Tagged Questions
1
vote
1answer
36 views
The set of compact linear operators is a subspace of the set of bounded linear operators
I know that a linear operator $T:X \to Y$ (where $X$ and $Y$ are normed vector spaces) is compact if for every sequence
$\left(x_{n}\right)\subseteq X$ s.t. $\left\Vert x_{n}\right\Vert \leq C$,
the ...
1
vote
1answer
28 views
Strong operator convergence and adjoint operator
Let $H$ be a Hilbert space and $(T_n)_{n \in \mathbb{N}}$ be a sequence of bounded linear operators on $H$.
The strong convergence of $T_n$ doesn't imply the strong convergence of $T_n^*$, i.e.
...
3
votes
1answer
56 views
Invertibility of compact operators
I'm a little confused about compact operators and whether or not they are invertible. Just hoping someone here can clear up my confusion:
Let $T$ be a compact operator on a Banach space $X$. Since ...
3
votes
1answer
63 views
$K$ is a linear compact operator on Hilbert space $H$. Will the image of $I-K$ on every closed subspace of $H$ be also closed?
Just as the title. We know the image of $I-K$ is closed, but if we restrict $H$ to a closed subspace $V$, will $(I-K)(V)$ be a closed subspace of $H$? Any hint is appreciated.
3
votes
1answer
109 views
Spectral theorem of compact operators in Hilbert space
I am reading the following theorem from my lecture notes (English translation of German text). But I don't understand exactly what is meant from this theorem and the proof.
Theorem.
Let $H$ ...
5
votes
1answer
170 views
Bounded operator and Compactness problem
Let $H$ be a Hilbert space with orthonormal basis $(e_{n})_{n\in\mathbb{N}}$. Furthermore, let $T\colon H\rightarrow C[a,b]$ be a bounded operator.
a) Let $x\in [a,b]$. Show that there is a ...
2
votes
1answer
52 views
How to prove the compactness of this Sobolev embedding?
I have a question on compactness of the following Sobolev embedding.
Let $W^{1,p}([0,1],\mathbb{R}^n)$ be the Sobolev space of functions $u:[0,1]\rightarrow \mathbb{R}^n$ for $1<p<\infty$. How ...
0
votes
1answer
50 views
1.4.5 Theorem of Murphy's book
See 1.4.5 Theorem of Murphy's book : I want to prove that if $u$ be compact operator on $X$ which is Banach space and $\lambda\in \mathbb{C}\setminus\{0\}$, then ...
4
votes
1answer
54 views
Continuous, selfadjoint and compact?
Hell0 there!
I have to show whether the operator
$$
T\colon L^2(\mathbb{R})\to L^2(\mathbb{R}), f\mapsto\chi_{[0,1]}f
$$
is continuous, selfadjoint and compact.
I have problems to show the ...
4
votes
2answers
44 views
$T:\ell^{2} \rightarrow \ell^{2}$ defined by $T(\{x_{n}\})=\{2^{-n}x_{n}\}$ is compact
Please help me to proof of problem :
Show that the operator $T:\ell^{2} \rightarrow \ell^{2}$ defined by $T(\{x_{n}\})=\{2^{-n}x_{n}\}$ is compact.
Tanks for your hint.
2
votes
2answers
66 views
Compactness of multiplication operator on $C[0,1]$
Find a condition on function $a\in C[0,1]$ such that the operator $A:C[0,1]\rightarrow C[0,1]$ $$(Ax)(t) = a(t)x(t)$$ is compact? We are taking uniform norm on $C[0,1]$.
2
votes
0answers
127 views
Fredholm and Compact Operators
Let $X$ and $Y$ be Banach spaces and $T\in B(X,Y)$ be Fredholm. Then there is $S\in B(Y,X)$ such that $ST=I+K_{1}$ and $TS=I+K_{2}$ where $K_{1},K_{2}$ are compact operators.Proof: Since $T$ is ...
