# Tagged Questions

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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### 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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### 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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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 $D(A) = \{g\in\mathcal X,\ g\text{ absolutely continuous}, g(0)=0\text{ ... 2answers 31 views ### How does the product of sets of complex numbers give a character? I'm working through this "Introduction to Banach Algebras" and just after proposition 8.2 they say: If$A$is a commutative Banach algebra,$a\in A$and$\phi\in M(A)$, then$\phi(a)\in sp(a)$. ... 2answers 52 views ### In an inner product space, if the matrix is symmetric, is an eigenspace necessarily orthogonal to the range space? Say I have 3 distinct eigenvalues for a symmetric matrix. By the Spectral Theorem, the three eigenspaces are mutually orthogonal. But, if I just wanted to compute the first eigenspace, ... 1answer 27 views ### Dense Operators: Spectrum This thread is Q&A. Given a Banach space$E$. Consider closed operators: $$T:\mathcal{D}(T)\subseteq E\to E:\quad T=\overline{T}$$ Then for the domain: $$\sigma(T)\neq\mathbb{C}\implies\... 1answer 148 views ### Spectrum of weighted shift operator The Banach space considered is the following: (l^{\infty}(\mathbb{Z}), \|\cdot\|_{*}) with \|x\|_{*}=\|(...,x_{-1},x_{0},x_{1},...)\|_{*}=|x_{0}|+\text{sup}_{k\neq 0}|x_{k}|. Define A, an ... 2answers 102 views ### Confusion between spectral radius of matrix and spectral radius of the operator The adjacency matrix A(G) of an infinite undirected graph G is considered as a bounded self-adjoint linear operator A on the Hilbert Space l^2(G) (last section of https://en.wikipedia.org/wiki/... 1answer 25 views ### Are they true these generalizations from matrices to operators about functional calculus? Motivation: If we have some real function f defined on an interval I and D=\operatorname{diag}(\lambda_1,\ldots,\lambda_n) is a diagonal matrix such that \lambda_i \in I for all 1\leqslant i \... 4answers 552 views ### Why isn't every element of the spectrum an eigenvalue? (Where is the error in my proof?) My book defines the spectrum like this: Let H be a complex Hilbert space, let I \in B(H) be the identity operator and let T \in B(H). The spectrum of T, denoted \sigma(T), is defined ... 2answers 136 views ### How can I tell that my matrix is nilpotent? I just computed a 15x15 matrix by hand :( It is not upper triangular as I hoped it would be. But my computations agree with what's offered in the student solution. My question is: the solution ... 1answer 33 views ### spectral measure and normal operators range Let N be a normal operator with spectral measure E. We want to show that if N=\int z\ dE(z) and ε>0, then \operatorname{ran} E(\{z∶ |z|>ε\})⊆\operatorname{ran}N. Is this true? Let \... 2answers 54 views ### Projections: Orthogonality Given a unital C*-algebra 1\in\mathcal{A}. Consider projections:$$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: ... 2answers 146 views ### 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 ... 1answer 77 views ### 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 ... 1answer 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]...
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 ...