All Questions
14 questions
1
vote
0
answers
135
views
Description of state space of $C(K,M_n)$?
Edit: closed convex hull added.
I am trying to understand the state space of $C(K,M_n)=C(K)\otimes M_n$ for $K$ a compact space.
My guess would be that these are the closed convex hull of states on $C(...
- 598
9
votes
1
answer
359
views
Relaxation of notion of positive definite function
A function $f:\mathbb{R}\to\mathbb{R}$ is called positive definite (in the semigroup sense) if for all $n\geq 1$ and $x_1,\ldots,x_n\in\mathbb{R}$ pairwise different the matrix $(f(x_i+x_j))_{i,j=1}^n$...
- 128k
2
votes
0
answers
81
views
Convergence of random operators
I'm a statistician not versed in functional analysis and operator theory. I wish that I might not find a wrong place for my question. All my questions are trivial in the scalar time series case, but ...
- 121
3
votes
0
answers
201
views
Eigenvalue estimates for kernel integral operator for Laplace kernel on unit-sphere in high-dimensions
Let $d$ be a large positive integer and let $S_{d-1}$ be the unit-sphere in $\mathbb R^d$ and let $K_\gamma:S_{d-1} \times S_{d-1} \to \mathbb R$ be defined by $K_\gamma(x,x') = e^{-\|x-x'\|_2^\gamma}$...
- 6,853
2
votes
1
answer
264
views
Bounded-pointwise continuity of Markov operators / semigroups
Let $B_b(E)$ be the space of bounded measurable functions on some Polish space $E$ endowed with the supremum norm. It seems quite classical that Markov semigroups $P_t:B_b(E)\to B_b(E)$ are in one to ...
- 63.9k
0
votes
2
answers
244
views
Spectrum of a Markov kernel acting on $L^2$
Let $P$ be a Markov kernel on a measurable space $(E,\mathcal E)$ admitting an invariant probability measure $\pi$. $P$ acts on $L^2(\pi)$ via $$Pf:=\int\kappa(\;\cdot\;{\rm d}y)f(y).$$ The invariance ...
- 17k
-1
votes
1
answer
114
views
Construct a probability function on the operator monotone functions, $g(t)=t g(t^{-1})$, fitting certain values
To immediately pose the question of interest to us, without first expanding upon its (quantum-information-theoretic) origin—we seek a univariate function $f$, for which we have the ("two-qubit ...
- 723
8
votes
2
answers
640
views
Does a random sequence of vectors span a Hilbert space?
Let $\mathcal{H}$ be a separable Hilbert space. Let $v$ be a random variable taking values in $\mathcal{H}$ such that $P(v \perp h) < 1$ for all $h \in \mathcal{H}.$ Suppose we sample an infinite ...
- 17.2k
1
vote
0
answers
127
views
Gradient bound for the Markov semigroup generated by the solution to an Langevin SDE
Let
$h\in C^2(\mathbb R)$ with $$h''\ge\rho\tag1$$ for some $\rho>0$ and $$\int\underbrace{e^{-h}}_{=:\:\varrho}\:{\rm d}\lambda=1$$
$\mu$ be the measure with density $\varrho$ with respect to the ...
- 167
7
votes
4
answers
946
views
On operator ranges in Hilbert & Banach spaces
Lemma 1 from Anderson & Trapp's Shorted Operators, II isLet $A$ and $B$ be bounded operators on the Hilbert space $\mathcal H$. The following statements are equivalent:
(1) ran($A$) $\subset$ ...
- 5,407
3
votes
1
answer
282
views
Are injective Hilbert Schmidt operators (measure theoretically) generic?
It's well known that when the elements of an $n \times n$ matrix $A$ are chosen independently from e.g. $U[0,1]$ distributions, then with probability $1$ the matrix $A$ will be injective (indeed, ...
- 29.8k
17
votes
2
answers
953
views
Convexity of spectral radius of Markov operators, Random walks on non-amenable groups
Let $P_1,P_2$ denote stochastic transition matrices on a countable set $I$.
Consider $P_1,P_2$ as operators on $\ell^2(I)$ given by multiplication.
Question
Under which conditions can we show that ...
- 4,361
1
vote
0
answers
87
views
Linear evolution equation $u'(t)=A(t,\omega)u(t)$ with time-dependent random operator
I have had some previous knowledge on evolution equations in a Banach space of the form $$u'(t)=Au(t),$$ where $A$ generates some strongly continuous operator semigroup. Now I am looking at a problem ...
- 151
4
votes
0
answers
1k
views
The spectrum of a Markov Operator and Invariant Measures
Suppose I have a discrete-time Markov Chain (in an infinite dimensional state space $\Omega$) with Markov operator $P$, a linear operator on the space of bounded measurable functions on $\Omega$. (Or ...
- 509