All Questions
10 questions
17
votes
1
answer
1k
views
How many values determine a norm?
It is well known that for a bilinear form over an n-dimensional vector space, $n^2$ values (on all pairs of basis-vectors) determine it uniquely.
How many values do we need to specify in order to ...
- 6,741
7
votes
2
answers
647
views
If I multiply the coefficients of a trace-class operator with bounded complex numbers is it still trace class?
Suppose that $T \in TC(l^2( \mathbb{Z}))$ is trace class.
Consider its kernel $ T(i,j) = \langle e_i, T e_j \rangle $ where $ \{e_i\}_{i \in \mathbb{Z}}$ is an ONB for $l^2( \mathbb{Z})$. Now, ...
- 1,054
6
votes
2
answers
405
views
$\|T\|_2 \le \sqrt{\|T\|_1\|T\|_\infty}$
Let $T$ be a linear operator acting on a finite-dimensional real or complex
vector space. As a direct consequence (or rather a particular case) of the
Riesz-Thorin theorem, we have
$$ \|T\|_2 \le \...
- 23k
5
votes
1
answer
3k
views
Operator norm vs spectral radius for positive matrices
I believe the following statement should be true but somehow I don't see an argument:
For every integer $d>1$ there exists a constant $C=C(d)>1$ such that whenever $A$ is a $d \times d$ matrix ...
- 421
4
votes
1
answer
219
views
Is a mixture of $\ell_p$-norms $\eta(x):=\lVert x\rVert_2 + r\lVert x\rVert_p$ always dimensionlessly equivalent to some $\ell_q$-norm?
$\newcommand\norm[1]{\lVert#1\rVert}$For any $p \in [1,2]$, $r \ge 0$, and integer $d \ge 1$, define a mixed-norm $\eta:\mathbb R^d \to \mathbb R$ by $\eta(x) := \norm x_2 + r\norm x_p$, for any $x \...
- 6,853
4
votes
0
answers
311
views
Estimates of the Frobenius norm of commutator
Let $A,B$ be two unitary matrices in $U(n)$, and $\|\cdot\|_{F}$ denote the Frobenius norm (or Hilbert Schmidt norm on the finite dimensional $M_n(\mathbb{C})$). I am looking for estimates of the ...
- 949
3
votes
0
answers
588
views
Norm in a product vector space induced by a norm in $\mathbb{R}^d$
I posted this question originally here (nobody answered there): https://math.stackexchange.com/questions/2066318/is-the-following-function-a-norm
Let $\| \|$ be any norm in $\mathbb{R}^d$. Consider ...
- 153
2
votes
1
answer
226
views
Tighest Gap $\|x\|_1/\|x\|_2$ between $\ell^1$ and $\ell^2$ norms
I'm looking specifically at the optimization problem
$$
\begin{align*}
\text{maximize: }& n - \frac{\|\lambda\|_1^2}{\|\lambda\|_2^2}\\
\text{subj. to: }& \lambda \succeq \epsilon\mathbf{1}
\...
- 153
2
votes
0
answers
648
views
Closed-form expressions for dual norms of real normed vector spaces
Didn't get any biters over at MSE, so I figure this place might be more appropriate...
Say that $V$ is a finite-dimensional real normed vector space, where for some $v \in V$ the norm is notated by $\...
- 4,897
1
vote
3
answers
684
views
Norm of an operator formed using a unitary operator
Suppose, $ A $ is a unitary matrix in $ M_n(\mathbb{C}) $ given by $ (a_{i,j})_{1\le i,j\le n} $ which has the property that, for all the basis elements $ e_i $, $ Ae_i\ne |\lambda| e_j $ for all $i,j ...
- 817