1
vote
0answers
73 views
Hilbert space proof
$X$ is a separable Hilbert space and $ A\in L(X,X)$ and compact.
I need to prove that $A$ is approximately of finite dimension.
4
votes
0answers
44 views
Concerning unbounded linear operators on a Hilbert space
Let $H$ be some Hilbert space and let $B:H\rightarrow H$ be a bounded linear operator and $T:H\rightarrow H$ an unbounded linear operator. Furthermore we assume that $T$ is closed ,i.e. it's graph in ...
4
votes
2answers
188 views
+50
What is my operator norm (cannot get good enough bounds).
Given a space of square integrable functions $x(t)$ over the interval $[0;1]$ one can introduce a norm
$$\|x(t)\|= \sqrt{\int_0^1 (x(t))^2 \, dt};$$
Then what is a norm of the transformation below ...
1
vote
1answer
58 views
Computing an explicit solution to an integral equation via the Neumann Series.
I am hoping that someone can provide guidance for solving the integral equation
$$u=f+\lambda Au$$
where $1/\lambda\notin\sigma(A)$, $f\in L^2[0,2\pi]$, and $A:L^2[0,2\pi]\to L^2[0,2\pi]$ is defined ...
0
votes
1answer
26 views
Is my proof of characterisation of self-adjoint operators on complex Hilbert spaces okay?
I wish to show the following theorem:
Let $T:H\to H$ be
a bounded linear operator on a complex Hilbert
space $H$. Then if $\left\langle Tx,x\right\rangle \in\mathbb{R}$
for all $x\in H$, then $T$ is ...
1
vote
0answers
35 views
Unbounded self- adjoint and von Neumann algebra
I am reading Conway's Functional Analysis. Here is one exercise problem.I don't know how to show the following fact. For unbounded self-adjoint $T$ in Hilbert space $H$
1) $T$ commutes with its Borel ...
1
vote
2answers
55 views
Sequence of operators in a Hilbert space
The question is:
Let $H$ be a Hilbert space and $\{T_n\}$ be a sequence in $B(H)$ such that $\lim_{n\rightarrow\infty}\langle x, T_n y \rangle = 0$ for all $x, y \in H$. Prove or disprove $\sup_n ...
6
votes
1answer
53 views
Approximating a Hilbert-Schmidt operator
Let $H$ be a separable Hilbert space. Recall that a bounded operator $A : H \to H$ is said to be Hilbert-Schmidt if $$\|A\|_{HS}^2 := \sum_{i=1}^\infty \|A e_i\|^2 < \infty$$
where ...
1
vote
1answer
67 views
Bounded linear operator in weak topology
Let $B$ be a bounded linear operator on $H$. Prove $B\colon (H,w)\to (H,w)$ is continuous. $(H,w)$ is a Hilbert space with its weak topology.
1
vote
1answer
22 views
Is the Strong Limit of a Linear Operator in a Hilbert Space the Same as the Norm Limit?
If $H$ is a Hilbert Space, and I have an operator $F:H \rightarrow H$ which is the limit of a sequence of operators $F_n$ with respect to the operator norm; and this same sequence of operators ...
2
votes
1answer
69 views
Matrix Representation of Trace Class Operators
Suppose we have a separable Hilbert space (thus with a countable basis) and that represent an operator in matrix form, i.e:
$A: H \rightarrow H $$$x \;\rightarrow \sum_{j \in \mathbb{N}}\left(\sum_{k ...
1
vote
0answers
30 views
How can projection operators be limits of powers of unitary operators?
Consider a (fixed) unitary operator $U$ acting on the Hilbert space $\mathcal{H}$. Because the unit ball is compact in the weak topology, it is not hard to see that there exists a (smallest) compact ...
1
vote
1answer
62 views
Calculating the Norm of an operator in $L^2(0,1)$
If I have the following operator for $H=L^2(0,1)$:
$$Tf(s)=\int_0^1 (5s^2t^2+2)(f(t))dt$$ and I wish to calculate $||T||$, how do I go about doing this:
I know that in $L^2(0,1)$ we have that ...
0
votes
0answers
48 views
Showing an operator is self adjont
I am trying to show that the operator:
$$Tf(s)=5s^2\int_0^1t^2f(t)dt+2\int_0^1f(t)dt$$ is self adjoint where $H=L(0,1)$ with real scalars and $t\in \mathcal{L}(H)$.
So I can re-write this operator ...
2
votes
0answers
40 views
Prove that the sequence is in $\ell^{2}$. [duplicate]
Let $(a_{n})$ be a sequence of complex numbers such that for every $(b_{n})\in \ell^{2}$the series $\sum_{1}^{\infty}a_{n}b_{n}$ converges. Prove that $(a_{n})\in \ell^{2}.$
What I've tried so far is ...
