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

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What defines a Pseudospectral Method?

I'm trying to understand pseudospectral methods in the context of solving PDEs. However, I can't seem to find a solid definition for this. Is it simply a general term for solving a problem in parts: ...
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Simple example of operator spectral analysis.

I'm studying on Rudin - Functional Analysis the part related to Banach algebra Bounded/Unbounded operator. Specifically i've studied the part of Banach algebra (where the concept of spectrum is ...
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Spectral theorem for momentum operator

I'm trying to apply the spectral theorem to the explicit example of the momentum operator: \begin{equation} p:= i\frac{d}{dx} \end{equation} on the domain $D=\{\psi\in H^1(0,2\pi)\ |\ \psi(0)=\psi(2\...
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Proof of Spectral Theorem

There are a number of results called Spectral Theorems. This question deals with the Linear Algebra result on normal operators, which has the self-adjoint case as a particular case. In class, we saw ...
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Looking for a general approach to proving compactness of linear operators.

Preparing for my exam in functional analysis, I often have to prove that certain explicitly given operators are compact. I now have a decent amount of operators of which I can prove the compactness, ...
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Some claims about spectra

I see the following equalities used sometimes, but couldn't find proofs. How is it done? $$\overline{Ran(L-\lambda\mathbb{1})}^{\perp}=ker(L^*-\overline{\lambda}\mathbb{1})$$ and $$\lambda\in\sigma_p(...
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Sobolev spaces and symmetric operators

I am slightly confused with regards to the way one obtains self - adjoint differential operators in spectral theory. The aspect that I'd like to understand better is the following: Suppose we are ...
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$C^k$ one-parameter family of metrics

Consider a smooth Riemannian manifold $M$ and a $C^k$ one-parameter family of Riemannian metrics $g_t$ on $M$. Here $k$ could be any integer, $k$ could be infinity, when the one-parameter family $g_t$ ...
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Do you need to know a lot of (regular?) graph theory to get into spectral graph theory?

Do you need to know a lot of (regular?) graph theory to get into spectral graph theory? What are the prerequisites?
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Inverse of a dissipative operator

In the spectral theory of linear operators it is often helpful in the development of proofs for many results to proceed with the inverse of a (boundedly invertible) dissipative operator. For example, ...
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The subgroup of $GL_{n}(\mathbb{R})$ inside $B(1, 1)$ is $\lbrace I \rbrace$

Let $G \subset GL_{n}(\mathbb{R})$ be a subgroup such that $G \subset B$ where $$B = \lbrace M \in \mathcal{M}_{n}(\mathbb{R}), ||M-I|| < 1\rbrace $$ Let $g \in G$. I'm asked to show: The ...
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Show that $\lim \frac{f_\epsilon -1}{\epsilon}$ is purely imaginary if each $|f_\epsilon| = 1$

For $0< \epsilon < 1$, suppose we have complex numbers $f_{\epsilon}$ such that each $|f_{\epsilon}| =1$ and $$ \lim_{\epsilon \to 0} \frac{f_\epsilon - 1}{\epsilon} := a $$ exists. Prove ...
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Using Fubini's Theorem in Contour Integrals proof

I have a few questions regarding the following proof: Suppose that $\mathcal{A}$ is a unital Banach algebra, and that $g$ is a complex-valued function which is analytic on $\sigma(a)$ while $f$ is a ...
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55 views

Proof involving invertible elements of Banach algebra

I want to prove for a unital Banach algebra $\mathcal{A}$, it follows that if $\|a-b \| < \frac{1}{\|a^{-1} \|}$ then $b \in \mathcal{A}^{-1}$ (where $\mathcal{A}^{-1}$ is the subset of invertible ...
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Proposed proof of operator theory result

Hi I am interested in checking my proposed solution to the following problem in Operator Theory: Please give me hints as to how to improve the proposed proof rather than the full correct solution. ...
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49 views

Is this possible to decompose 2 arbitrary unitary matrices as: $U_{1}=ADB^{\dagger},U_{2}=BDA^{\dagger}$?

I guess this statement is true, but I can't prove it: "For any 2 arbitrary unitary matrices $U_{1}$ and $U_{2}$, we can always decompose them into $U_{1}=ADB^{\dagger},U_{2}=BDA^{\dagger}$, where $A$ ...
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Spectrum of compact operators on an infinite dimensional normed space

The question is as follows: Let $T:X \to X$ be a compact linear operator on a normed space. If the $dim X= \infty$ then show that $0 \in \sigma(T)$. My attempt: Suppose on the contrary that $0 \...
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About finding the common diagonalizing similarity transformation.

