Operator theory is the branch of functional analysis that focuses on bounded linear operators, but it includes closed operators and nonlinear operators. Operator theory is also concerned with the study of algebras of operators.

learn more… | top users | synonyms

1
vote
1answer
22 views

Bounded operator on $L^{2}(a,b)$

Let $p\in]1,\infty[$ and consider the mapping $$ T : L^{2}(-2,2) \to L^{2}(-2,2), \quad (Tf)(x):=xf(x)$$ I want to show that $T$ is bounded, $||Tf||_L \leq T ||f||_L $. So, $$ ||Tf||_L \leq ...
1
vote
1answer
21 views

Pure state on a C*-algebra

Let $\tau$ be a pure state on a C*-algebra $A$, $(\pi_\tau, H_\tau, \eta_\tau)=(\pi,H,\eta)$ be the corresponding cyclic representation of $\tau$, and $\xi$ a unit vector in $H_\tau$ such that ...
0
votes
0answers
10 views

definition of the spectral measure for the multiplication operator? [duplicate]

Which is the definition of the spectral measure for the multiplication operator? I have seen the question, where the answer was given but I didn't know the proof. Adriana,Thanks in advance!
2
votes
1answer
27 views

A representation of a $ C^{*} $-algebra.

I have a quick question about the representation theory of $ C^{*} $-algebras. A representation of a $ C^{*} $-algebra $ A $ is a $ * $-homomorphism $ \pi: A \to B(\mathcal{H}) $, where $ \mathcal{H} ...
0
votes
0answers
25 views

Wot topology on $B(H)$ is not metrizable

Let $H$ be a infinite dimensional Hilbert space and $B(H)$ be the space of bounded and linear operators on $H$. I know that weak operator topology (wot) and strong operator topology (sot) are ...
0
votes
0answers
5 views

Sufficient conditions on integration kernel for continuity of the integral operator

Suppose that we have a measure $d\mu(v)=e^{-|v|^2}dv$ on $\Bbb R^d$. We define a linear operator $$T[f](u)=\int_{\Bbb R^d} |u-v|^\beta d\mu(v).$$ I want to establish conditons on $\beta\in\Bbb R$ so ...
2
votes
1answer
23 views

Normal Operators: Construction

Problem Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}(T)\to\mathcal{H}:\quad N^*N=NN^*$$ Construct the operator: $$A:=N(1+N^*N)^{-1}\in\mathcal{B}(\mathcal{H})$$ ...
1
vote
1answer
49 views

A rather abstract strongly continuous semigroup

Define $X$ as the Hilbert space $L^{2}(0,\infty)$ and let the operators $T(t):X\to X$, $t\ge 0$ be defined by $(T(t)f)(\zeta):=f(t+\zeta)$ I want to show that $(T(t))_{t\ge 0}$ is a ...
-2
votes
0answers
23 views

extension of bounded linear operators

If $T$ is a bounded operator from $H_1$ to $H_2$ with a dense domain, then there exists a unique bounded extension $S$ of $T$ which is defined on the whole of $H_1$. This extension is also a bounded ...
0
votes
1answer
28 views

When $\lim_{n\to \infty} \log(f_n(x)) =\log\left(\lim_{n\to \infty} f_n(x)\right)$?

Is there something equivalent to the dominated convergence theorem here? Is it a silly question revealing my ignorance of the commutitivity of the limit operator? Thanks in advance.
0
votes
0answers
16 views

First order elliptic pseudodifferential operator and Sobolev space

Let $P$ be an elliptic pseudodifferential operator of order 1 on a compact manifold $M$. Also let the spectrum of $P$ lie inside $(0, \infty)$. I am trying to see how one could prove that $P : H^1(M) ...
0
votes
1answer
29 views

Convergence of spectrum with multiplicity under norm convergence

This article by Joachim Weidmann claims that, if a sequence $A_n$ of bounded operators in a Hilbert space converges in norm topology, i.e., $\|A_n - A\| \rightarrow 0$, then "isolated eigenvalues ...
-1
votes
0answers
32 views

Books for Arvesion's extension theorem [on hold]

I am going through the proof of Arveson's extension theorem. I am following the book by Vern Paulsen. Please suggest some other references for this topic , any lecture notes or book or lecture with ...
0
votes
0answers
8 views

Determining spectral bounds variationally.

I'm learning C0-semigroup theory (mainly from Arendt et al. (vector-valued Laplace-transforms and Cauchy problems), Engel & Nagel (One par. semigroups for linear evolution eq.),Evans (partial ...
2
votes
1answer
28 views

Stochastic kernel as linear operator

Let $K$ be a stochastic kernel for a set $S$ equipped with a countably generated $\sigma$-Algebra $B(S)$, i.e. $K:S\times B(S)\rightarrow [0,1]$ such that $K(\cdot,A)$ is a measurable function for ...
0
votes
0answers
14 views

Does Schatten-p (quasi-)norm satisfy the norm inequality for 0<p<1?

