Musielak-Orlicz Sequence Spaces Research Articles

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.

Some vector valued sequence spaces of Musielak-Orlicz functions and their operator ideals

AbstractIn the present paper we introduce generalized vector-valued Musielak-Orlicz sequence spacel(A,𝓜,u,p,Δr,∥·,… ,·∥)(X) and study some geometric properties like uniformly monotone, uniform Opial property for this space. Further, we discuss the operators ofs-type and operator ideals by using the sequence ofs-number (in the sense of Pietsch) under certain conditions on matrixA.

On generalized García-Falset coefficient in Musielak-Orlicz sequence spaces

We introduce a new geometric coefficient which is related Garcia-Falset coefficient and weak star fixed point property. The Garcia-Falset coefficient that was introduced by Benavides in (Houst. J. Math. 22:835-849, 1996) is calculated in this paper for Musielak-Orlicz sequence spaces equipped with the Luxemburg norm. Specifically, in reflexive Banach spaces, the new geometric coefficient and the Garcia-Falset coefficient are the same.

On some geometric properties of generalized Musielak-Orlicz sequence space and corresponding operator ideals

Let $\bold{\Phi}=(\phi_n)$ be a Musielak-Orlicz function, $X$ be a real Banach space and $A$ be any infinite matrix. In this paper, a generalized vector-valued Musielak-Orlicz sequence space $l_{\bold {\Phi}}^{A}(X)$ is introduced. It is shown that the space is complete normed linear space under certain conditions on the matrix $A$. It is also shown that $l_{\bold{\Phi}}^{A}(X)$ is a $\sigma$- Dedikind complete whenever $X$ is so. We have discussed some geometric properties, namely, uniformly monotone, uniform Opial property for this space. Using the sequence of $s$-number (in the sense of Pietsch), the operators of $s$-type $l_{\bold{\Phi}}^{A}$ and operator ideals under certain conditions on the matrix $A$ are discussed.

The problem of central orthomorphisms in a class of [formula omitted]-lattices

We prove that if an F-lattice E is locally bounded (i.e., an open ball in E centered at 0 is topologically bounded) then every orthomorphism in E is central: Orth(E)=Z(E). This solves partially a problem raised recently by Chil and Meyer.For the Nakano (non-Banach) F-lattice E=ℓ(pn), 0<pn<1, the above implication becomes an equivalence.We also study the problem of central orthomorphism in the class of (non-Banach) Musielak–Orlicz sequence spaces ℓΦ. We give a sufficient condition on Φ for a non-central orthomorphism on ℓΦ to exist.

Asymptotically isometric copies of c_0 in Musielak-Orlicz spaces

Criteria in order that a Musielak-Orlicz function space \(L^\Phi\) as well as Musielak-Orlicz sequence space \(l^\Phi\) contains an asymptotically isometric copy of \(c_0\) are given. These results extend some results of [Y.A. Cui, H. Hudzik, G. Lewicki, Order asymptotically isometric copies of \(c_0\) in the subspaces of order continuous elements in Orlicz spaces, Journal of Convex Analysis 21 (2014)] to Musielak-Orlicz spaces.

One-Complemented Subspaces in Musielak-Orlicz Sequence Spaces with a General Smoothness Condition

The aim of this article is to obtain a characterization (in the spirit of Jamison-Kamińska-Lewicki) of one-complemented subspaces in Musielak-Orlicz sequence spaces defined by Musielak-Orlicz functions satisfying a general smoothness condition. The result obtained is an important extension of the Jamison-Kamińska-Lewicki Theorem. Some applications are also discussed.

Contractive and optimal sets in Musielak-Orlicz spaces with a smoothness condition

In this paper we use our recent generalization of a theorem of Jamison- -Kami«ska-Lewicki (characterizing one-complemented subspaces in Musielak-Orlicz sequence spaces defined by Musielak-Orlicz functions satisfying a general smoothness condition) in or- der to compare contractive and optimal sets in finite-dimensional Musielak-Orlicz ' (n) spaces in the spirit of Kami«ska-Lewicki. We also give an example illustrating the importance of the smoothness assumptions in our theorem.

EXPOSED POINTS AND STRONGLY EXPOSED POINTS IN MUSIELAK-ORLICZ SEQUENCE SPACES

In this paper we give criteria of exposed points and strongly exposed points in Musielak-Orlicz sequence spaces endowed with Luxemburg norm.

