1
vote
1answer
38 views

Inverse of operator is not continuous in Banach spaces

Let $X$ be a Banach space. If $A:X\to X$ is an invertible bounded operator (injective, surjective and continuous), then $A^{-1}$ is also bounded. Now can I have an example of an unbounded operator ...
1
vote
0answers
16 views

decomposition of spectrum in case of a Banach algebra

Let A a complex Banach algebra containing a unit e. For x in A you can define the spectrum $\sigma(x)$ and the resolvent set of x. But, if you consider a complex Banach space X and a bounded linear ...
4
votes
2answers
78 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 ...
0
votes
1answer
29 views

Inverse of an operator on two functions

I have the following operator, defined for two twice-differentiable functions $f,g$: $X(f,g):=\frac{(g')^3+fg'f''+g'(f')^2-ff'g''}{g'f''-f'g''}$ This operator has the following property: A curve ...
2
votes
1answer
54 views

Is a linear operator on $\ell^2$ defined by the inner product necessarily bounded? [duplicate]

If $a=\{a_n\}\in \ell^\infty(\mathbb{R})$ and $\langle a,x \rangle<\infty$ for all $x\in \ell^2(\mathbb{R})$, (where $\langle a, x\rangle=\displaystyle \sum_{k=1}^\infty a_kx_k$), then is $a\in ...
1
vote
1answer
41 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 ...
0
votes
0answers
31 views

TT* Duality argument

I am trying to obtain $L^p$ estimate for a system of nonlinear PDE. I do not have $L^2$ to work with for my problem, I heard $TT^*$ argument is useful tool if one wishes to mapp from $L^p$ to $L^p$. ...
0
votes
1answer
10 views

Question about Neighborhood basis

In the Simon Reed text, after defining the strong operator topology it is said: "A neighborhood basis at the origin is given by the sets of the form $\{S \ | \ S \in \mathcal{L}(X,Y), ...
3
votes
2answers
92 views

Two questions in spectral theory: the spectrum of the Fourier transform and the Hamiltonian of the hydrogen atom.

I have the following two questions: The Fourier transform defines a unitary (provided that it is normalized properly) map $\hat{\cdot}:L^2(\mathbf{R})\rightarrow L^2(\mathbf{R})$. I figured out its ...
0
votes
1answer
18 views

Why does T symmetric imply T* extends T?

This is a result I've seen stated a few times, but I can't seem to come up with a proof! Suppose $T$ is a densely defined linear operator with domain $D(T)\subset H$, where $H$ is a Hilbert space ...
2
votes
1answer
24 views

Definition and analyticity of $T^z$ where $T$ is a positive operator

Let $H$ be a Hilbert space. Suppose that $T\colon D(T) \to H$ is a positive selfadjoint operator where $D(T)$ is the domain of $T$. The spectrum $\sigma(T)$ of the operator $T$ is a subset of ...
1
vote
1answer
89 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 ...
5
votes
1answer
156 views
+50

Uniform continuity of the function $x(t)=e^{tA}x$

Let $A$ be a bounded operator on a Banach space $X$. Consider the exponential function $x(t)=e^{tA}x:=\sum_{n=0}^{+\infty}\dfrac{t^nA^n}{n!}x$, for all $t\in \mathbb{R}$, where $x\in X$. If the ...
2
votes
2answers
71 views

The trace of an operator

My question is derived from A. Deitmar's book: A First Course in Harmonic Analysis (second edition), p22, Exercise 1.17. Let me rewrite it again: Let $k:\mathbb{R}^2 \rightarrow \mathbb{C}$ be smooth ...
1
vote
1answer
25 views

Solutions depending on something continuously

Let $V$ be a a real Banach space, $K \subset V$ a closed convex set, $A: K \rightarrow V^{*}$ a (nonlinear) operator and $F \in V^{*}$. Then the variational inequality is the following problem: find ...
2
votes
0answers
49 views

Trace class operators problem

Let $\mathcal{B}_1(\mathcal{H})$ be the set of trace class operators in a Hilbert space $\mathcal{H}$ and $\mathcal{H}^{(d)} = \bigoplus_{i=1}^d \mathcal{H}$ with $1 \leq d \leq \infty$. If $C \in ...
2
votes
1answer
34 views

A question on linearity of inner product

The linearity of inner product on $(X,\langle.,.\rangle)$ is usually written as $$\langle x+\alpha y,z\rangle = \langle x,z\rangle + \alpha\langle y,z\rangle,\qquad \forall (\alpha,x,y,z)\in R\times ...
2
votes
1answer
23 views

Another proof of the iniectivity of a linear operator

Let $g(x)= \chi_{[-\frac{1}{2}, \frac{1}{2}]}(x) $, and $ T \colon L^2(\mathbb{R}) \longrightarrow L^2 (\mathbb{R})$ , $Tf= g \star f$. I was asked to prove that $T$ is injective, and I succedeed ...
1
vote
0answers
13 views

Equivalent Gaussian measures

Let $\mu$ be a gaussian measure with eigenpair $\{e_k,2^{-k}\}$ and $\nu$ with eigenpair $\{ Te_k,2^{-k}\}$. Here, $T$ is the unitary operator given by $Tx = x - 2\left\langle x,v \right\rangle v$. ...
0
votes
2answers
44 views

Complete ONS and pure point spectrum

In all that follows all operators are taken to be densely defined on a Hilbert space $H$. Some textbooks state that an operator $A$ on $H$ has pure point spectrum if $H$ admits a complete ONS (Hilbert ...
1
vote
2answers
39 views

Is it a compact operator?

