Let $F:X\to X$ be a $C^2_\loc$ map in a Banach space $X$, and $A$ be its Frèchet derivative at the element $w:=w_\ve$, which solves the problem $(\ast) \dotw=-A^{-1}_\ve(F(w)+\ve w)$, $w(0)=w_0$, where $A_\ve:=A+\ve I$. Assume that $\|A^{-1}_\ve\|\leq c \ve^{-k}$, $0<k\leq 1$, $0<\ve>\ve_0$. Then $(\ast)$ has a unique global solution, $w(t)$, there exists $w(\infty)$, and $(\ast\ast) F(w(\infty))+\ve w(\infty)=0$. Thus the DSM (Dynamical Systems Method) is justified for equation $(\ast\ast)$. The limit of $w_\ve$ as $\ve\to 0$ is studied.
Let $L$ be an unbounded linear operator in a real Hilbert space $H$, a generator of $C_0$ semigroup, and $g:H\to H$ be a $C^2_{loc}$ nonlinear map. The DSM (dynamical systems method) for solving equ$ $F(v):=Lv+gv=0$ consists of solving the Cauchy problem $\dot {u}=Φ(t,u)$, $u(0)=u_0$, where $Φ$ is a suitable operator, and proving that i) $\exists u(t) \quad \forall t>0$, ii) $\exists u(\infty)$, and iii) $F(u(\infty))=0$.
Consider an operator equation (*) $B(u)+\ep u=0$ in a real Hilbert space, where $\ep>0$ is a small constant. The DSM (dynamical systems method) for solving equation (*) consists of a construction of a Cauchy problem, which has the following properties: 1) it has a global solution for an arbitrary initial data, 2) this solution tends to a limit as time tends to infinity, 3) the limit solves the equation $B(u)=0$. Existence of the unique solution is proved by the DSM for equation (*) with monotone hemicontinuous operators $B$ defined on all of$ If $\ep=0$ and equation (**) $B(u)=0$ is solvable, …
Consider an operator equation (*) $B(u)-f=0$ in a real Hilbert space. Let us call this equation ill-posed if the operator $B'(u)$ is not boundedly invertible, and well-posed otherwise. The DSM (dynamical systems method) for solving equation (*) consists of a construction of a Cauchy problem, which has the following properties: 1) it has a global solution for an arbitrary initial data, 2) this solution tends to a limit as time tends to infinity, 3) the limit is the minimal-norm solution to the equation $B(u)=f$. A global convergence theorem is proved for DSM for equation (*) with monotone $C_{l…
The representation theorems of Gelfand and Kakutani for commutative C*-algebras and AM- and AL-spaces are the basis for the Koopman linearization of topological and measure-preserving dynamical systems. In this article we prove versions of these results for dynamics on topological and measurable Banach bundles and the corresponding weighted Koopman representations on Banach modules.
For a topological group G, we show that a compact metric G-space is tame if and only if it can be linearly represented on a separable Banach space which does not contain an isomorphic copy of $l_1$ (we call such Banach spaces, Rosenthal spaces). With this goal in mind we study tame dynamical systems and their representations on Banach spaces.
Let $F$ be a nonlinear map in a real Hilbert space $H$. Suppose that $\sup_{u\in B(u_0,R)}$ $\|[F'(u)]^{-1}\|\leq m(R)$, where $B(u_0,R)=\{u:\|u-u_0\|\leq R\}$, $R>0$ is arbitrary, $u_0\in H$ is an element. If $\sup_{R>0}\frac{R}{m(R)}=\infty$, then $F$ is surjective. If $\|[F'(u)]^{-1}\|\leq a\|u\|+b$, $a\geq 0$ and $b>0$ are constants independent of $u$, then $F$ is a homeomorphism of $H$ onto $H$. The last result is known as an Hadamard-type theorem, but we give a new simple proof of it based on the DSM (dynamical systems method).
