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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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Sum operator precedence

I'm trying to read some simple equations and in order to interpret them in the right way I need to know $\sum$ and $\prod $ operator range/precedence. $$ \sum p(s, a) +\gamma $$ is equal to $\sum(p(...
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Relation between spectrum of an operator and its cut down

Let $T$ be a self-adjoint operator in $\mathcal{H}$ with spectrum $\sigma(T)$, Let $P$ be a projection in the commutant of $\text{vN}\{T\}$, the von Neumann algebra generated by $T$, question what is ...
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(continuous or essential-) spectrum of wave equation with dirichlet b.c

I want to know if the continuous spectrum for the wave equation(I make it precise soon) has parts on the imaginary axis, where continuous spectrum is here defined for a closed linear operator $T$ in a ...
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1answer
23 views

a question on the definition of direct sum of $C^*$ algebras

According to the definition in the Olsen's book,if $A=B\bigoplus C$,the intersection of $B$ and $C$ should be zero.But in other reference books ,when talking about direct sum of matrix algebras,there ...
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1answer
26 views

If $T(t)$ is an immediately differentiable semigroup on $H$ with generator $A$, does $\frac{d}{dt}\|T(t)x\|_H^2=2⟨AT(t)x,T(t)x⟩_H$ hold for all $x∈H$?

Let $(T(t))_{t\ge0}$ be a semigroup on a $\mathbb R$-Hilbert space $H$ with $$\sup_{s\in[0,\:t)}\left\|T(s)\right\|_{\mathfrak L(H)}<\infty\tag1$$ for some (and hence all) $t>0$ and $(\mathcal D(...
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2answers
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Geometric proof for Composition bound property of operator norms?

This is just a curiosity. For linear transformations $A$ and $B$, $||AB|| \le ||A|| \cdot ||B||$ where$||\cdot||$ denotes the operator norm (Of course provided $AB$ exists.) This fact has a proof, but ...
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1answer
14 views

GNS representation of a nuclear $C^*$-algebra

Suppose $A$ is a nuclear $C^*$-algebra with a tracial state $\psi$, $(\pi_{\psi},H_{\psi})$ is the GNS reprsentation with respect to $\psi$. My question: Does there exist $A$ which satisfy the above ...
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25 views

Adjoint of Bounded Below Operator

Let $H$ be a complex, separable Hilbert space, and $T:H \rightarrow H$ a linear, bounded operator. Assume that $$\sigma(T) = \sigma(T^*) = \{ \lambda \in \mathbb{C}: a \leq |\lambda| \leq b \}$$ for $...
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1answer
10 views

find all ideals of direct sum of simple $C^*$ algebras

Suppose $A=\bigoplus A_n$ where each $A_n$ is a simple $C^*$ algebra(The direct sum is $c_0$ direct sum).I guess all the ideals of $A$ are precisely the direct sum of ideals of $A_n$ .It is easy that ...
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1answer
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An operator exponential/commutator question

There is "an important lemma" related to the Baker-Campbell-Haussdorff theorem which says that $$ e^XYe^{-X} = Y + [X,Y] + \frac{1}{2!}[X,[X,Y]]+\ldots $$ Clearly if $[X,Y]=0$ we get (noting that $e^...
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1answer
20 views

Circled minus sign in Operator theory

I found "circled minus" sign when I read one paper in Operator theory. What does that means? It was $H^{2}\ominus u H^{2}$. What does this mean and where can I find explanation for this?
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1answer
17 views

GNS representation

Suppose $A$ has a tracial state $\psi$, I want to prove $A/ker(\pi_{\psi})$ has a faithful tracial state,where $\pi_{\psi}$ is the $GNS$ respresentation with respect to $\psi$. My thought: define $\...
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32 views

What is the usual definition of the spectral measure for a nonnegative self-adjoint operator on a Hilbert space?

