2
votes
1answer
33 views

Spectrum of integral operator

Given $g\in C^1([0,1]\times[0,1])$, consider the operator $$Tu(x) = \int_0^1 g(x,t) u(t) dt$$ defined on $u\in C([0,1])$. Discuss the spectrum of T. My attempt: First I can show that $T$ is ...
1
vote
1answer
32 views

Exercise about an operator (adjoint and spectrum)

Let $y\in c_0$ and define the operator from $l^2 \rightarrow l^2$ as the following $$T\bigg(\sum x_n e_n\bigg) \mapsto \sum y_n x_n e_n.$$ I have shown that the operator is continuous, compact and ...
2
votes
1answer
46 views

Definition of exponential for operators

if I have a self-adjoint operator $T:D(T) \rightarrow L^2$, then I define its unitary exponential operator by $$e^{iT}(f) := \lim_{k \rightarrow \infty} e^{iT_{k}}(f),$$ where $T_k(f):=\frac{1}{2} ...
2
votes
1answer
60 views

Selfadjoint Restrictions of Legendre Operator $-\frac{d}{dx}(1-x^{2})\frac{d}{dx}$

Problem: Let $Lf =-((1-x^{2})f')'$ be the Legendre differential operator defined on the domain $\mathcal{D}(L)$ consisting of twice absolutely continuous functions on $(-1,1)$ for which $f, Lf \in ...
3
votes
1answer
58 views

Is $AA^*$ and $A^*A$ self-adjoint?

if I have a densely defined closed linear operator $A$ and $A^* = -A$(same domain also closed). Is this sufficient that $AA^*$ and $A^*A$ are proper self-adjoint operators, assuming that we can also ...
1
vote
1answer
55 views

Domain of densely-defined second derivative operator, and its factorization

Let $$-d_x^2: \{f \in L^2[0,1];f \in AC^1[0,1] , f(0)=f(1)\} \rightarrow L^2[0,1]$$ be the second derivative operator. Here $AC^1[0,1]$ is the space of functions whose first derivative is absolutely ...
2
votes
2answers
53 views

Decomposition of a positive semidefinite self-adjoint operator?

If I have a positive semi-definite self-adjoint operator $H:D(H) \rightarrow L^2$, is it true that there is always a decomposition $H=A^* A$ available? If this is true, what can we say about the ...
1
vote
0answers
41 views

Proving that a Sturm-Liouville problem is in the limit-point/-circle case

I would like to understand techniques anybody is able to detail to me on how one may actually prove that a particular Sturm-Liouville (S-L) problem, i.e., of the form \begin{equation} ...
4
votes
2answers
114 views

Spectral theory - continuous spectrum

imagine that I have some differential operator $D$ that is defined on an interval $[a,b]$. Now, assume that we take the boundary conditions in such a way that this operator is self-adjoint. Then, I ...
1
vote
1answer
44 views

Fredholm index for 1-d Schroedinger operator

if I look at a Schroedinger-operator $-\frac{d^2}{dx^2} +V$ on a compact intervall $[a,b] \subset \mathbb{R}$ and I take boundary conditions that this operator is self-adjoint (for example periodic ...
1
vote
1answer
104 views

Spectral theory

I have absolutely no idea about Spectral theory and want to ask some fundamental questions. 1.) What does it mean that the resolvent of an operator is Hilbert-Schmidt? (Cause I saw a theorem that was ...
1
vote
1answer
89 views

Mathieu differential equation

Given the operator $T (\psi)(x):= \psi''(x)-2q \cos(2x)\psi(x)$ with $T : D(T) \subset L^2[0,2\pi] $ I was wondering: What is the right domain $D(T)$ for this operator if we want to solve the ...
1
vote
2answers
61 views

Book on periodic Schrödinger operators

I am looking for good books about the spectral theory of periodic (1-dimensional) Schrödinger operators on a compact interval. A good reference I found was Reed/Simon Analysis of Operators (and a ...
1
vote
1answer
33 views

How to show whether this operator is normal? self-adjoint? unitary?

Consider the Hilbert space $H=l_2$ over the complex field and $A:H\rightarrow H.$ How to show whether this operator is normal? self-adjoint? unitary? ...
1
vote
1answer
25 views

How to show that the spectrum is equal to the range of $y$

How to show that the spectrum of $T_y$ is equal to the range of $y$ Given $y\in C[0,1]$ and $T_y: C[0,1] \rightarrow C[0,1]: x\mapsto x\cdot y$ Any help is appreciated, thanks.
1
vote
1answer
35 views

When $A_y$ is invertible?

Given $y\in C[0,1]$ Let $A_y:C[0,1]\rightarrow C[0,1]: x\mapsto xy$ When $A_y$ is invertible? Could you please help.
1
vote
1answer
30 views

How to find the point spectrum $\sigma_p(A)$ of $A$

How to find the point spectrum $\sigma_p(A)$ of $A$ Consider the Hilbert space $H=l_2$ over the complex field and $A:H\rightarrow H.$ ...
2
votes
2answers
72 views

How to show: $A_y$ has no eigenvectors if $y$ is not constant on any subinterval of $[0,1]$

Let $y\in C[0,1]$ and $A_y : C[0,1]\rightarrow C[0,1]: x\mapsto xy$ How to show: $A_y$ has no eigenvectors if $y$ is not constant on any subinterval of $[0,1]$. Could you please help.
0
votes
3answers
67 views

Positive unbounded operator with zero not as an eigenalue

I am currently doing Quantum Mechanics and I am supposed to show that zero is an eigenvalue of a positive operator. I have no knowledge of Functional Analysis at that kind of level, so I was wondering ...
0
votes
3answers
214 views

Invertibility of $I-A$ if the spectral radius of the operator $A$ is less than $1$

I want an explication of the following fact: If the spectral radius of a bounded operator $A$ on a Banach space is less than one, then $I - A$ is invertible.
1
vote
1answer
65 views

Proof that these Hessian matrix identities are similar matrices

I am wondering if $Q, P$ are similar matrices where for a function $f:\mathbb{R^n}\to\mathbb{R}$ and for a diagonal matrix $D$ $Q=I-D^{-1}\nabla^2f(x)$ and $P=I-D^{-1/2}\nabla^2f(x)D^{-1/2}$. ...
3
votes
1answer
270 views

Spectral radius, and a curious equality.

Given a $N\times N$ matrix $A$ over $\mathbb R$. Let $ \rho\left( A \right) = \max \left\{ {\left| \lambda \right|;\lambda \mbox{ eigenvalue of }A} \right\}$. Someone told me that, the following ...