# 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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### trace of inverse operator

Let $H$ be an RPK-Hilbert space and $K:X\times X\rightarrow \mathbb{R}$ be the reproducing Kernel s.t. $K$ is bounded by $1$. For some Probability Space $(X, \nu)$ It is assumed that all $f \in H$ ...
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### When the image of operator be subset of $l_p$

Let $f:l_{\infty} \rightarrow l_{\infty}$ defined by $f(x_1,x_2,...)=(x_1,\frac{1}{2}x_2,...,\frac{1}{2}x_n,...)$. Then we need to (1) prove that $f$ is continuous operator and (2) find $p\geq 1$ such ...
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### Essential spectrum of a product of self-adjoint operators which have positive essential spectrum

Given an infinite-dimensional Hilbert space $H$, a bounded linear self-adjoint operator $A:H\to H$ and a bounded linear invertible operator $B:H\to H$. If both $A$ and $B$ have positive essential ...
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### Composition of operators and their derivatives

I'm a newbie in mathematical physics and is currently reading Sadri Hassani's book entitled Mathematical Physics - A Modern Introduction to Its Foundations. I came upon this page in the book and I ...
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### Extension of functional calculus of continuous functions

On Reed & Simon's book, we can find the following theorem, which is called the continuous functional calculus. Notation: $\sigma(A)$ is the spectrum of the operator $A$, $\mathscr{L}(\mathscr{H})$ ...
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### Numerical range of a symmetric positive matrix as an operator acting on Hilbert space

Let $A$ be a $3\times 3$ real constant symmetric positive matrix, $\Omega\subset\mathbb{R}^3$ a bounded Lipschitz domain and $(L^2(\Omega))^3$ the space of square integrable functions on $\Omega$. I ...
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### Real determinant line bundle is trivial for complex linear operators

In Fredholm theory we can define the determinant line of a linear Fredholm operator $F$ to be $$\det F=\Lambda^\text{top}\operatorname{Ker}F\otimes(\Lambda^{top}\operatorname{Coker}F)^*$$ For a family ...
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### Essential spectrum of a constant matrix as an operator defined on a product of Hilbert spaces

Let $M$ be a $3\times 3$ constant real matrix, $\Omega\subset\mathbb{R}^3$ a bounded Lipschitz domain, and define $(L^2(\Omega))^3$ as the space of square integrable functions on $\Omega$. I want to ...
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### About an inequality in the proof of maximal function

In the proof of the fact that if $f\in L^p$, where $1<p\leq L^p$, then $Mf\in L^p$ and $$||Mf||_p\leq A_p ||f||_p$$ ($Mf$ is maximal function) Stein says $$|f(x)|\leq |f_1(x)|+\alpha /2 (*)$$ and ...
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### Why is the operator $P = - i \frac{d}{dx}$ not bounded in this domain?

the domain is: $$D(P) = \{f \in C^{\infty}(\mathbb{R}) \cap L^2(\mathbb{R})\}$$ I think that is because some functions that belong to $D(P)$ should not be bounded at infinity. The text I am studying ...
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### For given Pontryagin space the norm induced topology is not depending on the fundamental decomposition

For given Hilbert space $H$ consider its corresponding antispace, i.e. the vector space endowed with its negative inner product. $\Pi$ is now called a Pontryagin space if it can be written as a direct ...
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### How is $\mathcal L(E)$ graded? Question about Hilbert $B$-module

I am reading G.G.KASPAROV's THE OPERATOR AT-FUNCTOR AND EXTENSIONS OF C*-ALGEBRAS Let $B$ be a C*-algebra and let $E$ be a $B$-right-module. Assume there is a $B$-valued inner product on $E$ which (...
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### Graded algebra associated to finite groups [closed]

I would like to know many examples of graded algebra that arise or associated from a finite group. Please give me some reference. Thanks in advance
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### Contraction proof of averaging operator: $\operatorname{Var}_\nu (M f) \leq (1 - \kappa)^2 \operatorname{Var}_\nu (f)$

Let $(X, d, m)$ be an ergodic random walk on a metric space, with invariant distribution $\nu$. Suppose that the coarse Ricci curvature of $X$ is at least $\kappa > 0$ and that the average ...
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### Is the inverse Laplacian bounded in $\mathbb{R}^{2}$?

I'm searching for an inequality in the form $$\forall s>2,\quad\forall u\in H^{s-2}(\mathbb{R}^{2}),\quad\|\Delta^{-1}u\|_{H^{s}(\mathbb{R}^{2})}\lesssim\|u\|_{H^{s-2}(\mathbb{R}^{2})}$$ where $H^s$...
Let $A$ be a (possibly unbounded) linear operator on a Banach space $E$ such that $(0,\infty)\subseteq\rho(A)$ and $$\left\|R_\lambda(A)\right\|_{\mathfrak L(E)}\le\frac1\lambda\;\;\;\text{for all }\... 1answer 47 views ### What is the value of knowing the eigenfunctions of an operator? Let f_i be the set of functions of operator O such that Of_i = \lambda_if_i\;\;\lambda_i \in \mathbb{R} In linear algebra, there are a number of uses of the eigenvalues and eigenvectors. Are ... 0answers 20 views ### 1/2 Hölder estimate on the difference of Hilbert--Schmidt operators Let A,B be trace class, self-adjoint, and positive operators on some separable Hilbert space H. Then, does an inequality of the following form hold true:$$ \lVert A^{1/2}-B^{1/2}\rVert_{\mathrm{...
Let $v$ be a fixed non-zero vector of an $n$-dimensional real vector space $V$. Let $P(v)$ be the subspace of the vector space of linear operators on $V$ consisting of those operators that admit $v$ ...