Spectral theory is a study of generalized notions of eigenvalues and eigenvectors.

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As $\lambda \to \infty$, $||R_{\lambda}(T)|| \to 0$

"Let $T \in B(X,X)$. Prove that $||R_{\lambda}(T)|| \to 0$ as $\lambda \to \infty$." We have that $R_{\lambda}(T)x=(T-\lambda I)^{-1}(x)$. The problem doesn't specify but the book says that they will ...
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Bounded operator with empty spectrum acting on a non-complete normed space

It is well-known that a bounded operator $T$ acting on a complex Banach space $X$ has non-empty spectrum. Suppose that $X$ is a non-complete normed space. Is it possible to find a bounded operator ...
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Showing self adjointness

$\pi:$ $Lx=\sum_{j=0}^{n}(p_{n-j}x^{(j)})^{(j)}$,$\,\,$ $x^{(j)}(a)=x^{(j)}(b)=0,\, j=0,1,...,n-1.$ where $p_{n-j}\in C^{n-j}[a,b]$ are real and $p_0(t)\neq0$ on $[a,b]$. I want to show that the ...
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Show $\sigma(T)=\sigma{(\overline{T^{*}})}$

Let $T \in B(H)$ be a bounded operator. Is $\sigma(T)=\sigma{(\overline{T^{*}})}$ true for $T$? $\textbf{TRY-}$ I have proved it is true for normal operator but could not do it for bounded ...
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Spectral decomposition - generalized eigenspaces

Suppose we have a real $n\times n$ matrix $A$ with spectrum $\sigma(A)=\{\lambda_1,\lambda_2\}$ (with $\lambda_1, \lambda_2$ discrete). Also, we have $alg\,mult\, \lambda_i\neq geo\, mult \,\lambda_i ...
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Spectral Measures: Subspace Characterization

Disclaimer This thread is related to: Spectral Measures: Subspace Decomposition It is meant to record. See: Answer own Question It is written as question. Have fun! :) Question Given a Hilbert ...
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Spectrum of Multiplication Operator $T$ in $C[0,1]$

"Let $X=C[0,1]$ and $v \in X$ be a fixed function. Let $T$ be the multiplication operator by $v$, i.e. $Tx(t)=v(t)x(t)$. Find the spectrum of $T$." This is an exercise from a PDF of notes I found ...
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Spectral Measures: Subspace Decomposition

Attention This thread has been split into this one and: Spectral Measures: Subspace Characterization Problem Given a Hilbert space $\mathcal{H}$ and let the Lebesgue measure be $\lambda$. ...
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Faber-Krahn inequality for domain in Z^d with nearest-neighbor connections

In $\mathbb{R}^d$ there is a theorem that if you are looking for the first Dirichlet eigenvalue $\lambda_1$ of a domain $D \subset \mathbb{R}^d$ with a given volume $V$, then $\lambda_1$ will be ...
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21 views

Proof that the spectrum of an element of a Banach Algebra is non-empty

I don't see why the line indicated with ***** in the following proof is true in the proof that spectrum of an element of a Banach Algebra is non-empty (Arveson, p.27) : For every $\lambda_0 \not\in ...
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Spectrum of Unitary Operators

Let $T_1$ and $T_2$ be two unitary operators. Is it true that the spectrum of $T_1+T_2$ is contained in the closed disc of radius 2?
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Spectral methods with linear programming

Is it possible to model and solve some fundamental spectral methods (say Singular-Value Decomposition) with (Integer?) Linear Programming? Update: say you want to do SVD. Can you model it as a ...
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143 views

Spectrum of a nilpotent operator

Let $X$ be a Banach space and $A:X\to X$ be a bounded operator such that $A^n=0$ for some $n\in \mathbb{N}$. Is the spectrum of $A$ finite, countable ?
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pertubation of an operator by orthogonal projection

Let $G$ be an operator with compact resolvent on Hilbert space $H$ such that $\ker G$ is different from $\{0\}$. Let $P$ be the orthogonal projection onto $\ker G$, and let $G_{0} = G+P$ My question ...
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26 views

Reference for Spectral theory

I'm studying Bernard Aupetit: A Primer on Spectral Theory but the textbook we are using is a little bit heavy going for me. Is there a "gentler" book to learn about these things? Thank you.
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310 views

Affine transform of multivariate gaussian

If $X_1, \ldots, X_n$ are iid $N(0,1)$ or in other words $\mathbf{X}=(X_1, \ldots, X_n)$ is distributed $N(\mathbf{0}, \mathbf{I})$, then $A\mathbf{X}+\mu$ is distributed $N(\mu, AA^t)$. Showing that ...
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Eigenfunctions of a second derivative operator

Consider the operator $L :=\frac{-d^2}{dy^2}+ \alpha^2 - K(y)$ on the space of functions $f(y) $ on $H^2(-a,a) \cap H_0^1(-a,a)$. Here $K(y)$ is an even function and $\alpha >0$ is a positive real ...
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Eigenvalue of Compact Operators

To prove that the set of eigenvectors of a compact linear operator on a normed space $X$ is countable, I read "it suffices to show that for every real $k > 0$ the set of all eigenvalues whose ...
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72 views

Normal Operators: Spectrum vs. Numerical Range

Disclaimer: As I realized in the comments that this works for normal operators I decided to modify this question. Besides, I got the proof now - thanks to T.A.E.! Prove that for normal operators the ...
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wave propagation - Memory variables and weak formulation

Working on the acoustic wave equation (not in terms of displacement $u$ but in displacement potential $X$ s.t. $\rho u = \nabla X$), I'm trying to get a proper viscoacoustic model by the mean of ...
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48 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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24 views

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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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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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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Basic Criterion on Selfadjointness

How to prove the following in a neat subjective way: $$A\text{ symmetric}:\quad\mathcal{R}(A\pm\imath)=\mathcal{H}\implies A^*=A$$
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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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Eigenvectors of inverse complex matrix

For a non-singular matrix, its pretty straightforward to prove that $\lambda$ is eigenvalue of $A$ if and only if $\frac{1}{\lambda}$ is eigenvalue of $A^{-1}$. Let $A$ be a non-singular matrix, $x$ ...
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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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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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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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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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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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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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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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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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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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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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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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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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What is iterative integral operator?

I don't know what is iterated integral operator, and why $K^{(3)}$ is square-integrable implies $K$ is compact. Can anyone help?
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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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Spectral Measures: Support vs. Spectrum

Given a complex Hilbert space $\mathcal{H}$. Consider a Borel spectral measure: $$E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H})$$ and its associated normal operator: ...
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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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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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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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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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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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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 $\overline{B}$?