2
votes
2answers
52 views
If $T:X \to Y$ is a linear homeomorphism, is its adjoint $T^*$ a linear homeomorphism?
$X$ and $Y$ denote Hilbert spaces. If $T:X \to Y$ is a linear homeomorphism, is its adjoint $T^*$ a linear homeomorphism? Homeomorphism means continuous map with continuous inverse.
I think the ...
0
votes
0answers
25 views
Resolvent of a restriction of a dual operator
UPDATE: After a couple more days of thinking maybe this is however true in general? Is it a known theory?
Theorem(?). Let $A\colon D(A) \to X$ be a linear, densely defined operator in a Banach ...
-1
votes
0answers
18 views
Help with Toeplitz operators applications. [closed]
I am trying to find a physics problem which solution involves Toeplitz operators.
2
votes
1answer
21 views
Selfadjoint operator $\Rightarrow$ Idempotent Operator?
If $P\in\mathcal{L}(H,H)$, with $H$ a Hilbert space, such that $P = P^*$, Is possible to show that $P^2 = P$?
If that is possible, then $P$ is a projection operator, right?
Thanks in advance.
1
vote
0answers
27 views
To prove that an operator is bounded [duplicate]
I have this problem:
Let $(H, \langle\cdot,\cdot\rangle)$ a Hilbert space on $\mathbb{C}$
and $A:H\rightarrow H$ a linear operator such that $$\langle A(x),
y\rangle\ =\ \langle x, ...
5
votes
1answer
41 views
Adjoint operator, a condition for closed range
Let $X$ and $Y$ be two Hilbert spaces and $A\in\mathcal{L}(X,Y)$. Suppose that there's $\beta > 0$ such that
$$\inf_{z\ \in\ \text{Ker}(A)}\|x-z\|\ \leq\ \beta\|A(x)\|,\quad \forall\ x\in X.$$
Show ...
1
vote
1answer
29 views
Composition of Partial Isometry
Let $T$ be a linear operator in $H$, a Hilbert space.
An operator $T \in L(H)$ is said to be a partial isometry if the restriction of $T$ to $ker(T)^{\perp}$ is an isometry. I would like to prove that ...
4
votes
0answers
43 views
Concerning unbounded linear operators on a Hilbert space
Let $H$ be some Hilbert space and let $B:H\rightarrow H$ be a bounded linear operator and $T:H\rightarrow H$ an unbounded linear operator. Furthermore we assume that $T$ is closed ,i.e. it's graph in ...
3
votes
0answers
65 views
Conditions for a kernel of a bounded operator to be complemented
I am well aware of the problem of complementing subspaces in Banach spaces as it was discussed here and here .
Nevertheless, I wonder whether there are conditions for existence of a complement $M$ ...
1
vote
0answers
22 views
Conditions under which $A$ is a W*-algebra for a positive map between C*-algebras $\phi : A \rightarrow B$
Let $A$ and $B$ be a C*-algebra.
Let $\phi : A \rightarrow B$ be a positive map.
Suppose that $B$ is a W*-algebra. Under what conditions on $\phi$ can we ensure that $A$ is also a W*-algebra?
4
votes
2answers
163 views
+50
What is my operator norm (cannot get good enough bounds).
Given a space of square integrable functions $x(t)$ over the interval $[0;1]$ one can introduce a norm
$$\|x(t)\|= \sqrt{\int_0^1 (x(t))^2 \, dt};$$
Then what is a norm of the transformation below ...
1
vote
1answer
29 views
Density of the image and closedness of the inverse of a bounded linear operator
Let $A \colon X \to X$ be a bounded linear operator, where $X$ is a Banach space.
$(Q1)$ Is it true that if $A$ is injective then the image of $A$ is dense in $X$?
$(Q2)$ Is it true that $A^{-1} ...
0
votes
0answers
65 views
Bounded operator on dense subspaces
Give an operator like this or show it doesn't exist: Operator $T: X\rightarrow Y$ is bijective. $X,Y$ are dense subspaces of a Banach space $Z$, and $X$ is proper subset of $Y$. Both $T$ and $T^{-1}$ ...