Let $$C^{1}_{2\pi}=\{u\in C^{1}((0,2\pi),\mathbf{R}^n):u(0)=u(2\pi)\}$$ $$C_{2\pi}=\{u\in C^{0}((0,2\pi),\mathbf{R}^n):u(0)=u(2\pi)\}.$$ $C_{2\pi}$ is equipped with the norm $$\|u\|_0=max|u(s)|$$ ...
0
votes
1answer
76 views

Proof of functional analysis theorem

Can anyone see why it follows that $\Vert u_{j}-u_{l} \Vert^{2} = 2[\Vert u_{j} \Vert^{2}+ \Vert u_{l} \Vert^{2}] - \Vert u_{j} + u_{l} \Vert^{2} = 2[E(u_{j}) + E(u_{l})] - ...
3
votes
1answer
39 views

Norm of self-adjoint operator

i am trying to prove that $\|A\|=\sup_{||x||=1}|\langle x,Ax\rangle|$ for some selfadjoint bounded operator A on a Hilbertspace. Can anyone give me a hint how to prove it.
2
votes
2answers
29 views

Sequence of bounded linear operators implicating Cauchy sequence in $\mathbb K$

Let H be a Hilbert space and $(T_n)_{n \in \mathbb N}$ be a sequence in ${\rm BL}(H)$ (bounded linear operators) such that $(\langle y,T_nx \rangle)_{n \in \mathbb N}$ is a Cauchy sequence in $\mathbb ...
2
votes
1answer
46 views

Elliptic partial differential equations and elliptic operators

I'm starting to study elliptic partial differential equations and I just want to know if there are any connections between the following concepts: An elliptic partial differential equation is given ...
0
votes
0answers
26 views

Operator Equation?

A space of polynomials $\{f_n\}$ is given, where $f_n$ is of degree $n$. The operator $T$ in this space, satisfy the relation $$T^2(f_n)+a_nT(f_n)-f_n=0$$ where $a_n$ is a scalar depending on $f_n$. ...
1
vote
1answer
35 views

Projecting self-adjoint operator onto closed subspace

Let $H$ be a complex Hilbert space and let $(Q, D(Q))$ be a closed, densely defined, positive semidefinite, Hermitian quadratic form on $H$. (That is, $D(Q)$ is a dense subspace of $H$, $Q$ maps ...
0
votes
0answers
19 views

Weyl Operators: Discontinuity

Let $\mathcal{A}_{CCR}(\mathcal{H})$ be a CCR algebra over a Hilbert space $\mathcal{H}$. Then the Weyl operators are unitary and therefore $\sigma(W(f))\subseteq \mathbb{S}$ so by the spectral ...
2
votes
1answer
48 views

Implying a positive definite operator

If we are given that $A:V \rightarrow V$ is an operator where $V$ is a real Hilbert space. If we are given that $A$ is bounded, strictly positive $\big(\langle Au,u \rangle > 0$ for all $u \neq ...
-1
votes
1answer
36 views

Bogoliubov Transformation

Let $\mathcal{A}_{CAR}(\mathcal{H})$ be a CAR algebra over a Hilbert space $\mathcal{H}$. Consider a linear $S$ and an antilinear $T$ both bounded operators acting on $\mathcal{H}$ satisfying: ...
1
vote
1answer
29 views

Operator Tensor Product

Let $S$ and $T$ be bounded operators over a Hilbert space $\mathcal{H}$. Define their tensor product $S\otimes T$ as acting on $\mathcal{H}\otimes\mathcal{H}$ by $S\otimes T(x\otimes y):=Sx\otimes Ty$ ...
1
vote
1answer
23 views

Analytic Vectors (Nelson's Theorem)

Is there a (simple) proof for Nelson's theorem that a symmetric operator is essentialky selfadjoint if it contains a dense subset of analytic vectors?
0
votes
1answer
56 views

C*-Algebra: Positive Operator

Let $\mathcal{A}$ be a C*-algebra. If $A\in\mathcal{A}$ is a selfadjoint element then $A^*A=A^2$ has positive spectrum since: ...
1
vote
1answer
40 views

