Separable Banach Space Research Articles

Trace regularization problem for a fourth‐order differential operator on separable Banach space

In this study, we employ new results on the semi‐inner product, the adjoint operator, and the Schatten class of operators on a separable Banach space, which allow us to investigate a differential operator with unbounded operator‐valued coefficients. Specifically, we derive an asymptotic formula for the second regularized trace based on the asymptotic and spectral properties of the extended operator. For this purpose, we use the theory of continuous dense embeddings and known results regarding the regularized trace in Hilbert space. We conclude our paper by providing examples that support our findings.

Trichotomy for the orbits of a hypercyclic operator on a Banach space

We obtain a trichotomy for the orbits of a hypercyclic operator T T on a separable Banach space X X : (1) every vector is mean asymptotic to zero; (2) generic vectors are absolutely mean irregular; (3) every hypercyclic vector is mean divergent to infinity. Examples of weighted backward shifts on ℓ p \ell ^p show that all three cases can happen.

Equivalence of Sobolev Norms with Respect to Weighted Gaussian Measures

AbstractWe consider the spaces $${\text {L}}^p(X,\nu ;V)$$ L p ( X , ν ; V ) , where X is a separable Banach space, $$\mu $$ μ is a centred non-degenerate Gaussian measure, $$\nu :=Ke^{-U}\mu $$ ν : = K e - U μ with normalizing factor K and V is a separable Hilbert space. In this paper we prove a vector-valued Poincaré inequality for functions $$F\in W^{1,p}(X,\nu ;V)$$ F ∈ W 1 , p ( X , ν ; V ) , which allows us to show that for every $$p\in (1,\infty )$$ p ∈ ( 1 , ∞ ) and every $$k\in \mathbb {N}$$ k ∈ N the norm in $$W^{k,p}(X,\nu )$$ W k , p ( X , ν ) is equivalent to the graph norm of $$D_H^{k}$$ D H k (the k-th Malliavin derivative) in $${\text {L}}^p(X,\nu )$$ L p ( X , ν ) . To conclude, we show exponential decay estimates for the V-valued perturbed Ornstein-Uhlenbeck semigroup $$(T^V(t))_{t\ge 0}$$ ( T V ( t ) ) t ≥ 0 , defined in Section 2.6, as t goes to infinity. Useful tools are the study of the asymptotic behaviour of the scalar perturbed Ornstein-Uhlenbeck $$(T(t))_{t\ge 0}$$ ( T ( t ) ) t ≥ 0 , and pointwise estimates for $$|D_HT(t)f|_H^p$$ | D H T ( t ) f | H p by means of both $$T(t)|D_Hf|^p_H$$ T ( t ) | D H f | H p and $$T(t)|f|^p$$ T ( t ) | f | p .

(Looking for) the heart of abelian Polish groups

We prove that the category M of abelian groups with a Polish cover introduced in collaboration with Bergfalk and Panagiotopoulos is the left heart of (the derived category of) the quasi-abelian category A of abelian Polish groups in the sense of Beilinson–Bernstein–Deligne and Schneiders. Thus, M is an abelian category containing A as a full subcategory such that the inclusion functor A→M is exact and finitely continuous. Furthermore, M is uniquely characterized up to equivalence by the following universal property: for every abelian category B, a functor A→B is exact and finitely continuous if and only if it extends to an exact and finitely continuous functor M→B. In particular, this provides a description of the left heart of A as a concrete category.We provide similar descriptions of the left heart of a number of categories of algebraic structures endowed with a topology, including: non-Archimedean abelian Polish groups; locally compact abelian Polish groups; totally disconnected locally compact abelian Polish groups; Polish R-modules, for a given Polish group or Polish ring R; and separable Banach spaces and separable Fréchet spaces over a separable complete non-Archimedean valued field.

On the relations between Auerbach or almost Auerbach Markushevich systems and Schauder bases

We establish that the summability of the series ∑εn is the necessary and sufficient criterion ensuring that every (1+εn)-bounded Markushevich basis in a separable Hilbert space is a Riesz basis. Further we show that if nεn→∞, then in ℓ2 there exists a (1+εn)-bounded Markushevich basis that under any permutation is non-equivalent to a Schauder basis. We extend this result to any separable Banach space. Finally we provide examples of Auerbach bases in 1-symmetric separable Banach spaces whose no permutations are equivalent to any Schauder basis or (depending on the space) any unconditional Schauder basis.

