-1
votes
3answers
41 views

Selfadjoint Operator: Empty Spectrum

Can a selfadjoint operator have empty spectrum? (As far as I remember, yes; but just to be sure.) The point is that if so then the closure of its spectrum cannot equal the convex hull of its ...
1
vote
1answer
14 views

Commutant of a set of operators and norm topology.

In the references I have it's remarked that the commutant $S'$ of a set $S$ in $B(H)$, where $H$ is a Hilbert space, is closed in the weak operator topology. And this is true because if ...
1
vote
0answers
31 views

Spectral decomposition of a Hilbert space

I have this proof, but I don't understand how they do the spectral decomposition of $H$ into $H_-$and $H_+$? Please help me. Thank you.
0
votes
0answers
16 views

Eigenvalues of correlation matrices in the limit of infinite dimensions

Consider a continuous function $f(x,t)$ with $x\in X$ and $t\in[0,1]$, then one may define a series of functions $f_n\in\mathbb{R}^n$ defined naturally as $f_n(x)_i=f(x,i/n)$. Now compare the ...
1
vote
1answer
25 views

Skew-adjoint differential operator $B$ with spectrum $\sigma(B)=i(-\infty,-1]$

Consider the Hilbert space $X=L^{2}\left(\mathbb{R}^n\right)$ and the Schrödinger operator $A=i\Delta$ defined on the domain $D(A)=H^2(\mathbb{R}^n)$. It is known that the spectrum of $A$ is ...
1
vote
1answer
37 views

Relation between $\epsilon$-pseudospectrum of operators

If $H$ is a Hilbert space and $\sigma_{\epsilon}(T)$ denotes the space of all $\epsilon$-pseudospectrum of the operator $T$ and $S, T\in B(H)$ be such that $TS=ST=0$, why ...
1
vote
3answers
59 views

Prove that if $T=T^*$ and $\sigma(T)=\{\lambda\}$, then $T=\lambda I$

Show that if $T$ is a self adjoint linear operator on a Hilbert space such that the spectrum contains a single point $\lambda$, then $T=\lambda I$. Then, show this is false if $T$ is not self ...
1
vote
0answers
23 views

Helffer-Sjöstrand-Formula: Idea behind?

I have to present the Helffer-Sjöstrand-Formula. Now I'm wondering: Why does it include a factor $\chi(y\langle x\rangle^{-1})$ for some bump function $\chi$ and the chinese symbol ...
1
vote
0answers
39 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} ...
0
votes
1answer
33 views

Exercise about spectrum of selfadjoint operator.

I'm stuck on an exercise about the spectrum of a selfadjoint operator on a Hilbert space. The problem is the following: Let $(X,\langle \cdot, \cdot\rangle)$ a Hilbert space and let $A \in B(H)$ a ...
0
votes
1answer
50 views

Spectrum of the operator $A(f,g)=(g,\Delta f-f)$

Let $\Omega$ be an open set in $\mathbb{R^n}$. We consider the product Hilbert space $H=H^1_0(\Omega)\times L^2(\Omega)$ with the norm $$|(f,g)|^2=\int_\Omega (|\nabla f|^2+|f|^2+|g|^2 ) dx$$ We ...
1
vote
0answers
26 views

Skew adjoint operator with uncountable spectrum

Let $H$ be a Hilbert space. I just want an example of a skew adjoint operator $(A^*=-A)$ with uncountable spectrum. I also want an example for unbounded differential operators. The only example I ...
0
votes
0answers
10 views

Integral representation of joint projection valued measures

Given two positive $\sigma$-finite measures $\mu_{1/2}$ on the spaces $X_{1/2}$ one can define the product measure $\mu_1\otimes\mu_2$ on the product space $X_1\times X_2$. It can be proved that the ...
5
votes
2answers
89 views

What's the spectrum of this operator in $\ell^2$?

Suppose that $\ell^2 = \biggl\{(x_n)_n \in \mathbb{K}^{\mathbb{N}_0} \biggm| \sum_{n=1}^{\infty}|{x_n}^2| < +\infty \biggr\}$ is a Hilbert-space with the inproduct $\langle\cdot,\cdot\rangle_2: ...
0
votes
2answers
68 views

Symmetric Operator $\iff$ Real Spectrum

One the one hand not every symmetric operator has real spectrum: $$A\text{ symmetric}\nRightarrow\sigma(A)\text{ real}$$ (In fact this is true only for the self adjoint ones.) Does the converse fail ...
1
vote
1answer
55 views

Operator: not closable!

