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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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### How to prove this for the spectrum of a self-adjoint operator?

Let $T$ be a bounded self-adjoint operator. Prove: A number $\lambda \in \mathbb{R}$ belongs to the spectrum of $T$ if and only if $\mathbb{1}_{(\lambda - \epsilon, \lambda + \epsilon)} (T)$ is non-...
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### Norm of adjoint of operator of $l_1$

Let be $T: l_{1} -> l_{1}$, $T(x_1,x_2,x_3,...)=(0,0,0,...x_1,0,0,...)$ . ( Coordinate x_1 is on n-place ). I proved that T is linear and bounded. But, I don't know how can prove that T is ...
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### Is there a Deterministic Matrix with Restricted Isometry Property?

The Restricted Isometry Property (Low-Rank Matrices) Let $\mathcal{A}:\mathbb{R}^{n\times n}\rightarrow \mathbb{R}^m$ be a linear operator. The constant $\delta_r:=\delta_r(\mathcal{A})$ is defined ...
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### Example of a strongly dense set which is not ultrastrongly dense.

Let $H$ be a Hilbert space and let $B(H)$ denote the Banach space of bounded operators on $H$. Then there are several topologies we can endow on $B(H)$. I am interested in the case of the strong ...
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### Explicit solution of an operator equation

Suppose we have an operator $A:C([0,\infty )) \to C([0,\infty ))$ defined as $$(Af)(x)=\int _0 ^ x \dfrac{f(y)}{\sqrt{x^2-y^2}}dy$$ Now I want to prove that for arbitrary $g\in C([0,\infty ))$ the ...
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### existence of eigenfunctions in non-self adjoint operator

Remark: The existence of eigenfunctions is only guaranteed for self-adjoint operators. How does one justify this statement? An operator not being self-adjoint makes it not have eigenpairs?
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### Eigenvalue of Hermitian operator(curl) complex?

This question is in continuation to a previous question What are the Eigenvectors of the curl operator?. I know now that the curl operator \begin{bmatrix} 0 & -\frac{\partial}{\partial z} &...
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### Passive transformation by antiunitary operator and orientation of the complex plane.

I have recently tried to make sense of the concept on an antiunitary passive transformation on a complex Hilbert space $H = \mathbb{C}^N$. I still do not know whether the concept even makes sense ...
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### If $A$ is self-adjoint, then $\left\|A\right\|=\sup_{x\in H\setminus\{0\}}\frac{\langle Ax,x\rangle}{\left\|x\right\|^2}$

Let $H$ be a $\mathbb R$-Hilbert space and $A\in\mathfrak L(H)$ be self-adjoint. I want to show that \left\|A\right\|_{\mathfrak L(H)}=\sup_{x\in H\setminus\{0\}}\frac{\langle Ax,x\rangle_H}{\left\|...
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### Linear Operator in a Banach Space

I'm trying to get my head around linear operators and their usage with Banach spaces. Could someone help me understand how some properties relate to the following operator? We have Banach space $L^2$,...
Let $X$ be an infinite dimensional Banach space (over $\mathbb{C}$) and let $T\colon X\to X$ be a compact operator. Let $\sigma(T)$ denote the spectrum of $T$ and let $\sigma_{\text{p}}(T)$ denote the ...
Suppose $P$ is a rank one projection in $B(H)$,if we have $T\in B(H)$ such that $0\leq T\leq P$,does the following conclusion hold: There exists $\alpha \geq 0$ such that $T=\alpha P.$