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Questions tagged [operator-theory]

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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Representing Vector Norm Through Operator Norms

Let $V$ be a vector space over $\mathbb R$ and $v\in V$. We endow $V$ with a norm $\|\cdot\|_V$ and $\mathbb R$ with the standard norm. I am trying to prove \begin{equation} \sup \{\lvert Lv\rvert \...
Joseph Expo's user avatar
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Non-injective closure of injective unbounded operator [duplicate]

Let $A\colon H\to H$ be an unbounded operator on a Hilbert space $H$ with domain dom$(A)$. Suppose that $A$ is injective. Suppose that $A$ is closable (or symmetric, even). Question: Does this imply ...
geometricK's user avatar
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Nonunital weak operator closed self adjoit subalgebra has a projection that acts like an identity operator?

Let $A$ be weak operator closed self adjoint subalgebra of the bounded operators for a Hilbert space $H$(may not have identity so cannot use usual von neumann alegbra properties). I have shown that if ...
3j iwiojr3's user avatar
1 vote
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Image of the closed unit ball is weak* dense

I'm aware that there are a few posts about this question here, but none of the proofs make the last step of this argument make any sense to me. Here is the problem statement: Let $X$ be Banach and $\...
Isochron 's user avatar
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double commutant of unitary operators of a Von Neumann algebra.

Let $M$ be a Von Neumann algebra. I am having trouble showing that the double commutant of unitary operators for $M$ is equal to $M$. I have shown it with projections using borel functioncal calculus. ...
3j iwiojr3's user avatar
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Convergence of integral operators' inverses

This is a cross-post from mathoverflow, where I didn't receive an answer so I'm trying my luck here. Let $K(x, y)$ be a sufficiently nice real-valued continuous kernel on $[0, 1]^2$. Let $K$ denote ...
tsnao's user avatar
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Asking for reference about self adjointness

Is there someone that knows and can share any reference about the compactness and the self-adjointness of the operator \begin{equation} \sigma\cdot L \end{equation} on $H^1(\mathbb{R}^{3},\mathbb{C}^{...
Davide's user avatar
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Lefschetz fixed point theorem-type formula for a finite group acting on a Fredholm operator

Let $D:X\to Y$ be a Fredholm operator, so that its index $\dim\ker (D)-\dim \text{coker}(D)$ is defined. We can view $D$ as a complex $\mathfrak{D}:0\to X\xrightarrow{D} Y\to 0$, and for a chain map $...
user302934's user avatar
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Reconstruction of an operator given the eigenfunctions and eigenvalues

I am interested in operator theory, in particular if I know the sequence of eigenvalues $\{\lambda_n\}_{n=1}^\infty \subset \mathbb{R}$ and eigenfunctions $f_n \subset X$ of a self-adjoint ...
mathematurgist's user avatar
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Symmetry of an operator

Let $\pmb{\sigma}=(\sigma_1,\sigma_2,\sigma_3)$ where $\sigma_1,\sigma_2,\sigma_3$ are the Pauli matrices, i.e. \begin{equation} \sigma_1= \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix},\, \...
Davide's user avatar
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isometries and unitary operators, Specht theorem

I was looking at the properties of the trace operator $\operatorname{tr}$, in particular the properties of the trace regarding isometric conjugation. We say that $T\in \mathcal{H}$ is an isometry if $...
ana's user avatar
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Does this integral operator between Banach spaces have a non-trivial kernel?

In a question which was recently asked, the goal was to (dis)prove that the set of functions which satisfy a certain equation is trivial. There is already a very neat solution there, but I came up ...
Stratos supports the strike's user avatar
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1 answer
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Definition of closure of operator

Let $\mathcal{H}$ be a Hilbert space and $A:\mathrm{dom}(A)\to\mathcal{H}$ be an unbounded operator. I have seen the following definition of the closure in a lecture: $$\mathrm{dom}(\overline{A})=\{x\...
B.Hueber's user avatar
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Problem of Fourier transform and compact operator

Problem: Is the Fourier transform $F(f(x))=\int_{-\infty }^{\infty }f(y)e^{-ixy}dy$ a compact operator in the case of $F:L_1\left ( \mathbb {R} \right )\to \mathrm{BC}\left ( \mathbb{R}\right )$. $\...
Dmitry's user avatar
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Prove that the operator is not compact

