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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Exercise: The range of Compact Operators

Exercise: Suppose $K:X\to Y$ is compact operator. Show that $K(X)\subseteq Y$ is separable Assume $Y$ is a separable Banach space. Find a Banach space $Z$ and a compact operator $K:Z\to Y$ s.t. $K(...
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I don't see why $W^{1, 2}(\partial D)$ being compactly embedded in $L^2(\partial D)$ lets us show an operator is Fredholm of index zero.

Let $D$ be a bounded Lipschitz domain. Let $A$ be the single layer potential which maps $L^2(\partial D)$ into $W^{1, 2}(\partial D)$ boundedly. $A$ is given by: $$ A_D[\phi] = \int_{\partial D}G(x-y)...
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Sequence of *compact* operators that converges to a bounded linear operator $K_{\lambda}$

Proposition: Fix $1\leq p\leq \infty$ and a bounded sequence of real number $\lambda=(\lambda_i)_{i\in\mathbb{N}}$ and $e_i=\delta_{ij}\in l^p$. Defined the bounded linear operator \begin{equation} ...
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$Im(K)\subset Y$ closed, infinite-dimensional: $K:X\to Y$ is not a compact operator

Proposition: If $K:X\to Y$ is a bounded linear operator between two Banach spaces $X$ and $Y$ such that $\operatorname{Im}(K)\subset Y$ is an infinite-dimensional closed subspace, then $K$ is NOT ...
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Exercise involving sequence Palais-Smale and operator compact.

Let H be a Hilbert space and let K: H $\rightarrow{R}$ be $C^1$ and such that $\nabla K: H\rightarrow H$ sends bounded sets into precompact sets. Consider the functional $J: H\rightarrow R$ defined by ...
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Operator linear and continuous, show that it is compact

Exercise: Let $X$, $Y$ be normed spaces and $A:X\to Y$ be a linear and continuous operator, which has the property \begin{equation} \exists C>0 \text{ s.t. } \forall x\in X \quad \Vert Ax\Vert\geq ...
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Continuous linear operator that is NOT compact.

Exercise: Let $U:L^1(0,\infty)\to L^{\infty}(0,\infty)$, defined as \begin{equation}U(f)(x):=\int_0^x f(t)dt\end{equation} Prove that $U$ is linear, continuous but not compact. My solutions (or what ...
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Characterization of compact operators by their spectra

In any functional analysis book there is usually a section devoted to the study of the properties of the spectrum of compact operators. Is there any spectral characterization of compact (self-...
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About the Banach algebra $\ell^{\infty} (K( \ell^{1}))$

If $P_{n}: \ell^{1} \rightarrow \ell^{1}$ is the projection onto the first n coordinates, then it's well known that $ P_{n} K(\ell^{1}) P_{n}$ is isomorphic to $B(\ell^{1}_{n})$, the Banach space of ...
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$L:L^2([0,1])\rightarrow L^2([0,1])$, $Lf(x)=\frac1{x+1}f(x)$ not compact

Let $L:L^2([0,1])\rightarrow L^2([0,1])$ given by $Lf(x)=\frac1{x+1}f(x)$. I want to show that $L$ is continuous but not compact. The boundness follows by $\|Lf\|_2\leq \|f\|_2 $ since $\frac1{x+...
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Find the spectrum of an operator related to Fourier series

As an exercise, I was told to find the spectrum of the bounded operator $K\in B(L^2[-\pi,\pi])$ defined by $$K\varphi (t)=t\int_{-\pi}^\pi\varphi (x)\cos (x)dx+\cos t\int_{-\pi}^\pi x\varphi(x)dx.$$ ...
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When is orthogonal projection compact? [duplicate]

Let $M$ be a closed subspace of a Hilbert space $H$. Let $P$ be the orthogonal projection on $M$. I was told to find the eigenvalues and eigenvectors of $P$ and moreover say when it is compact. Since ...
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Simple norm inequality