We prove that the crossed product Banach algebra $\ell^1(G,A;α)$ that is associated with a ${\mathrm C}^\ast$-dynamical system $(A,G,α)$ is amenable if $G$ is a discrete amenable group and $A$ is a strongly amenable ${\mathrm C}^\ast$-algebra. This is a consequence of the combination of a more general result with Paterson's characterisation of strongly amenable unital $\mathrm{C}^\ast$-algebras in terms of invariant means for their unitary groups.
If we change the upper and lower density in the definition of distributional chaos of a continuous linear operator on Banach space by the Banach upper and Banach lower density, respectively, we obtain Li-Yorke chaos. Motivated by this fact, we introduce the notions of reiterative distributional chaos of types $1$, $1^+$ and $2$ for continuous linear operators on Banach spaces, which are characterized in terms of the existence of an irregular vector with additional properties. Moreover, we study its relations with other dynamical properties and give conditions for the existence of a vector subs…
We construct a crossed product Banach algebra from a Banach algebra dynamical system $(A,G,α)$ and a given uniformly bounded class $R$ of continuous covariant Banach space representations of that system. If $A$ has a bounded left approximate identity, and $R$ consists of non-degenerate continuous covariant representations only, then the non-degenerate bounded representations of the crossed product are in bijection with the non-degenerate $R$-continuous covariant representations of the system. This bijection, which is the main result of the paper, is also established for involutive Banach algeb…
We introduce a version of Voiculescu-Brown approximation entropy for isometric automorphisms of Banach spaces and develop within this framework the connection between dynamics and the local theory of Banach spaces discovered by Glasner and Weiss. Our fundamental result concerning this contractive approximation entropy, or CA entropy, characterizes the occurrence of positive values both geometrically and topologically. This leads to various applications; for example, we obtain a geometric description of the topological Pinsker factor and show that a C*-algebra is type I if and only if every mul…
Contraction rates of time-varying maps induced by dynamical systems illuminate a wide range of asymptotic properties with applications in stability analysis and control theory. In finite-dimensional smoothly varying inner-product spaces such as $\mathbb{R}^n$ and $\mathbb{C}^n$ with Riemannian metrics, contraction rates can be estimated by upper-bounding the real numerical range of the vector field's Jacobian. However, vector spaces with norms other than $L^2$ commonly arise in the stability analysis of infinite-dimensional systems such as those arising from partial differential equations and …
To every Banach space V we associate a compact right topological affine semigroup E(V). We show that a separable Banach space V is Asplund if and only if E(V) is metrizable, and it is Rosenthal (i.e. it does not contain an isomorphic copy of $l_1$) if and only if E(V) is a Rosenthal compactum. We study representations of compact right topological semigroups in E(V). In particular, representations of tame and HNS-semigroups arise naturally as enveloping semigroups of tame and HNS (hereditarily non-sensitive) dynamical systems, respectively. As an application we obtain a generalization of a theo…
We prove that, under certain conditions, uniform weak mixing (to zero) of the bounded sequences in Banach space implies uniform weak mixing of its tensor product. Moreover, we prove that ergodicity of tensor product of the sequences in Banach space implies its weak mixing. Applications of the obtained results, we prove that tensor product of uniquely $E$-weak mixing $C^*$-dynamical systems is also uniquely $E$-weak mixing as well.
