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.

6,346 questions
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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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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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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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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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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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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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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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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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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 $||B-M_{\... 1answer 12 views Computations of a counterexample in order to check that the sum and product of closed operators are not always closed While I was studying functional analysis I found in the script the following counterexample: Let$X = l^1$and consider the linear operator$$(Ax)_n\left\{ \begin{array}{ll} n x_{n-1} ... 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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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 $S$
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 ...