Dense Subspace Research Articles

Riesz* homomorphisms on pre-Riesz spaces consisting of continuous functions

In the theory of operators on a Riesz space (vector lattice), an important result states that the Riesz homomorphisms (lattice homomorphisms) on C(X) are exactly the weighted composition operators. We extend this result to Riesz* homomorphisms on order dense subspaces of C(X). On those subspace we consider and compare various classes of operators that extend the notion of a Riesz homomorphism. Furthermore, using the weighted composition structure of Riesz* homomorphisms we obtain several results concerning bijective Riesz* homomorphisms. In particular, we characterize the automorphism group for order dense subspaces of C(X). Lastly, we develop a similar theory for Riesz* homomorphisms on subspace of C_0(X), for a locally compact Hausdorff space X, and apply it to smooth manifolds and Sobolev spaces.

Spaceability and strong divergence of the Shannon sampling series and applications

In this paper the structure of the set of functions for which the peak value of the Shannon sampling series is strongly divergent is analyzed. Strong divergence is closely linked to the non-existence of adaptive reconstruction methods. Signals in the Paley–Wiener space PWπ1 of bandlimited functions with absolutely integrable Fourier transform are considered, and it is shown that the set of strong divergence is spaceable and dense-lineable, i.e., that there exist an infinite dimensional closed subspace and an infinite dimensional dense subspace, such that we have strong divergence of the peak value of the Shannon sampling series for all functions from these sets, except the zero function. Further, it is proved that this result is not restricted to the Shannon sampling series, but rather holds for an entire class of reconstruction processes.

On the Ascoli property for locally convex spaces

We characterize Ascoli spaces by showing that a Tychonoff space X is Ascoli iff the canonical map from the free locally convex space L(X) over X into Ck(Ck(X)) is an embedding of locally convex spaces. We prove that an uncountable direct sum of non-trivial locally convex spaces is not Ascoli. If a c0-barrelled space E is weakly Ascoli, then E is linearly isomorphic to a dense subspace of RΓ for some set Γ. Consequently, a Fréchet space E is weakly Ascoli iff E=RN for some N≤ω. If X is a μ-space and a kR-space (for example, metrizable), then Ck(X) is weakly Ascoli iff X is discrete. If X is a μ-space, then the space Mc(X) of all regular Borel measures on X with compact support is Ascoli in the weak⁎ topology iff X is finite. The weak⁎ dual space of a metrizable barrelled space E is Ascoli iff E is finite-dimensional.

Generalised weighted composition operators on Maeda–Ogasawara spaces

Every Archimedean Riesz space can be embedded as an order dense subspace of some C∞(X), the Riesz space of all extended continuous functions on a Stonean space X, called its Maeda–Ogasawara space. Furthermore, it is a fact that every Riesz homomorphism between spaces of ordinary continuous functions on compact Hausdorff spaces is a weighted composition operator. We prove that a generalised statement holds for Maeda–Ogasawara spaces and refine these results in case the homomorphism preserves order limits.

A Gelfand–Naimark type theorem

Let X be a completely regular space. For a non-vanishing self-adjoint Banach subalgebra H of CB(X) which has local units we construct the spectrum sp(H) of H as an open subspace of the Stone–Čech compactification of X which contains X as a dense subspace. The construction of sp(H) is simple. This enables us to study certain properties of sp(H), among them are various compactness and connectedness properties. In particular, we find necessary and sufficient conditions in terms of either H or X under which sp(H) is connected, locally connected and pseudocompact, strongly zero-dimensional, basically disconnected, extremally disconnected, or an F-space.

