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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Prove or disprove the compactness of an operator?

Consider $X=L^{2}(0,\pi, \mathbb{R})$. Let $X_{\frac{1}{2}}$ be the domain of $(\Delta)^\frac{1}{2}$ where $\Delta$ is the laplacien operator. We define the operator $K:C([0,a],X_{\frac{1}{2}})\...
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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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If-and-only-if condition on the kernel for an integral operator $T:L^2 \rightarrow L^2$ to be compact

Let $\Omega \neq \emptyset$ and $\mu$ be a finite measure on $\Omega.$ We are cosidering a kernel $k$ on $\Omega,$ i.e. a symmetric, non-negative definite jointly measurable function $$k: \Omega \...
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Compact, Self-Adjoint, Nonnegative operators have at least one Eigenvector

This is a statement presented in my class and I am having trouble to understand the proof given by the Professor: Let $T \in K(H)$, where $K(H)$ represents the space of compact operators on a Hilbert ...
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Brezis's Proof for compact operators have compact adjoint

This is the proof by Brezis on showing all compact operators have compact adjoints. I am a little bit lost in the last paragraph of the proof: This is how I understood the last paragraph: We know ...
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Proof of the Riesz-Schauder Theorem (for compact operators) using the Analytical Fredholm Theorem

First of all sorry for my bad English, I'm an Italian student, hope to let you understand! I'm having a little troubles with the proof of the Riesz-Schauder theorem for Compact Operators. Some infos ...
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Decomposition of $L^2(G)$

Let $G$ be a compact group. By Peter-Weyl, we have the following decomposition $$L^2(G) = \bigoplus_{\pi \in \hat{G}} V_\pi, $$ where $$V_\pi = \text{span} \{ x\mapsto \langle \pi(x)u,v \rangle \in L^...
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Why is the set of compact operators closed in the space of all bounded operators between Banach spaces?

Let $X$ and $Y$ be Banach space. $B(X,Y)$ is the vector space of all bounded linear maps from $X$ to $Y$. Also, $K(X,Y)$ is the set of all compact operators from $X$ to $Y$. Why is $K(X ,Y)$ ...
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Show the Volterra Operator is compact using only the definition of compact

The Volterra operator $V:L^{2}[0,1]\rightarrow L^{2}[0,1]$ is defined by $(Vf)(x)=\int_0^xf(t)dt$. I am wondering if it can be shown that $V$ is compact by definition - that is, either that $V$ ...
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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 \begin{equation} Tx = \bigg(x_2, \frac{x_3}{2}, \frac{x_4}{3}, \dots \bigg) \end{equation}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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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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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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Compactness of embeddings of Reproduing Kernel Hilbert Spaces with almost surely equal kernels

Let $\Omega \neq \emptyset$ and $\mu$ be a probability measure on $\Omega.$ Consider two reproducing kernels $k_1,k_2:\Omega \times \Omega \rightarrow \mathbb{R},$ such that they both represent the ...
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Rank of linear operator involving integral and inner product

I am studying for my exam in Functional Analysis but I'm confused about the following example: Consider $\mathcal{C}([-\pi,\pi],\mathbb{K})$ with the $\infty$-norm and define $T:X \to X$ by $$Tf(x)=\...
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Prove composition of bounded and compact operators is compact.

If $T:X\to Y$ is a compact linear operator on Banach spaces $X$ and $Y$and $S:Y\to Z$ is a bounded linear operator where $Z$ is a Banach space. Prove $ST$ is compact. Here's my proof, let $x_n \in X$ ...
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Decay rates of eigenvalues of Hilbert-Schmidt integral operator

Let $\Omega \subset \mathbb{R}^n$ be bounded. Suppose we have an integral kernel $K: \Omega^2\to \Omega$ with $\int_{\mathbb{R}^n}\int_{\mathbb{R}^n}|K(x,y)|^2dxdy < \infty$. We know that the ...
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Prove that $T$ is compact if and only if $\mu_n \to \infty$ when $n \to \infty$

Let $\{e_n\}$ be an orthonormal basis for a Hilbert space $H$, $T: H \rightarrow H$ be a bounded linear mapping and $$\mu_n = \sum_{x \perp \{e_1, \cdots, e_n\}, x \neq 0}\frac{\|T(x)\|}{\|x\|^2}$$ ...
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The equation $x- T(x) = y$ has no solution for some $y \in X$.

