2
votes
1answer
36 views

If a compact operator satisfies $T^nx\to0$ weakly for all $x$, then $\|T^n\|\to0$

Let $H$ be a real Hilbert space, $T:H\to H$ be a compact operator. Suppose that for every $x\in H$, sequence $(T^n x)_{n\in \mathbb{N}}$ converges weakly to $0$. How to prove that $ ...
2
votes
0answers
45 views

Sufficient condition for an operator to be compact in Hilbert space of holomorphic function with respect to Gaussion weight (Fock space).

What I read in a book I could not understand, some one please help. Let $\mathcal{F}=\{f:\mathbb{C^n}\rightarrow\mathbb{C}: \text{$f$ is holomorphic and}\int_{\mathbb{C}^n}\lvert ...
1
vote
0answers
43 views

Prove that operator is completely continuous

Let's consider Banach space $\ell^\infty$ of bounded sequences $x = \{ \xi_n\}_{n=1}^\infty$: $$ ||x|| = \sup_{n \in \mathbb N} |\xi_n|. $$ Suppose matrix $||a_{i j}||_1^\infty$ specifies operator $A$ ...
3
votes
1answer
40 views

Rellich's theorem for Sobolev space on the torus

From John Roe: Elliptic operators, topology and asymptotic methods, page 73: Let $H^{k}$ be the Soblev space defined on the torus $\mathbb{T}^{n}$ with the discrete $k$-norm: $$ \langle f_{1}, ...
0
votes
2answers
25 views

Space of bounded functions vs. bounded space of functions.

Suppose I have a bounded set of functions, say $B\subset C[0,1]$. What exactly does this mean? I.e. is a bounded set of continuous functions equivalent to a set of continuous bounded functions? For ...
2
votes
1answer
59 views

Show $T: C([0,1]) \rightarrow C([0,1])$ is compact

Consider $T: C([0,1]) \rightarrow C([0,1])$ defined by $$(Tf)(t) := \int_0^1 \kappa_t(s)f(s)ds,$$ where $\kappa:[0,1]^2 \rightarrow \mathbb{R}$ satisfies the following properties: for all $t\in ...
0
votes
1answer
34 views

Canonical inclusion $L^q(0,1) \to L^p(0,1)$ is compact?

Does there exist $q>p$ such that the canonical inclusion $L^q(0,1) \to L^p(0,1)$ is compact? My answer is no. Since we know that $L^\infty (0,1) \to L^p(0,1)$ is not compact, take $\{\sin(nx)\}$ ...
1
vote
1answer
63 views

Compactness of an operator involving the resolvent of laplacian

Let $w\in L^n(\mathbb{R}^n)$, ($n\geq 3$), and for $\tau\in\mathbb{C}$, $Im(\tau)\neq 0$, let $R_{\tau}=(-\Delta-\tau)^{-1}$ be the resolvent of the Laplacian. I need to show that $T:=wR_{\tau}w$ is a ...
4
votes
3answers
90 views

What is a predual of the Banach space of compact operators on $\ell^2$?

I am wondering if the space $K(\ell^2)$ of compact operators on $\ell^2$ can have a predual. Thank you in advance for your help.
0
votes
1answer
24 views

Normal Compact Operator: not diagonalizable!

To proposition 5.17 in Weidmann's 'Lineare Operatoren in Hilberträumen' (german version) it is noted that the expansion of compact operators that are normal rather than self adjoint doesn't apply in ...
2
votes
1answer
60 views

Sum of the Eigenvalues of a Compact Positive-Definite Linear Operator on a Hilbert Space

Let $ A $ be a compact positive-definite linear operator on a Hilbert space $ \mathcal{H} $. Let $ \{ v_{1},v_{2},\ldots,v_{n} \} $ be an orthonormal $ n $-subset of $ \mathcal{H} $. Let $ \lambda_{1} ...
2
votes
1answer
47 views

Question about compact operator

So here is my question, Let $H$ be a Hilbert-space and $K:H\rightarrow H$ a compact operator. I know that if $K$ is self adjoint, then it has one eigenvalue $\mu$ such that $|\mu|=||K||$. Can some ...
2
votes
2answers
36 views

