Infinite Dimensional Separable Banach Space Research Articles

Stability analysis in N-dimensional gravitational collapse with an equation of state

We study stability of occurrence of black holes and naked singularities that arise as a final state for a complete gravitational collapse of type I matter field in a spherically symmetric $N$ dimensional spacetime with equation of state $p = k \rho$, $0 \leq k \leq 1$. We prove that for a regular initial data comprising of pressure (or density) profiles at an initial surface $t = t_{i}$, from which the collapse evolves, there exists a large class of the velocity functions and classes of solutions of Einstein equations, such that the spacetime evolution goes to a final state which is either a black hole or a naked singularity. We further prove that in an infinite dimensional separable Banach space, the set of regular initial data leading the collapse to a black hole or a naked singularity, forms an open subset of the set of all regular initial data. In this sense, the gravitational collapse leading to either a black hole or a naked singularity is stable. These results are discussed and analyzed in the light of the cosmic censorship hypothesis in black hole physics.

Operators similar to their restrictions to invariant subspaces

An operator T on a separable infinite dimensional Banach space X is called crystal-like if for each nonzero invariant subspace M of T, there exists a positive number r such that T|M, the restriction of T to M, is similar to rT. In this paper, we investigate some elementary properties of crystal-like operators, such as the spectrum, cyclicity, strong irreducibility, essential similarity, reflectivity and the commutativity of the commutant.

On the Bishop-Phelps-Bollobás Property for Numerical Radius

We study the Bishop-Phelps-Bollobás property for numerical radius (in short, BPBp-nu) and find sufficient conditions for Banach spaces to ensure the BPBp-nu. Among other results, we show thatL1μ-spaces have this property for every measureμ. On the other hand, we show that every infinite-dimensional separable Banach space can be renormed to fail the BPBp-nu. In particular, this shows that the Radon-Nikodým property (even reflexivity) is not enough to get BPBp-nu.

On the Existence of Polynomials with Chaotic Behaviour

We establish a general result on the existence of hypercyclic (resp., transitive, weakly mixing, mixing, frequently hypercyclic) polynomials on locally convex spaces. As a consequence we prove that every (real or complex) infinite-dimensional separable Frèchet space admits mixing (hence hypercyclic) polynomials of arbitrary positive degree. Moreover, every complex infinite-dimensional separable Banach space with an unconditional Schauder decomposition and every complex Frèchet space with an unconditional basis support chaotic and frequently hypercyclic polynomials of arbitrary positive degree. We also study distributional chaos for polynomials and show that every infinite-dimensional separable Banach space supports polynomials of arbitrary positive degree that have a dense distributionally scrambled linear manifold.

Hypercyclic operators on countably dimensional spaces

According to Grivaux, the group GL(X) of invertible linear operators on a separable infinite dimensional Banach space X acts transitively on the set Σ(X) of countable dense linearly independent subsets of X. As a consequence, each A∈Σ(X) is an orbit of a hypercyclic operator on X. Furthermore, every countably dimensional normed space supports a hypercyclic operator. Recently Albanese extended this result to Fréchet spaces supporting a continuous norm.We show that for a separable infinite dimensional Fréchet space X, GL(X) acts transitively on Σ(X) if and only if X possesses a continuous norm. We also prove that every countably dimensional metrizable locally convex space supports a hypercyclic operator.

On the Borelness of the intersection operation

We study the intersection operation of closed linear subspaces in a separable Banach space. We show that if the ambient space is quasi-reflexive, then the intersection operation is Borel. On the other hand, if the space contains a closed subspace with a Schauder decomposition into infinitely many non-reflexive spaces, then the intersection operation is not Borel. As a corollary, for a closed subspace of a Banach space with an unconditional basis, the intersection operation of the closed linear subspaces is Borel if and only if the space is reflexive. We also consider the intersection operation of additive subgroups in an infinite-dimensional separable Banach space, and show that if this intersection operation is Borel then the space is hereditarily indecomposable.

The metric entropy of random dynamical systems in a Banach space: Ruelle inequality

AbstractConsider a random cocycle Φ on a separable infinite-dimensional Banach space preserving a probability measure, which is supported on a random compact set. We show that if Φ satisfies some mild integrability conditions on the differentials, then Ruelle’s inequality relating entropy and positive Lyapunov exponents holds.

Frequent hypercyclicity, chaos, and unconditional Schauder decompositions

We prove that if X is any complex separable infinite-dimensional Banach space with an unconditional Schauder decomposition, X supports an operator T which is chaotic and frequently hypercyclic. In contrast with the complex case, we observe that there are real Banach spaces with an unconditional basis which support no chaotic operator.

Regression operator estimation by delta-sequences method for functional data and its applications

In this paper, we introduce a somewhat more general class of nonparametric estimators (delta-sequences estimators) for estimating an unknown regression operator from noisy data. The regressor is assumed to take values in an infinite-dimensional separable Banach space, when the response variable is a scalar. Under some general conditions, we establish the uniform almost-complete convergence with the rates of these estimators. Moreover, we give some particular cases of our results, which can also be considered as novel in the finite-dimensional setting. Moreover, after giving some examples of the impact of our results, we show how to use them in some statistical applications (prediction procedure and curve discrimination).

