Tagged Questions
0
votes
3answers
56 views
Diagonalizable Operators: An Operational Extension
Let $T$ be a diagonalizable operator on a vector space $V$. Prove that the operator
$$a_nT^n + a_{n-1}T^{n-1}+\cdots+a_1T+a_0 Id_V$$
on $V$ is also diagonalizable for any scalars $a_1, ...
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 ...
0
votes
0answers
26 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
vote
2answers
54 views
What is the role of supremum in operator norm
An operator norm is defined as $\|A\|_S=\sup\{\|Av\|:v\in \Bbb R^n, \|v\|=1\}$. Where $\|\cdot\|$ is some norm on $\Bbb R^n$ and $A\in M_n(\Bbb F)$, space of square matrices of dimension $n$ over ...
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 ...
3
votes
1answer
63 views
$K$ is a linear compact operator on Hilbert space $H$. Will the image of $I-K$ on every closed subspace of $H$ be also closed?
Just as the title. We know the image of $I-K$ is closed, but if we restrict $H$ to a closed subspace $V$, will $(I-K)(V)$ be a closed subspace of $H$? Any hint is appreciated.
1
vote
2answers
60 views
Operator norm converging to 0 for certain condition
Let $X$ be a finite-dimensional normed space and $T_n : X \to X$ a sequence of linear operators such that $\lim_nT_nx = 0$ for all $x$ in $X$. Prove that $\lim_n\|T_n\|=0$.
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 $\dim(H)>1$?
1
vote
2answers
70 views
How to prove that two non-zero linear functionals defined on the same vector space and having the same null-space are proportional?
Let $f$ and $g$ be two non-zero linear functionals defined on a vector space $X$ such that the null-space of $f$ is equal to that of $g$. How to prove that $f$ and $g$ are proportional (i.e. one is a ...
0
votes
1answer
115 views
why isn't “functional operator” a contradiction?
Consider the term "functional operator". My understanding was that:
(a) An operator in this context refers to a mapping from one vector space to another vector space.
(b) A functional is a mapping ...
0
votes
1answer
71 views
what is mathematical difference between an hermitian operator $\hat A$ and a vector $\vec A$?
what is mathematical difference/relation between an hermitian operator $\hat A$ and a vector $\vec A$?
4
votes
0answers
63 views
Extensions of finite-rank operators
Let $V$ be a vector space and let $W$ be its subspace of infinite codimension. Let $\mathcal{F}_W$ be the family of all finite-rank operators on $V$ with range contained in $W$. Consider the ...
