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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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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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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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Rudin's functional analysis, theorem 4.23 (existence of a certain sequence)

If $X$ is a Banach space, $T \in \mathcal{B}(X)$, $T$ is compact, and $\lambda \neq 0$, then $T - \lambda I$ has closed range. Proof with questions below Proof: By (d) of Theorem 4.18, $\text{dim }...
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A question on linear integral equation about non degenerated bilinear form

Let $X$ be a Banach Space , $X\subseteq H,\bar{X}=H$,where $H$ is a Hilbert space $i:=X\to H$ defined by $i(x)=x$ and is continuous. Define $\langle x,y\rangle=\langle ix,iy\rangle$ then $\langle X,X,...
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Convergence of finite dimensional projection of trace class in trace norm

Assume $\mathbb{H}$ is a Hilbert space and $K$ is a trace-class operator on it. Given a fixed ONB $\{e_i\}$ and assume $$K=\sum_{i,j}c_{ij}e_i\otimes e_j.$$ Now, let $K_n = \sum_{1\leq i,j\leq n}c_{...
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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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How is $\Bbb K$ well defined, operator algebra

The letter $\Bbb K$ in Bruce Blackadar, on operator algebra denotes the algebra of compact operators on a separable infinite dimensional hilbert space, $H$. In my other post, it is shown that $M_\...
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Reference request on operators with compact powers

In this Wikipedia page about compact operators at the very bottom it says:"If $B$ is an operator on a Banach space X such that $B^n$ is compact for some $n$, then the theorem proven above also holds ...
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$A(D) \subseteq Y$ is compact if $A \in \mathcal{L}_c(X,Y)$, $X$ reflexive, $Y$ Banach and $D$ closed, convex and bounded.

Exercise : Let $X$ be a reflexive Banach space and $Y$ a Banach space. Also, let $A \in \mathcal{L}_c(X,Y)$ and $D \subseteq X$ be a closed, convex and bounded space. Show that $A(D) \subseteq Y$ ...
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$C^*$ algebra of compact operators as a direct limit of matrix algebras?

This was written in Page 92, of Higson's Analytic $K$-Homology book. Let $H$ be a hilbert space. The $C^*$ algebra $K(H)$ of compact operators is the direct limit of a sequence $$M_2(\Bbb C) \...
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Norm of AB+BA when A, B are well understood

Here is the situation: I have the operator/matrix expression AB+BA and want to know the norm. 1.) B is compact, so I believe AB+BA is compact (as are AB and BA, so any finite-dimensional matrix ...
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If $X$ is reflexive and $A \in \mathcal{L}(c_0,X)$ then show that $A$ is a compact operator.

Exercise : Let $X$ be a reflexive Banach space and $A \in \mathcal{L}(c_0,X)$. Show that $A$ is a compact operator. Thoughts : First of all, I know that if $X,Y$ are Banach then $A \in \mathcal{L}...
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Is weakly continuous operator weakly compact?

Suppose we are given an operator $T\colon X^*\to Y$ and both $X$ and $Y$ are Banach spaces. Assume that operator $T$ is continuous for the weak$^*$ topology of $X^*$ and weak topology of $Y$. Does it ...
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Schauder fixed point extended

The Schauder fixed point theorem states that if $X$ is a Banach space, $K\subset X$ is a convex, bounded and closed subset and $T:K\rightarrow K$ is compact, then $T$ has, at least, one fixed point in ...
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Operator norm of integral operator

Suppose we have $X=L^2([0,1];\mathbb{R})$ and \begin{equation} T:X\rightarrow X, \ Tf(x)=\int_0^1x^2yf(y)dy. \end{equation} Show that $T$ is compact and determine $||T||.$ I already have that $||T|...
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Showing that $\exists x \in H : \|A(x)\| = \|A\|_\mathcal{L}$ if $H$ is Hilbert and $A \in \mathcal{L}_c(X,Y)$.

Exercise : Let $H$ be a Hilbert space and $A \in \mathcal{L}_c(H)$. Show that $\exists x \in H : \|A(x)\| = \|A\|_\mathcal{L}$. Attempt - Thoughts : Note : The space $\mathcal{L}_c(H)$ is the ...
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Showing that $0 \in \overline{A(\partial B_1^X)}$ if $X,Y$ Banach with $\dim X = + \infty$ and $A \in \mathcal{L}_c(X,Y)$.