Let $H$ be a $\mathbb R$-Hilbert space and $(H_\lambda)_{\lambda\ge0}$ be a spectral decomposition of $H$ (see below). Now, let $$\mathcal D\left(A_\varphi\right):=\left\{x\in H:\int_0^\infty\varphi(\...
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Shift operator on the double-size Hilbert space $\ell^2(\mathbb{N}^*)\oplus \ell^2(\mathbb{N}^*)$

It is well known that, the right shift operator is given by \begin{align*} A_1\colon \ell^2(\mathbb{N}^*) & \rightarrow \ell^2(\mathbb{N}^*) \\ (x_1,x_2,\cdots)&\mapsto (0,x_1,x_2,\cdots), \...
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1answer
35 views

The tensor product of two blocks of positive operators is positive

Let $$T = \begin{bmatrix} T_{11} & T_{12}\\ T_{21} & T_{22} \end{bmatrix},\quad S = \begin{bmatrix} S_{11} & S_{12}\\ S_{21} & S_{22} \end{bmatrix}$$ be two positive operators on $E\...
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29 views

existence of maximal ideals in $C^*$ algebras [on hold]

If $A$ is a $C^*$ algebra,$A$ has a non-trivial ideal,can we conclude that there exists a maximal ideal in $A$?If it exists,is it closed?
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35 views

Show that the operator associated to a spectral decomposition on a Hilbert space is self-adjoint

Let $H$ be a $\mathbb R$-Hilbert space and $(H_\lambda)_{\lambda\ge0}$ be a spectral decomposition of $H$ (see below). Now, let $$\mathcal D\left(A_\varphi\right):=\left\{x\in H:\int_0^\infty\varphi(\...
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1answer
19 views

Finding the inverse of a linear transformation

Let $A:P_3 \to P_3$ be linear operator such that $$Ap(x)=\int_0^1p(x+t)dt$$ where $p \in P_3$. Find $A(e)$ if $(e)=\{1,x,x^2,x^3\}$ and $A^{-1}(2x-x^3)$ I just started learning about linear operators ...
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1answer
25 views

Show that the generator of a strongly continuous contraction semigroup on $L^2$ is nonpositive definite

Let $(E,\mathcal E,\mu)$ be a finite measure space, $(T(t))_{t\ge0}$ be a strongly continuous contraction semigroup on $L^2(\mu)$ and $(\mathcal D(A,A)$ denote the generator of $(T(t))_{t\ge0}$. ...
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1answer
63 views

If $T(t)$ is a semigroup on $E$ and $F$ is a subspace of $E$ such that $T(t)$ is $F$-preserving, how are the generators on $E$ and $F$ related?

Let $E$ be a $\mathbb R$-Banach space, $(T(t))_{t\ge0}$ be a semigroup on $E$ and $(\mathcal D(A),A)$ denote the generator of $(T(t))_{t\ge0}$. If $F$ is a closed subspace of $E$ and $T(t)F\...
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1answer
19 views

Proof of Lemma preceding Principle of Condensation of Singularities

Under the Wikipedia page for the Principle of Uniform Boundedness, we have the Corollaries of the Uniform Boundedness Principle. The third of these relates to the Principle of Condensation of ...
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1answer
35 views

What are the approximate eigenvalues of the right shift operator $R$ on $\ell_\infty$

I have shown that the spectrum of $R=\{z\in C||z|\leq 1\}$. Also, elements on the boundary of the spectrum are approximate eigenvalues, i.e. $\forall |z|=1$, $z$ is an approx. eigenvalue. However, ...
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2answers
29 views

Does a bounded linear operator on a Hilbert space conjugate to its adjoint?

Let $H$ be a Hilbert space over $\mathbb R$ or $\mathbb C$ and $f\in B(H)$. I wonder if there is a $g\in B(H)$ such that $fg=gf^*$, where $f^*$ is the adjoint of $f$. We know a matrix over a field is ...
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2answers
24 views

$PQ = 0 \Leftrightarrow P(H) \bot Q(H)$ if $H$ is Hilbert and $P,Q$ orthogonal projections.

Exercise : Let $H$ be a Hilbert space and $P,Q \in \mathcal{L}(H)$ orthogonal projections. Show that : $$PQ = 0 \Leftrightarrow P(H) \bot Q(H)$$ Attempt-Thoughts : $(\Rightarrow)$ Let $PQ = 0$. ...
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14 views

There exists a continuous inverse of $(\text{id}-A)$ in the set $(\text{id}-A)(X)$.

Exercise : Let $X$ be a Banach space and $A \in \mathcal{L}_c(X)$ (means that $A$ is a compact operator). Suppose that $(\text{id}-A)$ is $"1-1"$. Show that the operator $(\text{id}-A)$ has a ...
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If a closable linear operator satisfies the nonnegative maximum principle, does the same apply to its closure?

