All Questions
12 questions
6
votes
2
answers
979
views
Literature on behaviour of eigenfunctions under multiplication?
Dear community,
I would be happy about any literature or comments on the behaviour of the pointwise product of eigenfunctions of a self-adjoint operator with discrete spectrum, acting on a separable ...
- 199
5
votes
2
answers
491
views
Is independence meaningful for commutative $C^*$-algebras?
I don't know very much about spectral theory so probably the answer to my question has a basic reference which I would appreciate.
Let's say I have two self-adjoint operators on a Hilbert space and ...
- 2,243
5
votes
0
answers
242
views
Spectral gap for the Brownian motion with drift on a compact manifold
Let $M$ be a compact Riemannian manifold without boundary, $X$ a smooth vector field on $M$. Consider the Brownian motion $t\mapsto B_t$ on $M$ with drift $X$, so that its generator is $L=\Delta +X$. ...
- 3,669
4
votes
0
answers
289
views
Solving a Fredholm equation with a piecewise kernel : Karhunen-Loeve of a stopped Brownian motion
Is there a way to solve analytically the Fredholm integral equation of the second kind
$$
\int_0^{100} K(s, t) f(s) ds = \lambda f(t)
$$
where the kernel has the piecewise 'linear' form
\begin{align}
...
user24451
3
votes
1
answer
498
views
Spectral Radius and Spectral Norm for Markov Operators
My question concerns differences between the spectral radius $\rho$ and norm $\| \cdot \|$ of Markov operators in infinite-dimensional Banach spaces. This is far from my area of expertise, that is ...
- 560
3
votes
0
answers
461
views
Analytic formula for the eigenvalues of kernel integral operator induced by Laplace kernel $K(x,x') = e^{-c\|x-x'\|}$ on unit-sphere in $\mathbb R^d$
Let $d \ge 2$ be an integer and let $X=\mathcal S_{d-1}$ the unit-sphere in $\mathbb R^d$. Let $\tau_d$ be the uniform distribution on $X$. Define a function $K:X \times X \to \mathbb R$ by $K(x,y) := ...
- 6,853
2
votes
2
answers
242
views
iid random operator and its spectrum
consider an insteresting question:
given Banach Space $ \mathcal{B}$, independent identical distribution random operator on $ \mathcal{B}$: $ (T_i)_{i \ge 1} $, where operator space is endowed with ...
- 553
2
votes
0
answers
57
views
Right spectral gap of vector of two independent Markov chains
Let $(X_i)$ be a stationary Markov chain on $S$ (a potentially uncountable space with a Borel sigma algebra) with stationary distribution $\pi$ and transition kernel $P$. Let $(Y_i)$ be a stationary ...
- 203
2
votes
0
answers
238
views
Examples for Markov generators with pure point spectrum
I'm looking at symmetric diffusion Markov generators $L$ with pure point spectrum, i.e. infinitesimal generators of symmetric diffusion Markov semigroups, which are defined on $L^2(\mu)$ where $\mu$ ...
- 235
1
vote
1
answer
173
views
Spectral gap of a Markov chain on the nonnegative integers
Let $\lambda_k,\mu_k\in\mathbb R_{\ge0}$ $(k\ge1)$ be nonnegative real numbers such that $\sum_{k=1}^\infty k\lambda_k<\infty,$ let $S=\mathbb Z_{\ge0}$ be the nonnegative integers, let $T=\mathbb ...
- 276
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 ...
- 167
0
votes
1
answer
622
views
Is there any way to compare between diagonals of a resolvent and a Cauchy transform?
Say $A$ is a symmetric matrix of $n$ dimensions. Then let the ``resolvent" of $A$ be the matrix valued function $R_A(z) = \frac{1}{z-A}$ and its Cauchy transform be the real valued function $C_A(z) = ...
- 617