Exercise : Let $X,Y$ be Banach spaces with $\dim X = + \infty$ and $A \in \mathcal{L}_c(X,Y)$. Show that $0 \in \overline{A(\partial B_1^X)}$. Attempt : So, since $A$ is a Linear Compact operator ...
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If $\|A(x)\|_Y \geq c \|x\|_X$ and $\dim X = + \infty$, can $A \in \mathcal{L}(X,Y)$ be a compact operator?

Exercise : Let $X,Y$ be Banach spaces with $\dim X = + \infty$ and $A \in \mathcal{L}(X,Y)$ such that : $$\|A(x)\|_Y \geq c\|x\|_X \; \; \forall x \in X \; \text{and} \; c>0$$ Can the ...
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Compact operators and Fredholm's theory

Can you help me to find all real numbers $\beta \in \mathbb {R}$ for which the equation $x(t) + \int_0^1(1+\alpha ts)x(s)ds = \beta + t^2$ is solvable in space $L_2[0,1]$ for any real number $\alpha \...
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Spectral theory: Operator compact implies existence of convergent subsequence

So, I'm looking at the proof of the spectral theorem for self-adjoint compact operators in my functional analysis lecture notes (an introductionary class). We defined a compact operator $T$ as a ...
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$T :N \to N_1$ is compact iff closure of $T(S)$ is compact $\forall S$ bounded subset of $N$

In a normed linear space N, let $B=\{x \in N : |x| <1\}$ . Then define a linear map $T :N \to N_1$ ,( $N_1$ some normed linear space ) to be compact if closure of $T(\bar{B})$ is compact in $N_1$ ...
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$\tilde{T} : X/X_0 \to Y $ is also compact

In a normed linear space N, let $B=\{x \in N : |x| <1\}$ . Then define a linear map $T :N \to N_1$ ,( $N_1$ some normed linear space ) to be compact if closure of $T(\bar{B})$ is compact in $N_1$ ...
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Linear operator is compact if and only if its adjoint is compact

Let $H$ be a Hilbert space, and $A:H\rightarrow H$ a linear operator. Prove that $A$ is compact if and only if $A^*$ is compact. I saw the following proof in my book - What I don't understand ...
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Source for a particular proof of the spectral theorem.

Consider the following ''spectral decomposition'' for self-adjoint compact operators: If $T\neq 0$ is a self-adjoint compact operator on a Hilbert space $H$, then there exists a sequence $\{\...
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Help understanding proof that Hilbert-Schmidt operators are compact

I am trying to understand a proof that Hilbert-Schmidt operators are compact. From the book that I am reading it is stated that for a Hilbert Schmidt operator $K$ with kernel $$k=\sum_{n=1}^{\infty}\...
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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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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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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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Norm and compactness of the Operator $(Tu)(x)=\alpha(x)u'(x), u\in Y, x\in I$

Let $I=[0,1]$ and call $X$ the Banach space $C(I)$, endowed with the uniform norm. Introduce $Y=\{u\in X, u$ diffentiable on $I$ with $u'\in X\}$ and set $||u||_Y=||u||_\infty+||u'||_\infty, u\in Y, ...
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Integral Equation and Fredholm Alternative

I am learning about functional analysis at the moment and I have difficulties grasping the connection to integral equations or differential equations. For simplicity, let us consider the following ...
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Compact embedding of the domain and compact inverse

I have several problems in showing this point of a problem: we consider $X$ Banach space and $T: D(T) \to X$ a closed operator with domain $D(T) \subseteq X$. Let be $T$ bounded, invertible and ...
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Compact open operator between Banach spaces

Let $X,Y$ be Banach space, $Y$ infinite dimensional. Show that no $T \in \mathcal{K}(X,Y)$ is open. By definition $T$ is open if and only if $\exists r >0$ such that $B_Y(0,r) \subset T(B_X(0,1))$ ...
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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 \...
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Compact operator $L:\ell^2\to\ell^2$ with $\Vert L\Vert=1$ such that $\Vert L(x)\Vert<\Vert x\Vert$ for all $x$