Let $E$ be a locally compact Hausdorff space and $(\mathcal D(A),A)$ be a closable linear operator on $C_0(E)$. If $(\mathcal D(A),A)$ satisfies the nonnegative maximum principle$^1$, does the same ...
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1answer
23 views

Show that $\exists u_0 \in C : g(u_0) = u_0$, if $g$ is nonexpansive over a Banach subspace.

Exercise : Let $X$ be a Banach space, $C \subseteq X$ compact and convex and $g : C \to C$ a nonexpansive operator. Show that $\exists u_0 \in C : g(u_0) = u_0$. Thoughts : In a previous exercise, ...
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1answer
20 views

Heat semigroup representation

It is known that the Laplacian operator with Dirichlet boundary condition in $L^2(\Omega),$ (with $\Omega$ being a open subset of $R^n$) generates a $C_0-$semigroup in $L^2(\Omega)$). Moreover, in ...
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Weak compacity and reflexive subset

Is that true that any closed subspace of a weakly compact set is also weakly compact ? Can we have that : A closed subspace of a weakly compact is reflexive ?
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If $R_\lambda$ is the resolvent of a linear operator $A$ at a regular value $\lambda$, what is $R_\lambda(\lambda-A)$?

Let $E$ be a $\mathbb R$-Banach space, $(A,\mathcal D(A))$ be a linear operator and $\lambda\in\mathbb R$ such that $$A_\lambda:=\lambda\operatorname{id}_{\mathcal D(A)}-A$$ is injective, $A_\lambda\...
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Showing that $\inf \{\|u-g(u)\| : u \in C \} = 0.$

Exercise : Let $X$ be a Banach space and $C \subseteq X$ be closed, convex and bounded. Moreover, let $g:C \to C$ be a non-expansive operator, meaning that : $$\|g(u) - g(v) \| \leq \|u-v\| \; \...
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1answer
36 views

If $f\in C_0(E)$ with $\inf_{x\in E}f(x)<\infty$, then there is a $x\in E$ with $f(x)\le\min(f,0)$

Let $E$ be a locally compact Hausdorff space and $$C_0(E):=\left\{f\in C(E):\left\{|f|\ge\varepsilon\right\}\text{ is compact for all }\varepsilon>0\right\}.$$ Let $f\in C_0(E)$ with $$\inf_{x\...
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1answer
20 views

$PQ \; \text{orthogonal projection} \; \Leftrightarrow PQ = QP$

Exercise : Let $H$ be a Hilbert space and $P,Q \in \mathcal{L}(H)$ are orthogonal projections, then show that : $$PQ \; \text{orthogonal projection} \; \Leftrightarrow PQ = QP$$ Seeking a formal ...
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1answer
13 views

exponential null operator

In $A$ banach algebra with unit, and $X\in A$. if i define $e^X=\sum_{n=0}^{\infty} \frac{1}{n!}X^n$ why $e^0=Id$ , i am aassuming $O^0=Id $ with $0$ null operator thanks
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24 views

Generation theorem for Feller semigroups

Let $E$ be a locally compact Hausdorff space. I want to show that a linear operator $(\mathcal D(A),A)$ on $C_0(E)$$^1$ is closable and the closure $(\mathcal D(\overline A),\overline A)$ is the ...
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1answer
23 views

If $\limsup\limits_{n\rightarrow \infty} a_n \leq a$ and $a>b>0,$ is it true that $\limsup\limits_{n\rightarrow \infty} a_n \leq b?$

If $\limsup\limits_{n\rightarrow \infty} a_n \leq a$ where $a_n\geq 0$ for all $n\in \mathbb{N},$ and $a>b>0,$ is it true that $\limsup\limits_{n\rightarrow \infty} a_n \leq b?$ For some ...
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If $\mathcal D(\overline A)$ is dense, are we able to conclude that $\mathcal D(A)$ is dense?

Let $(\mathcal D(A),A)$ be a closable linear operator on a $\mathbb R$-Banach space $E$ and $(\mathcal D(\overline A),\overline A)$ denote its closure. If $\mathcal D(\overline A)$ is dense, are we ...
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1answer
23 views

How are the two versions of the Lumer-Phillips theorem given in the books of Engel/Nagel and Pazy related?

