1
vote
0answers
14 views

Eigenvalues of a Self-Adjoint Operator

It is easy to see that eigenvectors corresponding to distinct eigenvalues of a self-adjoint operator $T$ are mutually orthogonal. However, from this, it is supposed to be easy to see that for a given ...
2
votes
1answer
35 views

Hilbert Spaces - an application of the minimax principle.

Let $A$ be a compact, self-adjoint operator, $A \geq 0$. We need to prove that for any orthonormal system $\{e_i\}_1^{\infty}$ and for any $N$, $$\sum_1^N \langle Ae_i,e_i \rangle \leq \sum_1^N ...
1
vote
1answer
30 views

Hilbert Spaces; eigenvalues of $PBP$ vs. $B$ for $B$ compact selfadjoint and $P$ orthoprojection.

An exercise I have come upon while studying Hilbert Spaces: Let $A$ be a compact operator, and $P \in L(H)$ be an orthoprojection. Prove that $$\lambda_n (PA^*AP) \leq \lambda_n (A^*A)$$ (Where ...
1
vote
3answers
53 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 ...
0
votes
0answers
9 views

equivalence of statements involving compact operators

Let T be a compact operator on a hilbert space H.I want to show that the following 2 statements are equivalent. For each $\lambda \in {\bf C} $, let $V_\lambda = \{ x \in H: Tx = \lambda x \}.$ ...
0
votes
0answers
38 views

Show there exists a unique solution to $-u''(x)+\lambda \int^1_0 \sin(u(y))dy =f(x)$

Let $\lambda\in (-1,1)$. Show that for every $f\in C[0,1]$ there exists a unique solution $u\in C[0,1]$ to $$-u''(x)+\lambda \int^1_0 \sin(u(y))dy =f(x)$$ With $u(0)=u'(1)=0$. My work thus far: ...
2
votes
0answers
84 views

Spectrum of $Tu=\int^1_{-1} (1-|x-y|)u(y)dy $

Consider the operator $$ Tu(x)=\int^1_{-1} (1-|x-y|)u(y)dy $$ We want to find the spectrum of $T$. The kernel is certainly bounded and so this operator is Hilbert-Schmidt, so $T$ is compact. We ...
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
36 views

Why does s = z+1?

What exactly is Laplace transform? motivated me to ask why unit function is 1/s by Laplace transform and 1/(1-z) by Z-transform? Both seem to be integrals of delta-pulse and secondary integration ...
2
votes
1answer
50 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 ...
0
votes
0answers
21 views

Doubt on eigenvalues of normal operators

I'm trying to understand the solution of the following problem: $T$ is a normal operator. If $T( v)=\lambda v$, then $T^*(v)=\bar\lambda v$: The solution is: I didn't understand why we ...
0
votes
1answer
22 views

Common eigenvector of a sequence of compact operators

Let $H$ be a separable, infinite-dimensional Hilbert space and suppose we have a sequence of norm-one compact operators $(A_n)$ on $H$ which all have 1 as an eigenvalue. Can we pass to a subsequence ...
1
vote
1answer
65 views

Question about the essential spectrum of a negative difinite operator

please on an infinite dimensional Hilbert space how to difine the essential spectrum of an operator which is negative definite ??? Please help me Thank you.
1
vote
1answer
53 views

Operator Norm = 1

Let there be a linear map $T$ such that $T: \mathbb R^n\to\mathbb R^m$. The operator norm $\lVert \cdot\lVert_{op}$ of $T$ is then defined as the largest value of $c$ for which $\lVert T(\vec v)\lVert ...
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 ...
2
votes
2answers
37 views

Application of the spectrum of an operator

http://en.wikipedia.org/wiki/Spectrum_of_an_operator What is the application of the spectrum of an operator
2
votes
0answers
222 views

Eigenvalues of self-adjoint eigenvalue problem

I am stack with the following problem: Consider the following eigenvalue problem $$ u \in H_B(0,1), \; \langle Lu, Lv\rangle = \lambda (\alpha \langle u, v\rangle + \langle u', v'\rangle) \; \forall ...
0
votes
2answers
104 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 ...
5
votes
1answer
199 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
45 views

Property of sequence of eigenvalues of an operator.

For a positive (self adjoint) operator $A$ with eigenvalues $\lambda_k$, is it possible to have the case when neither $\lambda_k\to \infty$ or $sup_k \lambda_k<\infty$ for example if a subsequence ...
1
vote
1answer
260 views

norm equivalence

Exercise 7. of Terence Tao's blog on random matrix, specifically on eigenvalues of Hermitian matrices ...
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 ...
5
votes
1answer
383 views

Normal operator + only real eigenvalues implies self-adjoint operator?

Let say we are in a complex vector space, is there an example of a normal operator with only real eigenvalues(or without eigenvalues) that is not a self-adjoint operator? Cause of the spectral theorem ...
3
votes
1answer
673 views

Rayleigh-Ritz Theorem

Let $U$ be an $n$-dimensional subspace of $L:=L_2([-1,1])$. Let $F$ be an acting on $L$, given at $f \in L$ $$ (Ff)(x):=\int_{-1}^1 \frac{\sin a(x-y)}{(x-y)}f(y) dy, \quad x \in [-1,1], \quad a>0. ...
1
vote
2answers
414 views

Example of a normal operator which has no eigenvalues

Is there a normal operator which has no eigenvalues? If your answer is yes, give an example. Thanks.
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
342 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 $K(x,y) = K(y,x)$, $K(x,y)\geq0$ $ ...
5
votes
0answers
359 views

Min Max Principle and Rayleigh-Ritz-Method for eigenvalues of unbounded operators?

Finding eigenvalues of matrices using the Rayleigh-Ritz quotient is well-known, c.f. http://en.wikipedia.org/wiki/Min-max_theorem Does the following generalization of that fact also hold? Theorem: ...
6
votes
1answer
460 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
503 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
117 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*} ...
2
votes
1answer
52 views

eigenvalue check

I have a question regarding operator theory and would be glad if someone could help. I have a linear operator $A$ that is non-self-adjoint, unbounded and is densely defined in a Hilbert space $H$. I ...
2
votes
1answer
83 views

meaning of this operator ??

given the operator $ P_{\Lambda } = (f\in L^{2}(R)^{even}| f(q)=0 , |q| \ge \Lambda) $ what does it mean? the operator $ P_{\Lambda} $ acts over a function $f(q) $ by setting this (even) function to ...
3
votes
2answers
240 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 ...
2
votes
0answers
162 views

Find the minimum value of the maximum eigenvalue of operator A?

So we are given the following: Operator $A$ with $Au=-u''$; $u \in D_A = \{u\colon[a,b]\rightarrow R,u\in C^2([a,b]),u(a)=u(b)=0\}$; $D_A$ is dense in $L^2((a,b))$. Find the minimum value that is ...
3
votes
1answer
368 views

Eigenvalues of compact operators and his adjoint.

Let $T: H \to H$ be a compact operator with $H$ a Hilbert space. Let then $\lambda \neq 0$ be an eigenvalue of $T$ with eigenfunction $v$. Is then $\lambda$ an eigenvalue for the adjoint $T^*$ ...
1
vote
1answer
277 views

Sturm-Liouville Theorem

I was reading the Wiki page on the Sturm-Liouville theory. Why are those tenets true? Are there any (not too advanced) reference material? I have also read that "There are countably infinite ...
6
votes
1answer
391 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 ...
2
votes
0answers
115 views

Principal eigenvalue

How is the principal eigenvalue of elliptic differential operator defined? Is it just a spectral radius?