Fredholm Index: Finite Corank $\Rightarrow$Closed Range [duplicate]

Obviously closed subspaces turn quotient spaces into normed spaces rather than just merely vector spaces. However the dimension involved in Freholm's index are purely algebraic. Why do we thus ...
1
vote
1answer
41 views

Showing that a certain operator is compact

So here is my problem, I try to show that following operator is compact, \begin{align} J: h_1 & \rightarrow\ell^1 \\ (x_n) & \mapsto(x_n) \end{align} where $$h_1:=\left\{x_n\in ...
4
votes
2answers
61 views

Question about a counterexample concerning compact operators

Does anybody know if the following is true, Let $H$ be an infinite dimensional Hilbert-space and $K:H\rightarrow H$ a compact operator. Then if $|\mathrm{spec}(K)|<\infty$ i.e the spectrum is ...
0
votes
1answer
34 views

Question if an operator is compact

So here is my problem, Let $$J_p:\ell^p\rightarrow c_o$$ be the canonical embedding where $c_0:=\{x_n\subseteq\mathbb C:x_n\rightarrow 0\quad n \rightarrow\infty\}$. I have to decide whether the ...
3
votes
1answer
30 views

Question about an integral operator

So here is my question, I know that the operator $$T:L^2[0,1]\rightarrow L^2[0,1]$$ $$f\mapsto(Kf)(x)=\int_{[0,1]}k(x,y)f(y)\;dy$$ for a function $k$ continuous on $[0,1]^2$ is compact. Is this also ...
4
votes
1answer
81 views

What makes compact operators special?

I would like to understand why compact operators are considered so special to consider them as an extra class of operators. Over Hilbert spaces these (as far as I know) these are the ones with ...
0
votes
1answer
51 views

Is operator $T_n$ a compact operator?

Is operator $T_n$ a compact operator? $$T_n:l_2\rightarrow l_2$$ $$T_nx=(\underbrace{0,0,\ldots,0,}_{n\text{ zeros}}, x_1,x_2,x_3,\ldots)\text{ where }x=(x_1,x_2,x_3,\ldots)\in l_2,\ ...
2
votes
1answer
52 views

Two questions about a proof of the compactness of an operator

There are a few things that I don't understand about a proof and I'd appreciate any help. The theorem and its proof are the following: (1) Is the equality $$ \|v(\tau) -v(\tau_j)\| = \max_{1 \le ...
1
vote
1answer
36 views

Problems proving that a compact operator is completely continuous [duplicate]

I would like to prove that if $T:X\rightarrow Y$ is a compact operator, then for every weak convergent sequence $(x_n)_{n\in\mathbb N}$ with $x_n\rightharpoonup x$ for some $x\in X$ it follows that ...
3
votes
1answer
55 views

Compact operator as a limit of finite ranked operators

So here is my question, I had to show that the following operator is compact, $$T:C[0,1]\rightarrow C[0,1]$$ $$f\mapsto\int_0^tf(s)ds$$ with $||f||=\mathrm{sup}_{x\in[0,1]}|f(x)|$ I think I ...
1
vote
0answers
47 views

Question about compact operators

I would like to prove the following, Let $X$,$Y$ be infinite dimensional Banach-Spaces and $T$ a compact, linear and bounded operator. Then there exists a sequence $(x_n)_{n\in\mathbb N}$ with ...
0
votes
1answer
42 views

About the Volterra operator and the approximation property

I need some help with these questions. $\bullet\;$ First of all, if we define the Volterra operator $V:L^{1}[0,2\pi]\rightarrow L^{1}[0,2\pi]$ as $(Vf)(x)=\int_0^xf(t)dt$, Is this operator compact? ...
0
votes
0answers
59 views

Volterra operator with continuously differentiable Kernel has no Eigenvalue

First I'll describe the entire question, as it stated in the exercise: let $K(t,s)\in C([0,1]^2$), continuously differentiable in the first coordinate (meaning $K_t(t,s)\in C([0,1]^2$). And let ...
6
votes
4answers
303 views

is $T$ compact operator?

