)dy $$ We want to find the spectrum of $T$. The kernel is certainly bounded and so this operator is Hilbert-Schmidt, so $T$ is compact. We ...
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42 views

Just a translation issue.

I'm italian and my professor of spectral theory wrote the list to the arguments to be studied in italian. The problem is that all the literature is in english and often the translation are a bit ...
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Prove that for a bounded self adjoint operator, $\langle Tx,x\rangle \geq 0$ is equivalent to $\sigma(T)\subset [0,\infty)$

Prove that for a bounded self adjoint operator, the following are equivalent: A: $\langle Tx,x\rangle \geq 0$ B: $\sigma(T)\subset [0,\infty)$ What I have said so far: Since $T$ is self adjoint, ...
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Spectral decomposition of a Hilbert space

I have this proof, but I don't understand how they do the spectral decomposition of $H$ into $H_-$and $H_+$? Please help me. Thank you.
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Normal Matrices Unitarily Diagonazible

Disclaimer: This thread is just meant to record (Q&A). Are the unitarily diagonazible matrices precisely the normal ones? Surely, every normal matrix has an eigenbasis. Now I got asked by a ...
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36 views

Non-Unitarily Diagonalizable Matrices

When searching for matrices that are similar to a diagonal matrix but not in a unitary way then a first hint would be to exclude the normal ones. But apart from that is there a general form for such ...
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Prove that spec$(f(A)) = f$(spec$(A)).$

Can someone please explain this proof to me? Thanks! Let $A \in \mathbb{C}^n$ and let $f(x)$ be a polynomial. Prove that spec$(f(A)) = f$(spec$(A))$ (where if $S \subseteq \mathbb{C},$ $f(S) :=$ $\{ ...
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1answer
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Consider a symmetric matrix $X$ with eigendecomposition $X=UVU^T$, how to call $\sum_{v_{k,k}>0}v_{k,k}u_ku_k^T$?

Consider a symmetric matrix $X$ with eigendecomposition $X=UVU^T$ How do people call $\sum_{v_{k,k}>0}v_{k,k}u_ku_k^T$? Sum of positive components of $X$? The positive semi definite part of $X$? ...
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102 views

Skew-adjoint differential operator $B$ with spectrum $\sigma(B)=i(-\infty,-1]$

Consider the Hilbert space $X=L^{2}\left(\mathbb{R}^n\right)$ and the Schrödinger operator $A=i\Delta$ defined on the domain $D(A)=H^2(\mathbb{R}^n)$. It is known that the spectrum of $A$ is ...
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Spectra of operators on different spaces

Can the same operator when defined on two different spaces have different spectra? For example and operator defined on $C_0$ and on $\ell_2$?
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Resolvent set of extension

Suppose X is an incomplete normed space and X' is its completion. Let T be a bounded linear operator on T and T' the linear extension of T to X'. How do I prove that the resolvent set p(T) of T ...
2
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121 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 ...
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Definition of exponential for operators

if I have a self-adjoint operator $T:D(T) \rightarrow L^2$, then I define its unitary exponential operator by $$e^{iT}(f) := \lim_{k \rightarrow \infty} e^{iT_{k}}(f),$$ where $T_k(f):=\frac{1}{2} ...
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Relation between $\epsilon$-pseudospectrum of operators

If $H$ is a Hilbert space and $\sigma_{\epsilon}(T)$ denotes the space of all $\epsilon$-pseudospectrum of the operator $T$ and $S, T\in B(H)$ be such that $TS=ST=0$, why ...
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estimate the upper bound for the number of the nodes of length $l$

I have a graph $G=(V,E)$, where $V$ is a vertex set, and $E$ is an arc set. Now given a vertex $i\in V$, I want to find the (small) upper bound for the number of the nodes in $V$ that has the length ...
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303 views

Example: Algebraic Multiplicity vs Geometric Multiplicity

Is there a simple example of a matrix having an eigenvalue whose geometric multiplicity is strictly smaller than its algebraic multiplicity?
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Prove that if $T=T^*$ and $\sigma(T)=\{\lambda\}$, then $T=\lambda I$

Show that if $T$ is a self adjoint linear operator on a Hilbert space such that the spectrum contains a single point $\lambda$, then $T=\lambda I$. Then, show this is false if $T$ is not self ...
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Ways to calculate the spectrum of an operator

Friends, I am learning some very basic stuff of spectral theory and kind of lost, in some sense. I am trying to find ways to compute the spectra of different operators, when they work and don't work. ...
2
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1answer
101 views

Selfadjoint Restrictions of Legendre Operator $-\frac{d}{dx}(1-x^{2})\frac{d}{dx}$

Problem: Let $Lf =-((1-x^{2})f')'$ be the Legendre differential operator defined on the domain $\mathcal{D}(L)$ consisting of twice absolutely continuous functions on $(-1,1)$ for which $f, Lf \in ...
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Is $AA^*$ and $A^*A$ self-adjoint?

if I have a densely defined closed linear operator $A$ and $A^* = -A$(same domain also closed). Is this sufficient that $AA^*$ and $A^*A$ are proper self-adjoint operators, assuming that we can also ...
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65 views

Domain of densely-defined second derivative operator, and its factorization

Let $$-d_x^2: \{f \in L^2[0,1];f \in AC^1[0,1] , f(0)=f(1)\} \rightarrow L^2[0,1]$$ be the second derivative operator. Here $AC^1[0,1]$ is the space of functions whose first derivative is absolutely ...
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Decomposition of a positive semidefinite self-adjoint operator?

If I have a positive semi-definite self-adjoint operator $H:D(H) \rightarrow L^2$, is it true that there is always a decomposition $H=A^* A$ available? If this is true, what can we say about the ...
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Proving that a Sturm-Liouville problem is in the limit-point/-circle case

I would like to understand techniques anybody is able to detail to me on how one may actually prove that a particular Sturm-Liouville (S-L) problem, i.e., of the form \begin{equation} ...
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225 views

Spectral theory - continuous spectrum

imagine that I have some differential operator $D$ that is defined on an interval $[a,b]$. Now, assume that we take the boundary conditions in such a way that this operator is self-adjoint. Then, I ...