0
votes
1answer
142 views
There are compact operators that are not norm-limits of finite-rank operators
Given an example of a Banach space for which There are compact operators that are not norm-limits of finite-rank operators.
Tanks for answer
1
vote
0answers
53 views
If limit of $f(n)$ is zero then the operator is compact
I want to prove the following:
Suppose $\mathfrak{H}$ is the Hilbertspace $l^2(\Bbb{N})$ and $T_f$ the multiplication operator on $\mathfrak{H}$, thus $T_f\psi=f\psi$ for ...
0
votes
0answers
41 views
u is compact if and only if $\lim_{n \to \infty}{\lambda_n}=0$ [duplicate]
Please help me for solve of the following problem :
Let $H$ be a Hilbert space with an orthonormal basis $(e_n)_{n=1}^{\infty}$, and let $u$ be an operator in $B(H)$ diagonal with respect to $(e_n)$ ...
1
vote
1answer
73 views
Spectral radius of an operator .
I would like to know the spectral radius of $$T_k x (t)= \int_0^t k(t,s) x(s) ds$$
where $T_k$ is a map from $C[0,1] \to C[0,1]$ and $k(x,y)\colon [0,1]^2 \to \mathbb C$ is continuous.
And also ...
3
votes
1answer
144 views
Eigenvalues integral operator - general case
Let $T$ be an integral operator on $L^2([0,1])$, such that:
$$
(Tf)(x) = \int_0^1K(x,y)f(y)dy,
$$
with $K(x,y): [0,1]^2 \rightarrow \mathbb{R}$ continuous and $K(x,y) = K(y,x)$, $K(x,y)\geq0$ $ ...
1
vote
1answer
35 views
A Linear map $u : X \longrightarrow Y$ is not bounded below iff there is …
Do you help me to:
checking that a linear map $u : X \longrightarrow Y$ between Banach spaces is not bounded below if and only if there is a sequence of unit vector ...
0
votes
0answers
34 views
Spectrum of weighted shift operator [duplicate]
Possible Duplicate:
Compact operator? self adjoint operator? Stirling’s formula
Let $H$ be a Hilbert space and let $\{e_n, n \geq\}$ be an orthonormal basis in H.
Let $T \in B(H)$ be the ...
3
votes
1answer
183 views
Compactness and spectrum of integral operator
Show that the operator $C: L^2([0,1]) \rightarrow L^2([0,1])$ defined by
$$Cf(x) = \int_0^x\int_1^tf(s)dsdt$$
is compact and determine its spectrum.
Im not sure how to find the spectrum when we are ...
6
votes
1answer
147 views
Trace class for operators
Let $ \mathcal{H} $ be a Hilbert space and $ T: \mathcal{H} \to \mathcal{H} $ a bounded linear operator. The $ n $-th singular number $ {\mu_{n}}(T) $ of $ T $ is defined as the distance from $ T $ ...
4
votes
0answers
70 views
The control of norm in quotient algebra
Let $B_1,B_2$ be two Banach spaces and $L(B_i,B_j),K(B_i,B_j)(i,j=1,2)$ spaces of bounded and compact linear operator between them respectively. If $T \in L(B_1,B_1)$, we have a $S \in K(B_1,B_2)$ and ...
3
votes
2answers
198 views
Show that a finite-dimensional Banach space has a bijective compact operator
It is clear that if $ T: X \rightarrow X $ is a bijective compact operator, where $ X $ is a Banach space, then $ \dim(\text{Range}(T)) = \dim(X) $, which implies that $ \dim(X) $ must be $ < ...
2
votes
3answers
137 views
Compact integral and multiplication operator in Banach spaces
Let $ A\colon C[0,1] \to C[0,1] $
$$ A(x)(t) = f(t)x(t) + \int_0^t x(s)ds,\quad f \in C[0,1]: f(1) \neq 0, \forall t \in [0,1] $$
Is $A$ a compact operator or not?