is $T$ compact operator? $T:C[0,1]\rightarrow C[0,1]$: $x(t)\mapsto x(t^2)$ where $t\in[0,1]$ with supremum norm Could you please help.
6
votes
3answers
154 views

How to figure whether it is a compact operator

How to figure whether it is a compact operator: $$T:C[0,1]\rightarrow C[0,1] $$ $C[0,1]$:the space of all continous function on [0,1] with supremum norm $$(Tx)(t)=\int^t_0 x(s)ds, \ \ \forall ...
2
votes
1answer
67 views

How to find if it is a compact operator

How to find if it is a compact operator: $F\colon C[0,1]\rightarrow C[0,1]$ : $x(t)\mapsto \int^1_0 \cos(t^2+s^2)x(s)ds$ Could you please help with this question.
1
vote
0answers
41 views

If an upper semicontinuous multivalued map is compact on a set, is it compact on the boundary as well?

I have stumbled upon the following problem during my research: Let $X$ and $Y$ be Banach spaces, $K\subset X$ nonempty, $F:\overline{K}\rightarrow 2^{Y}$ an upper semicontinuous multivalued map with ...
0
votes
2answers
95 views

Compact operator whose range is not closed

I am asked to find a compact operator (on a Hilbert space) whose range is not closed, but I am having trouble coming up with one. My guess is that you need to have some sequence in the range that ...
1
vote
3answers
42 views

Examples of spectrum of compact operators on the sequence space $l_2$

Suppose $T$ is a compact operator on the sequence space $l_2$, and let $\sigma(T)$ be its spectrum. Is it possible to find a $T \ne 0$ such that $\sigma(T) = \{0\}$? Also, is it possible to find $T$ ...
1
vote
1answer
39 views

Show compactness of an operator with Arzelà–Ascoli

We have $K\colon L^{2}(a,b) \rightarrow L^{2}(a,b)$ such that $ Kf(t)=\sum_{j=1}^{n}\phi_{j}(t) \int_{a}^{b} \psi_{j}(S) f(s)ds$ where $\phi_{j} ,\psi_{j} \in L^{2}(a,b)$. We want to show that K is ...
1
vote
0answers
81 views

Compact operators is a linear subspace of bounded operators

Let $X,Y$ be Banach spaces. Let $B(X,Y)$ be the set of bounded linear operators and let $K(X,Y)$ be the set of compact linear operators. I want to prove that $K(X,Y)$ is a vector subspace of ...
2
votes
2answers
107 views

Proof of equivalent characterizations of compact operators

As an exercise I tried to prove the following theorem: If $X,Y$ are Banach spaces and $u \in B(X,Y)$ is a bounded linear operator then the following are equivalent: (1) $u$ is compact ...
1
vote
0answers
24 views

Showing a particular integral operator is trace class

Let $f$ and $P$ be continuous, integrable functions $\mathbb{R} \to \mathbb{C}$ vanishing at $\pm \infty%$. Concisely, $f,P \in C_0(\mathbb{R}) \cap L^1(\mathbb{R})$. Also, assume that $P$ is ...
2
votes
1answer
45 views

Equivalent conditions for composition to be compact operator

I did some exercises in Conway's functional analysis book and found the following problem: Let $\tau:[0,1]\to [0,1]$ be continuous and define $A:C[0,1]\to C[0,1]$ by $Af:= f\circ \tau$. Give ...
1
vote
0answers
22 views

Schauder's theorem: consequences and applications

I am about to give an informal talk about Schauder's theorem ($T:X\to Y$ linear operator between Banach spaces is compact if and only if its adjoint is). Does anyone know any derived ...
3
votes
1answer
43 views

Why is this estimate using a compact embedding in a sobolev space true?