Is there an operator between Banach spaces with the following properties: $$T:\mathcal{D}(T)\subseteq X\to Y:\text{ injective, dense range, continuously invertible, not closable!}$$ (Note that the ...
1
vote
0answers
32 views

Spectral theory - how to prove this lemma?

in Anver Friedman, Foundations of Modern Analysis I found a lemma (6.7.3): If A is a self-adjoint operator and $\{E_\lambda\}$ is a spectral family such that $A=\int_m^{M+\varepsilon} \lambda ...
0
votes
1answer
47 views

Positive Operator: Norm Estimate

In class we encountered the statement: $$H\geq C\mathrm{Id},C>0\Rightarrow\|\mathrm{e}^{-\beta H}\|<1,\beta>0$$ How does one prove this? Moreover what about the weakened version: $$H\geq ...
1
vote
1answer
19 views

Proof that solution of $\lambda$-affine, linear ODE is entire in $\lambda$

Suppose $F(\lambda)~(\lambda\in\mathbb{C})$ is a linear ordinary differential operator (with, say, domain $D$ dense in some Hilbert space), and is also affine-linear in $\lambda$. Is there a proof ...
4
votes
1answer
85 views

Spectral decomposition of normal operator

Define $T$ from $L_{2}(R)$ into itself by $T(f)(t)=f(t+1)$. Show that $T$ is normal and finds its spectral decomposition. I've shown that $f$ is normal (in fact it's unitary) but how do I find its ...
4
votes
0answers
99 views

Spectral decomposition of $TT^*$

On $l_{2}$ let $T$ be given by $Te_{n}=\frac{e_{n+1}}{n+1}$ where $(e_{n})_{n\ge1}$ is the canonical orthonormal basis. Find the spectral decomposition of $TT^*$. I find that ...
3
votes
0answers
36 views

In which cases the spectrum of an operator contains only eigenvalues?

Let $X\neq \{0\}$ be a complex normed spaces (not necessarily finite-dimensional) and $T:D(T)\subset X\to X$ a linear operator (not necessarily bounded). I would like to know under what conditions can ...
0
votes
1answer
66 views

Spectral theory and sequences: is this fact a general truth or does it depend on the operator?

Let $\lambda\in\mathbb{R}\setminus\{0\}$, $\textbf{i}$ the imaginary unit, $H$ a Hilbert space, $T:D(T)\subset H\to H$ a invertible densely defined linear operator such that $T^{-1}$ is bounded, ...
4
votes
2answers
61 views

Examples of spectral decompositions

I would like examples of spectral decompositions and how they are obtained for normal compact operators and normal non-compact operators on an infinite dimensional hilbert space. I have googled it, ...
0
votes
1answer
66 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
0answers
29 views

A couple of proofs on a spectrum

Let $T$ be a normal bounded operator. Let ${\lambda}$ be in $({\sigma}(T))$. Without invoking general algebra theories, show that: a) $p({\lambda},{\lambda}^*)$ is in $({\sigma}(T))$ for all ...
0
votes
0answers
16 views

characterisation up to unitary equivalence

My book says that the spectral theorem for compact normal operators characterises compact normal operators up to unitary equivalence. It doesn't expand on this so I was wondering what does this mean ...
0
votes
0answers
25 views

The eigenspace of 0 problem

In the spectral theorem for a compact normal operator, do we exclude the eigenspace corresponding to 0 (assuming its an eigenvalue) from the space decomposition. My reason for asking is this: Every ...
0
votes
2answers
84 views

Spectral Theorem for normal operators

I want to prove this in the infinite dimensional Hilbert space case. What is the easiest way to go about this (What do I need to know, what theorems do I need,etc). My aim is to show every normal ...
1
vote
1answer
238 views

Spectrum of operator

Like my previous question, I'm considering the same space and operator: Hilbertspace adjoint But this time I am trying to determine the spectrum of $T$. I feel like I'm messing up my definitions a ...
1
vote
1answer
77 views

Multiplication operator on Hilbert space

i looked to the question Spectrum and point spectrum of this operator. I will go further with asking. We know that $T$ is well-defined iff $(\lambda_n)\in\ell^{\infty}$. But if ...
2
votes
1answer
44 views

Approximate point spectrum and left topological zero divisors

Recall that a left topological zero divisor in a Banach algebra $A$ is an element $a\in A$ such that there exists a sequence of unit vectors $(a_{n})$ in $A$ with $\lim_{n\rightarrow\infty}aa_{n}=0$. ...
2
votes
1answer
102 views

Eigenvalues and adjoint of operator $T(x_k)_{k=1}^{\infty} = (x_{2k})_{k=1}^{\infty}$

Let $T$: $l^2 \rightarrow l^2$ denote the operator \begin{align} T(x_1,x_2,\dots, x_n,\dots) = (x_2,x_4,\dots,x_{2n},\dots). \end{align} There are several questions regarding this operator that I need ...
0
votes
2answers
91 views

Prove that if $\lambda$ is an isolated eigen-value of $T=T^*$, then $\ker(T-\lambda)=E_{\{\lambda\}}H$

Here we have a self-adjoint, densely-defined operator $T$ on a Hilbert space $H$, and $E_M$ is the usual spectral projector for any Borel set $M$, i.e., $E_M=\int_M\text{d}E_t$ (this means, by ...
1
vote
1answer
133 views

