Unconditional Basis Research Articles

Performance of the Thresholding Greedy Algorithm with larger greedy sums

The goal of this paper is to study the performance of the Thresholding Greedy Algorithm (TGA) when we increase the size of greedy sums by a constant factor λ⩾1. We introduce the so-called λ-almost greedy and λ-partially greedy bases. The case when λ=1 gives us the classical definitions of almost greedy and (strong) partially greedy bases. We show that a basis is almost greedy if and only if it is λ-almost greedy for all (some) λ⩾1. However, for each λ>1, there exists an unconditional basis that is λ-partially greedy but is not 1-partially greedy. Furthermore, we investigate and give examples when a basis is(1)not almost greedy with constant 1 but is λ-almost greedy with constant 1 for some λ>1, and(2)not strong partially greedy with constant 1 but is λ-partially greedy with constant 1 for some λ>1. Finally, we prove various characterizations of different greedy-type bases.

The (E, E)-approximation property for Banach spaces

For a given Banach space E with an 1-unconditional basis, we introduce the (E, E)-approximation property ((E, E)-AP) which is a natural generalization of the classical approximation property (AP) and the (p, p)-AP. It is shown that the AP always implies the (E, E)-AP. If E has an 1-unconditional and shrinking basis, then the (E*, E*)-AP for X* implies the (E, E)-AP for X. If in addition that the unconditional basis for E is subsymmetric, then X* has the (E*, E*)-AP implies that X has the duality (E, E)-AP.

О некоторых особенностях, связанных с приобретением права собственности на недвижимое имущество

The purpose of the study is to consider controversial positions existing in the law enforcement practice of the Supreme Court of the Russian Federation and related to registration actions for the transfer of ownership of vend in the Federal Service of State Registration, Cadastre and Cartography. Research objectives: to determine the necessity of state registration of the transfer of ownership, to identify the main ways to protect the rights of a bona fide acquirer of vend, to study the features of the division of real estate of spouses. General scientific methods (dialectical, logical, systematic), private scientific methods (historical, statistical, sociological), as well as special legal (comparative legal, formal legal) were used. The actual circumstances taken into account by the courts in disputes over the rights to real estate are indicated. It is noted that in the current legislation there are no clear criteria for classifying property vend. The ways of protecting the right of a bona fide buyer are considered, including confirmation of the actual transfer of the apartment under the contract of sale and obtaining the consent of the second spouse when concluding a preliminary contract of sale on the seller’s side. Such subjects of legal relations as spouses have been studied, since when dividing the joint acquired property, the question of determining the share in the ownership of vend is most acute. It is noted that the fact of registration is not always an unconditional basis for the emergence of ownership of vend.

Narrow operators on tensor products of Köthe spaces

We investigate narrow operators on tensor products of Köthe-Banach spaces. Our first main result asserts that for a bounded linear operator S:F→F on Köthe-Banach space F with an order continuous norm and a linear bounded operator T:L1(μ)→L1(μ), the narrowness at least one of the T or S implies the narrowness for an operator T⊗S:L1(μ)⊗ˆπF→L1(μ)⊗ˆπF defined on the positive projective tensor product of L1(μ) and F. The converse statement holds under an additional assumption of regularity for the operator S:F→F. As a consequence, we resolve an open problem concerning the existence of a Köthe-Banach space on [0,1] with an absolutely continuous norm without an unconditional basis in which the identity operator is a sum of two narrow operators, suggested in [45]. We also supply complementing results for the Köthe-Banach spaces E[X] with mixed norm where E and X are symmetric spaces and fully characterize the spaces E[X] having an unconditional basis.

Copies of $c_0(\tau)$ in Saphar tensor products

Let $X, Y$ be Banach spaces, τ an infinite cardinal and $1 \leq p < \infty $. We extend a result by E. Oja by showing that if $X$ has a boundedly complete unconditional basis and either $X \widehat{\otimes}_{g_p} Y$ or $X \widehat{\otimes}_{\varepsilon _p} Y$ contains a complemented copy of $c_0(\tau )$, then $Y$ contains a complemented copy of $c_0(\tau )$. We show also that if α is a uniform crossnorm, $X \widehat{\otimes}_\alpha Y$ contains a (complemented) copy of $c_0(\tau )$ and the cofinality of τ is strictly greater than the density of $X$, then $Y$ also contains a (complemented) copy of $c_0(\tau )$. As an application, we obtain a result concerning complemented copies of $\ell _1(\tau )$ in $X \widehat{\otimes}_\alpha Y$.