Real-variable characterizations of anisotropic product Musielak-Orlicz Hardy spaces

Let $$\vec A: = \left( {{A_1},{A_2}} \right)$$ be a pair of expansive dilations and φ: ℝ n ×ℝ m ×[0, ∞) → [0, ∞) an anisotropic product Musielak-Orlicz function. In this article, we introduce the anisotropic product Musielak-Orlicz Hardy space $$H_{\vec A}^\varphi \left( {{\mathbb{R}^n} \times {\mathbb{R}^m}} \right)$$ via the anisotropic Lusin-area function and establish its atomic characterization, the $$\vec g$$ -function characterization, the $$\vec g_\lambda ^*$$ -function characterization and the discrete wavelet characterization via first giving out an anisotropic product Peetre inequality of Musielak-Orlicz type. Moreover, we prove that finite atomic decomposition norm on a dense subspace of $$H_{\vec A}^\varphi \left( {{\mathbb{R}^n} \times {\mathbb{R}^m}} \right)$$ is equivalent to the standard infinite atomic decomposition norm. As an application, we show that, for a given admissible triplet ( $$\left( {\varphi ,q,\vec s} \right)$$ ), if T is a sublinear operator and maps all ( $$\left( {\varphi ,q,\vec s} \right)$$ )-atoms into uniformly bounded elements of some quasi-Banach spaces B, then T uniquely extends to a bounded sublinear operator from $$H_{\vec A}^\varphi \left( {{\mathbb{R}^n} \times {\mathbb{R}^m}} \right)$$ to B. Another application is that we obtain the boundedness of anisotropic product singular integral operators from $$H_{\vec A}^\varphi \left( {{\mathbb{R}^n} \times {\mathbb{R}^m}} \right)$$ to L φ (R n × R m ) and from $$H_{\vec A}^\varphi \left( {{\mathbb{R}^n} \times {\mathbb{R}^m}} \right)$$ to itself, whose kernels are adapted to the action of $$\vec A$$ . The results of this article essentially extend the existing results for weighted product Hardy spaces on ℝ n × ℝ m and are new even for classical product Orlicz-Hardy spaces.

On spaces weakly ‐analytic

AbstractA subset Y of the dual closed unit ball of a Banach space E is called a Rainwater set for E if every bounded sequence of E that converges pointwise on Y converges weakly in E. In this paper, topological properties of Rainwater sets for the Banach space of the real‐valued continuous and bounded functions defined on a completely regular space X equipped with the supremum‐norm are studied. This applies to characterize the weak K‐analyticity of in terms of certain Rainwater sets for . Particularly, we show that is weakly K‐analytic if and only if there exists a Rainwater set Y for such that is both K‐analytic and angelic, where denotes the topology on of the pointwise convergence on Y. For the case when X is compact, one gets classic Talagrand's theorem. As an application we show that if X is a compact space and Y is a ‐dense subspace, then X is Talagrand compact, i.e., is K‐analytic, if and only if the space is K‐analytic.

On dense subspaces of countable pseudocharacter in function spaces

A set A⊆Cp(X) is uniformly dense in Cp(X) if, for any f∈Cp(X) and ε>0, there is g∈A such that |g(x)−f(x)|<ε for all x∈X. We prove that Cp(X) has a uniformly dense subspace that condenses onto a second countable space if and only if Cp(X) has cardinality of the continuum. This answers a question of Tkachuk published in 2003. It is also proved that Cp(X) has a dense Fσ-metrizable subspace if X is metrizable or Eberlein compact. If X is Corson compact or Cp(X) is a Lindelöf Σ-space, then it is possible to find a dense set Y⊆Cp(X) with countable pseudocharacter.

Regularity of the Szegö projection on model worm domains

In this paper, we study the regularity of the Szegö projection on Lebesgue and Sobolev spaces on the boundary of the unbounded model worm domain . We consider the Hardy space . Denoting by the boundary of , it is classical that can be identified with the closed subspace of , denoted by , consisting of the boundary values of functions in , where is the induced Lebesgue measure. The orthogonal Hilbert space projection is called the Szegö projection. Let denote the Lebesgue–Sobolev space on . We prove that P, initially defined on the dense subspace , extends to a bounded operator , for and .