I'm reading the paper by ANGELIKA ROHDE AND ALEXANDRE B. TSYBAKOV, ESTIMATION OF HIGH-DIMENSIONAL LOW-RANK MATRICES. And in the paper, they provide an inequality of the Schatten-p (quasi-)norm, ...
0
votes
0answers
10 views

Selfadjoint Operators: Relative Boundedness

Problem Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Regard an operator: ...
1
vote
0answers
25 views

Self-adjoint operator and vertex conditions in quantum graphs

Let $\Gamma$ be a metric graph with finitely many edges. Consider the operator $H$ acting as $\frac{-d^2}{dx_e^2}$ on each edge $e$, with the domain consisting of functions that belong to $H^2(e)$ on ...
0
votes
0answers
11 views

Approximate unit for an ideal and its limit

Let $A$ be a C*-algebra and $I$ is a closed ideal of $A$. If $(\pi, H)$ is a cyclic representation of $A$, Could we show $$\pi(u_i)\xi \to \xi $$ where $\{u_i\}$ is an approximate unit for $I$ and ...
1
vote
3answers
36 views

Injectivity of $T:C[0,1]\rightarrow C[0,1]$ where $T(x)(t):=\int_0^t x(s)ds $

Prob. 2.7-9 in Erwin Kreyszig's "Introductory Functional Analysis with Applications": Is this map injective? Let $C[0,1]$ denote the normed space of all (real or complex-valued) functions defined and ...
0
votes
1answer
20 views

The linearness of extension of linear bounded operator

Let we have the following bounded linear operator $$T: D(T)\rightarrow Y$$ such that $D(T)$ is the domain and it is a vector space and $Y$ is a Banach space . Then it has an extension $$G: ...
-1
votes
2answers
40 views

Some question about extension of bounded linear operator

Let we have the following bounded linear operator $$T: D(T)\rightarrow Y$$ such that $D(T)$ is the domain and it is a vector space and $Y$ is a Banach space . Then it has an extension $$H: ...
0
votes
0answers
17 views

Diffusion semigroup generated by Laplacian [closed]

Let $M$ be a complete Riemannian manifold and $\Delta$ denote the Laplacian on it. Also assume that the spectrum of $-\Delta$ lies inside $[a, \infty)$. Let $P_t, t > 0$ denote the diffusion ...
0
votes
0answers
21 views

Commutative Operators from QM

In Theoretical Chemistry, there seems to be a lot of assumptions about mathematics that are incorporated without justification. One example that I found questionable is this: $$\int \Psi_1^*\ ...
-3
votes
1answer
48 views
2
votes
1answer
26 views

Prove operator is isometry

Let $(X,\mathcal{A},m,T)$ be a probability preserving transformation. Prove that the operator $U:f\mapsto f\circ T$ satisfies $$ \|Uf\|_{p}=\|f\|_{p} $$ for every $1\le p<\infty$. My idea: $$ ...
0
votes
2answers
37 views

Continuous Linear Operator in $\mathbb{R}$- normed spaces.

Let $E$ and $F$ $\mathbb{R}$-normed spaces and let $f:E\longrightarrow F$ satisfying: $f(x+y)=f(x)+f(y) \,\forall x,y\in E$; $f$ is bounded in the unit ball $B_E = \{x\in E: \Vert x\Vert ...
1
vote
0answers
16 views

Domain of $A^{1/2}$ on $L^2(\mathbb{T}^2)$

Let $A$ be a densely defined unbounded self-adjoint operator defined on $L^2(\mathbb{T}^2)$, where $\mathbb{T}^2$ stands for the 2-torus. It is known that $A$ is positive, that is, $\langle Au, ...
3
votes
1answer
25 views

Self-adjointness under relatively bounded perturbation

Let $T$ be a densely defined linear operator on a Banach space $X$. Another operator $A$ satisfying $\mathcal{D}(T) \subset \mathcal{D}(A)$ is called a relatively bounded perturbation of $T$ if ...
1
vote
2answers
45 views

What happend if the divergence of a vector field is zero?

I just want to be sure if I'm wrong or not, I want to know what happend for a vector fiel if his divergence is zero ? Are the vectors have all the same lengh ? Or maybe are they all time parallel ? ...
0
votes
1answer
23 views

Spectral projection and isolated point of spectrum

Let $u\in B(H)$ be a normal element with spectral resolution of the identity $E$ and $\lambda$ be an isolated point of spectrum $u$. Show that $E(\lambda)H = \ker(u-\lambda)$ . I can show that ...
-1
votes
0answers
15 views

Is this operator continuous? Evaluating norm of this operator.

a) $X = L^1([0,2])$, $Y = l^{\infty}([0,2])$, $(A[f])(t) = \int_0^t f(s) ds$, where $[f]$ is abstract class of functions which are equal almost ewerywhere and $0 \le t \le 2$. b) $X = C([0,1])$, $Y = ...
0
votes
0answers
40 views

Showing that if any non-zero $f \in X^*$ takes its maximum value on the unit sphere at most once, then X is strictly convex

Let $(X, \| \|)$ be a normed space. I'm trying to show that if all non-zero $f \in X^*$ take their maximum value on the unit sphere at most once, i.e. $\forall f \in X^* - \{0\}$ there is at most one ...
1
vote
0answers
31 views