Construct a normed space $X$ and a bounded linear operator $T \colon X \rightarrow X$ with $\|T\|<1$ such that the equation $x- T(x) = y$ has no solution for some $y \in X$. I know that if $B$ is a ...
2 votes
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Compactness of the trace operator

Is it true that for a set $\Omega$ with Lipschitz boundary the trace operator $T : H^1(\Omega) \to L^2(\partial \Omega)$ is compact? Can you please give a reference? I found a theorem in Necas' ...
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Motivation behind the term, "precompact."

Let $X$ and $Y$ be real Banach Spaces and let $$ K:X\rightarrow Y $$ be a bounded linear operator. Consider a sequence $\{u_k\}_{k=1}^\infty$ in $X$. We call a sequence $\{Ku_k\}_{k=1}^\infty$ to be ...
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Why does $E_2: = (I-T)E_1$ with $(I-T)$ being injective imply that $E_2 \neq E_1$?

In chapter 6 of Brezis: Functional Analysis, in the proof on the "Fredholm Alternative," there is the following: $T$ is a compact operator on $E$ where $E$ is infinite dimensional. Assume $(...
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The spectrum of $T+C$ is stable under the perturbation of the form $\frac An$, where $A$ and $T$ are skew-adjoint and $C$ is compact.

Let $\mathcal{H}$ be a Hilbert space. Let $T$ and $A$ be two skew-adjoint (unbounded) operators on $\mathcal{H}$, i.e., $T^*=-T, A^*=-A$. Also we have a bounded compact operator $C$ on $\mathcal{H}$. ...
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Trace of the projection operator

Usually the trace of the orthogonal projection onto a finite dimensional space equals to the dimension of the space: Trace$(P)=M$ where $M$ denotes its dimension. For an orthonormal basis $e_1,\cdots,...
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Is this operator compact in $L^2[0,d]$?

Consider $X=L^2[0,d]$ and define $A \colon X \to X$ a linear bounded operator by the integral function \begin{equation} {A}x=\int_0^\cdot x(\xi)d\xi \end{equation} This operator looks compact to me ...
1 vote
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Essential Spectrum of arbitrary compact operator

I need to find the essential spectrum of an abitrary compact operator. Let $T : X \to Y$ be a compact operator. The essential spectrum is the set of all $\lambda \in \mathbb{C}$ such that $T- \lambda ...
3 votes
2 answers
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Spectrum of sum of bilateral shift operator and a compact operator

Let $T$ be the bilateral shift operator, that is: $T: \ell^2(\mathbb{Z}) \to \ell^2(\mathbb{Z})$ such that $(T(x))_k=x_{k-1}$ (where $x_{k-1}$ means the $k-1$ coordinate of the sequence. I have been ...
2 votes
1 answer
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Spectrum of sum of bounded and compact map

Assume $X$ is a Banach space and that $T$ is a bounded linear map and $K$ is a compact linear map from $X$ to itself. I need to prove that that if $\lambda$ is in the spectrum of $T$ but is not an ...
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example in spectral theory

Consider $X=l_{2}.$ Let $T : l_{2}\longrightarrow l_{2}$ be defined by : $T(x_{1},x_{2},....)= (x_{1},\frac{x_{2}}{2},\frac{x_{3}}{3},...).$ And $S=I$ , the identity operator. Here $N(T)=N(S)=\lbrace{...
2 votes
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Prove that a given operator is compact

Let $H$ be an Hilbert space with scalar product $(\cdot,\cdot)$ and $T : H \to H$ a linear operator defined as follow: $$ Tx = \sum _{n=1} ^\infty (x,a_n)b_n $$ where $(a_n)_{n\in\mathbb{N}},(b_n)_{n\...
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If $\{\varphi_n\}$ is an eigenbasis of $K^\ast K$'s support and $h\perp K^m\varphi_n$ for all $n,m\ge1$, then how do you show $Kh=0$?