3
votes
1answer
198 views
Operators on $C([0,1])$ that is compact or not.
For $f\in C([0,1])$ set
$$Hf(x) = \frac{1}{x}\int_0^x f(t)dt.$$
a) Show that $H$ is a bounded operator from $C([0,1])$ into itself which is not compact.
b) From a) it follows that $H$ induces a ...
2
votes
1answer
99 views
Determine the operator T in a Hilbert space
Let $H$ be a Hilbert space and let $\{e_n, n \geq 1\}$ be an orthonormal basis for $H$.
a) Determine the operator $T\in B(H)$ that satisfies
$$ Te_1 = 0,\; Te_n = \frac{1}{n}e_{n-1}, n ...
1
vote
0answers
51 views
Finite Dimensional TVS
Let $E, F$ topological vector spaces, $E$ normable and $T: E \longrightarrow F$ linear, compact and surjective. Show that $\mbox{dim}(F)< \infty$.
3
votes
2answers
115 views
Proof that certain operators are compact
I want to examine which of the following operators $T \colon C[0,1] \to C[0,1]$. are compact, by some I think I got the argument, but others I have no idea.
a) $Tx(t) = x(t^2)$
Guess it is ...
3
votes
3answers
156 views
Showing that the orthogonal projection in a Hilbert space is compact iff the subspace is finite dimensional
Suppose that we have a Hilbert Space $H$ and $M$ is a closed subspace of $H$. Let $T\colon H\rightarrow M$ be the orthogonal projection onto $M$.
I have to show that $T$ is compact iff $M$ is finite ...
3
votes
1answer
305 views
Hilbert-Schmidt Operator
We have just covered Hilbert-Schmidt operators in class (which I missed) and I am having a hard time getting my head around them. I know the definition:
If $H$ is a Hilbert space and ...
0
votes
1answer
137 views
Hilbert space the trace
I need help from someone to solve this problem.
Given a bounded sequence $(\lambda_n)$ in $\mathbb С$ define an operator $S$ in $B(\ell_2)$ by $S(x_1) = 0$ and
$S(x_n) = \lambda_n x_{n-1}$ , ...
1
vote
0answers
49 views
Containment of an element to an operator system
This question will probably appeal to people in operator systems theory as it is very much related. However, I'm interested in down-to-earth concrete systems with finite dimensional Hilbert space ...
1
vote
1answer
158 views
No Nonzero multiplication operator is compact
Let $f,g \in L^2[0,1]$, multiplication operator $M_g:L^2[0,1] \rightarrow L^2[0,1]$ is defined by $M_g(f(x))=g(x)f(x)$. Would you help me to prove that no nonzero multiplication operator on $L^2[0,1]$ ...
3
votes
1answer
196 views
eigenvalue question
I think this question isn't that hard, but I am a bit confused.
Define the linear operator $T_k:H\mapsto H$ by
\begin{align}
T_ku=\sum^\infty_{n=1}\frac{1}{n^3}\langle u,e_n\rangle e_n+k\langle ...
4
votes
2answers
127 views
Compact and self-adjoint operator
It is true that if $T:H \to H$ is a compact operator ($H$ Hilbert space) then $T^\ast T$ is algo compact and indeed self-adjoint.
Conversely, is it true that every compact and self-adjoint operator ...
5
votes
2answers
114 views
Behaviour of the spectrum of a compact operator w.r.t. perturbations.
Suppose $A$ and $B$ are linear compact operators on a Hilbert space with $\sigma(A)$ and $\sigma(B)$ as their spectrum.
Is it possible to obtain some continuity result of $\sigma(A+\epsilon B)$ as ...
5
votes
1answer
448 views
Compact operators and completely continuous operators
A compact operator between Banach spaces is an operator that maps bounded sets into relatively compact sets, while a completely continuous operator maps all weakly convergent sequences into convergent ...
