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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 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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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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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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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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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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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 $u\in C^1_0(\Omega)$ ...
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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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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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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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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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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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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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Prove that operator is compact

Prove that if normal operator doesn't have nonzero limit points of spectrum and $\dim \ker(T-\lambda I) < \infty$ $\forall$ point of spectrum, then operator is compact.
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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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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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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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Why is the sequence $(\hat A_k)$ convergent in the strong$^*$ topology to $(\hat A)$?

I have questions regarding part of E. Christopher Lance. Ergodic Theorems for Convex Sets and Operator Algebras, Page 213. In the proof of the theorem $(5.7)$ the second line in the middle. Why is ...
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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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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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Continuity with respect to the norm [closed]

Let $T: C([0,1])$->$C([0,1])$ be linear map, such that: if $({f_{n}})_{n=1}^{\infty} \in C([0,1])$ pointwise converges to $f$, then $(T f_{n})_{n=1}^{\infty}$ pointwise converges to $Tf$. 1. Prove ...
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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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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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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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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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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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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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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 ...
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Hereditary C*-subalgebras of the unitization of a hereditary C*-subalgebra

Let $A$ be a unital C*-algebra with unit $1$, and $B$ a non-unital hereditary C*-subalgebra of $A$. Suppose $J$ is a proper hereditary C*-subalgebra of $B\oplus C1$ (the unitization of B). Is $J$ ...
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Showing that $0\leq A\leq B$ and $B \in \mathcal{L}_c(H)$ implies that $A \in \mathcal{L}_c(H)$.

Exercise : Let $H$ be a Hilbert space and $A,B \in \mathcal{L}(H)$ be self-adjoint operators with $0 \leq A \leq B$ and $B \in \mathcal{L}_c(H)$. Show that $A \in \mathcal{L}_c(H)$. Thoughts : ...
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Square root of positive matrix operator

Let $F_1,F_2$ be two complex Hilbert spaces. Consider \begin{equation*} T=\begin{pmatrix}A & B \\ C & D \end{pmatrix}\in \mathcal{B}(F_1\oplus F_2). \end{equation*} If $T$ is a positive ...
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Family of linear functionals on separable Banach space coinciding with norm when evaluated in one element

let $B$ be a real, separable and infinite-dimensional Banach space and let $x_1$, $x_2$, $x_3$, $\dots$ be a dense subset of $B$. In the proof of Lemma 2.1 on p. 2 in this article, the author takes ...
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Does the limit of this sequence of operators have infinitely many eigenvalues?

Suppose that I have a sequence of compact, injective operators $\{T_\delta\}_{\delta>0}$ on a Hilbert space $H$ such that each operator $T_\delta$ has infinitely many eigenvalues. My question is ...
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Prove or deny that $\ker\prod_{i = 1}^{k}(A - \lambda_i)^{d_i} = \sum_{i=1}^{k}\ker(A - \lambda_i)^{d_i}.$

Let $A$ be some linear operator(possibly over an infinite-dimensional space). Prove or deny that: $$\ker\prod_{i = 1}^{k}(A - \lambda_i)^{d_i} = \sum_{i=1}^{k}\ker(A - \lambda_i)^{d_i}$$ Here $d_i, ...
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Showing that $A(\overline{B_1^X}) \subseteq Y$ is closed when $A \in \mathcal{L}(X,Y)$.

Exercise : Let $X,Y$ be Banach spaces and $X$ be reflexive, $A \in \mathcal{L}(X,Y)$. If $\overline{B_1^X} = \{ u \in X : \|u\|_X \leq 1\}$, show that $A(\overline{B_1^X}) \subseteq Y$ is closed. ...
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stably finite $C^*$ algebras

I saw a conlusion:If $A$ is simple $C^*$ algebra,then $A\otimes \Bbb K$ contains no infinite projections? If $A$ is simple, $A\otimes \Bbb K $ contains no infinite projections,can we conclude that $A$...
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about the Beurling Theorem in Murphy's book

Here is a part of Murphy's book C*-Algebras and Operator Theory: In the fourth line of his proof, he claims that $1-\lambda a$ is invertible. But we do not have $||\lambda a|| <1$, how to show ...
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Fredholm operator [closed]

