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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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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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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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Show convergence of a sequence of resolvent operators

Let $E$ be a locally compact separable metric space $(\mathcal D(A),A)$ be the generator of a strongly continuous contraction semigroup on $C_0(E)$ $E_n$ be a metric space for $n\in\mathbb N$ $(\...
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1answer
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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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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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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
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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
42 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
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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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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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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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25 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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23 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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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
29 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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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(\...
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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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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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2answers
24 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
30 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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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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1answer
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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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27 views

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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1answer
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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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1answer
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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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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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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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1answer
47 views

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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1answer
26 views

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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1answer
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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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1answer
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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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1answer
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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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1answer
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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) ...