Questions tagged [compact-operators]

A compact operator is an operator from normed space $X$ to a normed space $Y$, such that image of every bounded subset of $X$ is relatively compact in $Y$. It's used with (functional-analysis) and (operator-theory) tags.

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Showing that a sequence of compact operators converge uniformly to their pointwise limit

I am working out of Phillipe G. Ciarlet's Linear and Nonlinear Functional Analysis with Applications and am struggling with exercise 4.9-5. The problem is as follows: Let $G$ be a function in the ...
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Why is a compact linear operator also a bounded linear operator?

I know that a linear operator is bounded iff continuous, with these definitions of "bounded" and "continuous": a linear operator T (on H Hilbert space) is bounded if ∃M>0: ||Tf|...
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dilatation analytic

I have seen that the spectrum of the operator $T=-\frac{d^2}{dx^2}+x^2$ is the $\{2n+1,n \in \Bbb{N}\}$ and by dilatation $x=r y$ the spectrum of the operator $T_r=-r^{-2}\frac{d^2}{dx^2}+r^2x^2$ is ...
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Commutant of corner of $*-$algebra on a hilbert space is corner of commutant

Let $A$ be a $*-$algebra on a Hilbert space $H$ and $p$ be a projection in $A'$, where $A'$ is the commutant of $A$, that is, $$A':=\{u \in B(H): ua=au~\text{ for all }~a \in A\}.$$ If also $p \in A''$...
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Proving that operator $x_n \to x_{n+1}/ n$ is compact

Prove that the operator $T:\ell^2\rightarrow\ell^2$ defined as $$Tx = \bigg(x_2, \frac{x_3}{2}, \frac{x_4}{3}, \dots \bigg)$$for $x = (x_1, x_2, x_3, \dots) \in \ell^2$ is ...
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operators that are compact and positive, their norm equal to the greatest eigenvalue.

I read this from a post Norm of integral operator in $L_2$ Can someone point me to formal proof of this statement? Thanks
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Proving an operator is contractive

I am having trouble with the following problem: a) Let $H_n (x), n \in \mathbb{N}$ be Hermite polynomials associated to the measure $$d \mu = \frac{1}{\sqrt{2\pi}} e^{-\frac{1}{2}x^2}d \lambda$$ ...
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Can the Volterra operator be norm-approximated by nilpotent operators?

Let $V$ be the Volterra operator on the Hilbert space $L_2[0,1]$. Then $V$ is bounded and quasinilpotent. Can $V$ be approximated in norm by nilpotent operators?
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Spectrum of a compact operator - proof clarification

I'm reading through the book Linear Analysis by Bollobás. I'm having a bit of trouble understanding part of the proof of theorem 7 in chapter 13, which states: Let $T$ be a compact operator and ...
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Compact integral operator on $H^1(\mathbb{R})$

Consider the operator $${\mathcal{L}}v=e^{-x}\int_{0}^x v(y)\, dy.$$ Is the operator ${\mathcal{L}}$ compact as an operator from $H^1({\mathbb{R}^+})$ to itself? To give some context to the problem ...
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Let $X$ be Banach space, $N$ a (closed) subspace of $X$ and $\pi\in X\to X/N$ the natural quotient map. Call $Y\subseteq\mathbb{Z}\to X\sqcup X/N$ the set of functions on $\mathbb{Z}$ such that $f\in ... 1 vote 1 answer 37 views Question on pointwise and uniform boundedness. I'm reading through C* algebras by Murphy, and I'm covering compact operators at the moment, and the following was stated: "Let$X = C([0,1])$, and define$u \in B(X)$by$u(f)(s) = \int_{0}^1 k(...
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I am studying functional analysis and got stuck with a textbook exercise. I greatly appreciate some hints/help or a push in the right direction! Define for $g\in C^0[-1,1]$ the integral operator \$T_g:...