Orthogonal systems of spline wavelets as unconditional bases in Sobolev spaces

AbstractWe exhibit the necessary range for which functions in the Sobolev spaces can be represented as an unconditional sum of orthonormal spline wavelet systems, such as the Battle–Lemarié wavelets. We also consider the natural extensions to Triebel–Lizorkin spaces. This builds upon, and is a generalization of, previous work of Seeger and Ullrich, where analogous results were established for the Haar wavelet system.

On a criterion for the existence of unconditional bases of reproducing kernels in Fock spaces with radial regular weight

We consider the spaces Fφ of entire functions f such that fe−φ∈L2(C), where φ(z)=φ(|z|) is a radial subharmonic function with some regularity property. It was proved that Fφ has Riesz basis of normalized reproducing kernels if and only if (φ(er))″ is bounded above.

On the Crawford number attaining operators

We study the denseness of Crawford number attaining operators on Banach spaces. Mainly, we prove that if a Banach space has the Radon–Nikodým property, then the set of Crawford number attaining operators is dense in the space of bounded linear operators. We also see among others that the set of Crawford number attaining operators is dense in the space of all bounded linear operators while they do not coincide, by observing the case of compact operators when the Banach space has a 1-unconditional basis. Furthermore, we show a Bishop–Phelps–Bollobás type property for the Crawford number for certain Banach spaces, and we finally discuss some difficulties and possible problems on the topic.

E-Operator Ideals Determined by Banach Spaces with Unconditional Bases

E-Operator Ideals Determined by Banach Spaces with Unconditional Bases

Addendum to “uniqueness of unconditional basis of infinite direct sums of quasi-Banach spaces”

After [Uniqueness of unconditional basis of infinite direct sums of quasi-Banach spaces, Positivity 26 (2022), Paper no. 35] was published, we realized that Theorem 4.2 therein, when combined with work of Casazza and Kalton (Israel J. Math. 103:141–175, 1998) , solves the long-standing problem whether there exists a quasi-Banach space with a unique unconditional basis whose Banach envelope does not have a unique unconditional basis. Here we give examples to prove that the answer is positive. We also use auxiliary results in the aforementioned paper to give a negative answer to the question of Bourgain et al. (Mem Am Math Soc 54:iv+111, 1985)*Problem 1.11 whether the infinite direct sum ell _{1}(X) of a Banach space X has a unique unconditional basis whenever X does.

On the permutative equivalence of squares of unconditional bases

We prove that if the squares of two unconditional bases are equivalent up to a permutation, then the bases themselves are permutatively equivalent. This settles a twenty-five year-old question raised by Casazza and Kalton in [13]. Solving this problem provides a new paradigm to study the uniqueness of unconditional basis in the general framework of quasi-Banach spaces. Multiple examples are given to illustrate how to put in practice this theoretical scheme. Among the main applications of this principle we obtain the uniqueness of unconditional basis up to permutation of finite sums of spaces with this property, as well as the first addition to the scant list of the known Banach spaces with a unique unconditional bases up to permutation since [14].

Factorisation in stopping-time Banach spaces: Identifying unique maximal ideals

Stopping-time Banach spaces is a collective term for the class of spaces of eventually null integrable processes that are defined in terms of the behaviour of the stopping times with respect to some fixed filtration. From the point of view of Banach space theory, these spaces in many regards resemble the classical spaces such as L1 or C(Δ), but unlike these, they do have unconditional bases. In the present paper, we study the canonical bases in the stopping-time spaces in relation to factorising the identity operator thereon. Since we work exclusively with the dyadic-tree filtration, this setup enables us to work with tree-indexed bases rather than directly with stochastic processes. En route to the factorisation results, we develop general criteria that allow one to deduce the uniqueness of the maximal ideal in the algebra of operators on a Banach space. These criteria are applicable to many classical Banach spaces such as (mixed-norm) Lp-spaces, BMO, SL∞, and others.

On Narrow Operators from L_p into Operator Ideals

It is well known that every l_2-strictly singular operator from L_p, 1<p<infty to any Banach space X with an unconditional basis is narrow. In this article, we extend this result to the setting of Banach spaces without an unconditional basis. We show that if 1 le p,r <infty , then every ell _2-strictly singular operator T from L_p into the Schatten–von Neumann r-class C_r is narrow. This is a noncommutative complement to results in Mykhaylyuk et al. (in Israel J Math 203:81–108, 2014).

