# Tagged Questions

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

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### Generator of an analytic semigroup with a compact resolvent --> pairwise conjugate eigenvalues?

I am reviewing a paper in which the authors claim that their operator has eigenvalues $\lambda$ that are either real or pairwise conjugate (meaning that if $\lambda$ is an eigenvalue, then also the ...
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### Lower bound on gap between consecutive eigenvalues on $L_2(\mathbb{R}^3)$

A similar version of this question was originally posted by me in the physics community, but it was suggested that I ask the mathematicians instead. So I have tried to strip off most of the physics ...
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### nth product of sequential matrices

$\forall n \in \mathbb{N}$, let: $$P_n = \left( \begin{matrix} a & 1-a \\ b_n & 1-b_n \end{matrix} \right).$$ Whereby $\{b_n\}_{n \in \mathbb{N}}$ is a monotonically increasing sequence of ...
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### A Question on Normal Operators

Let $T$ be a compact operator on a Hilbert space $H$. I want to prove the following: If there exists Orthonormal Basis from the eigen vectors of $T$, then $T$ is normal operator.
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### $\displaystyle\frac{\langle A x_{\min}, x_{\min}\rangle }{\langle x_{\min}, x_{\min}\rangle }=\lambda_1 \not = \lambda_2$? - Explanations

Related the question Min-Max Principle $\lambda_n = \inf_{X \in \Phi_n(V)} \{ \sup_{u \in X} \rho(u) \}$ - Explanations, I tried to complete the details of mickep's answer, but I have a little problem....
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### Question on spectral theorem for compact operators

I'm studying a proof of the spectral theorem for compact operators. The first part of it reads as follows: Let $X$ be an infinite dimensional inner product space and let $A: X \to X$ be a compact and ...
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### Operator commutes with spectral projection

Let $E$ be the spectral measure to an (unbounded) self-adjoint operator $A$. Is there a sufficient and necessary condition so that for a bounded interval $I$ we have $E_I A= AE_I$?
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### Ineqality involving operators

I need your help to solve the following problem: We define the operatorss $a$ and $b$ with domain $\mathcal{C}_0^{\infty}(\mathbb{R}^n)$ by: $a=\partial_x+\frac{1}{2}\partial_xV(x)$ (where $V$ in ...
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### Neumann vs Dirichlet eigenvalue problem - Intuition

What is the fundamental different between a Neumann eigenvalue problem and Dirichlet eigenvalue problem? I know that for DEP, we just fix the boundary (e.g. a drum), but what about the NEP. Now, ...
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### Min-Max Principle $\lambda_n = \inf_{X \in \Phi_n(V)} \{ \sup_{u \in X} \rho(u) \}$ - Explanations

In general, I am generally someone who like to solve questions with visual support. With this idea in mind, is it someone could explain to me, with a visual support if possible, how is it possible to ...
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The Langevin equation is given by: $dq=pdt,\ dp=-\nabla V(q) dt-pdt+\sqrt{2}dW$ I want to know what does the variables $p,\ q,\ t,\ V,\ W$ represent . Can someone help me ? Thanks.
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### Integrating a Linear Operator $A:H\longrightarrow H$ (Matrix)

I am trying to prove a functional analysis proposition, but I got stuck. I have to integrate a matrix. In my proof I use the following matrix: Let $A$ be a self-adjoint matrix on $H=\mathbb{C}^n$ ...
I recently came across this in my studies of functional analysis in C* algebras which got me stuck: For a completely positive map between C* algebras $\phi : A \to B$ we are to prove these two ...
Let $H_1$ and $H_2$ be two Hilbert spaces and $U \in B(H_1,H_2)$ be unitary. Assume that $A\in B(H_2)$ and $B \in B(H_1)$ satisfy $UB = AU$. How can I prove that $sp(A) = sp(B)$ and \$sp_p(A) = sp_p(B)?...