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.

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Generating a contraction semigroup on an energy space

Consider the system of partial differential equations $\displaystyle\frac{\partial Q}{\partial t}(\zeta, t)=-\frac{\partial}{\partial\zeta}\frac{\phi(\zeta, t)}{L(\zeta)}$ ...
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Schrodinger Operator with Finite Discrete Spectrum in $(-\infty, -1]$

I'm reading parts of Reed and Simon's Analysis of Operators and have come across a statement I find puzzling. They say that if $V$ is a bounded function of compact support on $\mathbb{R}^3$ then ...
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closed graph theory and unbounded operator

I am studying unbounded operators and the graphs of those operators. I found that the closure of a graph may not be the graph of any operator. Can someone provide an example of an operator and a ...
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Primitive ideal space of C*(Z2*Z2)

Find the primitive ideal space, the center, a continuous field of $C^*(Z_2*Z_2)$. Here, $C^*(Z_2*Z_2)$ is the full group $C^*$-algebra. I know the definitions of all of them, but I'm having hard ...
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27 views

Numerical range of inverse operator

Let $T$ be a bounded self-adjoint operator such that the numerical range is contained in $[a,b]$ with $0<a<b< \infty.$ Does it then follow that the numerical range of $T^{-1}$ is contained in ...
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Using Bounded Operator sequence Theorem

Let E$\subset L_1$ be a set of fourier series functions $e_n(t)=e^{int}$ for $n \in Z$. What is meant by saying to prove $Ge_n$ is a scalar multiple of $e_n$ and it is continuous? How can we prove it? ...
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Composition with a projection remain surjective in a neighborhood of the parameter

Let $H$ be an Hilbert space and $\varphi:H\to \mathbb R^m$ a smooth map. It is known that the map $u\mapsto d_u\varphi$ is continuous from $H$ to the space of linear operators $L(H,\mathbb R^m)$. ...
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Explicit inverse of $\lambda-U$ when $U$ is unitary and $|\lambda|<1$

Let $U$ be a unitary operator on a Hilbert space $\mathcal{H}$. By the spectral theorem, it is known that $\sigma(U)\subseteq \{z\in \mathbb{C}:|z|=1\}$. How can the explicit inverse of $\lambda-U$ be ...
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Universal properties of certain crossed products

So I was wondering if there are any nice universal properties that the crossed product $C^*$ algebra, $C(\mathbb{T})\times_\alpha \mathbb{Z}_2$ satisfies, where $\alpha$ is the action of conjugation. ...
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Normal Operators: Retransform

Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}(N)\to\mathcal{H}:\quad N^*N=NN^*$$ Construct the operator: $$Q:=(1+N^*N)^{-1}:\quad Z:=N\sqrt{Q}$$ By the previous ...
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42 views

Bounded Operator Norm: Special Element

Given a Banach spaces $X$ and $Y$. Consider a bounded operator: $$T:X\to Y:\quad\|T\|<\infty$$ Then theres an element: $$\|Tx\|=\|T\|\cdot\|x\|\quad(x\neq0)$$ Does it always exist?
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Reducing subspaces of a normal operator

If $A$ is a normal operator on an infinite dimensional Hilbert space $H$, then $H$ is the direct sum of a countably infinite collection of subspaces that reduce $A$, all with the same infinite ...
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55 views

Is the space of bounded linear operators from E (space with an inner product) to C (complex numbers) a Hilbert space?

In other words is there an inner product that produces the operator norm? Let $E$ be a space with an inner product. Show that its topological dual $E^*$ equiped with the operator norm is a Hilbert ...
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39 views

Norm of integral operator in $L^1(0,2)$

How exactly do I show that an integral operator is bounded. For example, consider the following operator $$ T: L^1(0,2) \to L^1(0,2)\\ (Tf)(x):=\int_0^x tf(t) dt$$ My first approach \begin{align} ...
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Normal Operators: Transform

Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}(N)\to\mathcal{H}:\quad N^*N=NN^*$$ Construct the operator: $$Q:=(1+N^*N)^{-1}:\quad Z:=N\sqrt{Q}$$ Then it is ...
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Weak continuous convergence of operators

Let $T_n$ and $T$ be linear maps from Banach space $X$ to a Banach space $Y$. Suppose $T_n$ satisfies $T_nx \to Tx$ (convergence in the $Y$ norm) for all $x \in X$. Let $x_n \rightharpoonup x$ in ...
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Weak continuous convergence of operators

Let $T_n$ and $T$ be linear maps from Banach space $X$ to a Banach space $Y$. Suppose $T_n$ satisfies $T_nx \to Tx$ (convergence in the $Y$ norm) for all $x \in X$. Let $x_n \rightharpoonup x$ in ...
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A calculation for an open ball in $\mathbb{R}^N$ and function space.

Let $B_r(x)$ denoting the ball of center $x$ and radius $r>0$. We denote by $\lambda_{1,\,B_\rho(y)}$ the first eigenvalue of $-\Delta$ in $W^{1,\,2}_0\left(B_\rho(y)\right)$ and by ...
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33 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 ...
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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 ...
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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!
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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} ...
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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 ...
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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 $$T[f](u)=\int_{\Bbb R^d} |u-v|^\beta f(v) d\mu(v).$$ I want to establish conditons on $\beta\in\Bbb ...
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51 views

Normal Operators: Draft

Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}(T)\to\mathcal{H}:\quad N^*N=NN^*$$ Construct the operator: $$A:=N(1+N^*N)^{-1}\in\mathcal{B}(\mathcal{H})$$ Then it ...
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90 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 ...
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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.
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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 ...
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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 ...
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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 ...
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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, ...
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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: ...
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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 ...
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19 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 ...
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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 ...
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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: ...
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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: ...
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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^*\ ...
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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: $$ ...
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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 ...
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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, ...
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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 ...
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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 ? ...
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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 ...
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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 ...
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81 views

Wave Operators: Calculus

Given Hilbert spaces $\mathcal{H}_0$ and $\mathcal{H}$. Consider Hamiltonians: $$H_\#:\mathcal{D}(H_\#)\to\mathcal{H}_\#:\quad H_\#=H_\#^*$$ Denote their evolutions: ...
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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 ...
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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 ...
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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 ...