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.

Filter by
Sorted by
Tagged with
1 vote
0 answers
22 views

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 ...
user avatar
  • 506
0 votes
0 answers
26 views

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|...
user avatar
  • 31
1 vote
0 answers
39 views

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 ...
user avatar
1 vote
0 answers
18 views

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 $p \in A''$...
user avatar
  • 1,571
0 votes
0 answers
36 views

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 \...
user avatar
0 votes
1 answer
36 views

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 ...
user avatar
0 votes
0 answers
45 views

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 ...
user avatar
0 votes
0 answers
29 views

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^...
user avatar
  • 31
1 vote
0 answers
50 views

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 ...
user avatar
  • 105
0 votes
0 answers
23 views

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
user avatar
  • 77
0 votes
0 answers
70 views

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 $$ d \mu = \frac{1}{\sqrt{2\pi}} e^{-\frac{1}{2}x^2}d \lambda $$ ...
user avatar
  • 225
1 vote
0 answers
44 views

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?
user avatar
1 vote
1 answer
29 views

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 ...
user avatar
  • 4,624
6 votes
1 answer
197 views

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 ...
user avatar
0 votes
1 answer
41 views

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)=\...
user avatar
  • 83
2 votes
1 answer
43 views

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 ...
user avatar
2 votes
1 answer
43 views

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 $x_n \in X$ ...
user avatar
0 votes
0 answers
30 views

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 ...
user avatar
  • 51
0 votes
1 answer
59 views

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 $$\mu_n = \sum_{x \perp \{e_1, \cdots, e_n\}, x \neq 0}\frac{\|T(x)\|}{\|x\|^2}$$ ...
user avatar
0 votes
1 answer
41 views

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 ...
user avatar
  • 899
2 votes
1 answer
58 views

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 ...
user avatar
2 votes
1 answer
27 views

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 $(...
user avatar
  • 645
0 votes
0 answers
36 views

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,...
user avatar
  • 141
0 votes
0 answers
35 views

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 ...
user avatar
3 votes
2 answers
89 views

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 ...
user avatar
2 votes
1 answer
34 views

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 ...
user avatar
  • 566
0 votes
1 answer
38 views

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{...
user avatar
2 votes
1 answer
54 views

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\...
user avatar
  • 268
1 vote
1 answer
67 views

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 ...
user avatar
  • 566
1 vote
1 answer
41 views

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 ...
user avatar
  • 8,561
0 votes
1 answer
19 views

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 ...
user avatar
  • 485
1 vote
0 answers
55 views

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 $\mathcal{H}$. ...
user avatar
  • 5,814
0 votes
0 answers
28 views

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 ...
user avatar
  • 1,615
0 votes
1 answer
27 views

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 ...
user avatar
  • 1
2 votes
0 answers
76 views

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 ...
user avatar
  • 1,082
4 votes
1 answer
57 views

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 ...
user avatar
0 votes
1 answer
34 views

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. (...
user avatar
4 votes
1 answer
81 views

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 ...
user avatar
  • 566
2 votes
0 answers
53 views

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 \...
user avatar
  • 21
3 votes
1 answer
46 views

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: $$Te_{2k-1}=\frac{1}{k}(e_{2k-1}-ie_{2k})$$ $$Te_{2k}=\frac{1}{k}(ie_{2k-1}+e_{2k})$$ ...
user avatar
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 ...
user avatar
  • 566
1 vote
1 answer
43 views

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 ...
user avatar
  • 435
0 votes
1 answer
40 views

Prove that $T$ is compact

$A \in B(X)$ and compact operator. Define a map $T: X/\mbox{ker}(A) \to X$. Where $T$ defined as $T(x + \mbox{ker}(A)) = A(x)$ for all $x\in X$. I want to prove that $T$ is compact. I using the fact ...
user avatar
0 votes
0 answers
18 views

Compact identity operator is of finite rank

It's well-known that the compact identity operator on Hilbert space is of finite rank. I'm wondering about the case on Hilbert $A$-module where $A$ is a $C^*$-algebra. I feel it should be the same, ...
user avatar
  • 327
0 votes
1 answer
52 views

Is either $A$ or $B$ a compact operator?

$X$ is a Banach space, consider two operator $A,B\in B(X)$. Now product of two operator, i.e. $AB$ is compact. Can we say that one of $A$ and $B$ is compact? I am just thinking in the following way; $...
user avatar
3 votes
1 answer
146 views

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 ...
user avatar
1 vote
0 answers
54 views

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 ...
user avatar
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(...
user avatar
4 votes
1 answer
36 views

What assumptions are needed for compactness and self-adjointness?

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:...
user avatar
1 vote
1 answer
37 views

For non compact operators, is countability of singular values equivalent to countability of eigenvalues?

Generally speaking, compact operators have countable eigenvalues and countable s-values (or singular values). What about the reverse? If I know that a (non-compact) operator has countable singular ...
user avatar

1
2 3 4 5
24