Application Closed Graph Theorem to Cauchy problem

Consider $E:=C^0([a,b])\times\mathbb{R}^n$ and $F:=C^n([a,b])$ equipped with the product norms. Consider $$ u^{(n)}+\sum_{i=0}^{n-1}a_i(t)u^{(i)}=f $$ with $$u(t_0)=w_1,\dots,u^{(n-1)}(t_0)=w_n \\ ...
1
vote
1answer
35 views

unbounded self-adjoint operator

Given an operator $T:D_1(T)\subset L^2 \rightarrow L^2$ and the same operator $T:D_2(T) \subset L^2 \rightarrow L^2$, such that the operator is both times self-adjoint and closed, with $D_1(T) \subset ...
1
vote
1answer
84 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
1answer
40 views

Kadison's Inequality

Let $\mathcal{A}$ be a C*-algebra and $\omega$ a positive linear functional. Is there a simple proof for Kadison's inequality: $$\overline{\omega(A)}\omega(A)\leq\omega(A^*A)$$
0
votes
1answer
75 views

Momentum Operator: Selfadjoint Extensions

This might be a possible duplicate - please let me know if there is already a proof in another thread. Consider the momentum operator on $\mathcal{L}^2[0,2\pi]$: ...
0
votes
1answer
32 views

Unbounded Operators: Notation?

For continuous a.k.a bounded operators we have $\mathcal{B}(X,Y)$ stressing on boundedness and $\mathcal{L}(X,Y)$ stressing on linearity entailing $\mathcal{C}(X,Y)$. Is there a notation for ...
1
vote
1answer
37 views

The spectral projection of a positive operator

Let $T_{n}\in B(H)$ be a positive operator on Hilbert space $H$ and $T_{n}\rightarrow 1_{H}$ in the strong operator topology. Now fix $\delta>0$ and let $P_{n}$ be the spectral projection of ...
3
votes
0answers
67 views

Positive Operators: Definition?

Let $A$ be a self adjoint element of a C*-algebra $\mathcal{A}$ resp. a self adjoint operator of the operator algebra $\mathcal{B}(\mathcal{H})$ of bounded operators over a Hilbert space ...
2
votes
1answer
30 views

Approximations of compact operators

Let $(\xi_n)_{n=1}^\infty$ be a sequence in a Hilbert space $K$ convergent to some $\xi$. Suppose we have a compact operator $T$ on $K$ such that $T\xi = 0$. Can we find a sequence of compact ...
-1
votes
1answer
33 views

Number Operator closable on Fock Space?

In Bratelli Robinson the number operator in Fock space is defined as: $$\mathcal{D}(N):=\{\phi\in\mathcal{F}:\sum_{n=1}^\infty n|\|\phi_n\|<\infty\}\\ N:\mathcal{D}(N)\to ...
4
votes
1answer
54 views

Surjection of norms

Let $V$ be an infinite dimensional $\mathbb{C}$ (or $\mathbb{R}$) vector space. Suppose there exists two norms on $V$ such that \begin{equation*} \| \cdot\|_1 \leq \| \cdot \|_2. \end{equation*} Is ...
2
votes
2answers
27 views

Bounded, surjective linear operator between Banach spaces

How can I show that for a given surjective linear operator $T: X \to Y$ between Banach spaces, if there exists an $\epsilon > 0$ such that $||Tx|| \geq \epsilon||x||$ for all $x \in X$, then $T$ is ...
1
vote
0answers
41 views

Find the adjoint operator.

Consider the sequence space $\ell_p$ and S defined by $(1\leq p<\infty)$$$ S:\ell_p\to\ell_p:(x_1,x_2,x_3,\ldots)\mapsto(0,x_1,x_2,\ldots) $$ Find the $S^*$ operator.
3
votes
2answers
44 views

Computing the norm of operator when space is equipped with sup norm and $L^1$ norm

Let $\phi $ be the linear functional $\phi (f)=f(0)-\int^1_{-1}f(t)\:\mathrm{d}t$ a.Compute the norm of $\phi$ as a functional on Banach space $C[-1,1]$ with sup norm. b.Compute the of $\phi$ as a ...
4
votes
3answers
84 views

What is a predual of the Banach space of compact operators on $\ell^2$?

I am wondering if the space $K(\ell^2)$ of compact operators on $\ell^2$ can have a predual. Thank you in advance for your help.
1
vote
0answers
39 views

definition of unitary operator

Wiki says " A bounded linear operator $U: H \to H$ on a Hilbert space $H$ is called a unitary operator if it satisfies $U^{*}U=UU^{*}=I$ , where $U^{*}$ is the adjoint of $U$, $I$ is the identity ...
0
votes
0answers
40 views

Spectrum of a bounded operator and Liouville's theorem

Let $X$ be a Banach space and $A:X\to X$ be a bounded operator. We know that the spectrum of $A$ is not empty, because otherwise we find a contradiction by using the holomorphy of the function ...