1
vote
1answer
72 views
Bounded and invertible operator on dense subspace
Who can give me an operator like this or show it doesn't exist: Operator T: X-->Y, is a bijection from normed linear space X to normed linear space Y. X, Y are equipped with the same norm, and X is a ...
2
votes
1answer
31 views
Range of adoint operator
We consider infinite dimension.
$X,Y$: Banach Spaces
$T:X→Y$ is a bounded linear operator.
I want to prove
$(\ker\, T)^\bot = \overline {R(T^*)}$.
$(\ker\, T)^\bot = \{f\in X^*|f(x)=0\ (x\in ...
6
votes
2answers
66 views
Coercive bilinear form on Hilbert space
I need to show the two following results. If true, it must be a simple proof but I do not seem to be able to make it work. Thank you in advance.
Consider a continuous symmetric bilinear form $B$ on a ...
3
votes
1answer
48 views
$AB - BA = I$ in Hilbert Space [duplicate]
Let H be a Hilbert space and $A$ and $B$ be bounded operators in $H$. How can I prove that $AB - BA = I$ is not possible ? Probably this is as easy as in the matrices case, but I couldn't prove it. ...
4
votes
1answer
44 views
A problem on bounded invertible linear operator in Banach space
Let $X$ be a Banach space. Let $T : X \to X$ be a invertible
linear operator and $M > 0$ be such that $\|T^{-k}\| \le M$ for all
$k \ge 1$. Prove that $\inf_ {n\ge1} \|T^n(x)\| > 0$ for all $x ...
3
votes
0answers
44 views
Sub-unital maps between C*-algebras: is there any relevant result?
"In this section, we deal with positive linear maps $\phi : A \rightarrow M$ between two unital C∗-algebra $A$ and $M$ with units denoted by $I$. In fact, we may assume that $A$ is the unital ...
0
votes
0answers
27 views
Normal endomorphism
I have a question about normal endomorphism. In class, we said that normal endomorphisms in finite dimensional real vector spaces are always of the form that we have some eigenvalues and further ...
-1
votes
0answers
49 views
How we can change a strongly continuous semigroup to a contraction semigroup?
If $T(t), t>0$ is a strongly continuous, bounded semigroup on a Banach space. how we can transform it to contraction semigroup?
0
votes
2answers
49 views
Self adjoint operator
I am looking in the space of test functions $ \{f \in C^\infty|f^{(n)}(a)=f^{(n)}(b)=0\};n \in \mathbb{N}_0\} $whether the n-th derivative is a self adjoint operator. the dot product is given by ...
1
vote
0answers
34 views
Continuous spectrum for a specific linear operator
The operator is for $A:L^2[-1,1]\to L^2[-1,1]$ defined via
$$Au(x)=xu(x)+\theta\int_{-1} ^1u(s)ds.$$
The question is actually find the spectrum, but I managed to find everything else, pending the ...
0
votes
1answer
69 views
Representation of a bilinear form on an Hilbert space
Given a bilinear symmetric form $b(u,v)$ on a Hilbert space. I need to know some very basic facts. A reference where these are discussed would be greatly appreciated.
1) There exists a symmetric ...
1
vote
1answer
57 views
Computing an explicit solution to an integral equation via the Neumann Series.
I am hoping that someone can provide guidance for solving the integral equation
$$u=f+\lambda Au$$
where $1/\lambda\notin\sigma(A)$, $f\in L^2[0,2\pi]$, and $A:L^2[0,2\pi]\to L^2[0,2\pi]$ is defined ...
2
votes
1answer
39 views
Spectrum of the unbounded operator $i\partial_x$
I've been puzzling over this for some time now, and can't quite make my intuitions precise. I need to find the resolvent set and spectrum of the operator
$$
Lu=i\frac{du}{dx}
$$ taken to be ...
1
vote
1answer
35 views
Residual spectrum is empty
I'm following Kreyszig's "Introductory Functional Analysis with Applications" and am trying to follow the proof of the following Theorem (9.2-4 on p. 468)
For a bounded self-adjoint linear operator ...
1
vote
0answers
26 views
Find spectrum of the operator and an explicit form for the solution
Let $A: L^2[0,2\pi] \to L^2[0,2\pi]$ be defined by $Au=v$ if
$v(x)=\int^{2\pi}_{0} cos(x+t)u(t)dt$
(a) Find the point spectrum, eigenspace, residual spectrum, continuous spectrum, and spectrum
(b) ...
