1
vote
3answers
52 views

Basis of eigenfunctions in Banach space

I have a question about proving the existence of a basis of eigenfunctions. Assume we have a compact operator $L:\mathcal{B}\rightarrow\mathcal{B}$, where $\mathcal{B}$ is a Banach space of analytic ...
1
vote
0answers
30 views

Eigenvalues that are functions

Let us have the Laplacian on a compact manifold $M$. Suppose I have some equation of the form $$-\Delta u(x) = f(x)u(x).$$ If $f \equiv c$ were a constant, this would be an eigenvalue problem ...
1
vote
2answers
68 views

Eigenvalues of symmetric elliptic operators

As stated in one of my previous questions already, I have not had so much exposure to theoretical linear algebra. This time, I'm reading a theorem and proof from PDE Evans, 2nd edition, pages 335-356. ...
0
votes
1answer
37 views

Prove that for a bounded self adjoint operator, $\langle Tx,x\rangle \geq 0$ is equivalent to $\sigma(T)\subset [0,\infty)$

Prove that for a bounded self adjoint operator, the following are equivalent: A: $\langle Tx,x\rangle \geq 0$ B: $\sigma(T)\subset [0,\infty)$ What I have said so far: Since $T$ is self adjoint, ...
0
votes
0answers
48 views

Infinite Dimensional Vector Space Equivalence of Positive Matrix

What is the corresponding linear operator on an infinite dimensional vector space, say a Banach space or Hilbert space, to the nonnegative matrix on a finite dimensional vector space? What is the ...
2
votes
0answers
60 views

Reference for simplicity of the principal eigenvalue of the Laplacian

i'm currently searching for a proper reference or proof to see that the first eigenvalue $\lambda \in \mathbb{R}$ of \begin{equation*} - \Delta u = \lambda u \text{ in } \Omega, \\ u \in ...
2
votes
1answer
49 views

Question about compact operator

So here is my question, Let $H$ be a Hilbert-space and $K:H\rightarrow H$ a compact operator. I know that if $K$ is self adjoint, then it has one eigenvalue $\mu$ such that $|\mu|=||K||$. Can some ...
2
votes
1answer
21 views

Estimation with an orthonormalbasis in some finite dimensional subspace of $L_2(\Omega)$

I'm currently trying to understand a step of a proof of the following estimation: $\displaystyle \sum_{j=2}^{N} \left(\left< \varphi_1-R_N\varphi_1, \varphi_{j,N} \right>_{L_2}\right)^2 \ ...
5
votes
2answers
202 views

Generalization of the Jordan form for infinite matrices

Under what conditions is it the case that for a matrix $M$ whose rows and columns are indexed by a countably infinite set $S$ one has a Hamel basis consisting of generalized eigenvectors (i.e. $v \in ...
0
votes
0answers
33 views

Eigenvectors of Laplace operator on a particular functiona space

Exercise 8.15 in [1] is: "Show that $\Delta u=\lambda u$ has no solutions of polynomial growth if $\lambda > 0$, but does have such solutions if $\lambda < 0$." How should I make sense of this? ...
2
votes
1answer
59 views

Establishing an inequality between the $2^{nd}$ largest eigenvalue of $A$ and a related matrix.

Let $A$ be an irreducible, aperiodic matrix with non-negative entries, with $1 \in \ker(A - I)$, $w \in \ker(A^\top - I)$, $w_i > 0$ $\forall i$. Define $W = \text{diag}(w)$. I am studying the ...
4
votes
1answer
123 views

Condition for degenerate eigenvalues for a matrix

Given a diagonalizable matrix $M$ (that is, a normal matrix), can we determine whether the matrix has degenerate eigenvalues without explicitly calculating all the eigenvalues and eigenvectors? 1) An ...
9
votes
3answers
171 views

What is spectrum for Laplacian in $\mathbb{R}^n$?