If $X$ is a compact Hausdorff space and $σ$ is a homeomorphism of $X$, then a Banach algebra $\ell^1(Σ)$ of crossed product type is naturally associated with this topological dynamical system $Σ=(X,σ)$. If $X$ consists of one point, then $\ell^1(Σ)$ is the group algebra of the integers. We study the algebraically irreducible representations of $\ell^1(Σ)$ on complex vector spaces, its primitive ideals and its structure space. The finite dimensional algebraically irreducible representations are determined up to algebraic equivalence, and a sufficiently rich family of infinite dimensional algebr…
A well-known result in the area of dynamical systems asserts that any invertible hyperbolic operator on any Banach space is structurally stable. This result was originally obtained by P. Hartman in 1960 for operators on finite-dimensional spaces. The general case was independently obtained by J. Palis and C. Pugh around 1968. We will exhibit examples of structurally stable operators that are not hyperbolic, thereby showing that the converse of the above-mentioned result is false in general. We will also prove that an invertible operator on a Banach space is hyperbolic if and only if it is expa…
Motivated from two decades old famous Feichtinger conjectures for frames, $R_\varepsilon$-conjecture and Weaver's conjecture for Hilbert spaces (and their solution by Marcus, Spielman, and Srivastava), we formulate Feichtinger conjectures for p-approximate Schauder frames, $R_\varepsilon$-conjecture, Weaver's conjectures and Akemann-Weaver conjectures for Banach spaces. We also formulate conjectures on p-approximate Schauder frames based on the results of Casazza for frames for Hilbert spaces. We state conjectures and problems for p-approximate Schauder frames based on fundamental inequality f…
In analogy with the Fourier-Stieltjes algebra of a group, we associate to a unital discrete twisted C*-dynamical system a Banach algebra whose elements are coefficients of equivariants representations of the system. Building upon our previous work, we show that this Fourier-Stieltjes algebra embeds continuously in the Banach algebra of completely bounded multipliers of the (reduced or full) C*-crossed product of the system. We also introduce a notion of positive definiteness and prove a Gelfand-Raikov type theorem allowing us to describe the Fourier-Stieltjes algebra of a system in a more intr…
We study different pointwise recurrence notions for linear dynamical systems from the Ergodic Theory point of view. We show that from any reiteratively recurrent vector $x_0$, for an adjoint operator $T$ on a separable dual Banach space $X$, one can construct a $T$-invariant probability measure which contains $x_0$ in its support. This allows us to establish some equivalences, for these operators, between some strong pointwise recurrence notions which in general are completely distinguished. In particular, we show that (in our framework) reiterative recurrence coincides with frequent recurrenc…
Generalized hyperbolicity for diffeomorphisms of Banach spaces
Pontificia Universidade Catolica do Rio de Janeiro (PUC-Rio)（天主教里约热内卢 Pontifical University）；Institute of Mathematics, Statistics and Computer Science, University of São Paulo (IME-USP)（圣保罗大学数学、统计和计算机科学学院）
In this paper we develop two types of tools to deal with differentiability properties of vectors in continuous representations $π\: G \to \GL(V)$ of an infinite dimensional Lie group $G$ on a locally convex space $V$. The first class of results concerns the space $V^\infty$ of smooth vectors. If $G$ is a Banach--Lie group, we define a topology on the space $V^\infty$ of smooth vectors for which the action of $G$ on this space is smooth. If $V$ is a Banach space, then $V^\infty$ is a Fréchet space. This applies in particular to $C^*$-dynamical systems $(\cA,G, α)$, where $G$ is a Banach--Lie gr…
We study non-recurrence sets for weakly mixing dynamical systems by using linear dynamical systems. These are systems consisting of a bounded linear operator acting on a separable complex Banach space X, which becomes a probability space when endowed with a non-degenerate Gaussian measure. We generalize some recent results of Bergelson, del Junco, Lemańczyk and Rosenblatt, and show in particular that sets \{n_k\} such that n_{k+1}/{n_k} tends to infinity, or such that n_{k} divides n_{k+1} for each k, are non-recurrence sets for weakly mixing linear dynamical systems. We also give examples, fo…
