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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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54 views

Showing that $(Ax)(t) = x'(t)$ is a Fredholm operator [closed]

Can you help me to prove that the operator $A \colon C^1[0,1] \rightarrow C[0,1]$, $$(Ax)(t) = x'(t)$$ is a Fredholm operator and find $\alpha(A) = \dim(\operatorname{Ker} A)$ and $\beta(A) = \...
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1answer
32 views

Matrix transformation and boundedness

I have seem in books where they use the fact that a matrix transformation defined everywhere must be bounded. Can someone help me in understanding why this is true
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2answers
29 views

Does this result holds regarding unitization of $C^*-$ algebra?

For any $C^*$-algebra $V$, let $V^+$ denotes the unitization of $V$. Let $V$ and $W$ be two $C^*$-algebras then is it true that $$( V \otimes^{max}W)^+\cong V^+ \otimes^ {max} W^+$$ I guess it’s true ...
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0answers
40 views

Trotter-Kato approximation theorem for transition semigroups of Feller processes

Let $E$ be a locally compact separable metric space; $(T(t))_{t\ge0}$ be a strongly continuous contraction semigroup on $C_0(E)$ with generator $C_0(E)$; $D$ be a core of $(\mathcal D(A),A)$; $E_n$ ...
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1answer
151 views

Free probability: motivation for Voiculescu's free gaussian functor

Given a Hilbert space, let us denote by $\mathcal{T}(\mathcal{H})$ the full fock space $$ \mathcal{T}(\mathcal{H}) = \mathbb{C} \Omega \oplus \bigoplus_{n \geq 1} \mathcal{H}^{\otimes n}. $$ If $\xi \...
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1answer
26 views

Spectrum of compact operator on infinite dimensional Hilbert space is countably infinite

So, I've been working on this question for a while now and cannot get it right. I have a compact, self adjoint, positive definite operator $K$ on infinite dimensional Hilbert space. I know that $0\in ...
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2answers
30 views

Bounded linear operator $Tx(t) = x(t-\Delta)$?

Let $X = (B(\mathbb{R},\mathbb{R}), \|.\|)$ be the space of all linear operators on $\mathbb{R} \rightarrow \mathbb{R}$ and define the operator $T:X \rightarrow X$ as $$Tx(t)= x(t - \Delta)$$ where $\...
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3answers
40 views

Show differential operator is not bounded using definition of bounded operators

Let $T:C^{1}_{[a,b]} \rightarrow C^{0}_{[a,b]}$ with $a<b$ be the differential operator defined as $Tx=x’$. The practice exercise asks for the kernel and range of such operator and also a ...
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0answers
69 views

Norm of a 2-by-2 matrix

Let $$A = \begin{pmatrix} \cos\theta & b \\ -b & c \end{pmatrix} \in M_2 $$ be a contraction, i.e., $\Vert A\Vert\le 1$, and $$\gamma := \frac{b}{\sin\theta}\leq 1$$ Show that there exists $a \...
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Monotone condition for multivariable functions and monotone operators

I am looking for a general definition of monotone condition for a function $G: \mathbb{R}^m \to \mathbb{R}^m$, and since I did not find a unique definition of monotone condition for multivariable ...
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1answer
15 views

Spectrum of an operator $\sigma(L)$

How can I find $\sigma(L)$, $\sigma_{p}(L)$ , $\sigma_{c}(L)$ i , $\sigma_{r}(L)$ for operator $L(x_{1},x_{2},\ldots)=(x_{2},\frac{1}{2}x_{3},\frac{1}{3}x_{4},\ldots)$ ?
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If $f_1,f_2,…$ are bounded continuous functions with $\sup_{B_n}|f_n|\to0$ and $T_n$ is a contraction on $C_b$, can we show $\sup_{B_n}|T_nf_n|\to0$?

