Orlicz Sequence Spaces Research Articles

Nikishin’s theorem and factorization through Marcinkiewicz spaces

Consider L 0 L^0 , the F F -space of all equivalence classes of measurable functions on a finite measure space equipped with the topology of convergence in measure. Inspired by Nikishin’s classical result on the factorization of sublinear continuous operators from a Banach space to L 0 L^0 , we prove a theorem that characterizes those maps from any quasi-metric space into L 0 L^0 that factor strongly through Marcinkiewicz weighted spaces. We show applications to sublinear operators on a certain class of quasi-Banach spaces with generalized Rademacher type generated by Orlicz sequence spaces.

Weakly compact sets in Orlicz–Bochner sequence spaces

AbstractIn this work, we give three kinds of criteria for weak sets in Orlicz–Bochner sequence spaces without constraints, conditions posited in each criterion are necessary and sufficient. As an application, we give criteria for weak sets in Orlicz sequence spaces. Well‐known conclusions are exhibited once more, such as Schur's theorem, Banach–Alaoglu's theorem, and the boundedly compact principle of finite dimension space. The results obtained show that the weak compactness may not be extrapolated straightforwardly from to , for example, .

On subspaces of Orlicz spaces spanned by independent copies of a mean zero function

We study the subspaces of the Orlicz spaces $L_M$ spanned by independent copies $f_k$, $k=1,2,…$, of a function $f\in L_M$, $\int_0^1 f(t) dt=0$. Any such a subspace $H$ is isomorphic to some Orlicz sequence space $\ell_\psi$. In terms of dilations of the function $f$, a description of strongly embedded subspaces of this type is obtained, and conditions guaranteeing that the unit ball of such a subspace consists of functions with equicontinuous norms in $L_M$ are found. In particular, we prove that there is a wide class of Orlicz spaces $L_M$ (containing the $L^p$-spaces, $1\le p< 2$), for which each of the above properties of $H$ holds if and only if the Matuszewska-Orlicz indices of the functions $M$ and $\psi$ satisfy $\alpha_\psi^0>\beta_M^\infty$.

Smoothness, very (strongly) smoothness of Orlicz sequence spaces equipped with p-Amemiya norms

Smoothness, very (strongly) smoothness of Orlicz sequence spaces equipped with p-Amemiya norms

GENERALIZED ORLICZ SEQUENCE SPACES

Orlicz spaces were first introduced by Z. W. Birnbaum and W. Orlicz as an extension of Labesgue space in 1931. There are two types of Orlicz spaces, namely continuous Orlicz spaces and Orlicz sequence spaces. Some of the properties that apply to continuous Orlicz spaces are known, as are Orlicz sequence spaces. This study aims to construct new Orlicz sequence spaces by replacing a function in the Orlicz sequence spaces with a wider function. In addition, this study also aims to show that the properties of the Orlicz sequence spaces still apply to the new Orlicz sequence spaces under different conditions. The method in this research uses definitions and properties that apply to the Orlicz sequence spaces in the previous study and uses the -Young function in these new Orlicz sequence spaces. Furthermore, the results of the study show that the new Orlicz sequence spaces are an extension of the Orlicz sequence spaces in the previous study. And with the characteristics of the -Young function, it shows that the properties of the Orlicz sequence spaces still apply.

MATRIK TRANSFORMASI PADA RUANG BARISAN ORLICZ

In this study will be characterized a matrix that maps the space of the Orlicz sequence space to the space of the bounded sequence and the convergence sequence. This study is carried out by generalized the matrix mapping on lp space, 1<=p<\infty, to the Orlicz sequence space. This results establishes a necessary and sufficient conditions for a matrix that maps the Orlicz sequence space to the bounded sequence space, convergence to sequence space and the convergence sequence space.