5
votes
3answers
350 views
A compact operator in $L^2(\mathbb R)$
Let $g \in L^{\infty}(\mathbb R)$. Consider the operator
$$
\begin{split}
T_g\colon & L^2(\mathbb R)\to L^2(\mathbb R) \\
& f \mapsto gf
\end{split}
$$
Prove that $T_g$ is compact ...
7
votes
2answers
197 views
Proof that operator is compact
Prove that the operator $T:\ell^1\rightarrow\ell^1$ which maps $x=(x_1,x_2,\dots)$ to $\left(x_1,\frac{x_2}{2},\frac{x_3}{3},\dots\right)$ is compact.
For an arbitrary sequence $x^{(N)}\in\ell^1$ ...
2
votes
2answers
295 views
Is convolution operator compact?
I know convolution is not a Hilbert–Schmidt integral operator, but it needs more to tell if convolution is compact or not.
2
votes
1answer
220 views
A compact operator is completely continuous.
I have a question.
If $X$ and $Y$ are Banach spaces, we have to prove that a compact linear operator is completely continuous.
A mapping $T \colon X \to Y$ is called completely continuous, if it maps ...
3
votes
2answers
100 views
Analog of Compact Operators
This is kind of vague question, but I'll try to make it more precise.
$T$ is a compact operator on a Hilbert Space, $H$, if $\overline{T(D)}$ is compact in $H$, where of course, $D$ is the closed ...
3
votes
1answer
94 views
For $T$ compact, $I-T$ left or right invertible implies $I-T$ invertible
Let $S\in B(X)$ be a bounded linear operator from $X$ onto $X$ and let $T\in K(X)$ be a compact linear operator from $X$ onto $X$. Then
$$
S(I-T)=I \iff (I-T)S=I.
$$
I don't know if we need the fact ...
4
votes
1answer
256 views
Compact multiplication operators
In class, we started talking about operators on Banach spaces after covering the Arzela-Ascoli Theorem. We defined a continuous operator $T\colon X \to Y$ to be compact if $\overline{T(B_X)}^{Y}$ is ...
0
votes
1answer
160 views
compact operator and its spectrum
I'm trying to study compact operators, but i'm having a little trouble with the 'practice'.. What are some tecniques to prove an operator compact. I know it can be shown that a limit of finite range ...
12
votes
1answer
551 views
How to prove that an operator is compact?
Consider $T\colon\ell^2\to\ell^2$ an operator such that
$Te_k=\lambda_k e_k$ with $\lambda_k\to 0$ as $k \to \infty$ how to prove that it is compact?
4
votes
0answers
136 views
Set of all compact operators $K(H)$ is the unique ideal in $B(H)$?
I want to show that the set of all compact operators $K(H)$ is the unique ideal in $B(H)$. Is there any relation between invertibility and compactness of an operator?
2
votes
0answers
164 views
Operator on Banach spaces
Let $X,Y$ be Banach spaces. $T\colon X\to Y$ be a bounded linear operator.
How can I prove that $T$ is compact if and only if there is $\lbrace x_n^*\rbrace\subset X^*$ such that $\|x_n^*\|\to 0$ and ...
12
votes
3answers
610 views
Compactness of a bounded operator $T\colon c_0 \to \ell^1$
Pitt Theorem says that any bounded linear operator $T\colon \ell^r \to \ell^p$, $1 \leq p < r < \infty$, or $T\colon c_0 \to \ell^p$ is compact.
I know how to prove this in case $\ell^r \to ...
5
votes
2answers
295 views
On the isometry between bounded linear operators and the dual of nuclear linear operators
Let $H$ be a separable Hilbert space. Let $(e_i)_i$ be an orthonormal basis. For any bounded linear map $T$ we write, whenever possible
$$\operatorname{tr} T := \sum_{i}^{\infty} \langle T e_i, e_i ...
