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

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spectrum of compact operators

Let $\phi\in\ell^\infty(\mathbb{N})$. For $p\in[1,\infty]$, define $$M_\phi:\ell^p\to\ell^p,\quad f\mapsto\phi f.$$ Use spectral theory to show that, if $M_\phi$ is compact, then $\phi\in c_0$. Here ...
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Proving Stone's Formula for Constructively obtaining the Spectral Measure for $A=A^\star$

Let $A$ be a bounded or unbounded selfadjoint linear operator on a complex Hilbert space $H$ with spectral representation $A=\int_{\sigma}\lambda \, dE(\lambda)$ given by the Spectral Theorem for ...
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36 views

Where is a mistake in my proof concerning the spectrum of elements of a unital Banach algebra?

I was going to prove: Let $A$ be a unital Banach algebra. Then $$\sigma(a) = \{\tau(a) \mid \tau \in \Omega (A)\}$$ and I started the following argument: Let $\lambda \in \sigma (a)$ and let ...
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144 views

Norm of the Resolvent of a Self-Adjoint Operator

Let $\mathcal H$ be a Hilbert space and $\mathcal L$ is a self- adjoint operator with a discrete spectrum $\{\lambda_{j}\}$. I read that it is well known that for, $\lambda \notin \sigma(\mathcal ...
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Spectrum of Self-Adjoint Operators

This is an exercise (5-i) from here. It has two parts as follows. For a self-adjoint operator $A$. Show that $A \geq k I, \ k \in \mathbb R$ if and only if $\lambda \geq k$ for all $\lambda$ ...
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How to use the Spectral Theorem to Derive $L^{2}(\mathbb{R})$ Fourier Transform Theory

Without using Fourier transforms, how do I derive the spectral measure for $A=\frac{1}{i}\frac{d}{dt}$ on the domain $\mathcal{D}(A)$ consisting of absolutely continuous functions $f\in ...
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33 views

A question about spectral measure

The following is a part of a theorem of Takesaki's Operator theory: Let $T$ be an positive operator. Suppose $T = \int_0^{\|T\|} \lambda \, de(\lambda)$ is the spectral measure of $T$. Also put ...
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Modifications to the definition of graph Laplacian?

Many people have defined various definitions for graph Laplacian. For example see here [1]. What is common between various definitions of Laplacian that makes all of them ``Laplacian"? For example ...
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52 views

A question about spectral theorem

The following is a discussion about spectral theorem of Folland's Harmonic analysis page 18. Suppose $A$ is a unital commutative C*- subalgebra of $B(H)$ and $u,v\in H$. Put $\Sigma = \sigma(A)$ . ...
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105 views

Spectral convergence of coefficients of a Fourier series

I have seen claims that if a smooth function $f(x)$ is represented by its Fourier series, $f(x)=\sum_{n=-\infty}^\infty a_ne^{i(nt)}$, then as $|n|\rightarrow\infty$, then $|a_n|\rightarrow 0$ ...
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105 views

Measurable functional calculus

I am struggeling with this exercise: Let $T \in L(H)$ be a self-adjoint operator and $\Psi$ be a measurable (Borel) functional calculus on the spectrum of $T$. For a Borel set $\Delta \subset \sigma ...
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Is there a cyclic vector for $-\frac{d^{2}}{dx^{2}}$ on $L^{2}[0,2\pi]$ with periodic conditions?

Let $\mathcal{H}=L^{2}[0,2\pi]$, and let $L=-\frac{d^{2}}{dx^{2}}$ on the domain $\mathcal{D}(L)$ consisting of twice absolutely continuous functions $f$ on $[0,2\pi]$ with $f''\in\mathcal{H}$ and ...
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64 views

How to show $e^{-x}$ is a cyclic vector for $-\frac{d^{2}}{dx^{2}}$ in $L^{2}[0,\infty)$?

