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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Sufficient conditions on integration kernel for continuity of the integral operator

Suppose that we have a measure $d\mu(v)=e^{-|v|^2}dv$ on $\Bbb R^d$. We define a linear operator $$T[f](u)=\int_{\Bbb R^d} |u-v|^\beta d\mu(v).$$ I want to establish conditons on $\beta\in\Bbb R$ so ...
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On closed ranges and sequences which converge to zero

I'm reading a proof of the Fredholm alternative, and there is a claim that goes like this: Let $K:X\rightarrow X$ be a compact linear map. Define $T=I-K$, then $Y=\ker(T)$ is a finite dimensional ...
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Hilbert-Schmidt and compact operators

I am new to this site and i dont really know how to ask questions properly, so i am really sorry if i did something wrong. My question is if there is a way to prove that a Hilbert-Schmidt operator is ...
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How is the Point Spectrum of a Compact Operator Countable?

I'm working on understanding a proof that if an operator $A$ on a Hilbert space $\mathcal{H}$ is compact, then show that $\sigma(A) - \{0\} \subseteq \sigma_p(A)$. If you're not familiar with this ...
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Show that a set is totally bounded

Let Y = $L^1 $($\mu$) where $\mu$ is counting measure on N. Let X = {$f$ $\in$ Y : $\sum_{n=1}^{\infty}$ n|$f(n)$|<$\infty$} Define T : X -> Y by $Tf(n)=nf(n)$ Equip X with the graph norm ...
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About the largest eigenvalue of Hilbert-Schmidt integral operators

Let $\Omega$ be an open set of $\mathbb{R}^d$ and $K \in L^2(\Omega\times \Omega)$ such that for almost all $x,y \in \Omega$ : $K(x,y)=K(y,x)$ $K(x,y)>0$ One can show that under these ...
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A sequence $L_n$ of compact bounded linear transformations on a hilbert space defines a convergent subsequence in each $L_n$ for a bounded sequence?

Let $L_n:\mathcal{H}\to\mathcal{H}$ be a sequence of compact bounded linear transformation on a Hilbert space $\mathcal{H}$, and $h_m$ be a sequence in $\mathcal{H}$. Since each $L_n$ is compact, ...
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Singular values of compact operators

Let $T$ be a compact operator.Then show that $\sum_{n=1}^\infty$$M_n(T)=\sup\{||PT||_1:P\mbox{ is a rank } N\mbox{ projection}\}$ where $M_n$'s are the singular values of $T$ and ...
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Do $L^p$ spaces have the approximation property?

A Banach space $X$ has the approximation property if every compact operator $T:X \to X$ is the norm-limit of a sequence of finite-rank operators. My question is if there is a simple proof that the ...
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Problem 8.11 in Young's book (An introduction to Hilbert space)

I've tried to prove this problem which appears in Young's Book: Let $K$ be a compact Hermitian operator on a Hilbert space $H$ and let the kernel of $K$ be $\{0\}$. Show that there is a sequence ...
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Unclear passage of a theorem concerning compact operators (Schauder fixed point theorem)

I'm looking at this proof of Schauder theorem and I am struggling with a passage. This is my problem: Let $X$ be a Banach space, $K \subset X$ a convex, close and bounded set and $F:K \rightarrow ...
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Compactness of translation operator in weighted spaces

Let $x,v\in\Bbb R^d$, $t\in \Bbb R$ and $m(x,v)$ be a smooth strictly positive function rapidly decaying on infinity - think $m(x,v) = \exp(-|x|^2-|v|^2)$. Define Banach spaces $X$ and $Y$ by ...
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Norm, adjoint operator and compactness os some operators

Let $A_i:\ell^2\rightarrow \ell^2$ be two operators given as follows: $A_1x=(0,x_1,0,\frac{x_2}{2},0,\frac{x_3}{3},...)$ and $A_2x=(x_1,x_1,x_2,x_2,x_3,x_3,...)$ Compute the norm and the adjoint ...
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compact projection

Show that the operator T: l^2→l^2 defined by T({x_n })={( x_n)/2^n } is compact. How one can show that the sequence{ََ T(x_n)} is contain convergent subsequence if {x_n } is bounded? I know that ...
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norms of operator

I am stuck on a question on the operator norm. If $L$ is a bounded linear operator $L:H\rightarrow H$ where $H$ is a Hilbert space. How would you show that $$\|L\|\leq \sup_{\|u\|=\|v\|=1}(Lu,v)$$ any ...
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Question about the Image of a compact transformation of a Hilbert space

$T$ is a compact operator on a Hilbert space. Show that $\operatorname{im}(T)$ does not contain a closed infinite dimensional subspace. Here is my attempt at the problem: Suppose that ...
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Inverse of laplacian operator