I know very well that Laplacian in bounded domain has a discrete spectrum. How about Laplacian in $\mathbb{R}^n$?(not in some fancy-shaped unbounded domain, but the whole domain) Where can I find ...
2
votes
1answer
132 views

Second Derivative of Eigenvalue

Consider the potential $V_t=tx$ and the corresponding Schrödinger operator $H_t=- \frac{\partial ^2}{\partial x^2}+V_t$ on $L^2([0,R])$ with Neumann or Dirichlet boundary conditions. Let ...
0
votes
1answer
51 views

Problem determining eigenvalues of a Hermitian matrix

Suppose that you've got an $n \times n$ irreducible matrix $A$ with strictly positive real entries and eigenvalues $\lambda_i$, $i=1,...,m$, arranged so that $|\lambda_1| > \cdots > ...
2
votes
1answer
52 views

Polar decomposition corollary

Let $T$ be a compact operator on an infinite dimensional Hilbert space. Let $|T|=(T^*T)^{0.5}$. By the polar decomposition theorem there is a partial isometry $S$ of the closure of Im$(|T|)$ such that ...
0
votes
1answer
67 views

The eigenvalues of a compact and self-adjoint operator on Hilbert space

Show that if $K$ is a compact self-adjoint operator on Hilbert space then it has either finitely many eigenvalues or a sequence of eigenvalues $\lambda_n\to 0$ as $n\to \infty$.
0
votes
1answer
75 views

Projections in spectral decomposition.

In my quantum mechanics book the spectral decomposition of operator $A$ is given as $A=\sum\limits_j\lambda_jP_j$ where $\lambda_j$ are the eigenvalues of matrix $A$ and $P_j$ is the orthogonal ...
1
vote
1answer
110 views

The Trace Class Operators Form a Banach Space

I want examining the trace class operators $L_1(H)$ of a separable Hilbert space $H$ with the norm $||A||_1=\sum\limits^{\infty}_{n=1}\lambda_n$ where $\lambda_n$ are the eigenvalues of ...
5
votes
0answers
69 views

Question about a (relatively simple looking) differential operator and its eigenvalues

A colleague and I are interested in a specific differential operator on the reals. The differential operator L is of the form $L=-(1+x^{2})\frac{d^{2}}{dx^{2}}+c_{1}x\frac{d}{dx}+c_{2}x^{2}$ for ...
4
votes
3answers
79 views

$A^2$ self-adjoint and Compact, prove $A$ has an eigenvalue

Suppose $H$ is a Hilbert space and $A \in L(H)$ is such that $A^2$ is compact and self-adjoint. Prove that $A$ has an eigenvalue. (Here $L(H)$ is the set of bounded linear operators on a Hilbert ...
0
votes
2answers
103 views

Functional Analysis, operator theory, eigenvalues of a operator

We have $$T_\alpha:C[a,b]\to C[a,b]$$ $$T_\alpha f= \alpha f$$ where $C[a,b]=\{ f:[a,b]\to \mathbb{R} \quad f$ is continuous} and $\alpha\in C[a,b]$ fixed. Show: Spectrum of $T_\alpha\equiv ...
0
votes
0answers
34 views

A domain in which the dirichlet laplacian has eigen values of all orders

I am trying to come up with an example to the following. Construct a domain $\Omega$ in all dimensions $n \in \mathbb{N}$, such that the spectrum of the Dirichlet Laplacian on such a domain (i.e., ...
1
vote
1answer
106 views

Eigenfunction expansion

Use the appropriate engenfunction expansion to represent the best solution. $$u''=f(x), u'(0)=\alpha, u'(1)=\beta$$ I use the function $$\phi''+\lambda\phi=0$$ to get the eigenfunction is ...
1
vote
0answers
84 views

Sturm-Liouville Eigenvalues

Consider Sturm-Liouville endpoint problems of the form $y''+\lambda y=0$ with the usual endpoint conditions. $c_1y(a)+c_2y'(a)=0$, $d_1y(b)+d_2y'(b)=0$. Here $(c_1,c_2) \neq \vec{0}$ and $(d_1,d_2) ...
2
votes
0answers
134 views