Long-term accuracy of numerical approximations of SPDEs with the stochastic Navier–Stokes equations as a paradigm

Abstract This work introduces a general framework for establishing the long time accuracy for approximations of Markovian dynamical systems on separable Banach spaces. Our results illuminate the role that a certain uniformity in Wasserstein contraction rates for the approximating dynamics bears on long time accuracy estimates. In particular, our approach yields weak consistency bounds on ${\mathbb{R}}^{+}$ while providing a means to sidestepping a commonly occurring situation where certain higher order moment bounds are unavailable for the approximating dynamics. Additionally, to facilitate the analytical core of our approach, we develop a refinement of certain ‘weak Harris theorems’. This extension expands the scope of applicability of such Wasserstein contraction estimates to a variety of interesting stochastic partial differential equation examples involving weaker dissipation or stronger nonlinearity than would be covered by the existing literature. As a guiding and paradigmatic example, we apply our formalism to the stochastic 2D Navier–Stokes equations and to a semi-implicit in time and spectral Galerkin in space numerical approximation of this system. In the case of a numerical approximation, we establish quantitative estimates on the approximation of invariant measures as well as prove weak consistency on ${\mathbb{R}}^{+}$. To develop these numerical analysis results, we provide a refinement of $L^{2}_{x}$ accuracy bounds in comparison to the existing literature, which are results of independent interest.

Existence and multiplicity of solutions for fractional p-Laplacian equation involving critical concave-convex nonlinearities

Abstract We investigate the following fractional p-Laplacian convex-concave problem: ( P λ ) ( − Δ ) p s u = λ | u | q − 2 u + | u | p s * − 2 u in Ω , u = 0 in R n \ Ω , $$\left({P}_{\lambda }\right) \begin{cases}\begin{aligned}\hfill {\left(-{\Delta}\right)}_{p}^{s}u& =\lambda \vert u{\vert }^{q-2}u+\vert u{\vert }^{{p}_{s}^{{\ast}}-2}u\hfill & \hfill & \quad \text{in} {\Omega},\hfill \\ \hfill u& =0 \hfill & \hfill & \quad \text{in} {\mathbb{R}}^{n}{\backslash}{\Omega},\hfill \end{aligned}\quad \hfill \end{cases}$$ where Ω is a bounded C 1,1 domain in R n ${\mathbb{R}}^{n}$ , s ∈ (0, 1), p > q > 1, n > sp, λ > 0, and p s * = n p n − s p ${p}_{s}^{{\ast}}=\frac{np}{n-sp}$ is the critical Sobolev exponent. Our analysis extends classical works (A. Ambrosetti, H. Brezis, and G. Cerami, “Combined effects of concave and convex nonlinearities in some elliptic problems,” J. Funct. Anal., vol. 122, no. 2, pp. 519–543, 1994, B. Barrios, E. Colorado, R. Servadei, and F. Soria, “A critical fractional equation with concave-convex power nonlinearities,” Ann. Inst. Henri Poincare Anal. Non Lineaire, vol. 32, no. 4, pp. 875–900, 2015, J. García Azorero, J. Manfredi, and I. Peral Alonso, “Sobolev versus Hölder local minimizer and global multiplicity for some quasilinear elliptic equations,” Commun. Contemp. Math., vol. 2, no. 3, pp. 385–404, 2000) to fractional p-Laplacian. Owing to the nonlinear and nonlocal properties of ( − Δ ) p s ${\left(-{\Delta}\right)}_{p}^{s}$ , we need to overcome many difficulties and apply notably different approaches, due to the lack of Picone identity, the stability theory, and the strong comparison principle. We show first a dichotomy result: a positive W 0 s , p ( Ω ) ${W}_{0}^{s,p}\left({\Omega}\right)$ solution of (P λ ) exists if and only if λ ∈ (0, Λ] with an extremal value Λ ∈ (0, ∞). The W 0 s , p ( Ω ) ${W}_{0}^{s,p}\left({\Omega}\right)$ regularity for the extremal solution seems to be unknown regardless of whether s = 1 or s ∈ (0, 1). When p ≥ 2, p − 1 < q < p and n > s p ( q + 1 ) q + 1 − p $n{ >}\frac{sp\left(q+1\right)}{q+1-p}$ , we get two positive solutions for (P λ ) with small λ > 0. Here the mountain pass structure is more involved than the classical situations due to the lack of explicit minimizers for the Sobolev embedding, we should proceed carefully and simultaneously the construction of mountain pass geometry and the estimate for mountain pass level. Finally, we show another new result for (P λ ) and all p > q > 1: without sign constraint, (P λ ) possesses infinitely many solutions when λ > 0 is small enough. Here we use the Z 2 ${\mathbb{Z}}_{2}$ -genus theory, based on a space decomposition for reflexible and separable Banach spaces, which has its own interest.

