Tagged Questions

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Spectral theory

I have absolutely no idea about Spectral theory and want to ask some fundamental questions. 1.) What does it mean that the resolvent of an operator is Hilbert-Schmidt? (Cause I saw a theorem that was ...
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Compact operator and limit

I was wondering about something related to compact operators. If we have a compact operator $T:X \mapsto Y$ and a bounded sequence $(x_n)n$, then we know that there is a convergent subsequence ...
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Are these operators and the fourier transform compact?

I do not want a proof but rather an explanation. I just read that $T_k:L^2(\mathbb{R}) \rightarrow L^2(\mathbb{R})$ such that $(T_kf)(s) = \int_{\mathbb{R}} k(s,t)f(t) dt$ is compact. (in this ...
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An example of a non-closable operator

I've encountered the following: Consider the usual Hilbert space $L^2([0,1],dx)$ and the dense subspace $\mathcal{D}=\mathcal{C}[0,1]$. Define $T$ on $\mathcal{D}$ by $T(f)=f(0)$. This is a ...
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Are these linear operators continuous?

For every polynomial $p(t)= \sum_{k=0}^{n} a_k t^k$ we declare its norm by $||a_k||=\sum_{k=0}^{n}|a_k|$. Now, I am supposed to check whether these maps are continuous and in case that they are I ...
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Let $A$ be a symmetric operator satisfying $\langle \phi,A\phi\rangle\geq C\lVert \phi\rVert^{2}$ for all $\phi\in \mathcal{D}(A)$ and some $C\in \mathbb{R}$. Show that the deficiency indiecs are ...
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Dyadic approach to Marcinkiewicz interpolation for Lorentz Spaces

In Exercise 21 , in a note for professor Terrence Tao on his own blog http://terrytao.wordpress.com/2009/03/30/245c-notes-1-interpolation-of-lp-spaces/ Exercise 21 Suppose we are in the ...
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How to prove $e^{\log(1+x)}= 1+x$ by series expansion?

as the title says, i want to prove $e^{\log(1+x)}= 1+x$, by substitute $\log(1+x) = \sum _{i=1} ^{\infty} \frac{(-1)^{i+1}x^i}{i}$ and $e^x=\sum _{i=0} ^{\infty} \frac{x^i}{i!}$. Can some one help ...
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Spectrum of operator in infinite dimensional hilbert space

We know that if a complex hilbert space $H$ is separable, then for every compact set $K$, there exists a bounded linear operator $T : H \to H$ s.t $\sigma (T) = K$. My question is if this still holds ...
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Nilpotents of order 2 are dense in the strong operator topology

I need some help with this homework problem. (The Hilbert space in question is $\ell^2(\mathbb{N})$.) I let $T \in \mathcal{B}(\mathcal{H})$ and looked at a basic nbhd in the strong operator topology ...
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Let $\delta$ be a linear functional equipped with the sup-norm. Show that $\delta$ is bounded and compute its norm.

Let $\delta:C([0,1])\rightarrow\mathbb{R}$ be the linear functional at the origin: $\delta(f) = f(0)$. If $C([0,1])$ is equipped with the sup-norm $$\|f\|_{\infty} = \sup_{0\leq x\leq 1}|f(x)|.$$ Show ...
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Composition of analytic functions is analytic in a general setting, and are they continuous?

Regarding the notion of analyticity discussed in this setting: A possible equivalence for holomorphicity I wonder if this is truly the correct definition (even though it is from Dunford-Schwarz) An ...
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Let $X$ and $Y$ be Complex Banach spaces with $U$ an open subset of $X$, and $f:U\rightarrow Y$. We say that f is analytic/holomorphic if for every $z_1, ... z_n \in X$ we have that the mapping $a_1, ... 0answers 94 views A problem concerning measures on locally compact spaces I am stuck on a question for quite sometime now, although in the text (http://www.latp.univ-mrs.fr/~chaabi/ARTICLES%20IMPORTANTS/Autres%20articles%20interessant/Jewett.pdf , Pg. 10, 2.3E ) it is said ... 2answers 793 views Commutator of$x$and$p^2$I have a question: If I have to find the commutator$[x, p^2]$(with$p= {h\over i}{d \over dx} $) the right answer is:$[x,p^2]=x p^2 - p^2x = x p^2 -pxp + pxp - p^2x = [x,p]p + p[x,p] = 2hip$But ... 2answers 261 views Shift Operator has no “square root”? Consider the left shift operator$T : \ell^1(\mathbb N) \to \ell^1(\mathbb N) $by $$T(x_1,x_2..... )=(x_2, x_3 ........),$$ and also the right shift operator$S : \ell^1(\mathbb N) \to \ell^1(\mathbb ...
This is related to this question see here Let $V= \mathbb{Q}^{\mathbb{Z}}$, considered as an infinite dimensional linear space over $\mathbb{Q}$. And assume $W=\mathbb{Q}^F$ is a finite dimensional ...
Let $V$ to be an infinite dimensional linear space over some field $k$. (you can take $k=\mathbb{C}$, or further assume $V$ is a complex Hilbert space). And assume $W$ is a finite dimensional ...