Say I have $2k$ matrices $M_{a_1b_1}$, $M_{a_2b_2}$,..,$M_{a_kb_k}$ and their negatives. Here $M_{a_ib_i}$ is such that it has $0$ everywhere except that it has $1$ at $(a_i,b_i)$ and $(b_i,a_i)$ ...
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Is the spectral radius of a matrix a convex norm of it?

I am wondering if the spectral radius of a matrix is may be some kind of a norm ($l_{\infty}$-norm?) of it and if that is convex. Any pointers to related ideas would be helpful too.
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A question about minimizing the $\lambda_{max}$ over a set of diagonal perturbations

Say I have an off-diagonal symmetric $0,1,-1$ entry matrix $B$ and a set of $2k$ diagonal matrices, $D_{11}, D_{12}, D_{21}, D_{22},..,D_{k1},D_{k2}$. (if it helps you can assume that $(1)$ all the ...
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Given a spectrum, what can we know about its function?

Say we are given a well-behaved function $f(t)$ (either discrete or continuous) and are able to compute its spectrum $F(k)$ using the (discrete) Fourier transform. Then say we lose $f(t)$ and know ...
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significance and importance of spectral theorem

I have started recently started Operator Theory and have been introduced to the Spectral Mapping Theorem: If $a \in \mathcal{A}$, where $\mathcal{A}$ is a unital Banach Algebra and $f \in \text{hol}(a)...
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Eigenvalues of Moore–Penrose Pseudo-Inverse of a Symmetric Matrix

I was wondering if there is any bound or inequality for the eigenvalues of Moore–Penrose pseudo-inverse of a real $n\times n$ symmetric matrix $A$ in terms of eigenvalues of $A$, namely $\lambda_i$'s ?...
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Spectral measures, supports, compact operators

Let $H$ be a Hilbert space, $K:H\rightarrow H$ a compact self-adjoint operator. The spectral measure of $K$ wrt $v\in H$ is uniquely determined by $$\langle K^n v,v \rangle=\int_{\mathbb{R}} x^n d\...
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Continuous spectrum is a subset of point spectrum

I have to prove that the continuous spectrum $\sigma_c(T)$ is a subset of the point spectrum $\sigma_p(T)$. I started off by supposing that there is some spectral value $\lambda$ such that $\lambda \...
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51 views

Real eigenvalues, similar symmetric matrix

I know that symmetric matrices have real eigenvalues, and that non-symmetric matrices that are similar to symmetric matrices must also have real eigenvalues, but is the converse true? That is, if a ...
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75 views

Eigenvalues of matrix sums

Under what conditions are the eigenvalues of $A+B$ equal to the eigenvalues of $A$ plus the eigenvalues of $B$, where $A,B$ are square matrices? From searching, it seems that the condition is that $...
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Matrix spectral space vs entry space in symmetric matrices

Is is known that the space of symmetric matrices $\mathbb{R}_{sym}^{n \times n}$ has $\binom{n}{2}$ dimensions. And according to the spectral theorem every symmetric matrix $A \in \mathbb{R}_{sym}^{...
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Absolutely Continuous Spectrum and Norm of Resolvent

Problem. Let $H$ be a Hilbert space, and let $A:H\rightarrow H$ be a bounded, linear operator. Suppose $A$ has purely absolutely continuous spectrum and $\sigma_{ac}(A)=[0,1]$. Find the set of $D\...
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Spectral resolution of multiplication operator

Kosaku YOSIDA claims in his book "Functional Analysis" that it is easy to see that the multiplication operator $Hx(t) = tx(t)$ in $L^2(-\infty,+\infty)$ admits the spectral resolution $H = \int_{...
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A property of normal matrices

The statement I am trying to understand is that if $A$ is nilpotent and normal, then $A$ is the zero matrix. Here, I believe we take $A$ to be a square matrix over $\mathbb{C}$. Is this related to the ...
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Operators whose spectrum has a finite number of connected component

Assume that $H$ is a separable Hilbert space. Let $Q$ be the set of all operators$T \in B(H)$ such that the spectrum of $T$ has a finite number of connected component. Is $Q$ a subvector space ...
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Construction of a Strongly Regular Graph which has regular Neighbourhood graphs in all iteration.