Asymptotic results for compound sums in separable Banach spaces

We prove large and moderate deviation results for sequences of compound sums, where the summands are i.i.d. random variables taking values in a separable Banach space. We establish that the results hold by proving that we are dealing with exponentially tight sequences. We present two moderate deviation results: in the first one the summands are centered, in the second one the compound sums are centered.

Yet another proof of the density in energy of Lipschitz functions

We provide a new, short proof of the density in energy of Lipschitz functions into the metric Sobolev space defined by using plans with barycenter (and thus, a fortiori, into the Newtonian–Sobolev space). Our result covers first-order Sobolev spaces of exponent p∈(1,∞)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p\\in (1,\\infty )$$\\end{document}, defined over a complete separable metric space endowed with a boundedly-finite Borel measure. Our proof is based on a completely smooth analysis: first we reduce the problem to the Banach space setting, where we consider smooth functions instead of Lipschitz ones, then we rely on classical tools in convex analysis and on the superposition principle for normal 1-currents. Along the way, we obtain a new proof of the density in energy of smooth cylindrical functions in Sobolev spaces defined over a separable Banach space endowed with a finite Borel measure.

An analysis on time optimal feedback control for Caputo fractional neutral systems with delay

ABSTRACT This paper aims to provide a systematic approach to investigate the optimal feedback control systems governed by the fractional neutral evolution system with fixed delay in the sense of Caputo fractional derivatives in separable reflexive Banach spaces. Initially, we study the existence result of mild solutions for the system by applying fixed point theorem for a condensing map due to Martelli. The existence of feasible pairs is obtained by using the Filippov theorem and the Cesari property. Furthermore, the existence of optimal feedback control and time optimal control are studied for the corresponding control problem. Finally, an example is given to validate our findings.

Weak multi-sensitive compactness and mean sensitivity for linear operators

Let ( X , T ) be a linear dynamical system defined on a Banach space X and L T : K X ∗ → K X ∗ be defined by L T S = TS , where K X ∗ = span { 〈 ⋅ , y ∗ 〉 x : y ∗ ∈ X ∗ , x ∈ X } ¯ . This paper first obtains the equivalence of the (syndetically) weak multi-sensitive compactness between ( X , T ) and ( K X ∗ , L T ) . Then, it is shown that (1) there exists a non-mean-sensitive linear dynamical system ( X , T ) satisfying that ( K X ∗ , L T ) is mean sensitive; (2) There exists a non-topologically-transitive and weakly multi-sensitive compact system, which is not syndetically weakly multi-sensitive compact. Besides, it is proved that every weakly mixing linear dynamical system defined on a separable Banach space is weakly multi-sensitive compact.

On the continuity of the Walras correspondence in distributional economies with an infinite-dimensional commodity space

Distributional economies are defined by a probability distribution in the space of characteristics where the commodity space is an ordered separable Banach space. We characterize the continuity of the equilibrium correspondence and an associated stability concept which allows us to give a positive answer to an open question about the continuity of the Walras correspondence in infinite-dimensional spaces. As a byproduct, we study a stability concept where differentiability assumptions are not required, as is usual in the literature on regularity. Moreover, since distributional economies do not specify a space of agents, our setting encompasses several results in the literature on large economies.

Disjoint subspace-hypercyclic operators on separable Banach spaces

Disjoint subspace-hypercyclic operators on separable Banach spaces

Constructing the Oseledets decomposition with subspace growth estimates

The semi-invertible version of Oseledets’ multiplicative ergodic theorem providing a decomposition of the underlying state space of a random linear dynamical system into fast and slow spaces is deduced for a strongly measurable cocycle on a separable Banach space. This work is a much shorter means of obtaining this general version of the theorem, using measurable growth estimates on subspaces for linear operators combined with a modified version of Kingman’s subadditive ergodic theorem.

Recurrent Subspaces in Banach Spaces

Abstract We study the spaceability of the set of recurrent vectors $\text{Rec}(T)$ for an operator $T:X\longrightarrow X$ on a Banach space $X$. In particular, we find sufficient conditions for a quasi-rigid operator to have a recurrent subspace; when $X$ is a complex Banach space, we show that having a recurrent subspace is equivalent to the fact that the essential spectrum of the operator intersects the closed unit disk, and we extend the previous result to the real case. As a consequence, we obtain that a weakly-mixing operator on a real or complex separable Banach space has a hypercyclic subspace if and only if it has a recurrent subspace. The results exposed exhibit a symmetry between the hypercyclic and recurrent spaceability theories showing that, at least for the spaceable property, hypercyclicity and recurrence can be treated as equals.