Trying to follow the comments to this question I am struggling very much to understand how to simplify $\|Ax\|_2=\sup_{\|x\|_2=1}\sqrt{\sum_i(\sum_ja_{ij}x_j)^2}$ to arrive at an $x$-free bound. Can ...
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Prove multiplication by sequence is a compact operator

Let $c_0(\mathbb N)$ be the space of sequence in $\mathbb C$ whose limit is zero, equipped with the $\ell^\infty$ norm. Let $u_n$ be a sequence in $\mathbb C$ and define the operator $A$ taking a ...
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Proving an operator is compact exercise

Suppose $(a_{ij})_{i,j\in \mathbb N}$ satisfy $\sum_{i,j}|a_{ij}|^2<\infty$ and define $A:\ell ^2 \rightarrow \ell ^2$ by $(Ax)_i)=\sum _j a_{ij}x_j$. I need to prove $A$ is compact. Unfortunately,...
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For which $\alpha$ is this integral operator compact?

I have $Q\subset\mathbb{R}^n$ $Af(x)=\int_QK(x,y)f(y)dy$ , with $K(x,y)=\frac{K_0(x,y)}{|x-y|^\alpha}$ and $K_0\in C(Q)$ I want to estimate using an operator $A_Mf(x)=\int_QK_M(x,y)f(y)dy$ where, $...
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Convergence of Compact Linear Opeators in $L^2([0,1])$

Let $A_n: L^2([0,1]) \to L^2([0,1])$ be $(A_nf)(x)= \int_0^1 sin(n\pi(x-y))f(y)dy$. Are they compact operators? Is there any kind of convergence? They are continous since $\|A_nf\| \leq \|f\| $. ...
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Finding the adjoint of a compact operator?

Let $H = L^2([0,1])$ and $A:H \to H$ be defined as $$ Af(t) = \int_0^t f(s) ds, \quad f \in H $$ By the Ascoli-Arzela theorem, $A$ is compact. Now the adjoint of $A$ is given as $$ A^\ast g(s) = \...
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Finding this operator's spectrum

In an exam, my professor gave the following exercise: State and prove the spectral theorem for compact operators. Let $K$ be the operator defined by: $$Kf(t)=\int_0^1\min(t,s)f(s)\mathrm{d}s.$...
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A $*$-closed algebra of compact operators is completely reducible

In page 13 of Lang's $SL_2$ there is a proof that for a $*$-closed algebra $\mathscr A$ of compact operators on a Hilbert space $H$, $H$ is completely reducible. The proof follows by taking the ...
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Is $p \vee q \leq p+q$ for $p,q$ projections?

I am wondering if $p \vee q \leq p+q$ for $p,q$ projections acting on some Hilbert space $H$. In particular, I wonder if the set of finite trace projections is upwards directed with the usual ...
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All nonzero singular values of $A$ are equal to $1$ iff $A^*=A^*AA^*$ and $A=AA^*A$

I want to show that all the non-zero s-numbers, i.e. singular values $s_j(A):=(\lambda_j(A^*A))^{1/2}$, of A (a bounded linear operator of finite rank acting on a separable Hilbert space $H$) are ...
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Compactness of an operator

let $e(j)$ denote the canonical Hilbert base of $l^2(\mathbb{C})$, then define a linear operator by $T(e(j))=\frac{1}{1+j} e(2j)$. I know that this operator is compact. My question occurs within the ...
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Sufficient conditions for $f(T)$ to be compact and self adjoint whenever $T$ is compact and self adjoint

Let $Tx=\sum_{n=1}^\infty \lambda_n \langle x,e_n\rangle e_n$ be bounded where $\{\lambda_n\}_n$ are the complex eigenvalues and $\{e_n\}_n$ are an orthonormal basis of the separable space $H$. For ...
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Operators, Riesz

Suppose that $\Omega\subset\mathbb{R}^n$ is open, $k\mapsto f_k\in H^{1,2}(\Omega)$ and $f\in L^2(\Omega)$ such that $f_k\rightarrow f$ in $L^2(\Omega)$ and $||\partial_if_k||_{L^2(\Omega)}\le K\ \ \...
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A restriction $A_2$ of a compact self-adjoint compact linear operator $A$ is also compact and self-adjoint?

