All Questions
Tagged with operator-theory normed-spaces
541
questions
0
votes
1
answer
30
views
Boundedness of a matrix operator in a norm
I would like to ask a simple question.
How do I show that a matrix is a bounded linear operator, for example this matrix
$$A=\begin{bmatrix}
t~~0\\
0~~\frac{1}{\sqrt{t}}
\end{bmatrix}$$
I know that ...
- 537
1
vote
0
answers
36
views
Multiplicity of bilateral shift on a Banach space
Let $\mathbb{X}$ be a Banach space. A bijective linear map $V: \mathbb{X} \to \mathbb{X}$ is said to be a bilateral shift if there is a closed subspace $\mathbb{L}$ of $\mathbb{X}$ such that
$\mathbb{...
- 73
0
votes
1
answer
86
views
Prove that for every $n \in \mathbb{N}$, the mapping $A_n$ is a bounded linear operator from $C[0,1]$ to $C[0,1]$ and calculate its norm.
Let $C[0,1]$ be a normed space equipped with the norm $\|\cdot\|_\infty$, and let for every $n \in \mathbb{N}$, the mapping $A_n$ be given by the prescription $
(A_n(f))(x) =
\begin{cases}
f(x), &...
user1316790
0
votes
1
answer
47
views
We are interested in finding for which $\lambda$ the operator $A - \lambda I$ is not surjective.
We are in the space $X = C[1/2, b]$ for some $b < 1$. We are interested in finding for which $\lambda$ the operator $A - \lambda I$ is not surjective. The operator $A: X \to X$ is given as $(Af)(x) ...
user1316790
-2
votes
1
answer
117
views
Thus $MV - VM = V^2$. So the spectrum of $V^2$ is $\sigma(V^2) = (\sigma(V))^2=0$. Why??
I am curious if this statement holds (it doesn't make much sense to me, but it was written in solutions in this form):
$\sigma(A)=0\implies\sigma(A^2)=(\sigma(A))^2=0.$
Can anybody explain to me why ...
user1315539
0
votes
0
answers
38
views
Reference for isomorphism theorem for Banach spaces
Let $X,Y$ be Banach spaces, and a linear continuous operator $T:X \rightarrow Y$ then we have that,
denote by $N =\text{ker }T$ and by $\text{Ran }T$ the closed range of T then,
$$
X / N \cong \text{...
1
vote
1
answer
23
views
Vector Norm $\vert\vert v\vert\vert_V$ expressed as supremum of $\vert Lv\vert$ over all bounded operators L with $\vert\vert L\vert\vert_{op}\leq 1$
Let $V$ be a normed vector space with norm $\vert\vert\cdot\vert\vert_V$. How can I show that for all $v\in V$ we have
$$\vert\vert v\vert\vert_V = \sup\{\vert Lv\vert \:\colon\: L\in \text{Hom}(V,\...
- 567
1
vote
1
answer
83
views
Inequality regarding Matrix Norm and Inverse Matrix
Currently, I'm stuck to one of a statement in a paper. Following is a brief summary of the paper regarding my question. (although the topic of the paper is mainly statistics, the question purely ...
- 769
0
votes
0
answers
24
views
Inequality for inverse of an unbounded self-adjoint operator
Given an unbounded self-adjoint operator $A$ on some Hilbert space $\mathcal{H}$, and $\mu$ a non zero real number: I want to show that
\begin{equation}
\lVert (\mathbf{A} + i\mu \mathbf{I})^{-1} \...
0
votes
1
answer
51
views
unbounded operator satisfying $||T(x_n)|| \to \infty$
Let $E,F$ be a normed space, and
$T:D(T) (\subset E) \rightarrow F$
be a densely defined unbounded linear operator.
By unbounedness,
for all $x \in E$, there is a sequence $(x_n) \subset D(T)$ such ...
- 71
2
votes
2
answers
114
views
Estimate norm of convolution operator
I'm trying to find the operator norm for $T: L^2([0,1])\to L^2([0,1])$, defined as $Tf(x)=\int_{[0,1]}|\sin(x-y)|^{-\alpha}f(y)dy$, where $0<\alpha<1$. Using an upper bound on $|\sin(x)|\geq |x|/...
- 529
3
votes
2
answers
152
views
How can you show that the trace class norm $\|A\|_1:=\mathrm{Tr}(|A|)$ satisfies the triangle inequality?
Exact wording of my question is a bit oxymoronic, since a norm by definition is a metric, and thus requires proper context. Let $H$ be a separable Hilbert space over the field $\mathbb{K}$. I am aware ...
- 1,221
1
vote
2
answers
103
views
Functional doesn't reach its norm [duplicate]
I'm trying to prove that for $X=L_1([0,1])$ and the linear functional $\phi(f)=\int_0^1tf(t)dt$, the linear functional doesn't reach its norm.
I already figured out that $\|\phi\|\leq1$, however I don'...
- 143
0
votes
1
answer
52
views
Necessary and Sufficient Conditions for an Operator to be Zero [closed]
Let $H$ be a Hilbert space, $<,>$ be its norm, and $A:H\to H$ be an operator. Is it true
$A=0 \Leftrightarrow\langle\Psi, A \Psi\rangle=0\ \left(^\forall \Psi\in H\right)$?
- 61
1
vote
1
answer
96
views
Compute the norm of the operator $A$.
Let the operator $A: X \rightarrow X(X=\{f \in C([0,3]) ; f(3)=0\})$ be given by the the formula
$$
(A f)(x)=(x+5) f(x)
$$
Compute the norm of the operator $A$.
I have done the following to solve the ...
- 13