I recently read a paper, the author treats $$\int_{\mathbb{R}^d}f(y)\cdot \frac{1}{|x-y|^{d-2}}\,dx = (- \Delta)^{-1} f(y)$$ up to a constant in $\mathbb{R}^d$. I am not familiar with unbounded ...
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Question on the range of values of compact operator

Suppose $X$ is a Banach space and $T$ is a compact operator from $X$ to $X$. Let $B_1$ is closed unit ball in $X$. Then, can we get a resual that $\overline{T(B_1)} \subseteq T(X)$ ? It is easy to ...
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Approximation Property: Decomposition

This is a real question of me. Given a Banach space $E$. Consider a finite rank operator $F\in\mathcal{F}(X,E)$. Introduce a basis on the finite dimensional range: ...
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spectrum of convolution integral operator

Let $A f(x)= \int_{-\pi}^{\pi} h(x-y) f(y) dy$ operator $L^2( {-\pi},{\pi})->L^2( {-\pi},{\pi}), h$ is continuous, periodic with period $2\pi$ and $h(x)=h(-x)$ on $ [ {-\pi},{\pi}] $. How can I ...
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Adjoint of Integral Operator in $L^p$

Let $E = L^p(0,1)$ with $1 ≤ p < ∞$. Given $u ∈ E$, set $$Tu(x):=\int_0^x u(t)dt$$ Find the adjoint of $T$. I know how to this in the case $p=2$ as shown here. But in general $L^p$ is not an ...
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Bound for Integrator Operator

Let $E = L^p(0,1)$ with $1 ≤ p < ∞$. Given $u ∈ E$, set $$Tu(x):=\int_0^x u(t)dt$$ Prove that $T$ is compact on $E$. I would like to use Ascoli-Arzela', but I need to prove: $$|T u(x) − T u(y)| ...
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Hilbert- Schmidt class is an ideal

Definitions: 1 - An operator $y\in B(H)$ is said to be of trace class if $y$ is compact, and also $\sum|\alpha_n| <\infty$ where $\alpha_n \in \sigma(y)$ and $y$ has a representation $\sum ...
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Compact operator space is the greatest ideal of $B(H)$

Suppose $H$ is a separable infinite dimensional Hilbert space. Show that if $A\in B(H)$ is noncompact, then there exist two operators $B,C$ such that $BAC=1$. Clearly if $A$ is invertible it holds, ...
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Schrödinger Semigroup is compact if potential goes to infinity

Several papers (e.g. this one: arXiv:0810.3275v1 [math.SP] 17 Oct 2008 ) claim that if $H=-\Delta+V$ and $V(x)\to\infty$ if $|x|\to\infty$, then the semigroup $e^{-tH}$ is eventually compact. Does ...
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Trace class operator

Let $A\in B(H)$ and $\sum_{E}|\langle A e,e\rangle|< \infty$ for every orthonormal basis $E$. Show that $A$ is a trace class (means $\sum_E \langle |A|e,e\rangle < \infty$). I can not prove it. ...
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Compact operator and a sot convergent sequence of operators

The following is an exercise of Conway's operator theory: I proved all parts of this exercise except $\|KT_n\| \to 0$. I can easily prove $\|KT_n^*\|\to 0$, but do not have any idea to prove ...
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Show that the operator $(x_n)_n\mapsto (\frac{x_n}{n}) $ is compact

I want to show that the following operator is compact: $$T:\mathbb \ell^p\rightarrow \mathbb \ell^p, \text{ }(x_n)_n\mapsto(\frac{x_n}{n})_n \text{ } 1\leq p<\infty$$ Its the first time that ...
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parameter operator $A_a$ is compact??

I need some help in this exercise. Let define operator on $ L^2[0,1]$: $$ A_af(x)=\int_{0}^{1}{|x-y|}^{a-1} f(y)dy $$ for f $\in L^2[0,1] $. Prove that A_a is compact for all $a>0$. I see that ...
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Compactness of the fractional integral operator.

Is the fractional integral operator $J_a^{\alpha} :L^2[a,b] \rightarrow L^2[a,b]$ ( or $J_a^{\alpha}:C[a,b] \rightarrow C[a,b]$) which is defined by $$J_a^{\alpha} f(x) = {1 \over ...
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Finite rank volterra operator

I am wondering when a Volterra integral operator $V_K:L_2(0,1)\to L_2(0,1)$ is a finite rank operator: $$V_Kf=\int_0^xK(x,y)f(y)dy$$ thanks in advance for your help
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Ask for the name of a condition on commutator of two operators

Let $T, S$ be two bounded linear operators on a Hilbert space. I wonder whether there is a standard way referring the following condition: $$ \text{The commutator $[T, S]$ is in the Hilbert-Schmidt ...
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Finite dimensional operator space is dense in trace class space

