Nonnegativity constraints for structured complete systems

Some generalizations of Besselian, Hilbertian systems and frames in nonseparable Banach spaces with respect to some nonseparable Banach space K of systems of scalars are considered in this work. The concepts of uncountable K-Bessel, K-Hilbert systems, K-frames and K *-Riesz bases in nonseparable Banach spaces are introduced. Criteria of uncountable K-Besselianness, K-Hilbertianness for systems, K-frames and unconditional K *-Riesz basicity are found, and the relationship between them is studied. Unlike before, these new facts about Besselian and Hilbertian systems in Hilbert and Banach spaces are proved without using a conjugate system and, in some cases, a completeness of a system. Examples of K-Besselian systems which are not minimal are given. It is proved that every K-Hilbertian systems is minimal. The case where K is an space of systems of coefficients of uncountable unconditional basis of some space is also considered.

Archiv der Mathematik, 2009

In the first part of the article we characterize automatic continuity of positive operators. As a corollary we consider complete norms for which a given cone E+ in an infinite dimensional Banach space E is closed and we obtain the following result: every two such norms are equivalent if and only if E+ ∩ (−E+) = {0} and E+ − E+ has finite codimension. Without preservation of an order structure, on an infinite dimensional Banach space one can always construct infinitely many mutually non-equivalent complete norms. We use different techniques to prove this. The most striking is a set theoretic approach which allows us to construct infinitely many complete norms such that the resulting Banach spaces are mutually non-isomorphic.

2012

We say that a real-valued function f defined on a positive Borel measure space (X, µ) is nowhere q-integrable if, for each nonvoid open subset U of X, the restriction f | U is not in L q (U ). When (X, µ) satisfies some natural properties, we show that certain sets of functions defined in X which are p-integrable for some p's but nowhere q-integrable for some other q's (0 < p, q < ∞) admit a variety of large linear and algebraic structures within them. The presented results answer a question from Bernal-González, improve and complement recent spaceability and algebrability results from several authors and motivates new research directions in the field of spaceability.

Proceedings of the American Mathematical Society, 1975

If every function f f in the range of a bounded linear operator on L p {L_p} is equal to zero on a set of measure greater than a fixed number ϵ \epsilon , it is shown that there is a common set of measure ϵ \epsilon on which every function is zero. A decomposition theorem for such operators is proved.

Illinois Journal of Mathematics

We construct a Schauder basis for L 1 consisting of non-negative functions and investigate unconditionally basic and quasibasic sequences of non-negative functions in L p , 1 ≤ p < ∞.

Mathematical Notes of the Academy of Sciences of the USSR, 1985

The study of these systems was proposed by A. G. Kostyucbenko, in connection with the study of the leading terms of the asymptotics of elgenfunctions of certain bundles of differential operators. Using the Paley-Wiener theorem (see [I]), he showed that for ~ lying outside some disk Izl~r, the system (1), (2) forms a Riesz basis in L2(0, ~). Further development was made by Lyubarskii [2], who reduced the study of the completeness and basicityof the system (I) to the study of the spectrum of some singular operator K, estimating its norm by a number which turns out to be equal to r. We also note that the completeness and uniform minimality in C[O, w] of the system {~ cos nx + ~ (x) e-on~}~, a > 0, for ~>max [% 9 to, hi (x) I-A was proved by ~Jmarkin [3]. In this article we describe the spectrum of the opera~or K, and we thus establish a necessary and sufficient condition for Riesz basicity with finite defect in L2(0, ~) for the system (I), (2). We also show that we have completeness with finite defect in L2(0, ~) for any ~=/=0. This result is applied to solve the problem of the basicity of part of the eigenfunctions of certain quadratic bundles of ordinary differential operators. Let L be the closure of the linear hull of {~}~ in the Hilbert space H. Definition. The system {~n}~ is called complete with finite defect in H, if the number dim coker L is finite. Moreover, if after we discard a finite number of elements the system becomes a Riesz basis ~n the subspace L, we say that the system {~,,}~ forms a Riesz basis with finite defect in H.

Mathematica Slovaca, 2021

Let F denote the factorable matrix and X ∈ {ℓp, c0, c, ℓ∞}. In this study, we introduce the domains X(F) of the factorable matrix in the spaces X. Also, we give the bases and determine the alpha-, beta- and gamma-duals of the spaces X(F). We obtain the necessary and sufficient conditions on an infinite matrix belonging to the classes (ℓp(F), ℓ∞), (ℓp(F), f) and (X, Y(F)) of matrix transformations, where Y denotes any given sequence space. Furthermore, we give the necessary and sufficient conditions for factorizing an operator based on the matrix F and derive two factorizations for the Cesàro and Hilbert matrices based on the Gamma matrix. Additionally, we investigate the norm of operators on the domain of the matrix F. Finally, we find the norm of Hilbert operators on some sequence spaces and deal with the lower bound of operators on the domain of the factorable matrix.

Mathematical Proceedings of the Cambridge Philosophical Society, 1993

In , Partington proved that if A is a Banach sequence space with a monotone basis having the Banach-Saks property, and (X n ) is a sequence of Banach spaces each having the Banach-Saks property, then the vector sequence space S A X n has this same property. In addition, Partington gave an example showing that if A and each X n have the weak Banach-Saks property, then S A X n need not have the weak Banach-Saks property.

Indagationes Mathematicae, 2007

We show that a sequentially (τ)-complete topological vector lattice X_τ is isomorphic to some L¹(μ), if and only if the positive cone can be written as X₊ = R₊·B for some convex, (τ)-bounded, and (τ)-closed set B ⊂ X₊\{0}. The same result holds under weaker hypotheses, namely the Riesz decomposition property for X (not assumed to be a vector lattice) and the monotonic σ-completeness (monotonic Cauchy sequences converge). The isometric part of the main result implies the well-known representation theorem of Kakutani for (AL)-spaces. As an application we show that on a normed space Y of infinite dimension, the "ball-generated" ordering induced by the cone Y₊ = R₊· B̅(u,1) (for ‖u‖ > 1) cannot have the Riesz decomposition property. A second application deals with a pointwise ordering on a space of multivariate polynomials.

Archiv der Mathematik, 1987