All Questions
Tagged with matrix-exponential fa.functional-analysis
8 questions
8
votes
2
answers
674
views
Question on whether, "An entire function, nowhere zero, has an entire logarithm," holds for matrix-valued entire functions as well
It is known that an entire function that is nowhere zero must be the exponential of another entire function.
Does this hold for matrix-valued functions as well? That is, given a matrix-valued entire ...
- 286
5
votes
2
answers
458
views
Logarithm of a bounded operator
Let $\mathbb H$ be a Hilbert space and let $A\in \mathcal B(\mathbb H)$ such that the spectrum of $A$ does not meet a closed half-line issued from 0 in the complex plane. Then I guess that
$
A=\exp L
$...
- 16.2k
3
votes
1
answer
521
views
Is the set of real matrices with at least one real logarithm closed under multiplication?
Let $S$ be the set of real matrices with at least one real logarithm. For some couple of its elements, for example those with at least (one real logarithm each with submultiplicative norm smaller than ...
- 286
1
vote
0
answers
68
views
Inequality of exponentials of Banach operators
(I have moved this question from Stackexchange).
Given the operators $\{A_j\}$ in a Banach algebra and a positive integer $p$, let
\begin{equation}
g=\exp\left(\frac{1}{n}\sum_{j=1}^p A_j\right)\quad\...
- 111
2
votes
0
answers
114
views
Growth bounds for the exponential of an operator
Let $X$ be a complex Banach space and $A:X \to X$ a compact operator. It spectrum is the set $\sigma(A)=\lbrace \lambda \in \mathbb{C}, \ A-\lambda I \text{ is not invertible}\rbrace$. Let $L=\sup\...
1
vote
1
answer
136
views
Conditions to obtain a real logarithm of a unitary unimodular complex matrix?
The problem statement is the following:
$$U=\exp\{iV\}$$
where $U$ is a unitary unimodular matrix of the following form:
$$U=\begin{bmatrix}u_1+iu_2&u_3+iu_4\\-u_3+iu_4&u_1-iu_2\end{bmatrix}...
- 113
9
votes
1
answer
542
views
Behavior of a Baker-Campbell-Hausdorff problem at infinity
The Baker-Campbell-Hausdorff problem is to obtain $\log(e^{X}e^{Y})$ where $X,Y$ are appropriate operators. The Dynkin series $$\log(e^{tX}e^{tY})=t(X+Y)+\frac{t^2}{2}[X,Y]+o(t^3)$$ gives an expansion ...
- 319
2
votes
0
answers
458
views
Random variable matrix exponential
I am trying to find out the distribution of a matrix exponential which is a function of a random variable. My mathematics background is very limited and I hope I can receive some help from here.
What ...
- 21