Problem: Prove that the operator $A\in\mathcal{L}\left ( C[0,1] \right ):(Af)(x)=f\left ( x^2 \right )$ is not compact My attempt at a solution: It is necessary that the image of the unit ball be ...
Dmitry's user avatar
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Vector Norm $\vert\vert v\vert\vert_V$ expressed as supremum of $\vert Lv\vert$ over all bounded operators L with $\vert\vert L\vert\vert_{op}\leq 1$

Let $V$ be a normed vector space with norm $\vert\vert\cdot\vert\vert_V$. How can I show that for all $v\in V$ we have $$\vert\vert v\vert\vert_V = \sup\{\vert Lv\vert \:\colon\: L\in \text{Hom}(V,\...
Apollo13's user avatar
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2 votes
1 answer
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Applying Spectral Mapping Theorem to determine if $f(T)$ is compact

Given a self-adjoint, compact operator $T$ and a continuous $f:\sigma(T)\to\mathbb{R}$, I'm trying to determine the conditions under which $f(T)$ is also compact. I know of the Spectral Mapping ...
mtcicero's user avatar
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Some questions about integration and operator theory.

As we all know there are multiple integral operators which all basically do the same thing in various contexts. I am talking about operators like the Lebegues integral, Riemann integral and more ...
Logarithmnepnep's user avatar
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1 answer
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continuouty of operators

I was given a task to understand, wheter operators $A$ and $B$ are compact, $$\displaystyle A:\ell_2 \rightarrow L_1(\mathbb{R}), (Ax)(t) = \sum\limits_{k=1}^{+\infty}\frac{x(k)}{\cosh^2(kt)},$$ $$B:...
GeoArt's user avatar
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Multiplication operator is a closed

Let $\phi$ be a holomorphic function on the unit disk $\mathbb{D}$. We define the multiplication operator $M_{\phi}$ on the following domain $D = \{f \in H^{2}(\mathbb{D}): \phi f \in H^{2}(\mathbb{D})...
liamsi Meean's user avatar
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2 answers
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Application of Stone's theorem to regular representation

Consider the left regular representation $$\lambda: \mathbb{R}\to U(L^2(\mathbb{R})), \quad (\lambda_x f)(y) = f(y-x), \quad x,y \in \mathbb{R}.$$ By Stone's theorem, there is a positive, self-adjoint,...
Andromeda's user avatar
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1 answer
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Spectral Permanence Remark in Murphy's C*-algebras

In Murphy's $C^{*}$-algebras book, he states theorem $2.1.11$ which is that if $\mathfrak{B} \subset \mathfrak{A}$ are $C^{*}$-algebras with $\mathfrak{A}$ unital such that $1_{\mathfrak{A}} \in \...
Isochron 's user avatar
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1 answer
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Derivative of an operator exponential.

The image is from the book Lie group by Bryan Hall. Question. There is some book where 5.10 is valid for operators? for example, I need 5.10 for $X=x$ (multiplication operator by $x$) and $Y=\...
eraldcoil's user avatar
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3 votes
1 answer
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uniqueness of the inversion to Riemann-Stieltjes integral equation

I believe that if, for a Riemann-Stieltjes integral with $h(s)$ of bounded variation, $$ \int_0^1 s^\alpha dh(s) = 0 \qquad\text{for any }\alpha\in(\alpha_0,\alpha_1) , \tag{1}\label{eq1}$$ then $h$ ...
Martin Lanzendörfer's user avatar
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1 answer
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Hilbert-Schmidt integral operator without second integrable kernel

From this webpage, we know that if $X$ is a measurable space ($\sigma$-algebra is omitted) and $\mu$ is the measure, $K(x,y) \in L_2(X\times X,\mu\times \mu)$, we can define the following operator ...
efsdfmo12's user avatar
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Moller Operator and the Determination of Bound States in Quantum Scattering Theory

I am trying to understand the Moller operator in quantum scattering theory. An important result concerning this operator is the following useful property that can be used to determine bound states. ...
Debbie's user avatar
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writing empirical covariance operator as the multiplication of sampling operator (elaboration on a paper)

I have been reading this paper (page 85) and have difficulty to understand that how the empirical version of the covariance operator is $\hat{C}_{XX} = \frac{1}{n} S_x^\ast S_x$ can be written as ...
domath's user avatar
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-1 votes
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Operator on a complex Hilbert space that has the same eigenvalues than its adjoint

Let $\mathbf T$ be a bounded linear operator on an infinite dimensional separable complex Hilbert space $\mathcal H$. I suppose that $\mathbf T$ and $\mathbf T^*$ have the same eigenvalues (point-...
Cyril Soler's user avatar
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The GNS represenation for a Von neumann algebra is a Von neumann algebra.

