All Questions
Tagged with operator-theory compact-operators
574
questions
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18
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$T$ compact operator, Let $\Delta^*_{\bar{\lambda}}$ subspace of $X^*$. Prove $\Delta^*_{\bar{\lambda}} = \bar{\Delta^*_{\bar{\lambda}}}$
We have proved the following claim:
Let T be compact operator and $\lambda \neq 0$. Then $\Delta_\lambda = \bar{\Delta _\lambda}$.
Now there is the corollary:
$T$ compact operator, Let $\Delta^*_{\...
- 607
2
votes
1
answer
49
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Is the notion of compactness of operators preserved under quasi-similarity?
Let $H$ be a Hilbert space and $T \in \mathcal L (H).$ Then $T$ is said to be quasi-invertible if it is one-one and has dense range. The operator $T$ is said to be quasi-similar to an operator $S \in \...
- 2,612
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61
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What is a conjugate unitary operator?
I'm studying operator theory and I have encountered the concept of a cojugate unitary operagtor several times. However, I cannot find any reliable references. There is one paper which claims that a ...
- 573
1
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1
answer
40
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An interesting family of seminorms $\mathcal F$ and comparison between the topology generated by this seminorms and the Weak Operator Topology.
I am learning functional analysis and I am stuck with the following questions from Strong Operator Topology and Weak Operator topology on $\mathcal B(H)=\{T:H\to H:T$ is Op-Norm continuous,linear $\}$....
- 3,619
4
votes
1
answer
150
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Proving that operator in $L^2[0,1]$ is compact
I need help with some functional analysis:
Let $A$ be a continuous linear operator on $L^2[0,1]$ and for any $f \in L^2[0,1]$ the function $Af$ is Lipschitz continuous. Show that $A$ is compact.
It is ...
- 51
2
votes
1
answer
42
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Equivalence of Kraus operators on a single element
Let's say that I'm working in a Hilbert space $\mathcal{H}$,suppose I have two bounded operators $A.B\in B(\mathcal{H})$ and a positive semidefinite operator $x$, for example take $x$ to be a density ...
- 75
3
votes
1
answer
67
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Problem of Fourier transform and compact operator
Problem:
Is the Fourier transform $F(f(x))=\int_{-\infty }^{\infty }f(y)e^{-ixy}dy$ a compact operator in the case of $F:L_1\left ( \mathbb {R} \right )\to \mathrm{BC}\left ( \mathbb{R}\right )$. $\...
- 1,433
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votes
3
answers
66
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Prove that the operator is not compact
Problem:
Prove that the operator $A\in\mathcal{L}\left ( C[0,1] \right ):(Af)(x)=f\left ( x^2 \right )$ is not compact
My attempt at a solution:
It is necessary that the image of the unit ball be ...
- 1,433
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votes
1
answer
21
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continuouty of operators
I was given a task to understand, wheter operators $A$ and $B$ are compact, $$\displaystyle A:\ell_2 \rightarrow L_1(\mathbb{R}), (Ax)(t) = \sum\limits_{k=1}^{+\infty}\frac{x(k)}{\cosh^2(kt)},$$ $$B:...
- 139
1
vote
1
answer
52
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Hilbert-Schmidt integral operator without second integrable kernel
From this webpage, we know that if $X$ is a measurable space ($\sigma$-algebra is omitted) and $\mu$ is the measure, $K(x,y) \in L_2(X\times X,\mu\times \mu)$, we can define the following operator ...
- 487
1
vote
1
answer
48
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compact and trace operator [closed]
I am study compact and trace classes of operators in Hilbert space. To clarify, I asked the following question, but have not been able to resolve it yet. Consider $L_p \left( \mathbb{R}^d \right) $ ...
- 41
0
votes
0
answers
36
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Compactness and boundedness of integral convolution operator
Doing my homework on functional analysis I faced with the following problem.
Let $\displaystyle (Uf) (s)=\int\limits_{-1}^{1}\frac{f(t)}{|s-t|^{5/6}}dt$.
Using Young inequality it can be shown that $U$...
- 313
1
vote
1
answer
36
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Range of $I-T$ is closed for $T$ being a compact operator in a Banach Space
I know from this question that this is true for Hilbert spaces. But is the result true for Banach spaces as well?
So suppose $x_{n}$ is a bounded sequence and $(I-T)(x_{n})$ converges. Then upto a ...
- 1,285
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33
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Sequence of range space of compact operator stabilizes. Sheldon Axler MIRA Ex 10C 11
Suppose $T$ is a compact operator from a Hilbert Space. Then for $a\neq 0$, we have that $\text{range}(T- aI)^{n-1}=\text{range}(T-aI)^{n}$ for some $n$ and $\ker(T-aI)^{m-1}=\ker(T-aI)^{m}$ for some $...
- 1,285
1
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1
answer
33
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Convergence in trace
I am having difficulties with this problem.
Given a non-negative compact self-adjoint operator $\gamma$ i.e. $\langle \gamma u, u\rangle \geq 0$ for all $u$. Denote the cut-off function $\chi \in C^\...
- 89