# 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.

745 questions
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### Relation between Schatten-$p$-norm and $l^p$ norm of operator matrix

Let $\mathcal H$ be a separable Hilbert space and let $(e_i)$ be some orthonormal basis. Let $K$ be a compact operator on $\mathcal H$ with matrix elements $K_{ij}=\langle K e_i,e_j\rangle$. My goal ...
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### convergence RATE of the square root of a self-adjoint operator.

I am assuming $T$ is a compact operator and $\{T_j\}$ is a sequence of compact operators such that $\|T-T_j\| < \epsilon_j$ where $\epsilon_j$ is a quantity that goes to zero as $j \to \infty$. It ...
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### Is $|x|^{-d+\alpha}$ square integrable in $\mathbb{R}^d$ given $\alpha>0$?

This is a problem in the S.-T. Yau College Student Mathematics Contests in 2013. Suppose $H=L^2(B)$, $B$ is the unit ball in $\mathbb{R}^d$. Let $K(x,y)$ be a measurable function on $B\times B$ that ...
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### Is the spectral decomposition of self-adjoint compact operators unique?

Given a compact, self-adjoint operator $T$ on a Hilbert space $H$, then there is the Spectral Theorem which says that $T=\sum_i \lambda_i P_i$ where the sum is over the number of eigenvalues of $T$, ...
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### Continuous but not Compact operator

Let $l^1=\{x_n:||x_n||_1<\infty\}$ and $l^2=\{x_n:||x_n||_2<\infty\}$ be equipped with usual norms. Let $T:l^1\to l^2$ be defined by $T(x_n)=x_n$. Show that $T$ is a continuous operator which is ...
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### If $A+B$ is a compact non-zero operator, which properties follow for $A$ and $B$?

Assume we are in $l^p$ spaces or at least Banach spaces. I'm trying to find out what the knowledge that the linear operator $A+B$ is compact tells me about the properties of the linear operators $A$ ...
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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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### 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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### If $A \in \mathcal{L}_c(X)$ and $X$ is Banach, then $\dim \ker (\text{id}-A) < + \infty$.

Exercise : Let $X$ be a Banach space and $A \in \mathcal{L}_c(X)$. Show that $\dim \ker ( \text{id} - A) < + \infty$. Attempt/Thoughts : The kernel of the operator $(\text{id}-A) : X \to X$ ...
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### Prove that the following map is compact

I'm studying by myself PDEs without having done Functional Analysis and I'm trying do the following exercise of the book "Partial Differential Equations I" by Michael E. Taylor on Appendix $A$ - ...
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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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### 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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### 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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### Compact operator on $L_2([0,1],m)$

Consider the Hilbert space $H=L_2([0,1],m)$ where $m$ is the Lebesgue measure on the interval $[0,1]$. Let $T \in \mathcal{L}(H,H)$ given by \begin{equation*} T\ f(x)=x \ f(x) \ \ \ \ f \in H,\ x \...
### Convergence of $A_nT$ to $AT$ in operator norm for compact $T$
$A_n:Y\rightarrow Z$ are operators that strongly converge to $A$. Also, $\|A_n\|_\text {op}\le c$ for $c>0$. Given a compact operator $T:X\rightarrow Y$, I need to show that $A_nT$ converges to $AT$...
### Show that $(Tu)(x)=\int_{\alpha(x)}^{\beta(x)} u(t)dt$ is Compact linear operator on $C([0,1])$
Show that $$(Tu)(x)=\int_{\alpha(x)}^{\beta(x)} u(t)dt$$ is Compact linear operator on $C([0,1],R)$ where $\alpha, \beta:[0,1]\rightarrow [0,1]$ are continuous. My ...