Eigenvalues of differential operator

If $L : C^2[a,b] \rightarrow C^0[a,b] $ is s.t. $L y(t) = \ddot y(t) +p \dot y + q y(t) $ and $L$ is invertible then $L^{-1}$ has at most countable eigenvalues and they accumulate in $0$. Why ...
2
votes
1answer
173 views

Adjoint matrix eigenvalues and eigenvectors

I just wanted to make sure that the following statement is true: Let $A$ be a normal matrix with eigenvalues $\lambda_1,...,\lambda_n$ and eigenvectors $v_1,...,v_n$. Then $A^*$ has the same ...
5
votes
1answer
131 views

Convergence of eigenvalues for sequence of compressions of a compact operator

Suppose $H$ is a separable Hilbert space, $A$ is a Hilbert Schmidt operator on $H$, and $P_n$ is an increasing sequence of finite rank orthogonal projections of $H$ (so $P_nx\rightarrow x$ for all ...
5
votes
1answer
191 views

Generalized eigenspaces of a compact operator are finite dimensional

Let $T : H\rightarrow H$ be a compact operator on a Hilbert space $H$. Say that $\lambda \in \mathbb C$ is a generalized eigenvalue of $T$ if there is some $n \geq 1$ such that $(\lambda - T)^n$ is ...
1
vote
1answer
157 views

the first eigenfunction of Dirichlet problem

Let $\Omega$ be a bounded planar domain which has a axis of symmetry and $T:\Bbb{R}^2\longrightarrow\Bbb{R}^2$ symmetry with respect to this axis. Let $u_{1}(x)$ be the first eigenfunction of ...
1
vote
1answer
90 views

Eigenvalues of discretized linear integral operator

Suppose I have the following kernel operator: $Af(x) = \int_{-1}^1 K(x-y)f(y)dy$ which is also positive and compact. Hence, it has a countable set of positive eigenvalues. Suppose those eigenvalues ...
2
votes
0answers
91 views

Is there a generalization of eigenvalues/eigenvectors to $L^p$ for $p\neq2$?

Given symmetric positive definite $m\times m$ matrix $\Sigma$, $1 \le p,$ and constant $c > 0$, consider the solution to the optimization problem: $$ \min_{v : \left\|v\right\|_p \ge c} ...
9
votes
1answer
298 views

An approximate eigenvalue for $ T \in B(X) $.

This is a problem from Conway’s Functional Analysis: Definition An approximate eigenvalue for $ T \in B(X) $ is a scalar $ \lambda $ such that there is a sequence of unit vectors $ x_{n} \in X $ ...
1
vote
1answer
100 views

Which one is the right definition of eigenvalues for a differential operator?

This question might be trivial, but I have problems understanding the definition of eigenvalues for the Laplacian \begin{equation} \Delta : C^2(U) \to C(U). \end{equation} on some open, bounded domain ...
3
votes
0answers
117 views

show that the function satisfies condition of the lemma

Let $(f_n)_{n\geq 0}$ be an eigenfunctions of the compact integral operator $F$, defined on $L^2([-1,1])$ by $$ F(f)(x)=\int_{-1}^1\frac{\sin a(x-y)}{x-y}\,\psi(y)\,\mathrm{d}y, \quad \mbox{here ...
2
votes
1answer
77 views

Why are these two operators similar?

Let $X$ be a Hilbert space with ON-basis $\lbrace e_n : ~ n \in \mathbb{N} \rbrace$. Furthermore let $A, ~ \Gamma : X \to X$ be linear operators with $A e_n = \alpha_n e_n$ and $\Gamma e_n = \gamma_n ...
3
votes
1answer
334 views

Eigenvalues integral operator - general case

Let $T$ be an integral operator on $L^2([0,1])$, such that: $$ (Tf)(x) = \int_0^1K(x,y)f(y)dy, $$ with $K(x,y): [0,1]^2 \rightarrow \mathbb{R}$ continuous and $K(x,y) = K(y,x)$, $K(x,y)\geq0$ $ ...
4
votes
1answer
541 views

How to find an orthonormal basis for $L^2(\mathbb{R},\mathbb{C})$?

