Decompositions of spaces of measures

Let M be the Banach space of sigma-additive complex-valued measures on an abstract measurable space. We prove that any closed, with respect to absolute continuity norm-closed, linear subspace L of M is complemented and describe the unique complement, projection onto L along which has norm 1. Using this fact we prove a decomposition theorem, which includes the Jordan decomposition theorem, the generalized Radon-Nikodym theorem and the decomposition of measures into decaying and non-decaying components as particular cases. We also prove an analog of the Jessen-Wintner purity theorem for our decompositions.

Isometric embedding of finite ultrametric spaces in Banach spaces

We prove that for any finite ultrametric space M and any infinite-dimensional Banach space B there exists an isometric embedding of M into B.

On Osgood theorem in Banach spaces

Let $X$ be a real Banach space, $\omega:[0,+\infty)\to\R$ be an increasing continuous function such that $\omega(0)=0$ and $\omega(t+s)\leq\omega(t)+\omega(s)$ for all $t,s\in[0,+\infty)$. By the Osgood theorem, if $\int_{0}^1\frac{dt}{\omega(t)}=\infty$, then for any $(t_0,x_0)\in R\times X$ and any continuous map $f: R\times X\to X$ and such that $\|f(t,x)-f(t,y)\|\leq\omega(\|x-y\|)$ for all $t\in R$, $x,y\in X$, the Cauchy problem $\dot x(t)=f(t,x(t))$, $(t_0)=x_0$ has a unique solution in a neighborhood of $t_0$ . We prove that if $X$ has a complemented subspace with an unconditional Schauder basis and $\int_{0}^1\frac{dt}{\omega(t)}

Analyticity of smooth eigenfunctions and spectral analysis of the Gauss map

We provide a sufficient condition of analyticity of infinitely differentiable eigenfunctions of operators of the form Uf(x) = integral a(x, y) f(b( x, y)) mu(dy) acting on functions f: [u, v] --> C ( evolution operators of one-dimensional dynamical systems and Markov processes have this form). We estimate from below the region of analyticity of the eigenfunctions and apply these results for studying the spectral properties of the Frobenius-Perron operator of the continuous fraction Gauss map. We prove that any infinitely differentiable eigenfunction f of this Frobenius-Perron operator, corresponding to a non-zero eigenvalue admits a (unique) analytic extension to the set C\(-infinity, 1]. Analyzing the spectrum of the Frobenius Perron operator in spaces of smooth functions, we extend significantly the domain of validity of the Mayer and Ropstorff asymptotic formula for the decay of correlations of the Gauss map.

Decay measures on locally compact abelian topological groups

Source: PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS Volume: 131 Pages: 1257-1273 Part: Part 6 Published: 2001 Times Cited: 5 References: 23 Citation MapCitation Map beta Abstract: We show that the Banach space M of regular sigma-additive finite Borel complex-valued measures on a non-discrete locally compact Hausdorff topological Abelian group is the direct sum of two linear closed subspaces M-D and M-ND, where M-D is the set of measures mu is an element of M whose Fourier transform vanishes at infinity and M-ND is the set of measures mu is an element of M such that nu is not an element of MD for any nu is an element of M \ {0} absolutely continuous with respect to the variation \mu\. For any corresponding decomposition mu = mu(D) + mu(ND) (mu(D) is an element of M-D and mu(ND) is an element of M-ND) there exist a Borel set A = A(mu) such that mu(D) is the restriction of mu to A, therefore the measures mu(D) and mu(ND) are singular with respect to each other. The measures mu(D) and mu(ND) are real if mu is real and positive if mu is positive. In the case of singular continuous measures we have a refinement of Jordan's decomposition theorem. We provide series of examples of different behaviour of convolutions of measures from M-D and M-ND.

Structure of spectra of linear operators in Banach spaces

Descriptive characterizations of the point, the continuous, and the residual spectra of operators in Banach spaces are put forward. In particular, necessary and sufficient conditions for three disjoint subsets of the complex plane to be the point spectrum, the continuous spectrum, and the residual spectrum of a linear continuous operator in a separable Banach space are obtained.

Extended spectral decompositions of the Renyi map

We present new general methods to obtain spectral decompositions of dynamical systems in rigged Hilbert spaces and investigate the existence of resonances and the completeness of the associated eigenfunctions. The results are illustrated explicitly for the simplest chaotic endomorphism, namely the Renyi map.