Let $\mathcal{H}=L^{2}[0,\infty)$. How can one easily show that $e^{-x}$ is a cyclic vector under the $C^{\star}$ subalgebra of operators on $\mathcal{L}(H)$ generated by all resolvents $(L-\lambda ...
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50 views

Cheeger constant for $S^2$

I want to calculate explicitly Cheeger constant for $S^2$, but I haven't found any sources or examples. I'm using this definition $$h(M)=\inf_A\{\frac{vol_{n-1}(\partial A)}{vol_n{(A)}}:vol_n(A)\leq ...
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Selfadjointness of the differential operator in a singular potential

The free Dirac operator is the differential operator of the following form $$ T_0 = i \alpha \nabla + \beta,$$ where $\alpha$ and $\beta$ are Hermitian $4 \times 4$ matrices, and $T_0$ is selfadjoint ...
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Mysterious Commutation in an Unbounded Operators Argument (Compact Resolvent) — Is there a typo?

While reading A.S. Markus, Introduction to the Spectral Theory of Polynomial Operator Pencils, Chapter 1, Section 4, I came across a passage that I really do not understand, and I am trying to see if ...
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Showing T intertwines $D_T$ and $D_{T^*}$ using Spectral Theorem

Suppose $T$ is a contraction on a Hilbert space $H$ (separable, if you wish). $D_T=(I-T^*T)^{1/2}$ and $D_{T^*}=(I-TT^*)^{1/2}$. I want to show that $TD_T=D_{T^*}T$. I had done this before using a ...
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93 views

How to show a Borel Operator Measure dilates to a Spectral Measure?

Does anyone know a simple proof of the following theorem stating that a positive Borel operator measure $P$ on $\mathbb{R}$ can be written as $V^{\star}EV$ for a Borel spectral measure $E$? ...
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141 views

Spectral Measures: Spectral Spaces (I)

Problem Given a Hilbert space $\mathcal{H}$. Let the Lebesgue measure be $\lambda$. Consider a Borel spectral measure: $$E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H})$$ Denote its ...
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A Direct Proof of Representation Theorem for Positive Harmonic Functions in the Half Plane?

Does anyone know a direct proof of this representation theorem for non-negative harmonic functions in the half-plane that doesn't appeal to a similar result in the unit disk? Also, does anyone who ...
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55 views

spectral projection of an element in a C*-algebra

I'm studying Takesaki's Operator theory and I preferred "spectral projection "in the page 43 of this book while he didn't speak about it before. I searched it, but I could not find it. Please explain ...
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On calculating spectral projections

Consider following operator from this paper; Let $h$ be any function in $L^1$ relative to the measure $g(w)dw$ and $K\in\mathbb{C}$ Consider the linear operator $B$ on $L^1$ defined by $$(Bh)(x) = ...
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Eigenvalues of tridiagonal matrix

on page 13 of the paper here there is a proof in theorem 4 that all eigenvalues of this tridiagonal matrix, which has strictly positive entries down the subdiagonals, are simple. Unfortunately, I ...
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Numerical range of closure of operator

Let $B$ be an unbounded densely defined and closable operator. If $\mathcal{N}(B)$ is the numerical range, what can be said about the numerical range of its closure $\overline{B}$?
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Multiplication operator on $L^2$ and spectral theorem.

Let's consider the multiplication operator by the independent variable in $L^2(\mu)$, where $\mu$ is a borel regular measure on $\mathbb{C}$: $Mf(z)=zf(z)$. I want to show that if $\phi$ is a borel ...
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Spectrum of multiplication operator by the independent variable in $L^2$

If $\mu$ is a regular Borel measure on $\mathbb{C}$ with compact support $K$, define $N_\mu$ on $L^2(\mu)$ by $N_\mu f=zf$ (the multiplication by the indipendent variable). An exercise in "Conway" ...
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Simple proof that $\|p(A)\|\le \sup_{|z|\le 1}|p(z)|$ for polynomials $p$ and $\|A\| \le 1$.

Let $\mathcal{H}$ be a complex Hilbert space, and let $A$ be a bounded operator linear operator on $\mathcal{H}$ with $\|A\| \le 1$. It is known that $\|p(A)\|\le \sup_{|z|=1}|p(z)|$ for all complex ...
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Spectral measure and commutativity.

I want to prove that if $A\in B(H)$ and $N\in B(H)$ is a normal operator, and $AE(\Delta)=E(\Delta)A$, where $E$ is the spectral measure given by $N$ and $\Delta$ is a Borel subset of $\sigma(N)$, ...
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Eigenvalues of Hankel matrices

Let $\mathbf{A}$ be a $4-$ dimensional symmetric matrix with real entries, whose elements are given as \begin{equation} \mathbf{A} = \left( \begin{array}{cccc} a & b & c & d \\ b & c ...
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Some doubts concerning spectral theory.

