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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$Tx=(\lambda_kx_k)_{k=1}^\infty$. For what $(\lambda_k)$ is the operator $T$: (a) well defined? (b) bounded? (c) compact? [duplicate]
Came across this exercise in a book I'm reading. The full exercise is as such:
For a sequence of real numbers $(\lambda_k)_{k=1}^\infty$, define a linear operator $T:l_2\rightarrow l_2$ by $Tx=(\...
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Proving a linear operator is compact if it maps the open unit ball (centered at the origin) to a precompact set
I came across this exercise while reading a book. For context, $X$ and $Y$ are normed spaces; a linear operator $T:X\rightarrow Y$ is called compact if it maps bounded sets in $X$ to precompact sets ...
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Relationship between compactness of a set and compactness of a linear operator
I am recently learning functional analysis and topology, and I am wondering whether there is some relationship between compactness of a set (or a topological space) and compactness of a linear ...
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Proof of $ran(E(\lambda)) = ker(\lambda-A)^\alpha$
Let $A: H \to H$ be a compact operator on the complex Hilbert space $H$. Let $\lambda \neq 0$ be an Eigenvalue of $A$. Since A is compact, it's Eigenvalues can only accumulate at $0$, so we can find a ...
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Let $I$ be an open bounded interval of $\mathbb R$. The injection $W^{1,p}(I) \subset L^q(I)$ is compact for any $p, q \in [1, \infty)$
Let $I$ be an open bounded interval of $\mathbb R$. I have recently proved in this thread that the injection $W^{1,p}(I) \subset L^q(I)$ is compact for all $1\le p \leq q<\infty$.
We fix $1\le r &...
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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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Is this comment about weak convergence in $W^{1,2}(\Omega)$ correct?
I'm reading @emily20's question in this thread
Let $\Omega\subset \mathbb{R}^n$ be open and bounded. Also let
$W^{1,2}(\Omega)=\left\{u \in L^{2}(\Omega)|\,\, \forall \alpha \in \mathbb{N}^{n}:|\...
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Sum of completely continuous operators is compact.
Self Study - NOT HOMEWORK OR GRADED IN ANYWAY
I just want to know if my proof is correct.
Statement :
Let $X$ and $Y$ be normed linear spaces and let $T:X \to Y$ and $S:X \to Y$ be completely ...
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Finite rank operator achieves its norm [closed]
I need some help showing that a finite rank operator on a Hilbert space achieves its norm. I would also like to know if such a result holds on Banach spaces.
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Determining whether an operator is trace-class
Let $H$ be a separable Hilbert space, and let $A$ and $B$ be trace-class operators on $H$ such that $A^{-1/2}B$ is a Hilbert-Schmidt operator. Then is it possible to know whether the operator $B'A^{-1}...
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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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Hankel Operator is compact
I am currently working on compact operators. I am trying to solve the following exercise
Problem
Let $(a_j)_{j \in \mathbb{N}}$ be a sequence of complex numbers in $\ell_1$, i.e. $\sum_j |a_j| < \...
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Strong convergence of compact operator
I am currently doing an exercise on compact operators.
Problem
Let $K: X \to Y$ be a compact linear operator. Suppose $(x_{n})_{n \in \mathbb{N}}$ is a sequence in $X$ with the property that there ...
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Finding the conjugate of an operator between the Banach spaces $\ell_{p}$
I am working with conjugate operators acting between Banach spaces. I am doing the following exercise.
Let $(\beta_{n})_{n \in \mathbb{N}}$ be a bounded sequence of complex numbers. Define the ...
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$L^2$-ness of the finite difference operator
Is there a proof of the fact that the operator $\mathfrak{D}$ defined by $\mathfrak{D}[f](x,y):=\dfrac{f(x)-f(y)}{x-y}$ is continuous from $H^n(\mathbb{R})$ to $H^m(\mathbb{R}^2)$ or even $L^2(\mathbb{...
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Codimension of a linear operator restricted to a finite dimensional subspace
Suppose that $K$ is a compact linear operator acting on a Banach space $X$ into itself. Then there exist closed subspaces $N$ and $Z$ with $N$ finite dimensional, $Z \subset \ker{K}$ and $$ X = N \...
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Norm of $Tf(x)=\int_{-\pi}^{\pi} \cos(x-y)f(y)\,dy$
I'm dealing with this operator $T\in \mathcal{L}(H)$ where $H=L^2([-\pi,\pi])$
$$
Tf(x) = \int_{-\pi}^{\pi} \cos(x-y)f(y)\,\mathrm{d}y
$$
One of the questions request to compute the norm of the ...
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Toeplitz Operators Generate Compacts
This exercise is from Higson's 'Analytic K - Homology'.
