# 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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### Rellich-Kondrachov theorem in dimension one

Let $I:=(a, b)$ be an open interval, possibly unbounded. I'm reading below theorem in Brezis' Functional Analysis: Theorem 8.8. There exists a constant $C$ (depending only on $|I| \leq \infty)$ such ...
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### Left or right invertible implies Invertibility of compact operators

Suppose $K \in \mathcal{B}(X)$ is a compact linear operator, where $X$ is a Banach space. Suppose $I-K$ is either left or right invertible. Show that $I-K$ is invertible and $I-(I-K)^{-1}$ is a ...
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### Brezis' exercise 6.25(ii): the existence of $\widetilde M, \widetilde P \in \mathcal L(E)$ such that $(I+K) \circ \widetilde M = I-\widetilde P$

Let $E$ be a real Banach space. Let $\mathcal L(E)$ be the space of bounded linear operators on $E$ and $\mathcal K(E)$ its subspace consisting of compact operators. Let $I:E \to E$ be the identity ...
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### Brezis' exercise 6.25(i): the existence of $M,P \in \mathcal L(E)$ such that $M \circ(I+K)=I-P$

Let $E$ be a real Banach space. Let $\mathcal L(E)$ be the space of bounded linear operators on $E$ and $\mathcal K(E)$ its subspace consisting of compact operators. Let $I:E \to E$ be the identity ...
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### Check a point if it is in the interior of $T(B)$ or not
Define the bounded linear operator $T : L^2[0,1] \to L^2[0,1]$ by $$Tf(x) = xf(x), \quad \forall f \in L^2[0,1].$$ Let $B$ the open unit ball in $L^2[0,1]$. I need to determine whether $0$ is in the ...