2
votes
1answer
97 views
Matrix Representation of Operators in Infinite Dimensional (Separable) Hilbert Spaces
Suppose we have a separable Hilbert space (thus with a countable basis) and that we to represent an operator in matrix form, i.e:
$$A: H \rightarrow H \\ \; \; \; \; \; \;x \;\rightarrow \sum_{j \in ...
2
votes
3answers
44 views
The Kernel of unbounded operator in Hilbert space
If $T$ is a densely defined operator from a subspace of a Hilbert space $H$ to a Hilbert space $K$, how to prove that $\mbox{Ker}(T)=\mbox{Ker}(T^*T)$?
2
votes
1answer
54 views
Unbounded operator $T $ is bounded below when $\overline T$ is bounded
How to prove the following?
A densely defined symmetric operator $T$ in Hilbert space $H$ has a closure $\overline T$ which is bounded iff both $T,-T$ are bounded below (there exist constants $c,c' ...
2
votes
2answers
82 views
Are WOT/SOT topologies hereditarily separable?
Just out of curiosity,
Are weak and strong operator topologies on $B(H)$ hereditarily separable?
In other words, if $S$ is a subset of $B(H)$, where $H$ is a separable Hilbert space, is $S$ ...
2
votes
1answer
103 views
Point spectrum in Hilbert spaces
Let $H$ be a Hilbert space and and $T\in B(H)$ be normal and $\sigma_p(T)$ be the point spectrum of $T$ (i.e the set of all eigenvalues of T) and let $E$ denote the spectral measure. I'm trying to ...
1
vote
1answer
36 views
A basic result about operators on Hilbert space.
I am studying following result.
Let $H$ and $K$ be Hilbert spaces and an operator $A \in B(H, K)$, which has closed range.
The spaces $H$ and $K$ have the following orthogonal decompositions:
$H = ...
0
votes
0answers
34 views
A simple projection operation on a linear operator
Let $P_N$ be a projection operator of $\mathscr{H}$, a Hilbert space, onto the $N$-dimensional subspace $\mathscr{B}_N$.
I am reading a text that states for a linear differential operator $L$ ("whose ...
4
votes
0answers
119 views
Inverse of Identity plus Volterra operator
consider the following operator or $L_2(0,1)$,
$(Pw)(x)=w(x)+\int_0^x K(x,y)w(y)dy+\int_x^1 K(y,x)w(y)dy$, where the integral kernel is a polynomial.
I am trying to construct the inverse of this ...
4
votes
2answers
75 views
Show that linear Operator on $\ell^2$ is unbounded
Currently, I am preparing for a next semester course and trying to figure out some basic concepts in functional analysis.
Let $T:\mathcal{D}(T)\to \ell^2$ be defined by
...
5
votes
1answer
175 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
1answer
36 views
Norm of oblique projector and angle between subspaces
Take $V$ and $W$ closed subspaces of $H$ a Hilbert space with $V\oplus W=H$ (we'll assume this holds in the sequel, it may not be required everywhere but in the context of interest, it is always ...
1
vote
1answer
28 views
Property of sequence of eigenvalues of an operator.
For a positive (self adjoint) operator $A$ with eigenvalues $\lambda_k$, is it possible to have the case when neither $\lambda_k\to \infty$ or $sup_k \lambda_k<\infty$ for example if a subsequence ...
4
votes
1answer
124 views
Norms involving positive operators
Let's say we have $A \leq B$. Is it then true that $||Ax|| \leq ||Bx||$ (where $x, A, B$ all belong to the same finite-dimensional Hilbert space $H$)?
2
votes
1answer
50 views
Help with proving: If $X$ is a Hilbert $A$-$B$-module, then $ \| _A \langle x,x \rangle \| = \| \langle x,x \rangle _B \| $ for all $x\in X $.
Sorry, I posted a related question last week on here, but I'm still having trouble and this is a little different, I hope it's OK. Thank you!
( proof that this is an isometric map (on a $C^*$-module) ...
0
votes
0answers
82 views
When are two operators simultaneously diagonalizable?
I am reading a paper and they have diagonalised both operators in an equation, on a separable Hilbert space, with respect to the same basis. My question is, when can two operators be simultaneously ...
4
votes
1answer
69 views
Normal $T\in B(H)$ has a nontrivial invariant subspace
I am wondering if the following is true:
Every normal $T\in B(H)$ has a nontrivial invariant subspace if $\dim(H)>1$?
2
votes
1answer
58 views
The span of the orthorgonal projections is norm dense in $B(H)$
This is a question in my functional analysis book.
How to use the spectral theorem to prove that the span of the orthogonal projections is norm dense in $B(H)$?