Notation and Definition: $G$ is a Strongly Regular Graph (not complete or a cycle) and is denoted by $\mathrm{SRG}(n,r, \lambda, \mu)$ if it has the following properties: Every two adjacent ...
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Trace evaluation via complex analysis

We are given $U$, $V$ unitary matrices of size $N \times N$ whose spectral decomposition is known (in my specific problem, $N=4$, and $U$, $V$ are matrices with real coefficients but we can keep it ...
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spectrum of weighted translation semigroup

My Banach space is $\mathcal X=\rm{L}^1(\mathbb R_+)$. I would like to know the spectrum of $A\phi(x)=-\phi'(x)-f(x)\phi(x)$ on $D(A) = \{g\in\mathcal X,\ g\text{ absolutely continuous}, g(0)=0\text{ ...
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How does the product of sets of complex numbers give a character?

I'm working through this "Introduction to Banach Algebras" and just after proposition 8.2 they say: If $A$ is a commutative Banach algebra, $a\in A$ and $\phi\in M(A)$, then $\phi(a)\in sp(a)$. ...
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In an inner product space, if the matrix is symmetric, is an eigenspace necessarily orthogonal to the range space?

Say I have 3 distinct eigenvalues for a symmetric matrix. By the Spectral Theorem, the three eigenspaces are mutually orthogonal. But, if I just wanted to compute the first eigenspace, ...
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Dense Operators: Spectrum

This thread is Q&A. Given a Banach space $E$. Consider closed operators: $$T:\mathcal{D}(T)\subseteq E\to E:\quad T=\overline{T}$$ Then for the domain: $$\sigma(T)\neq\mathbb{C}\implies\...
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Spectrum of weighted shift operator

The Banach space considered is the following: $(l^{\infty}(\mathbb{Z}), \|\cdot\|_{*})$ with $\|x\|_{*}=\|(...,x_{-1},x_{0},x_{1},...)\|_{*}=|x_{0}|+\text{sup}_{k\neq 0}|x_{k}|$. Define $A$, an ...
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Confusion between spectral radius of matrix and spectral radius of the operator

The adjacency matrix $A(G)$ of an infinite undirected graph $G$ is considered as a bounded self-adjoint linear operator $A$ on the Hilbert Space $l^2(G)$ (last section of https://en.wikipedia.org/wiki/...
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Are they true these generalizations from matrices to operators about functional calculus?

Motivation: If we have some real function $f$ defined on an interval $I$ and $D=\operatorname{diag}(\lambda_1,\ldots,\lambda_n)$ is a diagonal matrix such that $\lambda_i \in I$ for all $1\leqslant i \...
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Why isn't every element of the spectrum an eigenvalue? (Where is the error in my proof?)

My book defines the spectrum like this: Let $H$ be a complex Hilbert space, let $I \in B(H)$ be the identity operator and let $T \in B(H)$. The spectrum of $T$, denoted $\sigma(T)$, is defined ...
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How can I tell that my matrix is nilpotent?

I just computed a 15x15 matrix by hand :( It is not upper triangular as I hoped it would be. But my computations agree with what's offered in the student solution. My question is: the solution ...
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spectral measure and normal operators range

Let $N$ be a normal operator with spectral measure $E$. We want to show that if $N=\int z\ dE(z)$ and $ε>0$, then $\operatorname{ran} E(\{z∶ |z|>ε\})⊆\operatorname{ran}N$. Is this true? Let $\...
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Projections: Orthogonality

Given a unital C*-algebra $1\in\mathcal{A}$. Consider projections: $$P^2=P=P^*\quad P'^2=P'=P'^*$$ Order them by: $$P\perp P':\iff\sigma(\Sigma P)\leq1\quad(\Sigma P:=P+P')$$ Then equivalently: ...
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Spectral theorem for representations proof.

Let $H$ be a separable Hilbert space, and $U$ a unitary representation of $\mathbb{Z}^d$ on $H$. Let $\chi_m$ be the characters of the Torus $T^d$, and $m$ the Haar measure on $T^d$. I would like to ...
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what is a spectral function?

My knowledge in spectral theory is very limited, but lately I heard talking about the spectral function of an operator and how it's important. By curiosity I tried to look for a definition and a ...
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65 views

Proof that a random measure with orthogonal increments is a measure

Let me first state what I mean by a random measure with orthogonal increments. Definition: A random measure with orthogonal increments $Z$ is a collection $\left(Z(B): B \in \mathcal{B}_{(-\pi,\pi]...
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How exactly does one define the “spectral measure” of an operator?

I am seeing kind of different definitions of "spectral measure" at different places and its not clear to me as to what is the universal idea. It would be great to get some "standard" definition. In ...
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Subspace Perturbation

For two positive semidefinite matrices $A,B\in\mathbb{R}^{n\times n}$, with dominant $r$ dimensional subspaces $U,V\in\mathbb{R}^{n\times r}$ and eigenvalues $\Sigma_A, \Sigma_B$, what can we say ...