# 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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### There exists a continuous inverse of $(\text{id}-A)$ in the set $(\text{id}-A)(X)$.

Exercise : Let $X$ be a Banach space and $A \in \mathcal{L}_c(X)$ (means that $A$ is a compact operator). Suppose that $(\text{id}-A)$ is $"1-1"$. Show that the operator $(\text{id}-A)$ has a ...
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### Finite rank operators on Hilbert spaces

Let $H$ be a Hilbert space. Question 1: Are all rank one operators from $H$ to $H$ is of the form $$T:H\rightarrow H, x \mapsto \langle x,u\rangle v$$ For some $u,v \in H$. Question 2:...
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### Reference request on operators with compact powers

In this Wikipedia page about compact operators at the very bottom it says:"If $B$ is an operator on a Banach space X such that $B^n$ is compact for some $n$, then the theorem proven above also holds ...
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### $A(D) \subseteq Y$ is compact if $A \in \mathcal{L}_c(X,Y)$, $X$ reflexive, $Y$ Banach and $D$ closed, convex and bounded.

Exercise : Let $X$ be a reflexive Banach space and $Y$ a Banach space. Also, let $A \in \mathcal{L}_c(X,Y)$ and $D \subseteq X$ be a closed, convex and bounded space. Show that $A(D) \subseteq Y$ ...
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### A self-adjoint operator with essential spectrum={0} is compact

Does every self adjoint operator (on a Hilbert space) with essential spectrum={0} is a compact operator ?
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### Question on Compact operators: Suitable definition of a sequence of compact operators

Let $H$ denote a separable Hilbert space with an orthonormal basis $\{e_k\}_k\in \mathbb N$ and consider a linear, bounded operator $A:H \to H$ such that: $Ae_k=\lambda_k e_k$. Show that $T$ is a ...
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### Show compactness of an operator

Let $T: C^0[0,1] \rightarrow l^1$, $(Tf)_n=a_n \int_0^{1/n} f(x)dx$, for an $f$ in $C^0[0,1]$. Prove that $T$ is compact when $\{\frac{a_n}{n} \}_n \in l^1$. I know the definitin of compact operator, ...
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### Compactness of an integral operator from $L^2$ to $L^2$

I want to prove that The operator (linear and bounded) $T: L^2(0,1) \rightarrow L^2(0,1)$, defined by: $Tu(x)=\int_0^1\sin(x^2+y^2)u(y)dy$, is compact. Just by using theory, it's an Hilbert ...
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### Show that this operator is not compact using Arzela-Ascoli

Let $T:C[0,1]\longrightarrow C[0,1]$ defined as $Tx(t) =tx(t)$. I need to prove that this operator is not compact using Arzela-Ascoli (using the Sup norm). I already prove that if X is a bounded ...
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### Show that $(Tu)(x)=\int_{\alpha(x)}^{\beta(x)} u(t)dt$ is Compact linear operator on $C([0,1])$

Show that \begin{equation} (Tu)(x)=\int_{\alpha(x)}^{\beta(x)} u(t)dt \end{equation} is Compact linear operator on $C([0,1],R)$ where $\alpha, \beta:[0,1]\rightarrow [0,1]$ are continuous. My ...
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### Compact operators and weak convergence

Let $X$ and $Y$ be Banach spaces. (a) Let $T \in \mathcal{L}(X, Y )$. For each sequence $(x_n)_{n \geq 1}$ in $X$ and each $x \in X$, show that $x_n →x$ weakly, as $n \rightarrow \infty$ ,implies ...
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I was reading and trying to understand the proof that the space $\mathcal{L}^2 (\mathcal{H})$ (Hilbert-Schmidt operators) is made by all the $T_K:L^2(X,\mu) \rightarrow L^2(X,\mu)$ with $K \in L^2(X \... 1answer 31 views ### Compact operator on$L^2[0,1]^2$Let$K\in L^2([0,1]\times[0,1])$, and we define the operator$T_k$on$L^2[0,1]$. $$(T_kf)(x)=\int_{0}^{1}K(x,y).f(y).dy \quad \quad \forall f\in L^2[0,1]$$ How to prove that$T_k$is a compact ... 1answer 61 views ###$||T-T_n|| \rightarrow 0$and$T_n$are compact but$T$is not a compact operator. It is a result that if$||T-T_n|| \rightarrow 0$in the norm operator an that the$T_n \in \mathcal{L}(X,Y)$(were$Y$is a Banach space) are compact operators, then$T$is compact. I found from here ... 3answers 73 views ### Integral Operator in$L^2$I was trying to do this exercise and I'm wondering if I figured it out well: I have$\mathcal{H} := L^2(0,1)$and$T$the operator with integral kernel$K(x,y) = \min\{x,y\}$,$x,y \in [0,1]$. I have ... 0answers 14 views ### Compact Integral Operators induced by positive Kernels Let$K$be a compact operator induced by the kernel$k(s,t)\in L^2([0,1])^2$with$k(s,t)>0$. Prove that$\|K\|<1$if and only if$(I-K)$has a bounded inverse$(I-K)^{-1}$which is induced by a ... 1answer 27 views ### Matrices of bounded linear operators Let$X,Y$be Banach spaces and let$A=(A_{n,k})$be an infinite matrix of bounded linear operators$A_{n,k}:X \to Y$. Suppose$\sup_n \sum_k \|A_{n,k}\|<\infty$. Property: For each sequence$x=(...
Let $k \in {L^2}((0,4) \times (0,1))$, $g \in {L^2}(0,1)$. We consider the following first kind Fredholm equation $$\int\limits_0^4 {k(s,x)f(s)ds=g(x), x\in(0,1).}$$ Where $f$ is the unknown. How ...