Some representation theorems for sesquilinear forms

The possibility of getting a Radon–Nikodym type theorem and a Lebesgue-like decomposition for a not necessarily positive sesquilinear Ω form defined on a vector space D, with respect to a given positive form Θ defined on D, is explored. The main result consists in showing that a sesquilinear form Ω is Θ-regular, in the sense that it has a Radon–Nikodym type representation, if and only if it satisfies a sort Cauchy–Schwarz inequality whose right hand side is implemented by a positive sesquilinear form which is Θ-absolutely continuous. In the particular case where Θ is an inner product in D, this class of sesquilinear form covers all standard examples. In the case of a form defined on a dense subspace D of Hilbert space H we give a sufficient condition for the equality Ω(ξ,η)=〈Tξ|η〉, with T a closable operator, to hold on a dense subspace of H.

The weight and Lindelöf property in spaces and topological groups

We show that if Y is a dense subspace of a Tychonoff space X, then w(X)≤nw(Y)Nag(Y), where Nag(Y) is the Nagami number of Y. In particular, if Y is a Lindelöf Σ-space, then w(X)≤nw(Y)ω≤nw(X)ω.Better upper bounds for the weight of topological groups are given. For example, if a topological group H contains a dense subgroup G such that G is a Lindelöf Σ-space, then w(H)=w(G)≤ψ(G)ω. Further, if a Lindelöf Σ-space X generates a dense subgroup of a topological group H, then w(H)≤2ψ(X).Several facts about subspaces of Hausdorff separable spaces are established. It is well known that the weight of a separable Hausdorff space X can be as big as 22c. We prove on the one hand that if a regular Lindelöf Σ-space Y is a subspace of a separable Hausdorff space, then w(Y)≤2ω, and the same conclusion holds for a Lindelöf P-space Y. On the other hand, we present an example of a countably compact topological Abelian group G which is homeomorphic to a subspace of a separable Hausdorff space and satisfies w(G)=22c, i.e. G has the maximal possible weight.

Dirichlet Forms Constructed from Annihilation Operators on Bernoulli Functionals

The annihilation operators on Bernoulli functionals (Bernoulli annihilators, for short) and their adjoint operators satisfy a canonical anticommutation relation (CAR) in equal-time. As a mathematical structure, Dirichlet forms play an important role in many fields in mathematical physics. In this paper, we apply the Bernoulli annihilators to constructing Dirichlet forms on Bernoulli functionals. Let w be a nonnegative function on N. By using the Bernoulli annihilators, we first define in a dense subspace of L2-space of Bernoulli functionals a positive, symmetric, bilinear form Ew associated with w. And then we prove that Ew is closed and has the contraction property; hence, it is a Dirichlet form. Finally, we consider an interesting semigroup of operators associated with w on L2-space of Bernoulli functionals, which we call the w-Ornstein-Uhlenbeck semigroup, and, by using the Dirichlet form, Ew we show that the w-Ornstein-Uhlenbeck semigroup is a Markov semigroup.

An Improved Negative Selection Algorithm Based on Subspace Density Seeking

Negative selection algorithm (NSA) is an important method for generating detectors in artificial immune systems. Traditional NSAs randomly generate detectors in the whole feature space. However, with increasing dimensions, data samples aggregate in some specific subspaces, not uniformly distributed in the whole space. The detectors randomly generated by traditional NSAs cannot exactly fall into these specific subspaces, which results in a low coverage of detectors and a poor performance in a high-dimensional space. To overcome this defect, an improved real NSA based on subspace density seeking (SDS-RNSA) is proposed in this paper. In an SDS-RNSA, a subspace density seeking algorithm is adopted to procure the dense subspace regions of samples. Then, detectors are generated in each subspace region to cover up nonself-region efficiently and improve the performance of the algorithm. During the process of detector generation, the redundancy of candidate detectors is calculated, and the redundant is eliminated to minimize the time expense of the algorithm. Experimental results demonstrate that, compared with the classic NSAs, the SDS-RNSA can significantly improve the detection rate with an approximative false alarm rate and a smaller time expense. At the best case, the detection rate of the SDS-RNSA is increased by 14.7%, while the time expense is decreased by 78.1%.