The equivalence between CPCP and strong regularity under Krein-Milman property

We obtain a result in the spirit of the well-known W. Schachermayer and H. P. Rosenthal research about the equivalence between Radon-Nikodym and Krein-Milman properties, by showing that, for closed, bounded and convex subsets C of a separable Banach space, under Krein-Milman property for C, one has the equivalence between convex point of continuity property and strong regularity both defined for every locally convex topology on C, containing the weak topology on C. Then, under Krein-Milman property, not only the classical convex point of continuity property and strong regularity are equivalent, but also when they are defined for an arbitrary locally convex topology containing the weak topology. We also show that while the unit ball B of c0 fails convex point of continuity property and strong regularity (both defined for the weak topology), there is a locally convex topology τ on B, containing the weak topology on B, such that B still fails convex point of continuity property for τ, but B surprisingly enjoys strong regularity for τ-open sets. Moreover, B satisfies the diameter two property for the topology τ, that is, every nonempty τ-open subset of B has diameter two even though every τ-open subset of B contains convex combinations of relative τ-open subsets with arbitrarily small diameter, that is, B fails the strong diameter two property for the topology τ. This stresses the known extreme differences up to now between those diameter two properties from a topological point of view.

A new result on the fractal dimension estimates of random attractor for non-autonomous random 2D stochastic dynamical type systems

We prove some conditions for bounding the fractal dimension of random invariant sets of non-autonomous random dynamical systems on separable Banach spaces. Then we apply these conditions to prove the finiteness of fractal dimension of random attractor for stochastic 2D hydrodynamical type equations with linear additive white noise in bounded domains or unbounded domains satisfying the Poincaré inequality.

Strong maximum a posteriori estimation in Banach spaces with Gaussian priors

This article shows that a large class of posterior measures that are absolutely continuous with respect to a Gaussian prior have strong maximum a posteriori estimators in the sense of Dashti et al (2013 Inverse Problems 29 095017). This result holds in any separable Banach space and applies in particular to nonparametric Bayesian inverse problems with additive noise. When applied to Bayesian inverse problems, this significantly extends existing results on maximum a posteriori estimators by relaxing the conditions on the log-likelihood and on the space in which the inverse problem is set.

Operator ranges in Banach spaces with weak star separable dual

We provide several extensions for Banach spaces with weak⁎-separable dual of a theorem of Schevchik ensuring that for every proper dense operator range R in a separable Banach space E, there exists a one-to-one and dense-range operator such that T(E)∩R={0}. These results lead to several characterizations of Banach spaces with weak⁎-separable dual in terms of disjointness properties of operator ranges, which yield a refinement of a theorem of Plichko concerning the spaceability of the complementary set of a proper dense operator range, and an affirmative solution to a problem of Borwein and Tingley for the class of Banach spaces with a separable quotient and weak⁎-separable dual. We also provide an extension to these spaces of a theorem of Cross and Shevchik, which guarantees that for every proper dense operator range R in a separable Banach space E there exist two closed quasicomplementary subspaces X and Y of E such that R∩(X+Y)={0}. Finally, we prove that some weak forms of the theorems of Shevchik and Cross and Shevchik do not hold in any nonseparable weakly Lindelöf determined Banach space.

Construction of 𝑝-energy and associated energy measures on Sierpiński carpets

We establish the existence of a scaling limit E p \mathcal {E}_p of discrete p p -energies on the graphs approximating a generalized Sierpiński carpet for p > d A R C p > d_{\mathrm {ARC}} , where d A R C d_{\mathrm {ARC}} is the Ahlfors regular conformal dimension of the underlying generalized Sierpiński carpet. Furthermore, the function space F p \mathcal {F}_{p} defined as the collection of functions with finite p p -energies is shown to be a reflexive and separable Banach space that is dense in the set of continuous functions with respect to the supremum norm. In particular, ( E 2 , F 2 ) (\mathcal {E}_2, \mathcal {F}_2) recovers the canonical regular Dirichlet form constructed by Barlow and Bass [Ann. Inst. H. Poincaré Probab. Statist. 25 (1989), pp. 225–257] or Kusuoka and Zhou [Probab. Theory Related Fields 93 (1992), pp. 169–196]. We also provide E p \mathcal {E}_{p} -energy measures associated with the constructed p p -energy and investigate its basic properties like self-similarity and chain rule.