Let $\Omega\subset\mathbb{R}^3$ be a bounded Lipschitz-domain. We then have, for $s\in[1,6)$ the compact embedding $H^1(\Omega)\stackrel{c}{\hookrightarrow}L^s(\Omega)$ ensuring the existence of a ...
1
vote
1answer
36 views

I want to show that some subset of $C([0,1])$ is equicontinous

First why the problem appeard. I want to show that the linear and continuous operator $T:C([0,1])\rightarrow C([0,1])$ , $ (Tf)(t)=\int_{[0,1]}k(t,s)f(s)ds$ where $k:[0,1]^2\rightarrow\mathbb R$ is ...
1
vote
1answer
108 views

Show the Volterra Operator is compact using only the definition of compact

The Volterra operator $V:L^{2}[0,1]\rightarrow L^{2}[0,1]$ is defined by $(Vf)(x)=\int_0^xf(t)dt$. I am wondering if it can be shown that $V$ is compact by definition - that is, either that $V$ ...
0
votes
0answers
56 views

Show these operators converge to a particular limit

Let $H$ be a Hilbert space, and $T$ be a operator on $H$ of the form $T=\sum_{n=1}^{\infty}{\lambda}_{n}<x,e_{n}>e_{n}$ where $e_{n}$ are the eigenvectors of $T$ and an orthonormal basis of H ...
3
votes
2answers
82 views

Subspaces in the image of compact operator

Let $X$ and $Y$ be some infinite dimensional Banach spaces. Let $T:X\longrightarrow Y$ be some compact linear operator. It is easy to understand that $T$ cannot be surjective: the Open Mapping Theorem ...
0
votes
1answer
66 views

The eigenvalues of a compact and self-adjoint operator on Hilbert space

Show that if $K$ is a compact self-adjoint operator on Hilbert space then it has either finitely many eigenvalues or a sequence of eigenvalues $\lambda_n\to 0$ as $n\to \infty$.
0
votes
1answer
25 views

Finite dimension and total boundedness

Let $T:X\to Y$ be a bounded operator between Banach spaces $X$ and $Y$. Assume that for any $\epsilon >0$ there is a finite-dimensional subspace $Y_\epsilon\subset Y$ so that $\|Q_\epsilon ...
0
votes
1answer
74 views

Does Hilbert Transform commute with Function Multiplication modulo Compact on $L^p(R)$?

Define Hilbert Transform (HT) as the convolution with the function $1/x$. E. Stein proves in his book Singular Integrals and Differentiability Properties of Functions that HT, when understood as a ...
2
votes
1answer
50 views

Show that the span of eigenvalues of a compact operator is a closed

Let $(X, ||\cdot||)$ be a real Banach space and $T: X \to X$ a compact operator (so $\{x_n\}_{n=1}^\infty$ bounded implies that $\{Tx_n\}_{n=1}^\infty$ has a convergent subsequence). Let $x_1, \dots, ...
0
votes
2answers
87 views

Totally boundedness of a compact operator [closed]

Let $T:\ell_2(\mathbb N)\longrightarrow \ell_2(\mathbb N)$ bounded linear operator such that $$T(\{x_n\})=\{x_n/n\}.$$ I need to prove that $TB(\ell_2(\mathbb N))$, that is closed unit ball in ...
1
vote
1answer
35 views

Compact embedding

Prove that the embedding $j\colon (C^1[0,1],\|\cdot\|)\to(L^1[0,1],\|\cdot\|_{L^1})$ where $\|f\|=\max\{\|f\|_\infty,\|f'\|_\infty\}$ and $\|f\|_\infty$ denotes the supremum norm, ...
2
votes
1answer
98 views

Direct sum of eigenspaces of a compact operator has finite codimension

In an infinite dimensional Hilbert space the orthogonal complement of the (closure) of the direct sum of eigenspaces of a compact normal operator is finite dimensional. Why is this the case? thanks.
0
votes
1answer
57 views

About a compact imbedding of Sobolev spaces

I am studying the Compactness lemma ( on page 570) of the article http://projecteuclid.org/euclid.cmp/1103922134. The lemma says (Compactness lemma ): for $0 < \sigma < \frac{2}{N-2}$, $(N ...