Let $\ell^2$ denote the space of square summable sequences of complex numbers. Let $L:\ell^2\to\ell^2$ be a linear operator with $\Vert L\Vert=1$ such that for all $x\in\ell^2\setminus\{0\}$, $\Vert L(...
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Show that $(Tu)(x)=\int_{\alpha(x)}^{\beta(x)} u(t)dt$ is Compact linear operator on $C([0,1])$

Show that \begin{equation} (Tu)(x)=\int_{\alpha(x)}^{\beta(x)} u(t)dt \end{equation} is Compact linear operator on $C([0,1],R)$ where $\alpha, \beta:[0,1]\rightarrow [0,1]$ are continuous. My ...
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Compact operators and weak convergence

Let $X$ and $Y$ be Banach spaces. (a) Let $T \in \mathcal{L}(X, Y )$. For each sequence $(x_n)_{n \geq 1}$ in $X$ and each $x \in X$, show that $x_n →x$ weakly, as $n \rightarrow \infty$ ,implies ...
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Isometric isomorphism between $L^2$ and $\mathcal{L}^2$

I was reading and trying to understand the proof that the space $\mathcal{L}^2 (\mathcal{H})$ (Hilbert-Schmidt operators) is made by all the $T_K:L^2(X,\mu) \rightarrow L^2(X,\mu)$ with $K \in L^2(X \...
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Compact operator on $L^2[0,1]^2$

Let $K\in L^2([0,1]\times[0,1])$, and we define the operator $T_k$ on $L^2[0,1]$. $$(T_kf)(x)=\int_{0}^{1}K(x,y).f(y).dy \quad \quad \forall f\in L^2[0,1]$$ How to prove that $T_k$ is a compact ...
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$||T-T_n|| \rightarrow 0$ and $T_n$ are compact but $T$ is not a compact operator.

It is a result that if $||T-T_n|| \rightarrow 0$ in the norm operator an that the $T_n \in \mathcal{L}(X,Y)$ (were $Y$ is a Banach space) are compact operators, then $T$ is compact. I found from here ...
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Integral Operator in $L^2$

I was trying to do this exercise and I'm wondering if I figured it out well: I have $\mathcal{H} := L^2(0,1)$ and $T$ the operator with integral kernel $K(x,y) = \min\{x,y\}$, $x,y \in [0,1]$. I have ...
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Compact Integral Operators induced by positive Kernels

Let $K$ be a compact operator induced by the kernel $k(s,t)\in L^2([0,1])^2$ with $k(s,t)>0$. Prove that $\|K\|<1$ if and only if $(I-K)$ has a bounded inverse $(I-K)^{-1}$ which is induced by a ...
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Matrices of bounded linear operators

Let $X,Y$ be Banach spaces and let $A=(A_{n,k})$ be an infinite matrix of bounded linear operators $A_{n,k}:X \to Y$. Suppose $\sup_n \sum_k \|A_{n,k}\|<\infty$. Property: For each sequence $x=(...
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Non existence of solution of a special first kind Fredholm integral equation

Let $k \in {L^2}((0,4) \times (0,1))$, $g \in {L^2}(0,1)$. We consider the following first kind Fredholm equation $$\int\limits_0^4 {k(s,x)f(s)ds=g(x), x\in(0,1).} $$ Where $f$ is the unknown. How ...
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Isomorphisms in the proof of the Fredholm alternative/Theorem of Riesz-Schauder (for compact operators)

In a proof of the Fredholm alternative/Theorem of Riesz-Schauder, I came across the following: Let $X$ be a Banach space, $T:X \to X$ be a compact operator and $A:=T-I$. We proved that $\mbox{ker}(A)...
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A linear operator is compact if and only if the image of any bounded set is relatively compact

Let $U$ and $V$ be normed spaces. Show that a linear operator $T:V \to U$ is compact if and only if the image of any bounded set is relatively compact. (Here $T$ is compact if the image of any ...
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A non compact operator on $L^2[0,1]$

Let $H$ be the Hilbert space $L^2[0,1]$. and the operator $T : H\rightarrow H$, such as $T(f)(x)=x.f(x)$ Why $T$ isn't compact ?
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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 ...