Let $E_n$ be a metric space, $B_n$ be a Borel subset of $E_n$ and $f_n:E_n\to\mathbb R$ be bounded and continuous for $n\in\mathbb N$ with $$\sup_{x\in B_n}\left|f_n(x)\right|\xrightarrow{n\to\infty}0....
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1answer
24 views

about Bounded Operator [closed]

Let be $l_p$ ( $1 \le p \le \infty$ ) space of sequences in $\mathbb{C}$ or $\mathbb{R}$. How can I prove that operator $L(x_1,x_2,x_3,\ldots)=(a_{1}x_{1},a_{2}x_{2},a_{3}x_{3},\ldots)$ is bounded if ...
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25 views

Proving that a specific Volterra integral operator is not positive

I want to prove that the operator $$ A: L^2[0,1] \to L^2[0,1], \quad A(u)(s) = \int_0^1 |t-s| u(t) dt $$ is not positive, i.e. $\langle Au, u \rangle \geq 0$ does not hold for every $u \in L^2[0,1]$. ...
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33 views

Formal solution diffusion equation

I'm not a mathematician, so please bear with me if I write things down in a non-rigorous manner. I read in either a mathematical finance or physics book (can't remember) that the formal solution of a ...
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1answer
33 views

Reference Request: Group C* Algebra

Currently I am finishing the reference called "C* Algebra By Example" written by Kenneth Davidson and looking for another reference related to Group C* Algebra. I tried to read the "C* Algebras and ...
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1answer
32 views

Spectral theorem for compact operators on Banach space.

Let $X$ be a Banach space. Let $A$ be a compact operator on $X$ and let's denote $\sigma(A)$ is spectrum of operator $A$. Let $f$ be holomorphic function in some neighbourhood of $\sigma(A)$ Out ...
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1answer
23 views

Example of an unbounded operator whose adjoint is not densely defined

In his book "Quantum Theory for Mathematicians", B. C. Hall mentions that there are some pathological examples of unbounded operators on separable Hilbert spaces whose adjoint is not densely defined (...
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0answers
33 views

When does a multiplication operator on $L^2$ have closed range?

I'm working on the following problem in Conway's Functional Analysis. Here $\phi$ is a bounded measurable function on $(X, \Omega, \mu)$. I was able to answer the first part of the problem but I am ...
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0answers
39 views

Find the norm of this operator

Consider the operator $A:\mathcal L_{2,w}(\mathbb R)\to\mathcal L_{2,w}(\mathbb R)$, which maps from the weighted $\mathcal L_2$-type space to itself. The operator acts in the following way: $$(Af)(t)=...
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1answer
26 views

Kronecker product on a matrix with structured blocks

I'm currently attempting to write a symmetric matrix with structured blocks into Kronecker-factorized form, but I'm not sure if the task is possible at all. My matrix takes the following form: $$ M= \...
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1answer
43 views

Showing that $\alpha\geq \beta$

Let $\mathcal{B}(F)$ the algebra of all bounded linear operators on a complex Hilbert space $F$. Let $A,B\in \mathcal{B}(F)$. Consider the following numbers: $$\alpha=\sup_{\substack{a,b\in \mathbb{C}...
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1answer
28 views

Generalized Laplacian?

I was wondering if any of you had ever encountered operators on $L^2(\mathbb{R}^d)$ of the form $$ - \nabla \cdot A(x)\nabla $$ where $A(x)$ is some matrix field (viewed as $L^2(\mathbb{R}^{d^2}$)), ...
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1answer
36 views

What's the difference between the operator norm and the sup norm

What's the difference between the operator norm and the sup norm over $C[0,1]$. a.k.a $\left\lVert x\right\rVert_\infty$ vs $\left\lVert x\right\rVert_{op}$
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0answers
58 views

Convergence of Feller processes implies convergence of the resolvent operators of their generators

Let $E$ be a locally compact separable metric space $(T_n(t))_{t\ge0}$ and $(T(t))_{t\ge0}$ be strongly continuous contraction semigroups on $C_0(E)$ with generator $(\mathcal D(A_n),A_n)$ and $(\...
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1answer
27 views