Maximal densely countably compact topologies

A topological space X is densely countably compact if it possesses a dense subspace D with the property that every infinite subset of D has an accumulation point in X. We study topologies which are maximal with respect to this property; in particular we show that a T1 densely countably compact space is maximal densely countably compact if and only if it is a scattered Frechet SC-space of scattering order 2.

Nonlinear Frames and Sparse Reconstructions in Banach Spaces

In the first part of this paper, we consider nonlinear extension of frame theory by introducing bi-Lipschitz maps F between Banach spaces. Our linear model of bi-Lipschitz maps is the analysis operator associated with Hilbert frames, p-frames, Banach frames, g-frames and fusion frames. In general Banach space setting, stable algorithms to reconstruct a signal x from its noisy measurement $$F(x)+\epsilon $$ may not exist. In this paper, we establish exponential convergence of two iterative reconstruction algorithms when F is not too far from some bounded below linear operator with bounded pseudo-inverse, and when F is a well-localized map between two Banach spaces with dense Hilbert subspaces. The crucial step to prove the latter conclusion is a novel fixed point theorem for a well-localized map on a Banach space. In the second part of this paper, we consider stable reconstruction of sparse signals in a union $$\mathbf{A}$$ of closed linear subspaces of a Hilbert space $$\mathbf{H}$$ from their nonlinear measurements. We introduce an optimization framework called a sparse approximation triple $$(\mathbf{A}, \mathbf{M}, \mathbf{H})$$ , and show that the minimizer $$\begin{aligned} x^*=\mathrm{argmin}_{\hat{x}\in {\mathbf M}\ \mathrm{with} \ \Vert F(\hat{x})-F(x^0)\Vert \le \epsilon } \Vert \hat{x}\Vert _{\mathbf M} \end{aligned}$$ provides a suboptimal approximation to the original sparse signal $$x^0\in \mathbf{A}$$ when the measurement map F has the sparse Riesz property and the almost linear property on $${\mathbf A}$$ . The above two new properties are shown to be satisfied when F is not far away from a linear measurement operator T having the restricted isometry property.

Contraction of Hamiltonian $K$-Spaces

In the spirit of recent work of Harada-Kaveh and Nishinou-Nohara-Ueda, we study the symplectic geometry of Popov's horospherical degenerations of complex algebraic varieties with the action of a complex linearly reductive group. We formulate an intrinsic symplectic contraction of a Hamiltonian space, which is a surjective, continuous map onto a new Hamiltonian space that is a symplectomorphism on an explicitly defined dense open subspace. This map is given by a precise formula, using techniques from the theory of symplectic reduction and symplectic implosion. We then show, using the Vinberg monoid, that the gradient-Hamiltonian flow for a horospherical degeneration of an algebraic variety gives rise to this contraction from a general fiber to the special fiber. We apply this construction to branching problems in representation theory, and finally we show how the Gel'fand-Tsetlin integrable system can be understood to arise this way.

⊤-diagonal conditions and Continuous extension theorem

In this paper, we obtain a generalization of Kowalsky diagonal condition and that of Fischer diagonal condition respectively, namely Kowalsky ⊤-diagonal condition and Fischer ⊤-diagonal condition. We show that our Fischer ⊤-diagonal condition assures a complete-MV-algebra-valued convergence space, proposed in this paper, is strong L-topological, and Kowalsky ⊤-diagonal condition assures a principle (or pretopological) complete-MV-algebra-valued convergence space is strong L-topological also. As applications, we give a “dual form” of our Fischer ⊤-diagonal condition and obtain a concept of regular ⊤-convergence space. In addition, we present an extension theorem for continuous maps from a dense subspace to a regular ⊤-convergence space to show that our ⊤-diagonal conditions works indeed.