Møller Operators: Functional Calculus

Problem Given Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$. Consider Hamiltonians: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ $$K:\mathcal{D}(K)\to\mathcal{K}:\quad K=K^*$$ and a bounded ...
2
votes
1answer
23 views

On closed ranges and sequences which converge to zero

I'm reading a proof of the Fredholm alternative, and there is a claim that goes like this: Let $K:X\rightarrow X$ be a compact linear map. Define $T=I-K$, then $Y=\ker(T)$ is a finite dimensional ...
1
vote
0answers
11 views

Strictly positive element in a C*-algebra

Searching about strictly positive elements, I found this exercise. I tried to solve it, and the following is my attempt. Please check my proof. Is it correct? Suppose $a$ is strictly positive. By ...
-1
votes
0answers
40 views

Bounds for spectrum of self-adjoint operator on Hilbert space

$A$ is an self-adjoint bounded operator on Hilbert Space $H$, that is for all $x,y\in H$, $(Ax,y)=(x,Ay)$. $(~,~)$ is inner product of H. $$ m=\inf\limits_{||x||=1}(Ax,x) ~~~~~ ...
5
votes
0answers
31 views

Spectrum of periodic schrödinger operators

In many articles it's stated, as if it's common knowledge, that any Schrödinger operator with periodic potenial has purely absolutely continuous spectrum. I've tried to actually find a theorem ...
-1
votes
1answer
23 views

Distance between a density operator and a pure quantum state.

Given density operators $\rho_1$ and $\rho_2$ and a pure quantum state $|\psi>$. It is promised that $|\psi>$ is in only one of the given density operators. How to find which density operator ...
2
votes
1answer
24 views

Norm of Operator Proof

I'm stuck on this problem that I can't seem to figure out. Here's the problem. To note, equation 2.42 says that $$||T|| = \sup \{ ||Tu||: u \in C([a,b]), ||u|| = 1 \}$$ where $T$ is defined, ...
0
votes
0answers
39 views

how to solve Dirac Equation numerically?

The effective Hamiltonian for my system is: \begin{equation} H=\nu_{F} {\bf \sigma}\cdot\left(q-By\hat x\right) \end{equation} where ${\bf \sigma}$ and $q$ are the Pauli matrices and the momentum ...
0
votes
0answers
15 views

States: Density

Problem Given a Hilbert space $\mathcal{H}$. Regard the CAR-algebra: $$\{a(\eta),a(\zeta)\}=0\quad\{a(\eta),a(\zeta)^*\}=\langle\eta,\zeta\rangle$$ Consider a density: ...
1
vote
1answer
47 views

Linear and nonlinear operator on normed space and its properties

We know every bounded linear operator is continuous operator , and every continuous linear operator is bounded operator , in other words the continuousness and boundedness are equivalent in linear ...
0
votes
0answers
16 views

Example of an ideal in $C(\Bbb D)$ that is not self adjoint

Give an example of an ideal in the C*-algebra $C(\Bbb D)$ that is not self adjoint. My attempt: The function $f: \Bbb D \to C$ such that $f(t) := t+i$ belongs to $C(\Bbb D)$. Let I be the ideal ...
1
vote
2answers
31 views

Distance preserving function on a Hilbert space

Let $\Bbb F = \Bbb R$. Show that every preserving function $f$ on Hilbert space $H$ has the form $f(x) = f(0) + Tx$ for some isometry $T$ in $B(H)$. If $f$ is linear then $f$ is an isometry. Suppose ...
4
votes
0answers
67 views

Are “most” operators on an infinite-dimensional complex Banach space “diagonalizable”? [closed]

This is true for finite-dimensional spaces, of course. To be precise, let $T$ be an operator on a complex Banach space $X$ which is not finite-dimensional. For each $\lambda \in \mathbb{C}$, let ...
0
votes
0answers
14 views

restriction of irreducible representation to an ideal is irreducible

Let $A$ be a C*-algebra and $I$ a closed left ideal of $A$. Show that if $\{\pi,H\}$ is an irreducible representation of $A$, then the restriction of $\pi$ to $I$ is either zero representation or ...
0
votes
0answers
29 views

Sobolev spaces on compact manifolds

Let us consider a self-adjoint elliptic pseudodifferential operator $P \in OPS^2$ on a compact manifold $M$ such that $spec(P) \subset (0, \infty)$. Is the norm $(Pu, u)^{1/2}$ on $H^1(M)$ equivalent ...
1
vote
1answer
24 views

Domain of square root of a self-adjoint positive operator

Let $A \geq 0$ be a densely defined self-adjoint positive operator on a Hilbert space $H$ obtained by Friedrichs extension, and let $Q$ be the densely defined quadratic form associated to $A$, that ...
0
votes
1answer
26 views

Inversion of differential operator

My goal is to solve the differential equation, written in the following form $$\Big(\frac{d}{dx}+I\Big)^{2n}V(x)=x+C$$ where $C$ is some constanst. I want to do it by the operator method. Namely one ...