Probably I'm saying something wrong (that's why the conclusions are strange) so please correct me! There is the continuous functional calculus for a normal element $N$ of a C*-Algebra. This means ...
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The spectrum of a product of operators

Suppose $A,B\in\mathcal{B}(\mathcal{H})$, where $\mathcal{H}$ is an infinite dimensional Hilbert space. In general, we know that there is no relationship between $\sigma(AB)$ and $\sigma(A)$ and ...
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A question about the article 'You can't hear the shape of a drum'

I have read the article http://www.math.upenn.edu/~kazdan/425S11/Drum-Gordon-Webb.pdf, , where Gordon and Webb describe in a simple a way the contruction of a pair of isospectral but non isometric ...
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Stability conditions for a non-negative AR(k) model

Given the sequence $\vec{x}_{n} = [A\vec{x}_{n-1}]_+$ where $[.]_+$ denotes the ramp function, i.e., $[.]_+ = x$ for $x>0$ and $0$ otherwise. $A$ is a $k$-by-$k$ matrix given by $$ A= ...
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Kac's question 'Can one hear the shape of a drum' and Sunada method, a clarification

I'm reading the article http://www.math.upenn.edu/~kazdan/425S11/Drum-Gordon-Webb.pdf, , where Gordon and Webb describe in a simple a way the contruction of a pair of isospectral but non isometric ...
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A Representation of $C(X)$ is a positive map.

I quote this excerpt from Conway: "A representation $\rho:C(X) \rightarrow \mathcal{B(\mathcal{H}})$ is a $\ast$-homomorphism with $\rho(1)=1$. Also, $\|\rho\|=1$. If $f\in C(X)_+$, then $f=g^2$ ...
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an exercise about the spectrum of an element in Banach algebra.

An exercise of Banach algebra, section of spectrum has wanted the proof of this statement: Let $A$ be a Banach algebra and $x\in A$. Show that for every open set $U$ in $\mathbb{C}$ that contains ...
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Representations and mutually singular measures

I'm finding some difficulties with an exercise from Conway and I ask for some help in understanding it: "Let X be a compact space and let $\{\mu_n\}$ be a sequence of measures in X. For each $n$ let ...
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Why is the spectrum usually defined for operators between Banach spaces?

The spectrum of a linear operator $L: \mathcal{D}(L) \rightarrow \mathcal{X} $ is generally defined for $\mathcal{X}$ a Banach space (as seen for example wikipedia on link above, or spectral ...
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Has the degree to which a partial eigensystem of a large sparse matrix approximates the complete eigensystem been determined?

Does anyone know of any studies or results regarding the degree of approximation or the error in estimating the complete spectrum of a large sparse matrix by means of its first $n$ eigenvalues and ...
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In Hilbert-space theory, is there a name for an operator “erasing” a projection?

Let $L$ be a self-adjoint operator with discrete spectrum $S=\{\lambda_1 < \lambda_2 < \dots \}$ on a Hilbert space $H$ such that the spectral theorem holds, i.e. for any $F \in H$ we have the ...
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Necessary and sufficient conditions for when spectral radius equals the largest singular value.

One well known fact about matrix norms is the following: If $\lambda_1\geq \dots\geq \lambda_n$ are eigenvalues of a square matrix $A$, then: $$\frac{1}{||A^{-1}||} \leq |\lambda|\leq ||A||$$ If we ...
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Dense invariant domain stable under resolvent?

I have thought about the following problem: Let $A_1\dots A_n$ a family of (unbounded) essentially selfadjoint operators on some Hilbert space $\mathcal{H}$ and $\Phi\subset\mathcal{H}$ the maximal ...
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How Many Negative Eigenvalues of $-\frac{d^{2}}{dx^{2}}$ on $[0,L]$?

What is the maximum number of eigenvalues $\lambda < 0$ for the trigonometric problems?: $$ \begin{array}{c} -\frac{d^{2}f}{dx^{2}}=\lambda f,\\ ...
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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 $$\sigma_p(T):=\{\lambda\in\mathbb C~|~T-\lambda\text{ has nonempty kernel}\}$$ ...
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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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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 ...
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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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61 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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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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216 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 ...