Let $M$ be a Von Neumann algebra and $\omega$ be a faithful $\sigma$-weakly continuous positive tracial state on $M$. I know that we have an inner product space on $M$ because $\omega$ is faithful. ...
3j iwiojr3's user avatar
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0 answers
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Multiplication Operator Version of Spectral Theorem in a Rigged Hilbert Space

(This is a reformulation of a deleted post which didn't quite reflect the confusion i have) Consider the setting of a Hilbert space $\mathscr{H}$. A function $f \in \mathscr{H}$ can be represented in ...
JohnAnt's user avatar
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1 answer
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Showing that $\lim_{h\to 0}\frac{1}{h}e^{\mu h}\int_0^h e^{-\mu t}A(t)xdt = A(0)x$

Let $A(t)\in\mathbb{C}^{n\times n}$ be a continuous matrix-valued function in $t$ (in matrix operator norm) for $t\geq 0$ such that $\|A(t)\| \leq Ce^{b t}$ for $b \in\mathbb{R}$ and some $C\geq 1$. ...
Epsilon Away's user avatar
1 vote
0 answers
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$\|B\|_{op} \le \|A\|_{op}$ when $B_{i,j} = v_i^TAv_j$

I'm currently stuck to the following statement. Let $A\in\mathbb R^{n\times n}$ be a diagonal matrix whose diagonal elements are only 0 or 1, and let $B$ be $n\times n$ matrix such that $B_{i,j} = ...
jason 1's user avatar
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1 answer
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Proof by induction in the Baker–Campbell–Hausdorff formula

Whilst studying Commutators, I stumbled across this post in which the Baker–Campbell–Hausdorff formula is being proven. In one of the answers, it is stated that a certain step can be proven by ...
haifisch123's user avatar
4 votes
1 answer
63 views

Solving $\mathrm{e}^{\partial_x}u(x)=f(x)$

\begin{align} \mathrm{e}^{\partial_x} u(x)=f(x),\quad x\geq 0 \end{align} The formal solution is $u(x)=\mathrm{e}^{-\partial_x}f(x)$ the fact $[ {-\partial_x}, {\partial_x}]=0$ (Lie bracket) implies ...
eraldcoil's user avatar
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1 vote
1 answer
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compact and trace operator [closed]

I am study compact and trace classes of operators in Hilbert space. To clarify, I asked the following question, but have not been able to resolve it yet. Consider $L_p \left( \mathbb{R}^d \right) $ ...
ets_ets's user avatar
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2 votes
1 answer
80 views

Extending Injectivity of a Fourier Multiplier Operator from $C^\infty_c$ to $L^2$ Using Density Arguments

Suppose an operator $T$ is well-defined on $L^2(\mathbb{R})$, but its injectivity is explicitly established for functions in the subspace $C^\infty_c$. Given the dense embedding of $C^\infty_c$ in $L^...
APIs's user avatar
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Affiliation of an unbounded operator with a von Neumann algebra

Let $\mathscr{B}([0,+\infty))$ denote the $*$-algebra of all Borel-measurable functions $f: [0, \infty)\to \mathbb{C}$ that are bounded on compact subsets. Given such a function $f$ and an unbounded, ...
Andromeda's user avatar
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1 answer
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Multiplication of two unbounded operators and functional calculus

Let $A$ be a positive, self-adjoint unbounded operator defined on a Hilbert space $H$. Let $f,g: [0, \infty]\to \mathbb{R}$ be Borel measurable functions that are bounded on compact subsets. We can ...
Andromeda's user avatar
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strong operator functional equvialent to weak operator continuous. A bit confused about a specific part in the proof.