1
vote
0answers
33 views
Find spectrum of the operator
I need some help to find point spectrum, residual spectrum, continuous spectrum and for this problem.
Let $0 \in\mathbb{C}$. On the complex space $L^2[-1,1]$ consider the operator
$Au=v,$ where ...
3
votes
1answer
68 views
Convergence of operator norm
I have a linear bounded operator $A:L_2(0,1) \rightarrow L_2(0,1)$ satisfying $\|A^n\|^{1/n} \rightarrow 0$. Thus, for some sufficiently large $N$, $\|A^N\| < 1$ and then from Gelfand's formula, I ...
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
1answer
37 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
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 ...
4
votes
0answers
47 views
When are two commuting linear operators functions of each other
I've computed that the following is valid for certain functions but I've hit a slight bump in my proof. I was wondering if someone could clear it up.
If we formally consider the integral operator $E ...
2
votes
2answers
28 views
Limit of bounded operators
Suppose $T_n$ is a sequence of self-adjoint bounded operators on a Hilbert space, and $T_n \rightarrow T$ in operator norm, $T$ being also bounded and self-adjoint.
Do we then have: $T_n^m\rightarrow ...
1
vote
0answers
16 views
Find a symbol for pseudodifferential operator
$\DeclareMathOperator{\Mel}{M}
\newcommand{\Rn}{\mathbb R^n}
\newcommand{\dd}{\,\mathrm{d}}$
Consider a pseudodifferential operator (Mellin operator) in positive orthant with symbol $\sigma(z)$:
$$
...
1
vote
1answer
53 views
Orthogonality & Adjoint Operator
I am trying to prove this simple statement left to the reader in Brézis's book.
Let $A \colon D(A)\subset E \longrightarrow F$ be an unbounded operator. Let $G:=\operatorname{Graph}(A)$ and $L=E ...
6
votes
1answer
51 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
22 views
product of bounded linear operators
If I have 2 bounded linear operators $T_1,T_2$ such that $T_2:X\rightarrow Y$ and $T_1:Y\rightarrow Z$. I know that by boundedness, $||T_2(x)||\leq||T_2||\,||x||$ and using the norm of $T$ defined as ...
2
votes
1answer
85 views
spectrum of two bounded linear operators
Suppose that L and B are bounded linear operator on H, assume $0\in \rho(L) \cap \rho(L+B)$ and that $L^{-1}$ is compact. Prove that L+B also has a compact inverse.
1
vote
1answer
66 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 ...
1
vote
2answers
39 views
Linear functional $\mathscr{L}(E,F)$
Let $\mathscr{L}(E,F)$ denote the space of all linear functionals from $E \to F$.
Let $\mathscr{C}(E,F)$ denote the space of continuous linear functionals from $E \to F$. My question:
How to prove ...
4
votes
2answers
157 views
Interchanging closed operators and integrals
I am dealing with a problem in Evans PDE without measure theory knowledge...
We have contraction semigroup $\{S_t\}_{t \geq 0}$ on real Banach space $X$, i.e family of bounded linear operators from $ ...
7
votes
1answer
58 views
How does $\lim A_n$ being not invertible imply $\sup_n\|A_n^{-1}\|=\infty$?
Consider a sequence of operators $\{A_n\}_{n=1}^{\infty}\subset B(X,Y)$, where $X,Y$ are normed vector spaces and $B(X,Y)$ denotes the space of bounded linear operators from $X$ to $Y$. Assume that ...
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 ...
2
votes
0answers
61 views
Relation between noncommutative geometry and functional analysis
Recently I came across the subject of noncommutative geometry via my interest in functional analysis. My very little exposure to this subject gives me a sense that part of it is built on the theory of ...
2
votes
0answers
112 views
How to decompose a representation into direct sum of cyclic representation?
Let $U$ be the bilateral shift operator on $l^2 (\mathbb Z)$, let $T=U+U^*$. How to calculate $\sigma(T)$? And how to show there is no cyclic vector for the action of $C^*(T,I)$. Further how to ...
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 ...