Linear operators with wild dynamics

If X X is a separable infinite-dimensional Banach space, we construct a bounded and linear operator R R on X X such that \[ A R = { x ∈ X , ‖ R t x ‖ → ∞ } A_R=\{x \in X, \|R^tx\| \rightarrow \infty \} \] is not dense and has a non-empty interior with the additional property that R R can be written I + K I+K , where I I is the identity and K K is a compact operator. This answers two recent questions of Hájek and Smith.

A new class of frequently hypercyclic operators

We study a hypercyclicity property of linear dynamical systems: a bounded linear operator T acting on a separable infinite-dimensional Banach space X is said to be hypercyclic if there exists a vector x in X such that {T^{n}x : n>0} is dense in X, and frequently hypercyclic if there exists x in X such that for any non empty open subset U of X, the set {n>0 ; T^n x \in U} has positive lower density. We prove that if T is a bounded operator on X which has sufficiently many eigenvectors associated to eigenvalues of modulus 1 in the sense that these eigenvectors are perfectly spanning, then T is automatically frequently hypercyclic.

Nonparametric density estimation for functional data by delta sequences

We consider the problem of estimation of density function by the method of delta sequences for functional data with values in an infinite dimensional separable Banach space.

On Peano's theorem in Banach spaces

We show that if X is an infinite-dimensional separable Banach space (or more generally a Banach space with an infinite-dimensional separable quotient) then there is a continuous mapping f : X → X such that the autonomous differential equation x ′ = f ( x ) has no solution at any point.

Li–Yorke and distributionally chaotic operators

We study Li–Yorke chaos and distributional chaos for operators on Banach spaces. More precisely, we characterize Li–Yorke chaos in terms of the existence of irregular vectors. Sufficient “computable” criteria for distributional and Li–Yorke chaos are given, together with the existence of dense scrambled sets under some additional conditions. We also obtain certain spectral properties. Finally, we show that every infinite dimensional separable Banach space admits a distributionally chaotic operator which is also hypercyclic.

Local automorphisms of standard operator algebras

Let X be an infinite-dimensional separable real or complex Banach space and A a closed standard operator algebra on X. Then every local automorphism of A is an automorphism. The assumptions of infinite-dimensionality, separability, and closeness are all indispensable.

Epsilon-hypercyclic operators

AbstractLet X be a separable infinite-dimensional Banach space, and T a bounded linear operator on X; T is hypercyclic if there is a vector x in X with dense orbit under the action of T. For a fixed ε∈(0,1), we say that T is ε-hypercyclic if there exists a vector x in X such that for every non-zero vector y∈X there exists an integer n with $\|T^nx-y\|\leq \varepsilon \|y\|$. The main result of this paper is a construction of a bounded linear operator T on the Banach space ℓ1 which is ε-hypercyclic without being hypercyclic. This answers a question from V. Müller [Three problems, Mini-Workshop: Hypercyclicity and linear chaos, organized by T. Bermudez, G. Godefroy, K.-G. Grosse-Erdmann and A. Peris. Oberwolfach Rep.3 (2006), 2227–2276].

On the spectrum of frequently hypercyclic operators

A bounded linear operator T on a Banach space X is called fre quently hypercyclic if there exists x E X such that the lower density of the set {n E N: T'x E U} is positive for any non-empty open subset U of X. Bayart and Grivaux have raised a question whether there is a frequently hypercyclic operator on any separable infinite dimensional Banach space. We prove that the spectrum of a frequently hypercyclic operator has no isolated points. It follows that there are no frequently hypercyclic operators on all complex and on some real hereditarily indecomposable Banach spaces, which provides a negative answer to the above question.

The Kitai criterion and backward shifts

It is proved that for any separable infinite dimensional Banach space $X$, there is a bounded linear operator $T$ on $X$ such that $T$ satisfies the Kitai criterion. The proof is based on a quasisimilarity argument and on showing that $I+T$ satisfies the Kitai criterion for certain backward weighted shifts $T$.

Coefficient Quantization in Banach Spaces

Let (e i ) be a dictionary for a separable infinite-dimensional Banach space X. We consider the problem of approximation by linear combinations of dictionary elements with quantized coefficients drawn usually from a ‘finite alphabet’. We investigate several approximation properties of this type and connect them to the Banach space geometry of X. The existence of a total minimal system with one of these properties, namely the coefficient quantization property, is shown to be equivalent to X containing c 0. We also show that, for every e>0, the unit ball of every separable infinite-dimensional Banach space X contains a dictionary (x i ) such that the additive group generated by (x i ) is (3+e)−1-separated and 1/3-dense in X.

Existence and nonexistence of hypercyclic semigroups

In these notes we provide a new proof of the existence of a hypercyclic uniformly continuous semigroup of operators on any separable infinite-dimensional Banach space that is very different from-and considerably shorter than-the one recently given by Bermudez, Bonilla and Martinon. We also show the existence of a strongly dense family of topologically mixing operators on every separable infinite-dimensional Frechet space. This complements recent results due to Bes and Chan. Moreover, we discuss the Hypercyclicity Criterion for semigroups and we give an example of a separable infinite-dimensional locally convex space which supports no supercyclic strongly continuous semigroup of operators.