I never really completely understood the first part of the proof for (iii) implies (i) even though I understood the rest of it. The thing that confuses me is that I know that a function $f:X \...
3j iwiojr3's user avatar
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2 answers
60 views

Positive operator and invertibility on Hilbert space

If $B(H)$ denotes the set of linear bounded operators, where $H$ be the infinite dimensional Hilbert space, then I have two following questions. a) If $A,B\in{B(H)},$ where $B$ is invertible and $A^*...
B.B.S's user avatar
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1 vote
0 answers
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When is $\Vert A+B \Vert = \Vert X + X^\dagger \Vert$ for $[A, X; X^\dagger, B] \succcurlyeq 0$?

Given $$ H = \begin{pmatrix} A & X \\ X^\dagger & B \end{pmatrix} \succcurlyeq 0 , $$ with $A \succcurlyeq 0$ meaning $A$ is positive semi-definite (and Hermitian) and $A, B, X$ are $n\...
Aritra Das's user avatar
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Sum of bounded linear operators on a Hilbert space is bounded

I am trying to prove that $B(\mathcal{H})$ is closed under operator addition using only the notion of boundedness that we have for topological vector spaces, and I am running into difficulties. I ...
lanf's user avatar
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0 answers
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Douglas-Rudin theorem: Approximation by inner functions [closed]

Douglas-Rudin theorem: Let $\phi\in L^{\infty}(\mathbb{T})$ with $|\phi|=1$ a.e. on $\mathbb{T}$. Then, for all $\varepsilon>0$ there exist inner functions $u$ and $v$ such that $$\|\phi-\bar{u}v\|...
Identicon's user avatar
1 vote
1 answer
73 views

What is the importance of the largest eigenvalue / spectral radius of a symmetric positive matrix being equal to 1? Particularly in attention.

It is often said, that if the spectral radius of a matrix $\boldsymbol{A}$ is equal to $1$, the matrix has "regularizing" properties for the matrix product $\boldsymbol{Ax}$ for a vector $\...
Philipp123's user avatar
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31 views

Bounded homomorphism

Let $H,K$ be Hilbert spaces and $X,Y$ be subsets of $\mathcal{B}(H,K)$ and $\mathcal{B}(K,H),$ respectively. By $[Y X]$ we denote the w*-closure of the linear span of the set consisting of operators $...
E.Papapetros's user avatar
1 vote
1 answer
32 views

Finding the conjugate operator of the following operator

Let $A$ be linear operator from $l^2$ to $l^2$ such that $Ax =y^0 \cdot \sum_{1}^{\infty}{x_k} $ , where $y^0 \in l^2$ — fixed element. Show, that conjugate operator $A^*$ exists and find it. Show, ...
Metal Sonic's user avatar
2 votes
1 answer
43 views

Eigendecomposition of the direct sum of two operator on Hilbert spaces

Let the (finite dimensional) Hilbert space $\mathcal{H}$ be the direct sum of $\mathcal{H}_A$ and $\mathcal{H}_B$. Let $A$ be a linear operator on $\mathcal{H}_A$ and $B$ be a linear operator on $\...
incud's user avatar
  • 107
2 votes
1 answer
59 views

Applying linearity of a differential operator to solve ODE

I'm attempting to solve the following ODE: $$xy''+y'-y=0$$ According to Frobenius' theorem, there exists a solution to this ODE in the form of a series given by the linear combination of two solutions ...
AlanFox86's user avatar
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0 answers
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When is a Symmetric Block Toeplitz Matrix Invertible?

Let $$ Q = \begin{bmatrix} Q_0 & Q_1 & Q_2 & \cdots\\ Q_{-1} & Q_{0} & Q_1 & \cdots\\ Q_{-2} & Q_{-1} & Q_0 & \cdots\\ \vdots & \vdots & \vdots & \ddots ...
Benjamin Tennyson's user avatar
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1 answer
30 views

Seeking Counterexamples: Bilinear Maps Continuous in Components but Not Globally in Non-Complete Normed Spaces

It can be shown that when X, Y, and Z are all Banach spaces (or at least when X or Y are Banach spaces) over the number fields R or C, and when B : X×Y →Z is a bilinear function, the continuity of B ...
Matrix AC's user avatar

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