Tagged Questions
2
votes
1answer
61 views
Show for compact operator $K$, if $||Kf|| < ||f|| \forall f$, then $||K|| < 1$.
I wanted to check my reasoning on proving this statement, and see if anyone had suggestions for other proofs of this fact.
Again, the statement is, if $K$ is a compact operator on a Hilbert space ...
0
votes
0answers
44 views
Hilbert Schmidt decomposition
Usually, for example in Reed and Simon, the Hilbert Schmidt (singular value) decomposition of a compact operator $T$ on a Hilbert Space is written as
$$T = \sum_{n=1}^{N} \lambda_n ...
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 ...
1
vote
2answers
126 views
compact and self adjoint square root of an operator
Let H a Hilbert space and $T:H\rightarrow H$ a linear bounded, self-adjoint, positive and compact operator. How can i prove that the square root of T, $\ T^{1/2}:H\rightarrow H$ is also compact and ...
3
votes
2answers
63 views
Given $F:X \to X$ on $X$:Hilbert space satisfying some properties, prove that $F$ is surjective.
Given a map $F:X \to X$ where $X$ is a Hilbert space, $F$ satisfying
$f(x):=x-F(x)$ is a compact map.
$\lim_{\|x\|\to \infty} \frac{(F(x),x)}{\|x\|} = \infty$
I'm seeking to prove that $F$ is ...
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)$ ...
6
votes
1answer
146 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 $ ...
1
vote
1answer
53 views
How do I prove that a particular linear operator has an orthonormal basis?
I have to show that if $T$ is a linear operator such that $T: L^2(\mathbb(R)^n) \to L^2(\mathbb(R)^n)$ and $T(f)(x) = \int_{R^n}f(y)g(x,y)dy$, where $g(x,y)$ is an $L^2$ function, that there is an ...
1
vote
1answer
78 views
Symmetric bounded linear maps can be approximated by compact symmetric linear maps.
Let $H$ be a separable Hilbert space and let $T:H \rightarrow H$ be a symmetric bound linear map.
a) Show that for every orthogonal projection $P$ on $H$ ($P' = P$, $P^2 = P$) PTP is symmetric.
b) ...
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 ...
3
votes
1answer
336 views
How to prove this integral operator is compact?
$T_kf=\int K(x,y)f(y)dy$
where $K(x,y)=\frac{\phi(x)\phi(y)}{|x-y|^{n-\alpha}}$
$\phi(x)$ is a smooth function on a compact support. $f$ is defined on $R^n$ and $K$ is defined on $R^n\times R^n$
...
5
votes
1answer
120 views
Show $T$ is compact
$H$ and $K$ are Hilbert Spaces, $(u_n)$ and $(v_n)$ are sequences in $H$ and $K$ respectively. $\sum_{n=1}^{n=\infty} \|u_n\|\|v_n\| $ converges.
$T\colon H\rightarrow K$ is defined by ...
3
votes
3answers
155 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
303 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 $\mathbb ะก$ define an operator $S$ in $B(\ell_2)$ by $S(x_1) = 0$ and
$S(x_n) = \lambda_n x_{n-1}$ , ...
1
vote
1answer
157 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 ...
2
votes
1answer
82 views
Compact operator on $l^2$
Let A be a bounded linear operator on $l^2$ defined by A($a_n$)=($\frac{1}{n} a_n$). Would you help me to prove that A is compact operator. I guess the answer using an approximation by a sequences of ...
0
votes
1answer
61 views
Compact Operator defined by inner product
Let $H$ be a Hilbert space and $y,z \in H$. Define bounded linear operator $Ax=\langle x,y\rangle z$ where $\langle,\rangle$ is inner product. Would you help me to prove that $A$ is compact operator.
2
votes
3answers
238 views
Compact operators and uniform convergence
Suppose $T: H \rightarrow H$ is a compact operator, $H$ is a Hilbert space, and let $(A_n)$ be a sequence of bounded linear operators on $H$ converging strongly to $A$. Show that $A_nT$ converges in ...
1
vote
1answer
82 views
Direct sum of compact operators
I am having some trouble proving this:
Let $T_1\in H_1$ and $T_2\in H_2$ where $H_1,H_2$ are Hilbert spaces. Let $T=T_1\oplus T_2$ on $H_1\oplus H_2$. I need to show $T$ is compact iff $T_1$ and $T_2$ ...
3
votes
1answer
138 views
Compact operators between Hilbert spaces
I have the suspect that the following statement is true, but I don't how to prove it. Any suggestion? Thanks to all!
Let $X$, $Y$ be Hilbert spaces and let $T \colon X \to Y$ be a linear continuous ...
3
votes
2answers
198 views
The image of orthonormal basis under compact operator
I need a help to prove that statement: if $\{e_n\}$ an orthonormal basis in Hilbert space $H$ and $A$ is a compact operator from $H$ to $H$, then $Ae_n\rightarrow 0$. Thx for any help.
0
votes
1answer
210 views
Square root of compact operator
I'm trying to solve a functional analysis problem
A self-adjoint non-negative operator $A$ on a Hilbert space $H$ is compact if and only if its $\sqrt{A}$ is compact.
12
votes
1answer
550 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?
5
votes
2answers
294 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 ...