$\newcommand{\span}{\operatorname{span}}\newcommand{\im}{\operatorname{Im}}$EDIT: According to the below answer, Royden's construction is wrong. However, the key detail which was omitted by the ...
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Zero is an eigenvalue of a compact operator on Non-Seperable spaces

Let $X$ be a non seperable Banach Space. If $T$ is a compact operator on $X$, is it true that $Ker(T)\neq \{0\}$, i.e., $0$ is an eigenvalue of $T$. This question was from my exercise, where $X$ was ...
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Compact linear operator definition

It is well known what we mean by a compact linear operator $A:X\to Y$ where $X,Y$ are Banach spaces (see https://en.wikipedia.org/wiki/Compact_operator#Compact_operator_on_Hilbert_spaces). I wonder ...
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How do I deal with the integration limits of the integrals that show up when showing that$Kx(t)=\int_0^tK(t,\tau)x(\tau)d\tau, x\in X$ is compact?

Let $X=C(I)$ with the sup norm $\|\cdot\|_\infty$, where $I=[0,1]$. Let $K$ be a Volterra integral operator: $$Kx(t)=\int_0^tK(t,\tau)x(\tau)d\tau, x\in X$$ Show that K: $X\rightarrow X$, is compact ...
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How can I prove that the spectrum of the operator $A\in B(C(K))$ defined by $Af = g∙f$ is equal to $\text{Im}(g)$, for $g\in C(K)$.

I want to prove that $\sigma(A)$, the spectrum of the linear operator $A \in B(C(K))$ which is defined by $Af = g∙f$, is equal to $\text{Im}(g)$, for $g\in C(K)$. We may assume that $K$ is a compact ...
4 votes
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Kernel of Bounded Operator on Hilbert Space

I encountered this question on an exam recently and was not able to solve it. Suppose you have a bounded operator $T$ on a Hilbert space $\mathcal{H}$ such that $I-T$ is compact, where $I$ is the ...
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Equivalence of compact operators

I'm reading about compact operators and I'm trying to prove the following statement: Let $X,Y$ be Banach spaces and $T:X \to Y$ a linear operator. Then the following are equivalent: (a) T is compact. (...
4 votes
1 answer
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Property of an operator that is Fredholm and compact?

I have been asked this question at my course on Functional Analysis, to tell something about an operator that is both compact and Fredholm. The answer needs to be related to the spaces between which ...
2 votes
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Infinite Matrices from $l^p$ to $l^{\frac{p}{p-1}}$ that are compact operators

I wanted to ask if my proof (sketch) of the following statement is correct. Namely, let $p>1$ and define $q= \frac{p}{p-1}$ we are given an operator $K : l^{p} \rightarrow l^{q}$ defined as $x \...
3 votes
1 answer
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Convergence in the operator norm

Given the following operator: $T:\ell^2\to\ell^2$, which acts in the following way on the standard basis vectors: $$Te_{2k-1}=\frac{1}{k}(e_{2k-1}-ie_{2k})$$ $$Te_{2k}=\frac{1}{k}(ie_{2k-1}+e_{2k})$$ ...
2 votes
1 answer
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Every compact operator on a Banach space with the approximation property is a norm-limit of finite rank operators

Let $X$ be a Banach space and suppose there is a net $\{F_i\}$ of finite-rank operators on $X$ such that (a) $\sup_i\|F_i\|<\infty$, (b) $\|F_ix-x\|\to 0$ for all $x$ in $X$. Show that if $A$ is ...
1 vote
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Proof of a Douglas counterexample

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
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Nuclear Operator with Finite Dimensional Range

I have a question regarding nuclear operators. Let me first give the definition: Let $\Xi$ and $H$ be real Hilbert spaces, and let $T\in L(\Xi,H)$. $T$ is a nuclear operator if there exists a sequence ...
3 votes
1 answer
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Check whether integral operator is compact

Let $T: X \to Y$ be the following integral operator, where $f$ is just assumed to be integrable (not neccessarilly continuous): $$(Tf)(x)=\int_0^x f(t)dt$$ In an exercise I have to check whether this ...
1 vote
2 answers
71 views

Check whether operator $T: L^1[0,1] \to L[0,1]$ is compact. [duplicate]

I need to check whether the integral operator $T:L^1[0,1] \to L^1[0,1]$ that sends $f \to \int_0^x f(y)dy$ is compact or not. I am guessing it is not but I am struggling on proving it. My best shot I ...

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