Let $X$ be an inner product space and let $A$ be a compact and self-adjoint linear operator. Let $p_1$ be an eigenvector of $A$. Let $A_2$ be the restriction of $A$ to $X_2$ where $X_2$ is given by $$...
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Question on spectral theorem for compact operators

I'm studying a proof of the spectral theorem for compact operators. The first part of it reads as follows: Let $X$ be an infinite dimensional inner product space and let $A: X \to X$ be a compact and ...
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Why is a normal operator with certain spectral properties compact?

Given a compact operator, it is well-known that the Spectrum consists only of eigenvalues and possibly 0. Now I'm thinking about the inverse implication with additional conditions. So, given a normal ...
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Is K(H) separable?

Let $H$ be an inifinite dimensional separable K-Hilbert space with $K$ could be $\mathbb{R}$ or $\mathbb{C}$. Are the compact operators $K(H)$ separable? It's well known that $\overline{F(H)}=K(H)$, ...
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Compact operator and norm

Let $E,F$ and $G$ be normed spaces, $f\in \mathcal{L}(E,F)$ and $g\in \mathcal{L}(F,G)$. Suppose that $g$ is injective and $f$ is compact. Show that, $\forall \varepsilon > 0$, there exists $M&...
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Show that the following is a bounded linear operator on $L^2(R_+)$ Calculate the adjoint operator.

The following is a question I have been working on for some time with help from my teacher. Unfortunately we have a solution but are not 100% confident with it. Some guidance on if part(s) of our ...
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I want to calulate the range of an operator $S$ which maps $L^{2}$ into $H^{1}(\Omega)$?

I have an equation $s(\lambda,\mu)=l(\mu)$, where s(.,.) is a symetric positive definite bilinear form in $L^{2}(\Omega)$, and $l(.)$ is in $H^{1}(\Omega)$. I want to show that the range of the ...
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Is $Ax = (\alpha_1 x_1, \alpha_2 x_2, \alpha_3 x_3, \dots, \alpha_k x_k, 0, 0, \dots)$ a compact operator?

Is the operator $A$ defined by $$Ax = (\alpha_1 x_1, \alpha_2 x_2, \alpha_3 x_3, \dots, \alpha_k x_k, 0, 0, \dots)$$ a compact operator? It only has finitely non-zero dimensions, so does this mean ...
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How to show $\sigma(T_q) = \overline{\{q(t) : t \in [0,1]\}}$ where $T_q$ is the multiplication operator?

Let $B$ be the Banach space of bounded complex functions on $[0,1]$ with sup-norm. For $q \in B$, define the (multiplication) operator $T_q : B\rightarrow B$ by $(M_q f)(t) = q(t)f(t)$. How do you ...
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Determine whether the differential operator is compact in the following cases

Given the differential operator $\displaystyle Tx(t)=\frac{dx}{dt}$, I need to determine (and be able to justify) whether it is compact in the following three cases: $T: C^{1}[0,1]\mapsto C[0,1]$ $T:...
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Complex Air operator

Help me to do this exrcice Consider the differential operator $A=-\partial^{2}_{x}-ix$ on $\mathbb{R}$ with $D(A)=\{f\in L^{2}(\mathbb{R},dx), Au\in L^{2}(\mathbb{R},dx)\}$ Check that $A$ is colsed ...
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Is the shift operator on square integrable periodic functions compact?

I'm considering the space $L^2[0,2\pi]$ of square integrable $2\pi$-periodic functions equipped with the inner product $$\langle{f},g\rangle=\int_0^{2\pi}f(x)\overline{g(x)}\,dx$$ and the shift ...
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Why is $\ker(T-\lambda I)^n$ finite-dimensional?