Direct and inverse problems for nonlocal heat equation with boundary conditions of periodic type

A mathematical model of the process of heat diffusion in a closed metal wire is considered. This wire is wrapped around a thin sheet of insulation material. We assume that the insulation is slightly permeable. Because of this, the temperature at the point of the wire on one side of the insulation influences the diffusion process in the wire on the other side of the insulation. Thus, the standard heat equation will change and an extra term with involution will be added. When modeling of this process there arises an initial-boundary value problem for a one-dimensional heat equation with involution and with a boundary condition of periodic type with respect to a spatial variable. We prove the well-posedness of the formulated problem in the class of strong generalized solutions. The use of the method of separation of variables leads to a spectral problem for an ordinary differential operator with involution at the highest derivative. All eigenfunctions of the problem are constructed. In the case when all eigenvalues of the problem are simple, the system of eigenfunctions does not form an unconditional basis. A criterion when this spectral problem can have an infinite number of multiple eigenvalues is proved. Corresponding root subspaces consist of one eigenfunction and one associated function. We prove that the system of root functions forms an unconditional basis and can be used for constructing a solution of the heat conduction problem by the method of separation of variables. We also consider an inverse problem. This is the problem on restoring (simultaneously with solving) of an unknown stationary source of external influence with respect to an additionally known final state. The existence of a unique solution of this inverse problem and its stability with respect to initial and final data are proved.

Uniqueness of unconditional basis of [formula omitted] and [formula omitted] for 0 < p < 1

Our goal in this paper is to advance the state of the art of the topic of uniqueness of unconditional basis. To that end we establish general conditions on a pair (X,Y) formed by a quasi-Banach space X and a Banach space Y which guarantee that every unconditional basis of their direct sum X⊕Y splits into unconditional bases of each summand. As application of our methods we obtain that, among others, the spaces Hp(Td)⊕T(2) and Hp(Td)⊕ℓ2, for p∈(0,1) and d∈N, have a unique unconditional basis (up to equivalence and permutation).

On Reflexivity and Point Spectrum

In this note, we prove a characterization of reflexive Banach spaces with unconditional basis and study the set theoretical complexity of certain sets associated to linear bounded operators acting on reflexive separable Banach spaces.

Legal gender recognition in times of change at the European Court of Human Rights

The COVID-19 pandemic has made the role of identity papers for the enjoyment of human rights once more obvious. It is thus a suitable moment to analyse the current implementation of the right to change the gender and/or name on official documents and civil registries in Europe. This article specifically examines the jurisprudence on the right to gender recognition of the European Court of Human Rights. It concludes that the Court is moving towards recognising the right to change one’s legal gender and/or name on an unconditional basis, and that it will need to deliberate on the right to be free from any state-imposed gender label in the future.

Uniqueness of unconditional basis of infinite direct sums of quasi-Banach spaces

This paper is devoted to providing a unifying approach to the study of the uniqueness of unconditional bases, up to equivalence and permutation, of infinite direct sums of quasi-Banach spaces. Our new approach to this type of problem permits to show that a wide class of vector-valued sequence spaces have a unique unconditional basis up to a permutation. In particular, solving a problem from Albiac and Leránoz (J Math Anal Appl 374(2):394–401, 2011. https://doi.org/10.1016/j.jmaa.2010.09.048) we show that if X is quasi-Banach space with a strongly absolute unconditional basis then the infinite direct sum ell _{1}(X) has a unique unconditional basis up to a permutation, even without knowing whether X has a unique unconditional basis or not. Applications to the uniqueness of unconditional structure of infinite direct sums of non-locally convex Orlicz and Lorentz sequence spaces, among other classical spaces, are also obtained as a by-product of our work.

Unconditional Bases for Homogeneous $$\alpha $$-Modulation Type Spaces

We construct orthonormal bases compatible with bi-variate homogeneous \(\alpha \)-modulation spaces and the associated spaces of Triebel–Lizorkin type. The construction is based on generating a separable \(\alpha \)-covering and using carefully selected tensor products of univariate brushlet functions with regard to this covering. We show that the associated systems form an unconditional bases for the homogeneous \(\alpha \)-spaces of Triebel–Lizorkin type.

Unconditional bases in radial Hilbert spaces

We prove necessary and (separate) sufficient conditions for the existence of unconditional bases of reproducing kernels in abstract radial Hilbert function spaces that are stable under division, in terms of the norms of monomials.