Adjoint of resolvent of self-adjoint, densely-defined operator on a Hilbert space

Let $H$ be a Hilbert space, $T=T^*$ a densely-defined linear operator on $H$. Denote the resolvent set of $T$ as $\rho(T)=\{\lambda\in\mathbb{C}~|~T-\lambda$ has bounded, everywhere-defined inverse}, ...
4
votes
0answers
86 views

For $A$ self-adjoint, $\sup_{|x|=1}\langle Ax,x\rangle = \max \sigma(A)$

For a self-adjoint operator $A$ on a Hilbert space $H$, one has $\sup_{|x|=1}\langle Ax,x \rangle = \max\sigma(A)$. I want to prove this using the spectral theorem. My idea is: Let $a = ...
4
votes
1answer
226 views

An alternate proof of Fuglede's theorem

To prove Fuglede's Theorem for normal operators on a separable Hilbert space, why does it suffice to show that $E(S_1)T E(S_2)=0$ for all disjoint Borel sets $S_1$ and $S_2$, where $E$ is the spectral ...
1
vote
0answers
42 views

How many projectors do two commuting self-adjoints have in their common spectral decomposition?

If $A$ and $B$ are two commuting observables on a Hilbert space of dimension $n$ say. So, $$A = \sum_{j \leq a} \lambda_j P_j $$ $$B = \sum_{i \leq b} \mu_i Q_i $$ $$I_n = \sum_{i \leq b} Q_j = ...
0
votes
0answers
137 views

Symmetry adapted basis function to make the Hamiltonian matrix Block Diagonal.

Can anybody give me a tip to solve this problem? I have large quantum mechanical Hamiltonian, to solve it numerically I have to decompose it into the block diagonal form. To convert the hamiltonian ...
2
votes
1answer
104 views

Spectrum of linear operators

I can't solve the following: i) Let $T:l^2 \rightarrow l^2$ , $Tx=\{ (Tx)_n\}_{n=1}^{\infty}$ given by $$(Tx)_n = \dfrac{1}{2}x_{n-1} + \dfrac{1}{2}x_n.$$ Find $\sigma(T)$. ii) Let $S : l^2 ...
3
votes
1answer
455 views

Spectral Theorem for bounded compact, self-adjoint operators as corollary of Hilbert-Schmidt theorem

I'm following Debnath and Mikusinksi's "Introduction to Hilbert Spaces with Applications" and am trying to understand how the spectral theorem for compact self-adjoint operators is a corollary of the ...
4
votes
1answer
131 views

Spectrum proofs

Let $T$ be a densely defined closed unbounded operator on a Hilbert space $H$. Show that if $\lambda$ is a point in the residual spectrum of $T$, then $\bar{\lambda}$ is in the point spectrum of the ...
2
votes
1answer
119 views

Show for compact operator $K$, if $||Kf|| < ||f|| \forall f$, then $||K|| < 1$.

I wanted to check my reasoning on proving this statement, and see if anyone had suggestions for other proofs of this fact. Again, the statement is, if $K$ is a compact operator on a Hilbert space ...
9
votes
1answer
292 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 $ ...
3
votes
1answer
185 views

Proof that the spectrum of the Dirichlet Laplacian is discrete

Let $\Omega\subset\mathbb{R}^n$ a open bounded set. The Dirichlet laplacian can be defined via it's closed semi-bounded form on $H^1_0(\Omega)$. The fact that it's spectrum is discrete is as far as I ...
3
votes
1answer
193 views

Spectrum in an separable Hilbert space

Let $H$ be a separable Hilbert space with orthonormal basis $\{e_i\}$. Let $(c_n)$ be a bounded sequence of complex numbers and consider the bounded linear operator $T$ on $H$ defined by $$Tx = ...
3
votes
2answers
319 views

Spectrum of an Orthogonal Projection Operator

I want to show that $ \sigma(p) = \{ 0,1 \} $ for any orthogonal projection operator $ p \notin \{ 0,I \} $ on a Hilbert space $ \mathcal{H} $. Recall that an orthogonal projection operator $ p $ on $ ...
2
votes
2answers
101 views

If $Lat(\mathcal{A})$ is trivial then $\mathcal{A}'$ consists of scalars.

This is related to Exercise 3 of Section 2.5 of Arveson's book on spectral theory. For those who are interested, we were asked to show the following $\mathcal{A}$ is a Banach *-algebra. ...
4
votes
1answer
389 views

Spectral theorem for unitary operators

I saw in several texts, as a part of the spectral theorem for unitary operators, that given a unitary operator $U$ on a Hilbert space $H$ (say it is separable), $H$ can be decomposed as an orthogonal ...
1
vote
0answers
239 views

Diagonal Dominance and Spectral Radius

For positive semi-definite matrices, $A$ and $B$ with real entries, Let: $X=I-(2Diag(A)-B)^{-1}(A-B)$ The spectral radius $\rho(X) \leq ||X||$. As, $(2 Diag(A)-B)$ becomes a better approximation ...