Let $X$ be a normed space and $T$ be a compact operator on $X$ and $\lambda \in \sigma(T)\setminus\{0\}$. A closed unit ball in $\ker(T-\lambda I)$ always admit a convergent subsequence of a sequence,...
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Operator $f \mapsto u(f)$ solution of non-homogeneous Laplace equation is compact and self-adjoint

Let $u : L^2_0(D) \to L^2_0(D): = \lbrace f \in L^2 : \int_D f = 0 \rbrace $ be the linear operator which associates $f$ to $u(f)$ the solution of $$ \begin{cases} \Delta u = f & \text{in } D \\ \...
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What role does the range of an operator play in showing the operator is compact?

To show an operator is compact I understand you have to show the operator is the limit of finite rank operators. However the proof I have doesn't do this. I have an operator $$k:C0[,\pi]) \to C([0, \...
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Proving a linear operator is compact: understanding the statement “norm limit of a sequence of finite rank operators”.

I am having serious trouble understanding the proof that an operator is compact. This is the original question I asked and the proof is included very helpfully in the answer. Show if $\lim_{n \to \...
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Integral operator with sub-exponential eigenvalue decay

I am looking for a positive-definite symmetric function $K: \Omega \times \Omega \to \mathbb{R}$ ($\Omega \subset \mathbb{R}$), such that $$ K(x,y) = \sum_{k=0}^\infty \exp(-\sqrt{k}) \phi_k(x) \...
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A question on Bounded Approximation property

Let $V$ be a Banach space and we say that $V$ has the $C$-BAP if there exists a net of bounded finite rank operators $T_\alpha$ in $B(V,V)$ and a constant $C$ such that $\|T_\alpha\| \leq C$ for each $...
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Showing a C* Algebra contains a compact operator

In my functional analysis class we are currently dealing with C* Algebras, and I just met this problem: Let $ \mathbb{H} $ be a separable Hilbert space, and suppose we have $ A \subset B(\mathbb{H}...
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Problem about a compact operator $T:l^p\rightarrow l^p$

I have to solve this problem. Let $\{\lambda_n\}$ be a sequence of real number such that $\lim_{n\rightarrow\infty}\lambda_n=0$ and consider the operator $T:l^p\rightarrow l^p$, $1\leq p\leq \...
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Singular integral operator

i got the following problem to solve. Let $0 < \alpha < 1$, $L \in L_\infty([0,1]^2)$, $D = \{(x,y) \in \mathbb{R}^2: x = y\}$ the diaagonal of $\mathbb{R}^2$ and $k:[0,1]^2 \setminus D \to \...
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Hilbert space …

i got the following operator. Let H be a Hilbert space, $(\lambda_n)_{n \in \mathbb{N}}$ a bounded sequence in $\mathbb{K}$ and $T^: H \to H, x \to \sum_{n\in\mathbb{N}} \lambda_n \left<x, e_n\...
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The Hilbert-Schmidt Theorem for Compact, Self-adjoint operators

Suppose $T$ is a compact, self-adjoint operator on a Hilbert space $\mathcal{H}$. Then there exists an orthonormal set $\{e_n\}_{n=1}^{\infty}$ of eigenvalues of $T$ such that every $x \in \mathcal{H}$...
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Spectrum of an unbounded operator

Consider a densely defined unbounded operator $A_0:D(A_0)(\subset{H})\to H$ which has the following properties: 1- Symmetric, $\langle A_0x,y\rangle=\langle x,A_0y \rangle$ 2- Positive, $\langle ...
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Show that there is an operator on $H^{2}$ and it's compact.

Let $H^{2}=W_{0}^{2,2}(\Omega)$. Define $(u,v)=\int_{\Omega} (\triangle u\triangle v+2v\triangle u)\mathrm{d}S$ as an inner product on $H^{2}$. Define $a(u;v)=\int_{\Omega} (\nabla u\cdot \nabla v-...