Let $H$ be the Hardy space $H^2(S^1)$. Show that the $C^*$-subalgebra of $\mathcal{B}(H)$ (bounded operators) generated by the Toeplitz ...
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Existence of proper invariant subspaces for compact operators
Around 1950, J. von Neumann proved the existence of proper invariant subspaces for compact operators in a Hilbert space. Does anyone know a reference for a simplified proof of this fact?
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Question on compactness of generator of unitary group
Let $\mathcal{H}$ be complex Hilbert space. Suppose we have a unitary group $\{U_t\},t\in\mathbb{R}$ and by Stone's Theorem we have a unique infinitesimal generator $A:\mathcal{D}(A)\rightarrow\...
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Are Cauchy sequences for Hilbert space an expression of a compact (multiplication) operator?
Background: I'm reading about Hilbert spaces that require a complete metric space using inner product, where every Cauchy sequence of points $x_m$,$x_n$ on the metric space, $X$, has a limit, also in $...
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Compact operator on Hilbert space: clarification of Wikipedia article
I have been trying to work through the Wikipedia article titled compact operator on a Hilbert space. I have made it though to the section 'Spectral theorem', subsection 'the idea'. This section seems ...
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Compact set in sequence of resolvet set of convergent sequence of linear and bounded operators
I'm working in the following excercise:
Let $(T_n)_{n\in\mathbb{N}} \subseteq\mathcal{L}(E)$ and $T\in \mathcal{L}(E)$ such that $T_n\to T$. Show that for every compac set $K\subseteq R(T)$ (resolvent ...
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Is $f\mapsto \int\limits_{\mathbb{R}^n} K(x,y)f(x)dy$ with kernel $K\in C(\mathbb{R}^n\times \mathbb{R}^n)$ compact?
I know if $K\in C([0,1]\times [0,1])$ then
$(Tf)(y):=\int\limits_{[0,1]} K(x,y)f(x)dy$ defines a compact operator on $L^2([0,1])$ i.e. $T$ is Hilbert Schmidt.
Is this also true if we replace $[0,1]$ ...
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Proof of theorem 6.6 in Functional Analysis of Haim Brezis (Fredholm alternative)
I'm reading the proof of the following theorem:
My question is in part d), the last paragraph says that:
I'm not sure why
$$
N(I-T) \subseteq N(I-T^{**})
$$
because
$$
N(I-T) \subseteq E \quad\mbox{...
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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 ...
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How to show following is weakly sequentially compact.
I have defined the integral operator on a finite measure space $(X,\Sigma,\mu)$ Orlicz space $T: L^{\Phi}\to L^{\Psi}$, suppose we have the result that says for any bounded sequence $\{f_n\}$, $\{Tf_n\...
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Compact operator in squence spaces $T(u) = \left(\sum_{k=1}^\infty \frac{u_k}{n+k}\right)_{n\in\mathbb{N}}$
I'm trying to prove the following statement with $E = \ell^2$ with usual norm $||x||_{\ell^2}$ and complex coefficients. Define $T:E \to E$ such that
$$
u = (u_n)_{n\in\mathbb{N}} \mapsto T(u) = \left(...
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Brezis' exercise 6.21.3: prove that there is $K_2 >0$ such that $K_2 d(u, N) \le p (u)$ for all $u \in V$
I'm trying to solve an exercise in Brezis' Functional Analysis, i.e.,
Let $(V, \| \cdot \|)$ and $(H, | \cdot |)$ be real Banach spaces satisfying $V \subset H$ with compact injection. Let $p(\cdot)$ ...
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Eigenvalues of a resriction of a compact operator [closed]
Let $H$ be a real separable Hilbert space, let $A$ be a compact operator on $H$ whose spectrum does not contain $1$, i.e., $1-A$ is invertible, and let $P$ be an orthogonal projection on some proper ...
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Brezis' exercise 6.21.1: prove that $N$ is finite-dimensional
I'm trying to solve an exercise in Brezis' Functional Analysis, i.e.,
Let $(V, \| \cdot \|)$ and $(H, | \cdot |)$ be real Banach spaces satisfying $V \subset H$ with compact injection. Let $p(\cdot)$ ...
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Does $x_n \rightharpoonup 0$, where $\|x_n \| \leq 1$ imply $Ax_n\rightarrow 0$, when $A$ is a compact, self-adjoint projection? [duplicate]
Let $(x_n)$ be a sequences in a Hilbert space $H$, with $\|x\|\leq 1$, where $x_n \rightharpoonup 0$. Let $A: H\rightarrow H$ be a linear, compact, self-adjoint projection i.e. $A^2=A$ and $A=A^*$. ...