I am quite new to Fredholm operators, can anyone help me with the following problem, thank you. For any given $ g \in C[0,1] $ let $ A: C^{(1)}[0,1] \to C[0,1] $ be an operator such that $$ Af(x) = f'...
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Spectral Theorem: Realization of a direct sum of $L^2$ spaces as a single $L^2$ space

The following is motivated by an attempt to understand the Spectral Theorem for Bounded operators on a none separable Hilbert space. One version of the theorem states that for a bounded (say normal) ...
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transformation of the Gaussian function

consider a gaussian function: $$ \frac{1}{\sqrt{\pi}} \exp \left\{-\frac{1}{2}\left(x^{2}+y^{2}\right)\right\} $$ And I have to prove that transformation of this function: $$ \exp \left\{-\eta\left(...
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Hyperfinite II$_1$-factor

1.What is the definition of hyperfinite II$_1$-factor? Can anyone show me concrete examples? 2.If $R$ is a hyperfinite II$_1$-factor ,we can define the ultraproduct of $R^{\omega}=l^\infty(R)/I_{\...
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Decomposition of Hilbert Space into a orthogonal direct sum of an eigenspace and the adjoint eigenspace

Let $\lambda_0$ is a discrete (isolated) spectrum of a densely defined closed linear operator $\mathcal{L}: H \to H$, where $H$ is a Hilbert space. Suppose that $\lambda_0$ has a finite algebraic ...
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Quick question regarding C*-algebras/W*-algebras

I'm reading about C*-algebras and W*-algebras, and I want to know the differences between the two,in other words properties you can find in W*-algebras that aren't in C*-algebras and the other way ...
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Spectra of Closure and Adjoint for Symmetric Operator

I noticed something when working on a problem in Hall's Quantum Theory for Mathematicians. The original problem is that if $A \equiv -i\hslash \frac{d}{dx}$ on $\mathcal{H} = L^2[0,1]$ and $$ Dom(A)...
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trace on quotient of a $C^*$ algebra

Does there exist a $C^*$-algebra $A$ such that $A$ has a faithful tracial state,but the quotient of $A$ has no tracial states(there exists a nontrivial ideal $I$ of $A$ such that $A/I$ has no traces )...
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different essential ideals

Does there exist a $C^*$ algebra which has more than one essential ideal?If there exists such a $C^*$ algebra ,suppose $I,J$ are two different essential ideals,$I\subset J$,can we compare the ...
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Spectrum of operator $(Tx)_{i}=\sum_{j=0}^{n}\alpha_{j}x_{i+j}$

Note: Please do not give a solution; I would prefer guidance to help me complete the question myself. Thank you. Let $\alpha_{0},\ldots,\alpha_{n}\in\mathbb{C}$ be given. Compute the spectrum of $T:\...
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Understanding of Weighted Shift Operators, Analytic function Theory by Allen Shields

I was given a paper by Allen Shields on Weighted shift operators and analytic function theory to study. I have background in functional analysis and complex analysis but I would say, I have spent more ...
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About the adjoint of some differential operator on $L^2(0,1)$

If $T=-d^2/dx^2$ is defined on the domain $D(T)=\{f\in C^2[0,1]: f(1)=f'(0)=f'(1/2)=0\}\subset L^2(0,1)$. What's the Hilbert adjoint operator $T^*$? Many many thanks for your answers. Math.
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Notation question: $T[X]=\bigcup\limits_{n=1}^{\infty} nT[B]$.

I'm reading the following text: https://www.ucl.ac.uk/~ucahad0/3103_handout_7.pdf I am reading the proof of Lemma 7.17 on page 16. $B$ is the open unit ball in X. What I am wondering is what is $nT[...
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1answer
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$A \in \mathcal{L}(X,Y) \implies A^* \in \mathcal{L}(Y^*_{w^*}, X^*_{w^*})$

Exercise : Let $X,Y$ be Banach spaces and $A \in \mathcal{L}(X,Y)$. Show that $ A^* \in \mathcal{L}(Y^*_{w^*}, X^*_{w^*})$. Attempt : The linearity is trivial. Τo show that $A^*$ is $w^*$ to $w^*$...