Sum of closed and bounded linear operators

Let $T_1: X \rightarrow Y$ be a closed linear operator and $T_2: X \rightarrow Y$ a bounded linear operator and $X$ and $Y$ normed spaces over the same field. Is the sum of such operators also closed?...
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1answer
27 views

sum of ideals in $C^*$ algebra

Suppose $I_1$ is a maximal ideal in $C^*$ algebra $A$,$I_2$ is an ideal of $A$,then $I_1+I_2$ is an ideal of $A$,can we conclude that $I_2\subset I_1$? My thought:if there exists an element $x\in I_2$...
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1answer
16 views

Linear operator in banach spaces: multiplication of open balls by positive scalar

I would like to demonstrate the following. Let $X,Y$ be Banach spaces and $T:X \rightarrow Y$ be a linear operator. Hyposthesis: Suppose you can take an open ball in Y such that $B(y, \epsilon) \...
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1answer
36 views

Momentum operator in the position basis

J.J Sakurai shows in the section of ' Momentum operator in the position basis' as $P$$\lvert\alpha\rangle$=$\int dx^{'}\lvert\ x{'}\rangle\Bigl(-i{h\over 2\pi}$ $\partial\over\partial x{'}$$ \...
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0answers
139 views

How can we proof this implication in Corollary 4.8.7 in the book of Ethier and Kurtz?

I'm trying to understand the proof of the implication "(g) $\Rightarrow$ (f)" in Corollary 8.7 of Chapter 4 in the book Markov Processes: Characterization and Convergence by Stewart N. Ethier, Thomas ...
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0answers
19 views

Decomposition of an unbounded operator

Suppose $H=H_1\oplus H_2\oplus...$ is an orthogonal decomposition of a Hilbert space $H$. Let $T:\mathrm{dom}(T)\subseteq H\to H$ a densely defined linear operator (it could be not bounded). For each $...
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1answer
33 views

$L^p$-contractivity implies $L^p$-dissipativity?

Does $L^p$-contractivity of an operator semigrop imply the $L^p$-dissipativity of the operator? Please note the definition of $L^p$-dissipativity: $(Au, |u|^{p-2}u)\leq 0$ for all $u\in C^1_0(\...
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1answer
21 views

Construction of an operator $A$ on $\ell^2(\mathbb{N}^*)$ satisfying a property

Let $\ell^2(\mathbb{N}^*)$ be the Hilbert space with the inner product $$\langle x\mid y\rangle_2:=\sum_{i=1}^{+\infty}x_i\overline{y_i},\;\,\forall\,x, y \in \ell^2(\mathbb{N}^*).$$ Consider the ...
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1answer
21 views

positive functionals on the full group C*-algebra

It it true that every positive linear functional on the full group C*-algebra is Completely positive? I am reading Brown and Ozawa's book and they seem to use this at some point, yet I'm not sure how ...
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1answer
32 views

Does the spectrum at a point vary continuously in this case?

Let $A$ be a C$^{*}$-algebra. Let $\hat{A}$ denote the set of all irreducible representations of $A$. Suppose $\pi\in\hat{A}$ has the following property: for all $a\in A$, the map from $\hat{A}\to\...
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0answers
55 views

Linear bounded operator from $L^p[0,1]$ to itself whose range consists of continuous functions.

Let $T\colon \mathbb L^p[0,1]\to \mathbb L^p[0,1]$, $1<p<+\infty$, be a linear bounded operator such that $\operatorname{Im}(T)$ is contained in the space of continuous functions. It was shown ...
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0answers
33 views

Example of operator with $\lVert T \rVert^ i = 1$ in normed spaces

Let X be a normed vector space of your choice with its norm $\lVert.\rVert$. I am looking for an operator of norm $\lVert T \rVert ^ i = 1$. Defined on as $T:X \rightarrow X$ s.t. its powers $T^{i} =...
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1answer
32 views

Maximal modular ideal

I met with some troubles with the two concepts:maximal ideal and maximal modular ideal in $C^*$ algebras. If $I$ is a maximal modular ideal in a $C^*$ algebra $A$,does this imply that for any other ...
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2answers
28 views

What are the eigenstates of $X^N$ operator?