Let $E$ be a $\mathbb R$-Banach space and $(\mathcal D(A),A)$ be a densely-defined dissipative linear operator on $E$. In the book of Engel and Nagel, I've found the following verison of the Lumer-...
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1answer
35 views

Show that there is a solution of the Laplace equation $(\mu-A)p=f$

Let $C_0(\mathbb R)$ denote the space of continuous functions vanishing at infinity equipped with the supremum norm, $B$ be a contractive linear operator on $C_0(\mathbb R)$ and $$Af:=\lambda(Bf-f)\;\;...
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1answer
48 views

Finite rank operators on Hilbert spaces

Let $H$ be a Hilbert space. Question 1: Are all rank one operators from $H$ to $H$ is of the form $$T:H\rightarrow H, x \mapsto \langle x,u\rangle v $$ For some $u,v \in H$. Question 2:...
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22 views

A question on the Non-degenerated bilinear form

Prove that $<C(G), C(G)>$ is dual system with bilinear form $$\int_G \phi(x)\psi(x)dx $$ and $\psi(x),\psi(x)\in C(G)$ so here i was trying bilinear form im not getting how to prove non-...
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2answers
39 views

Weak continuity of the addition and scalar multiplication

Let $X$ be an infinite dimensional normed vector space. Show that vector addition and scalar multiplication are weakly continuous. $$+:X×X \rightarrow X; +(x,y)=x+y$$ $$•:\mathbb {R}×X \rightarrow X; •...
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0answers
15 views

Eigenvalue of algebraic multiplicity $m$ is a pole of the resolvent of order $m$.

Let $X$ be a Banach space and $T \in \mathcal{L}(X)$ be a bounded linear operator. Suppose that for some isolated point $\lambda \in \sigma(T)$ and some $m \in \mathbb{N}$ we have $\ker(T-\lambda I)^m ...
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11 views

Matrix version of Scheffé's lemma

I am reading the paper An invitation to quantum tomography by R. Gill, L. Artiles and M. Guta. At some point they mention a "matrix version of Scheffé's lemma" and its statement should be something ...
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37 views

Why do we need projection in the definition of the Stokes operator?

$\DeclareMathOperator{\div}{div}$ $\def\bu{\mathbf{u}}$ Let $D$ be the square $[0,1]^2$ and consider the following space: $$ V:=\{\bu: \bu\in H^2(D)^2, \div \bu=0, u|_{\partial D}=0 \}. $$ Introduce ...
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0answers
35 views

Every Banach limit on $l_{\mathbb{C}}^{\infty}(\mathbb{N})$ is an extension of some Banach limit on $l_{\mathbb{R}}^{\infty}(\mathbb{N})$

Let $l_{\mathbb{C}}^{\infty}(\mathbb{N})$ be the space of bounded complex-valued sequences, $l_{\mathbb{R}}^{\infty}(\mathbb{N})$ the subspace of real-valued sequences. Given any Banach limit $L_1: l_{...
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35 views
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If $\kappa$ is a contractive operator on $C_0(ℝ)$, is $λ(κ-\text{id})$ the generator of a Feller semigroup?

Let $E$ be a locally Hausdorff space, $$C_0(E):=\left\{f\in C(E):\left\{|f|\ge\varepsilon\right\}\text{ is compact for all }\varepsilon>0\right\},$$ $\kappa$ be a Markov kernel on $(E,\mathcal B(E))...
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1answer
28 views

Free probability and projections

Let $(\mathcal{A},\varphi)$ be a free probability space, where $\mathcal{A}$ is a von Neumann algebra and $\varphi$ a finite and faithful trace. Let furthermore $p\in\mathcal{A}$ be a projection. ...
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1answer
23 views

Convergence of finite dimensional projection of trace class in trace norm

Assume $\mathbb{H}$ is a Hilbert space and $K$ is a trace-class operator on it. Given a fixed ONB $\{e_i\}$ and assume $$K=\sum_{i,j}c_{ij}e_i\otimes e_j.$$ Now, let $K_n = \sum_{1\leq i,j\leq n}c_{...
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15 views

Non-Negative Vs Positive Semi Definite

A matrix is PSD if $$\langle Ax, x\rangle \ge 0, \forall x \in H$$ Where, H is a hilbert space and A is a mapping $H \rightarrow H$. Is it the same as being Non-negative? I couldn't seem to find a ...