Exposed Points of Orlicz Sequence Spaces Equipped with p-Amemiya $$(1\le p\le \infty )$$ Norms

Exposed Points of Orlicz Sequence Spaces Equipped with p-Amemiya $$(1\le p\le \infty )$$ Norms

A characterization of $$\ell ^p$$-spaces symmetrically finitely represented in symmetric sequence spaces

For a separable symmetric sequence space X of fundamental type we identify the set $${{\mathcal {F}}}(X)$$ of all $$p\in [1,\infty ]$$ such that $$\ell ^p$$ is block finitely represented in the unit vector basis $$\{e_k\}_{k=1}^\infty$$ of X in such a way that the unit basis vectors of $$\ell ^p$$ ( $$c_0$$ if $$p=\infty$$ ) correspond to pairwise disjoint blocks of $$\{e_k\}$$ with the same ordered distribution. It turns out that $${{\mathcal {F}}}(X)$$ coincides with the set of approximate eigenvalues of the operator $$(x_k)\mapsto \sum _{k=2}^\infty x_{[k/2]}e_k$$ in X. In turn, we establish that the latter set is the interval $$[2^{\alpha _X},2^{\beta _X}]$$ , where $$\alpha _X$$ and $$\beta _X$$ are the Boyd indices of X. As an application, we find the set $${{\mathcal {F}}}(X)$$ for arbitrary Lorentz and separable Orlicz sequence spaces.

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.

Nonexpansive Mappings on New Premodular Special Space of Sequences

For different premodular, which is a generalization of modular, defined by weighted Orlicz sequence space and its prequasi operator ideal, we have examined the existence of a fixed point for both Kannan contraction and nonexpansive mappings acting on these spaces. Some numerous numerical experiments and practical applications are presented to support our results.

Strict convexity of Orlicz sequence spaces equipped with p-Amemiya norms

Strict convexity of Orlicz sequence spaces equipped with p-Amemiya norms

Entropy numbers of diagonal operators on Orlicz sequence spaces

AbstractLet M1 and M2 be functions on [0,1] such that and are Orlicz functions for some . Assume that is non‐decreasing for . Let be a non‐increasing sequence of nonnegative real numbers. Under some conditions on , sharp two‐sided estimates for entropy numbers of diagonal operators generated by , where and are Orlicz sequence spaces, are proved. The results generalise some works of Edmunds and Netrusov in [8] and hence a result of Cobos, Kühn and Schonbek in [6].

English

We combine the techniques of sequence spaces and general Orlicz functions that are broader than the classical cases of N-functions. We give three criteria for the weakly compact sets in general Orlicz sequence spaces. One criterion is related to elements of dual spaces. Under the restriction of $$\mathop {\lim }\limits_{u \to 0} M(u)/u = 0$$ , we propose two other modular types that are convenient to use because they get rid of elements of dual spaces. Subsequently, by one of these two modular criteria, we see that a set A in Riesz spaces lp (1 < p < ∞) is relatively sequential weakly compact if and only if it is normed bounded, that says sup $$\mathop {{\rm{sup}}}\limits_{u \in A} \sum\limits_{i = 1}^\infty {{{\left| {u(i)} \right|}^p} < \infty } $$ . The result again confirms the conclusion of the Banach-Alaoglu theorem.

The essential norm of multiplication operators acting on Orlicz sequence spaces

We calculate the measure of non-compactness or the essential norm of the multiplication operator Mu acting on Orlicz sequence spaces lφ. As a consequence of our result, we obtain a known criteria for the compactness of multiplication operator acting on lφ.

Compact Operators Under Orlicz Function

In this paper we investigate the compact operators under Orlicz function, named noncommutative Orlicz sequence space (denoted by Sϕ(ℌ)), where ℌ is a complex, separable Hilbert space. We will show that the space generalizes the Schatten classes Sp(ℌ) and the classical Orlicz sequence space respectively. After getting some relations of trace and norm, we will give some operator inequalities, such as Holder inequality and some other classical operator inequalities. Also we will give the dual space and reflexivity of Sϕ(ℌ) which generalizes the results of Sϕ(ℌ). Finally, as an application, we will show that the Toeplitz operator $${T_{1 — {{\left| z \right|}^2}}}$$ on the Bergman space $$L_\alpha ^2\left(\mathbb{R} \right)$$ belongs to some Sϕ(ℌ), and the norm satisfies $$1 = \sum\limits_{n \ge 0} {\varphi \left({{1 \over {\left({n + 2} \right){{\left\| {{T_{1 — {{\left| z \right|}^2}}}} \right\|}_\varphi}}}} \right)} $$ . Especially, if ϕ(T) = ∣T∣p, p > 1, the norm is $${\left\| {{T_{1 — {{\left| z \right|}^2}}}} \right\|_p} = {\left[{\sum\limits_{n \ge 0} {{1 \over {{{\left({n + 2} \right)}^p}}}}} \right]^{{1 \over p}}} = {\left({\zeta \left(p \right) — 1} \right)^{{1 \over p}}}$$ , where ζ(p) is the Riemann function.