NOTE: I have first asked this on physics.stackexchange, they advised me to ask on math.stackexchange The operator $X$ is called the position operator in physics with it's conjugate being the momentum ...
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25 views

Refrence for closedness of image of a compact operator

Let $\mathcal{H}$ be a Hilbert space, and let $T\in K(\mathcal{H})$ be a compact operator. There exists a theorem in the following way: "$T(\mathcal{H})$ is closed in $\mathcal{H}$ if, and only if, $\...
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1answer
33 views

Restriction of a compact operator on a finite-dimensional subspace

I have a self-adjoint compact operator $\Gamma : L^2[0,1] \to L^2[0,1]$ with positive eigenvalues $\lambda_j$, which of course tend to zero,and a general finite dimensional linear subspace $S \subset ...
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3answers
222 views

If $T:L^p[0,1] \to L^p[0,1]$ bounded for $1 < p < \infty$ with continuous image, then it's compact

Is the following statement true? Let $T:L^p[0,1] \to L^p[0,1]$ be a bounded operator for $1 < p < \infty$ and suppose that $\operatorname{Im}(T) \subset C[0,1]$ consists of continuous functions....
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1answer
20 views

Is multiplication by continous functions in $B(L^2)$ norm closed?

If you consider the multiplication by a continuous function as a subset of $B(L^2(\mathbb{T}))$, is this norm closed in operator norm? i.e. if B is an operator in $B(L^2(\mathbb{T}))$ with $||B-M_{\...
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1answer
12 views

Computations of a counterexample in order to check that the sum and product of closed operators are not always closed

While I was studying functional analysis I found in the script the following counterexample: Let $X = l^1$ and consider the linear operator $$ (Ax)_n\left\{ \begin{array}{ll} n x_{n-1} ...
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1answer
24 views

Every von Neumann algebra admits nontrivial trace

Let $M$ be a von Neumann. A functional $\tau\colon M_+\to [0,\infty]$ is called trace if it satisfies the following conditions: 1) $\tau(x+y)=\tau(x)+\tau(y), \forall x,y\in M_+$; 2) $\tau(\lambda x)=\...
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1answer
34 views

Almost adjointness propertie for distributions

Suppose $h\in S(\mathbb{R}^d)$ (Schwartz space) and a family $\mathcal{F}=\{f(\,\cdot\,;s)\}_{s\in\mathbb{R}^d}\subseteq S^\prime(\mathbb{R}^d)$ of tempered distributions. Then, for each fixed $s$ we ...
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1answer
23 views

find a smallest nuclear $C^*$ algebra containing set S [closed]

Suppose $S$ is a set ,can we find a smallest nuclear $C^*$ algebra containing $S$
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0answers
29 views

Why the space have to be complex in the spectral theorem for bounded self-adjoint operators?

In the spectral theorem for compact self-adjoint linear operators $T:H\to H$ (as stated in Conway's book), the Hilbert space $H$ can be real or complex. However, in the spectral theorem for bounded ...
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0answers
17 views

Showing that $C=\{h \in L^1(\Omega) : u_1(z) \leq h(z) \leq u_2(z)\}$ is weakly compact.

Exercise : Let $\Omega \subseteq \mathbb R^n$ be open and bounded, $u_1, u_2 \in L^1(\Omega)$ with $u_1(z) \leq u_2(z)$ almost everywhere in $\Omega$. We let $C$ be the set $C=\{h \in L^1(\Omega) : ...
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0answers
25 views

Bounded set and norm bounded set in a Banach lattice space

I am reading about Banach lattice space and confuse a little bit about two concepts "bounded" and "norm bounded set". Could you please help me to declare them? More precisely, let $E$ be a Banach ...