Consider the Hilbert space $X:=L^2(\mathbb{R},\mathbb{C})$ Now consider the operator that takes the second derivative, i.e. $A := \partial_{x}^2$, i.e. $A: H^2(\mathbb{R},\mathbb{C}) ...
7
votes
3answers
172 views

why do successive eigenfunctions have more oscillation?

I was told the following argument as to why successive eigenfunctions tend to have more oscillations: Suppose (without worrying about why) that the first eigenfunction has the least oscillation. ...
3
votes
1answer
126 views

Eigenvalues of Hilbert-Schmith operator

I am having trouble determining the eigenvalues and eigenvectors of the operator $Kv(x)= \int_0^1((x+t)v(t)dt$, where the kernel is $k=x+t$. I have tried to solve the equation $Kv(x)=\lambda v(x)$, ...
1
vote
1answer
183 views

spectrum of convex combination

$A,B$ are $n\times n$ Hermitian matrices. If the eigenvalues of $A$ and $B$ are all in an interval $I$, then the eigenvalues of any convex combination of $A,B$ are also in $I$. In the book Bhatia, ...
6
votes
1answer
459 views

Distinct eigenvalues for integral operator?

Is there some sufficient condition on the kernel $K$ of a (say, finite rank, to simplify) integral operator $$ \mathcal K:f(x)\in L^2(\mathbb R)\mapsto \int K(x,y)f(y)dy $$ so that it has all its ...
5
votes
1answer
489 views

What are the Eigenvectors of the curl operator?

The curl operator $\vec\nabla\times\mathbb{1}$ can be written as a skew-symmetric 3x3 matrix $$\mathrm{curl} = \begin{pmatrix}0 & -\partial_z & \partial_y \\ \partial_z & 0 & ...
2
votes
1answer
116 views

Has this operator $0$ as an eigenvalue / where is my error?

I know of a theorem that tells me, that every compact linear operator on an infinitedimensional Hilbert space has to have the eigenvalue $0$. On the other hand I have the operator \begin{eqnarray*} ...
4
votes
0answers
290 views

Continuous spectral value of right shift operator $\ell^2(\mathbb{N})$

Let $T:\ell^2(\mathbb{N} \to \ell^2(\mathbb{N})$ be the operator that sends$(x_1,x_2,x_3,...) \to (0,x_1,x_2,x_3,....)$. I want to show that $\lambda = 1$ is in the continuous spectrum. To approach ...
3
votes
0answers
204 views

Does matrix convergence in $L^p$ imply convergence of the eigenvalues in $L^p$?

Let $A_n(x)$ be a sequence of symmetric matrix functions that converges in $L^p(\Omega)$ to $A(x)$. Is it true that the eigenvalues of $A_n(x)$, or a subsequence of these, converge to the eigenvalues ...
3
votes
2answers
234 views

eigenfunctions of the adjoint of an operator

If the eigenfunctions of a linear operator are known, is there a way to calculate the eigenfunctions of the corresponding adjoint operator based on the known eigenfunctions? In other words, what's the ...
1
vote
1answer
497 views

What is the index of an eigenvalue?

I'm working out of Deuflhard and Bornemann's Scientific Computing with ODEs (Springer 2000). The statement is: Moreover, $\tau_c = \tau_c^+$ holds if all eigenvalues $\lambda \in \sigma(A)$ ... ...
6
votes
1answer
385 views

Eigenvalues, kernel and rank of a compact operator: how to start?

I'm trying to solve the following exercise: Let $f\in\mathcal{C}([0,1])$ and let $T$ an operator such that $Tf(x)=\int_0^1(x-t)f(t)dt$. I have proved that $T$ is a bounded linear operator and, by ...
3
votes
1answer
538 views

Eigenvalues of an operator

I think this question isn't that hard, but I am a bit confused: Define $$(Af)(x):=\int_{0}^{1}\cos(2\pi(x-y))f(y)dy.$$ Then $A$ is an operator on functions. Find the eigenvalues and the ...