Complex interpolation of families of Orlicz sequence spaces

We study the complex interpolation and derivation process induced by a family of Orlicz sequence spaces. We present a concrete example of an interpolation family of three spaces inducing a centralizer that cannot be obtained from complex interpolation of two spaces.

A class of a non-Banach F-lattices E such that every orthomorphism on E is central with applications to Musielak–Orlicz sequence spaces

Let E=(E,‖⋅‖) be a non-Banach F-lattice non-containing a subspace linearly homeomorphic to the F-lattice ω:=RN. The symbol Orth(E) denotes the linear lattice of all orthomorphisms on E, and Z(E) denotes the principal ideal of Orth(E) generated by the identity IE on E. In this paper, we give a sufficient condition for the equality Orth(E)=Z(E) to hold. In particular, this identity holds true for every symmetric sequence space E. We also obtain a useful series-criterion for the class of discrete F-lattices E with order continuous norm for the identity Orth(E)=Z(E) to hold. These results apply non-trivially for the class of Musielak-Orlicz sequence spaces. We give a few examples illustrating our main results. At the end of this paper, we set a few open questions about orthomorphisms.

On subspaces spanned by freely independent random variables in noncommutative Lp-spaces

This paper deals with new phenomena appearing in the structure of Banach spaces arising in noncommutative integration theory. Let E be a fully symmetric Banach function space on [0, 1] and M be a finite von Neumann algebra. Let [xk] be the closed subspace spanned by a sequence (xk) of freely independent mean zero random variables from E(ℳ). The subspace [xk] is complemented in E(ℳ) if and only if the closed subspace spanned by the pairwise orthogonal sequence (xk ⊗ ek) is complemented in a certain symmetric operator space $$Z_E^2\left({{\cal M}\overline \otimes {\ell_\infty}} \right)$$ . We obtain noncommutative (free) analogues of classical results of Dor and Starbird as well as those of Kadec and Pelczynski. We show that [xk] is complemented in L1(ℳ)provided (xk) is equivalent in L1(ℳ) to the standard basis of l2, while this never happens in the classical case. We prove that a sequence of freely independent copies of a mean zero random variable x in Lp(ℳ), 1 ≤ p ≤ 2, is equivalent to the standard basis in some Orlicz sequence space ℳΦ and give a precise description of the connection between the Orlicz function Φ and the distribution of the given random variable x. Finally, we prove that [xk] spanned by a sequence of freely independent copies of a mean zero random variable is complemented in E(ℳ) if and only if (xk) is equivalent in E(ℳ) to the standard basis of l2.

Monotonicities in Orlicz Spaces Equipped with Mazur-Orlicz F-Norm

Some basic properties in Orlicz spaces and Orlicz sequence spaces that are generated by monotone function equipped with the Mazur-Orlicz F-norm are studied in this paper. We give some relationships between the modulus and the Mazur-Orlicz F-norm. We obtain an interesting result that the norm of an element in line segments is formed by two elements on the unit sphere less than or equal to 1 if and only if that the monotone function is a convex function. The criterion that Orlicz spaces and Orlicz sequence spaces that are generated by monotone function equipped with the Mazur-Orlicz F-norm are strictly monotone or lower locally uniform monotone is presented.

Diameter two properties and the Radon–Nikodým property in Orlicz spaces

Some necessary and sufficient conditions are found for Banach function lattices to have the Radon–Nikodým property. Consequently it is shown that an Orlicz function space Lφ over a non-atomic σ-finite measure space (Ω,Σ,μ), not necessarily separable, has the Radon–Nikodým property if and only if φ is an N-function at infinity and satisfies the appropriate Δ2 condition. For an Orlicz sequence space ℓφ, it has the Radon–Nikodým property if and only if φ satisfies the Δ20 condition. In the second part a relationship between uniformly ℓ12 points of the unit sphere of a Banach space and the diameter of the slices are studied. Using these results, a quick proof is given that an Orlicz space Lφ has the Daugavet property only if φ is linear, so when Lφ is isometric to L1. Another consequence is that Orlicz spaces equipped with the Orlicz norm generated by N-functions never have the local diameter two property, while it is well-known that when equipped with the Luxemburg norm, it may have that property. Finally, it is shown that the local diameter two property, the diameter two property, and the strong diameter two property are equivalent in Orlicz function and sequence spaces with the Luxemburg norm under appropriate conditions on φ.