Weak and strong structures and the $${T_{3.5}}$$ T 3.5 property for generalized topological spaces

We investigate weak and strong structures for generalized topological spaces, among others products, sums, subspaces, quotients, and the complete lattice of generalized topologies on a given set. Also we introduce \({T_{3.5}}\) generalized topological spaces and give a necessary and sufficient condition for a generalized topological space to be a \({T_{3.5}}\) space: they are exactly the subspaces of powers of a certain natural generalized topology on [0,1]. For spaces with at least two points here we can have even dense subspaces. Also, \({T_{3.5}}\) generalized topological spaces are exactly the dense subspaces of compact \({T_4}\) generalized topological spaces. We show that normality is productive for generalized topological spaces. For compact generalized topological spaces we prove the analogue of the Tychonoff product theorem. We prove that also Lindelofness (and \({\kappa}\)-compactness) is productive for generalized topological spaces. On any ordered set we introduce a generalized topology and determine the continuous maps between two such generalized topological spaces: for \({|X|, |Y| \geqq 2}\) they are the monotonous maps continuous between the respective order topologies. We investigate the relation of sums and subspaces of generalized topological spaces to ways of defining generalized topological spaces.

A note on Mackey topologies on Banach spaces

There is a maybe unexpected connection between three apparently unrelated notions concerning a given w⁎-dense subspace Y of the dual X⁎ of a Banach space X: (i) The norming character of Y, (ii) the fact that (Y,w⁎) has the Mazur property, and (iii) the completeness of the Mackey topology μ(X,Y), i.e., the topology on X of the uniform convergence on the family of all absolutely convex w⁎-compact subsets of Y. To clarify these connections is the purpose of this note. The starting point was a question raised by M. Kunze and W. Arendt and the answer provided by J. Bonet and B. Cascales. We fully characterize μ(X,Y)-completeness or its failure in the case of Banach spaces X with a w⁎-angelic dual unit ball—in particular, separable Banach spaces or, more generally, weakly compactly generated ones—by using the norming or, alternatively, the Mazur character of Y. We characterize the class of spaces where the original Kunze–Arendt question has always a positive answer. Some other applications are also provided.

Regular biorthogonal pairs and pseudo-bosonic operators

The first purpose of this paper is to show a method of constructing a regular biorthogonal pair based on the commutation rule: ab − ba = I for a pair of operators a and b acting on a Hilbert space H with inner product (⋅| ⋅ ). Here, sequences {ϕn} and {ψn} in a Hilbert space H are biorthogonal if (ϕn|ψm) = δnm, n, m = 0, 1, …, and they are regular if both Dϕ ≡ Span{ϕn} and Dψ ≡ Span{ψn} are dense in H. Indeed, the assumptions to construct the regular biorthogonal pair coincide with the definition of pseudo-bosons as originally given in F. Bagarello [“Pseudobosons, Riesz bases, and coherent states,” J. Math. Phys. 51, 023531 (2010)]. Furthermore, we study the connections between the pseudo-bosonic operators a, b, a†, b† and the pseudo-bosonic operators defined by a regular biorthogonal pair ({ϕn}, {ψn}) and an ONB e of H in H. Inoue [“General theory of regular biorthogonal pairs and its physical applications,” e-print arXiv:math-ph/1604.01967]. The second purpose is to define and study the notion of D-pseudo-bosons in F. Bagarello [“More mathematics for pseudo-bosons,” J. Math. Phys. 54, 063512 (2013)] and F. Bagarello [“From self-adjoint to non self-adjoint harmonic oscillators: Physical consequences and mathematical pitfalls,” Phys. Rev. A 88, 032120 (2013)] and give a method of constructing D-pseudo-bosons on some steps. Then it is shown that for any ONB e = {en} in H and any operators T and T−1 in L†(D), we may construct operators A and B satisfying D-pseudo bosons, where D is a dense subspace in a Hilbert space H and L†(D) the set of all linear operators T from D to D such that T*D⊂D, where T* is the adjoint of T. Finally, we give some physical examples of D-pseudo-bosons based on standard bosons by the method of constructing D-pseudo-bosons stated above.