It is well known that, given an equivariant and continuous (in a suitable sense) family of selfadjoint operators in a Hilbert space over a minimal dynamical system, the spectrum of all operators from that family coincides. As shown recently similar results also hold for suitable families of non-selfadjoint operators in $\ell^p (\ZM)$. Here, we generalize this to a large class of bounded linear operator families on Banach-space valued $\ell^p$-spaces over countable discrete groups. We also provide equality of the pseudospectra for operators in such a family. A main tool for our analysis are tec…
We give formulas for the spectral radius of weighted endomorphisms $aα: C(X,D)\to C(X,D)$, $a\in C(X,D)$, where $X$ is a compact Hausdorff space and $D$ is a unital Banach algebra. Under the assumption that $α$ generates a partial dynamical system $(X,φ)$, we establish two kinds of variational principles for $r(aα)$: using linear extensions of $(X,φ)$ and using Lyapunov exponents associated with ergodic measures for $(X,φ)$. This requires considering (twisted) cocycles over $(X,φ)$ with values in an arbitrary Banach algebra $D$, and thus our analysis can not be reduced to any of mutliplicative…
Motivated by Rosenthal's famous $l^1$-dichotomy in Banach spaces, Haydon's theorem, and additionally by recent works on tame dynamical systems, we introduce the class of tame locally convex spaces. This is a natural locally convex analogue of Rosenthal Banach spaces (for which any bounded sequence contains a weak Cauchy subsequence). Our approach is based on a bornology of tame subsets which in turn is closely related to eventual fragmentability. This leads, among others, to the following results: $\bullet$ extending Haydon's characterization of Rosenthal Banach spaces, by showing that a lcs $…
In this work we study certain invariant measures that can be associated to the time averaged observation of a broad class of dissipative semigroups via the notion of a generalized Banach limit. Consider an arbitrary complete separable metric space $X$ which is acted on by any continuous semigroup $\{S(t)\}_{t \geq 0}$. Suppose that $§(t)\}_{t \geq 0}$ possesses a global attractor $\mathcal{A}$. We show that, for any generalized Banach limit $\underset{T \rightarrow \infty}{\rm{LIM}}$ and any distribution of initial conditions $\mathfrak{m}_0$, that there exists an invariant probability measure…
For a nontrivial locally compact group $G$, and $p\in [1,\infty)$, consider the Banach algebras of $p$-pseudofunctions, $p$-pseudomeasures, $p$-convolvers, and the full group $L^p$-operator algebra. We show that these Banach algebras are operator algebras if and only if $p=2$. More generally, we show that for $q\in [1,\infty)$, these Banach algebras can be represented on an $L^q$-space if and only if one of the following holds: (a) $p=2$ and $G$ is abelian; or (b) $|\frac 1p - \frac 12|=|\frac 1q - \frac 12|$. This result can be interpreted as follows: for $p,q\in [1,\infty)$, the $L^p$- and $…
Suppose that $Q$ is a weak$^{\ast }$ compact convex subset of a dual Banach space with the Radon-Nikodým property. We show that if $(S,Q)$ is a nonexpansive and norm-distal dynamical system, then there is a fixed point of $S$ in $Q$ and the set of fixed points is a nonexpansive retract of $Q.$ As a consequence we obtain a nonlinear extension of the Bader-Gelander-Monod theorem concerning isometries in $L$-embedded Banach spaces. A similar statement is proved for weakly compact convex subsets of a locally convex space, thus giving the nonlinear counterpart of the Ryll-Nardzewski theorem.
A bounded operator on a real or complex separable infinite-dimensional Banach space $Z$ is universal in the sense of Glasner and Weiss if for every invertible ergodic measure-preserving transformation $T$ of a standard Lebesgue probability space $(X,\mathcal {B},μ)$, there exists an $A$-invariant probability measure $ν$ on $H$ with full support such that the two dynamical systems $(X,\mathcal {B},μ;T)$ and $(H,\mathcal {B}_{H},ν;A)$ are isomorphic. We present a general and simple criterion for an operator to be universal, which allows us to characterize universal operators among unilateral or …
An extension of Chernoff's product formula for one-parameter functions taking values in the space of bounded linear operators on a Banach space is given. Essentially, the $n$-th one-parameter function in the product formula is mapped by the $n$-th iterate of a contraction acting on the space of the one-parameter functions. The motivation to study this specific product formula lies in the growing field of dynamical control of quantum systems, involving the procedure of dynamical decoupling and also the Quantum Zeno effect.