To show that $F(H)$ (the space of finite dimensional operators on a Hilbert space $H$) is dense in $L^1(H)$ (the space of trace class operators), suppose that $x\in L^1(H)$. Without loss of generality ...
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Trace class operator is an ideal

To show that trace class operators space is an ideal, we need to show that $\|uv\|_1\leq \|u\|\|v\|_1$ where $u,v \in B(H)$ and $\|u\|_1 = tr(|u|)$. Murphy in his book (C*-algebras and operator ...
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$*-$ isomorphism between two compact spaces $K(H)$ and $K(H')$

The following is a theorem of Murphy's C*-algebras and operator theory: I think we can write the proof more easily than Murphy's. After show that $E'$ is an orthonormal basis for $H'$, define ...
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Proof of the Riesz-Schauder Theorem (for compact operators) using the Analytical Fredholm Theorem

First of all sorry for my bad English, I'm an Italian student, hope to let you understand! I'm having a little troubles with the proof of the Riesz-Schauder theorem for Compact Operators. Some infos ...
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weakly continuous linear map

The following is a Theorem of Murphy's C*-algebra and operator theory: To prove the theorem, the author claims compact linear map $u$ is weakly continuous. I know that every bounded linear map is ...
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If $H$ is a one-dimensional Hilbert space then the zero representation of a C*-algebra on $H$ is irreducible.

It says on page 143 of Murphy's $C^*$-algebras and operator theory that if $H$ is a one-dimensional Hilbert space then the zero representation of any C*-algebra on H is irreducible. What is the zero ...
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Proving that each compact operator is bounded

I'm trying to prove that each compact operator is bounded, from the definition: On the internet I find solutions for this problem, but they always start from a different definition. I would be glad ...
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$s \in L^{1}(H)$ $\iff$ $s=\sum_{i=0}^\infty x_{i} \otimes y_{i} $

Let $H$ be a separable Hilbert space, and let $L^1(H)$ be the space of trace-class operators on $H$. I'd like to prove that $s\in L^{1}(H)$ if and only if there exists $\{ x_{i} \} , \{ y_{i} \} ...
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The ideal generated by a non-compact operator

I wanted to find a quick proof of the following well-known fact. Since I couldn't easily find a reference, I provide a proof below. Let $H$ be a separable Hilbert space, and $J\subset B(H)$ be a ...
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Spectrum of a bounded operator on a (not necessarily Banach) normed vector space

It's well known that on a Banach space, the spectrum of each bounded operator is compact in $\mathbb C$. What about a general normed vector space? Is there a counterexample if we don't assume ...
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Compactness of solution space of semi-linear parabolic PDE

Under what conditions a closed and bounded subset of solution space of following parabolic PDE is compact? $$x_{t}=x_{zz}+f(x,z)$$ Thank you!
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Toeplitz Operator is compact if and only if it has finite rank

A referee has pointed out to me that it is "well known that a Toeplitz operator is compact if and only if it has finite rank" and pointed me to R. Douglas: Banach algebra techniques in the ...
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sup norm of operator

Let $T$ be a compact linear operator defined as $$ T\circ u = \int_a^b k(x,y)\,u(y)\,dy, $$ where $k(x,y)\in C([a,b]\times[a,b])$ and $k(x,y)\ge0$ for all $x,y$, and $u\in C([a,b])$. Suppose that the ...
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Green-Operator for Sturm-Liouville Differential equation compact on Sobolev space?

Let $g$ be Green's Function for a Sturm-Liouville differential equation. Is the operator $G: H_{0}^{1}(0,1) \rightarrow H_{0}^{1}(0,1)$ defined by $(Gf)(x) := \int_{0}^{1} g(x,y)f(y) dy, \quad f \in ...
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Product of two positive compact, self adjoint operators

If we have two positive compact , self adjoint operators; $A$, $B$. Is the product $AB$ a positive operator?
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Is $L^\infty$ compactly embedded in $L^1$?

I'm trying to find a contraction example to show that the space $L^\infty$ is not compactly embedded in $L^1$ with the Lebesgue measure. Please help me!
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Prove the operator on hilbert space is compact

My question is actually the same as the first part of this one, Prove that T is compact which has not been answered. I am thinking about two ways, 1) use a bounded sequence $\{g_n\}$, and try to ...
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51 views

Understanding connectedness argument in proof of Analytic Fredholm Theorem

Let $X$ be a complex Banach space, and let $D \subset \mathbb{C}$ be a domain. Let $\mathcal{L}(X)$ denote the Banach space of bounded linear transformations $X \to X$. The Analytic Fredholm Theorem ...