# 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 smoothing operators are compact

Suppose I have a bounded, linear map $T: H^1(X) \to H^1(X)$ such that $T(H^1(X)) \subset C^\infty(X)$. Is $T$ a compact operator? I'm guessing this depends on whether or not $X$ is (pre)compact, and ...
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### Is every bounded derivation from compact to finite rank operators inner?

This question is related to Derivation into dense ideal of Banach algebras. Depending on which way it goes, an answer to one might answer the other (this is elaborated below). Let $H$ be a Hilbert ...
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### trace $(ADA^{-1})=$ trace $(D)$ in infinite dimensions?

Let $X$ be a separable Hilbert space, $D$ nonnegative definite (by which I also mean self-adjoint) and trace class operator on $X$. Let $A$ be a compact and injective operator with dense range $R$, ...
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### Linear bounded operator from $L^p[0,1]$ to itself whose range consists of continuous functions.

Let $T\colon \mathbb L^p[0,1]\to \mathbb L^p[0,1]$, $1<p<+\infty$, be a linear bounded operator such that $\operatorname{Im}(T)$ is contained in the space of continuous functions. It was shown ...
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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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### Why is this compact operator a Fredholm operator?

Let $X$ be a Banach space with the $L^{\infty}$ norm and let $A$: $X \rightarrow X$ be an integral operator of the following form, \begin{equation} Ax(s) = \lambda\int^{b}_{a}K(s,t)x(t)dt, \end{...
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### Compact embedding between $H^{m+1}(\Omega)$ and $H^{m}(\Omega)$ for $\Omega$ bounded

I know that we have Rellich-Kondrachov Theorem that says that there is a compact embedding between $H^{1}(\Omega)$ and $H^{0}(\Omega)$, or more generally as Adams states (pag 168 theorem 6.3) we have ...
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### Fredholm and Compact Operators

Let $X$ and $Y$ be Banach spaces and $T\in B(X,Y)$ be Fredholm. Then there is $S\in B(Y,X)$ such that $ST=I+K_{1}$ and $TS=I+K_{2}$ where $K_{1},K_{2}$ are compact operators.Proof: Since $T$ is ...
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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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### 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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### Third kind Fredholm integral equation

Let us consider the following integral equation $$a(x)u(x) + \int\limits_0^1 {K(s,x)u(s)ds} = f(x)$$ Let f in $L^p(0,1)$ for some $p \in [1,\infty] and let$ $K \in L^q((0,1) \times (0,1))$. Assume ...
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### What is known about the reciprocal $1/f$ of a holomorphic Banach-valued function $f$?

Let $U \subseteq \mathbb C$ be open, $A$ a (unital, associative) complex Banach algebra and $f : U \to A$ holomorphic and invertible in a punctured neighborhood of $0 \in U$, so that $0$ is an ...
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