3
votes
1answer
67 views
Operator norm and spectrum
Let $L$ be a bounded linear operator on a Hilbert space. Without assuming finite dimensions, can we express the operator norm of $L$ in terms of the spectrum of the positive operator $L^{\dagger}L$?
...
2
votes
1answer
26 views
Is the adjoint of a quasinormal operator quasinormal as well?
I am trying to make sense of the various properties of operators on Hilbert spaces that generalise the notion of normality. It is known that for a (bounded) operator $A$ there are the following ...
4
votes
1answer
45 views
Selfadjoint and continuous operator on a complex Hilbert space
Let $T\colon H\to H$ be a selfadjoint continuous operator on a complex Hilbert space. Show:
$$
\lVert (T\pm i\mbox{Id})x\rVert^2=\lVert Tx\rVert^2+\lVert x\rVert^2~\forall~x\in H.
$$
--
How can I ...
2
votes
0answers
70 views
Find the adjoint operator
I would like to find the adjoint operator in the Hilbertspace $L^2(0,\infty)$ of the operator
$$
(Ax)(t)=x(at), x\in L^2(0,\infty), a>0.
$$
My calculation is the following; I use the ...
2
votes
1answer
85 views
Estimate on the norm of a self-adjoint operator
EDIT: thks to Martin's comment I realize the previous version was wrong. Here is the correct version of what I need to show:
I am trying to show that if $A$ is a self - adjoint operator in a Hilbert ...
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
47 views
Picard Condition (searching for an idea)
The so-called Picard-condition is:
Let X,Y be Hilbertspaces and $T\colon X\to Y$ is a compact operator with singular value decomposition system $\left\{(\sigma_j,u_j,v_j)\right\}$. An element ...
4
votes
1answer
62 views
Smallness/ Rigidity of $\kappa(\mathcal{H})$ without using minimal projections?
Let $\mathcal{H}$ be a Hilbert space and $\kappa(\mathcal{H})$ the $C^*$-algebra of compact operators on $\mathcal{H}$. By smallness/ rigidity of $\kappa(\mathcal{H})$ I am referring to the following ...
2
votes
1answer
75 views
A generalization of the Cauchy-Schwarz inequality to linear operators
If $A$ is an operator and $A \in \mathcal{B_{+}(X)}$ (the set of the positive operators) then the generalization of the Cauchy-Buniakowsky-Schwarz inequality holds:
$$|\langle Ax,y\rangle| \leq ...
2
votes
0answers
51 views
A set of trajectories as a linear subspace of Hilbert space
Let $\left\{S(t)\right\}_{0 \leqslant t \leqslant \theta}$ be a strongly continuous semigroup of linear continuous operators in Hilbert space $H$, $S(0) = I$. Let $x$ be some element of $H$. Then its ...
3
votes
2answers
125 views
Find adjoint operator of an operator T
I would like to find the adjoint operator of
$$
T\colon L^2([0,1])\to H^1([0,1]),\quad x\mapsto\int\limits_0^t x(s)\, ds.
$$
Here $H^1([0,1])$ is the Sobolev space $W^{1,2}([0,1])$.
I tried to find ...
-1
votes
1answer
106 views
Functional analysis - bounded linear transformation
Let $ \mathcal{H} $ be a Hilbert space, and let $ T: \mathcal{H} \to \mathcal{H} $ be such that $ \langle x,Ty \rangle = \langle Tx,y \rangle $ for all $ x,y \in \mathcal{H} $.
How can one show that ...
3
votes
1answer
117 views
Bounded operators on separable Hilbert spaces
Let $H$ be a separable Hilbert space. Show that every bounded operator from $H$ to itself can be approximated in the strong operator topology by a sequence of finite rank operators.
Im not sure what ...
6
votes
1answer
148 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
2answers
179 views
Norm inequality for sum and difference of positive-definite matrices
If $X_{1}$ and $X_{2}$ are positive definite matrices, how to show
that $\left\Vert X_{1}-X_{2}\right\Vert \le\left\Vert X_{1}+X_{2}\right\Vert$ for the spectral norm? and how about for the nuclear ...
2
votes
1answer
87 views
Normal Operator surjective
I have difficulty proving:
If $T$ is a normal operator in a Hilbert space, $T$ is surjective if and only if $T^*$ surjective.
Please give me some help. Thank you.
4
votes
1answer
73 views
Multiplication operator and trace class
Suppose we work in $H=l^2(\Bbb{N})$ and suppose the multiplication operator $T_f$ such that $T_f\psi=f\psi$ and $f:\Bbb{N}\rightarrow \Bbb{C}$. We denote by $B_1(H)$ the trace class of operators.
...
