# Tagged Questions

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

49 views

### Prove or disprove: if $\lambda$ is an eigenvalue of square matrix T then so is $\bar\lambda$

I think this is true. I know that complex eigenvalues come out as conjugate pairs, but can't think of a formal proof. any lead?
31 views

### The composition of a dissipative operator and a positive opeartor is dissipative?

Let the real Hilbert space $H^1(\Omega)$ endowed with its usual inner product, denoted by $\langle ., . \rangle$ and let $A : H^1(\Omega) \rightarrow H^1(\Omega)$ be a dissipative ...
27 views

### 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 ...
37 views

### 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.
31 views

### 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 ...
14 views

### 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 ...
82 views
+50

### How linear map transform the unit ball?

Let $f:\mathbb{R}^n \to \mathbb{R^n}$ be a linear application, we suppose that $f$ is symmetric ($\langle f(x),y\rangle=\langle x, f(y)\rangle$), without using spectral theorem how we can see that $f$ ...
44 views

19 views

167 views

### $\displaystyle\frac{\langle A x_{\min}, x_{\min}\rangle }{\langle x_{\min}, x_{\min}\rangle }=\lambda_1 \not = \lambda_2$? - 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....
69 views

### If $\|p-q\|<{1\over2}$ then $p$ is homotopy equivalent to $q$

Let $A$ be a $C^*$ algebra, $p,q \in A$ projections, such that $\|p-q\|< {1 \over 2}$. Show that $p$ homotopy equivalent to $q$. Proof. Let $a_t=(1-t)p+tq$, then $a_t$ is positive (self-adjoint ...
19 views

27 views

### 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 ...
26 views

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.
57 views

### $H$ self-adjoint with mass gap, $P \ge 0,\Omega \in D(P)$, $H + \lambda P$ self-adjoint $\implies$ for $\lambda$ small, $H+ \lambda P$ has gap?

Suppose $H$ is a self-adjoint operator on a Hilbert space having a simple isolated least eigenvalue $0$ with gap $1$ ( $H\Omega = 0$, $\Vert \Omega\Vert = 1$ ), $P$ is a non-negative symmetric ...
191 views

### 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 ...
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 ...