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Brezis' exercise 6.20.1: prove that $T \in \mathcal{K}(E)$
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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Conditions for an integral operator to be a compact operator on L^2(R)
I think Conditions for an integral operator to be a compact operator on L^2(R). I know there are some conditions such as Hilbert-Schmidt integral operator is compact. However, do there exist other ...
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Compact operator in $\ell_2$ [duplicate]
Let $T: \ell_2 \to \ell_2$ be
$$
T((x_n)_{n \in \mathbb{N}}) = \left(\frac{x_n}{n}\right)_{n \in \mathbb{N}}.
$$
We have to determine if $T$ is a compact operator, self-adjoint and if it has ...
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Is it true that $T$ is surjective IFF $T-K$ is surjective?
Let $E$ be an infinite-dimensional 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 $T\in \...
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Atkinson Theorem for unbounded Fredholm Operators
Atkinsons Theorem states that a bounded linear operator $T\in L(X,Y)$ where $X,Y$ are Banach spaces is a Fredholm operator if and only if there exist bounded linear operators $S,S'\in L(Y,X)$ and ...
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Trace theorem for Bochner space.
Let $ \Omega $ be a smooth bounded domain in $ \mathbb{R}^n $, it is well-known that the trace theroem implies that there is a compact embeding $ H^1(\Omega)\hookrightarrow L^2(\partial\Omega) $. ...
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Strictly positive compact operator commuting with a given Fredholm operator
In the following snippet from a paper by Jody Trout on the converse functional calculus, they mention the existence of a strictly positive compact operator $T$ commuting with a given self-adjoint, odd ...
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Brezis' exercise 6.18.1: does $S_r$ or $S_{\ell}$ belong to $\mathcal{K}(E)$?
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. For $T \in \mathcal L(E)$,
we ...
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Proof of Ringrose Theorem
Here is the Ringrose Theorem regarding invariant subspace on Peter Lax's Functional Analysis (Maximal invariance nest is a sequence of nested subspace in inclusion which cannot insert subspace into ...
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Brezis' exercise 6.13.1: $\forall \varepsilon>0\,\exists C_\varepsilon > 0$ s.t. $|Tu|_F \le \varepsilon |u|_E + C_\varepsilon |u|$
I'm trying to solve an exercise in Brezis' Functional Analysis, i.e.,
Let $E$ and $F$ be two real Banach spaces with corresponding norms $|\cdot|_E$ and $|\cdot|_F$. Assume that $E$ is reflexive. Let ...
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Existence of compact operator with eigenvelues determined by a given sequence
Could anyone help me with the following exercise? Any help will be very welcome:
Given a sequence $\{\alpha_n\}$ so that $\alpha_n \to 0$, show that exists a compact operator $T$ with spectrum $\sigma(...
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Injective compact linear operator from infinite dimensional normed space
Let $V$ be an infinite dimensional normed space. How can I show that there exist a normed space $W$ and an injective compact linear operator $T$ from $V$ to $W$?
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Brezis' exercise 6.12: a lemma of J.-L. Lions
I'm trying to solve an exercise in Brezis' Functional Analysis, i.e.,
Let $X,Y,Z$ be real Banach spaces with corresponding norms $|\cdot|_X, |\cdot|_Y, |\cdot|_Z$. Assume that $X \subset Y$ with ...
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Brezis' exercise 6.12: $\forall \varepsilon>0\,\exists C_\varepsilon > 0$ s.t. $\|u\|_\infty \le\varepsilon \|u'\|_\infty + C_\varepsilon \|u\|_{L^1}$
I'm trying to solve an exercise in Brezis' Functional Analysis, i.e.,
Let $X,Y,Z$ be real Banach spaces with corresponding norms $|\cdot|_X, |\cdot|_Y, |\cdot|_Z$. Assume that $X \subset Y$ with ...
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Show that the operator $Ax(t)=\int_0^1 \frac{\cos(t\tau)}{|t-\tau|^\frac{1}{5}}x(\tau)d\tau$ is compact on $L_2[0,1]$.
I have to show that an operator $A:L_2[0,1]\rightarrow L_2[0,1]$ given by
$$Ax(t)=\int_0^1 \frac{\cos(t\tau)}{|t-\tau|^\frac{1}{5}}\ x(\tau)\ d\tau$$
is compact.
My idea is to show that it is ...
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Brezis' exercise 6.10.3: $\dim N(I-T) = \dim N(I-T^*)$
Let $E,F$ be real Banach spaces. Let $\mathcal L(E, F)$ be the space of bounded linear operators from $E$ to $F$, and $\mathcal K(E, F)$ its subspace consisting of compact operators. Let $\mathcal L(E)...
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