Explosiveness of Musielak-Orlicz Sequence Spaces

In this talk,we make a survey for criteria of the exposed points and the exposedness for the sequence spaces of Orlicz type,such as the classical Orlicz spaces,general Orlicz spaces and Musielak-Orlicz spaces,endowed with Luxemburg norm and Orlicz norm respectively.It is well known that explosiveness is basic concept in the geometric theory of Banach spaces.They have numerous application in separation theory and control theory.Criteria for exposed points and strongly exposed points in all classical Orlicz spaces were given in[10-12].In recent years,Zhao and Cui in[6]discussed the problem in Musielak-Orlicz sequence spaces under some restriction. By counterexamples,we show that the criteria of extreme points in[8]and exposed points in[6] are not true,and we give the criteria for exposed points and strongly exposed points in arbitrary Musielak-Orlicz sequence spaces equipped with Luxemburg norm and Orlicz norm by getting rid of the restriction on Musielak-Orlicz function in[6].

Strongly exposed points in Musielak-Orlicz sequence spaces endowed with Orlicz norm

Strongly exposed points in Musielak-Orlicz sequence spaces endowed with Orlicz norm

On the points of local uniform rotundity and weak local uniform rotundity in Musielak–Orlicz sequence spaces equipped with the Orlicz norm

In this paper, a criterion for locally uniformly rotund points and weakly locally uniformly rotund points in Musielak–Orlicz sequence spaces equipped with the Orlicz norm is given. As a corollary, we get the criterion in order that Musielak–Orlicz sequence spaces equipped with the Orlicz norm are locally uniformly rotund and weakly locally uniformly rotund.

Optimal and one-complemented subspaces

Let X be a real Banach space and let V ⊂ X be a closed linear subspace. In [4, Prop. 5] it has been proven that if X is strictly convex, reflexive and smooth and V is an optimal subset of X then V is one-complemented in X. In this note we would like to extend this result to non-smooth Banach spaces. In particular, we show that any existence subspace of c,co and l1 is one-complemented. Also some results concerning non-smooth Musielak-Orlicz sequence spaces equipped with the Luxemburg norm will be presented.

On Riesz–Musielak–Orlicz Sequence Spaces

In this paper, we define the Riesz–Musielak–Orlicz sequence space and exhibit some geometric properties for this space. The results herein proved are analogous to those by R. Wangkeeree (Thai J. Math. 2003; 1:119–130) for the Cesàro sequence space cesp.

Local uniform rotundity and weak local uniform rotundity of Musielak–Orlicz sequence spaces endowed with the Orlicz norm

Criteria for local uniform rotundity and weak local uniform rotundity of Musielak–Orlicz sequence spaces equipped with the Orlicz norm are given. Criteria in order that the subspace of order continuous elements equipped with the norm induced from the whole space is locally (weakly) uniformly rotund are also presented.

Schur property and lP isomorphic copies in Musielak–Orlicz sequence spaces

The author shows that if the dual of a Musielak–Orlicz sequence space lΦ is a stabilized asymptotic l∞, space with respect to the unit vector basis, then lΦ is saturated with complemented copies of l1 and has the Schur property. A sufficient condition is found for the isomorphic embedding of lp spaces into Musielak–Orlicz sequence spaces.

Extreme Points and Strong U-Points in Musielak–Orlicz Sequence Spaces Equipped with the Orlicz Norm

We give some criteria for extreme points and strong U-points in Musielak–Orlicz sequence spaces equipped with the Orlicz norm. It follows from these results that the notion of the strong U-point is essentially stronger than the notion of the extreme point in these spaces.

Extreme Points and Strongly Extreme Points of Musielak–Orlicz Sequences Spaces

Criteria for extreme points and strongly extreme points in Musielak–Orlicz sequence spaces, equipped with both the Luxemburg norm and the Orlicz norm, are given.

One-complemented subspaces of Musielak–Orlicz sequence spaces

The aim of this paper is to characterize one-complemented subspaces of finite codimension in the Musielak–Orlicz sequence space l Φ . We generalize the well-known fact (Ann. Mat. Pura Appl. 152 (1988) 53; Period. Math. Hungar. 22 (1991) 161; Classical Banach Spaces I, Springer, Berlin, 1977) that a subspace of finite codimension in l p , 1 ⩽ p < ∞ , is one-complemented if and only if it can be expressed as a finite intersection of kernels of functionals with at most two coordinates different from zero. Under some smoothness condition on Φ = ( φ n ) we prove a similar characterization in l Φ . In the case of Orlicz spaces we obtain a complete characterization of one-complemented subspaces of finite codimension, which extends and completes the results in Randrianantoanina (Results Math. 33(1–2) (1998) 139). Further, we show that the well-known fact that a one-complemented subspace of finite codimension in l p , 1 ⩽ p < ∞ , is an intersection of one-complemented hyperplanes, is no longer valid in Orlicz or Musielak–Orlicz spaces. In the last section we characterize l p -spaces, 1 < p < ∞ , and separately l 2 -spaces, in terms of one-complemented hyperplanes, in the class of Musielak–Orlicz and Orlicz spaces as well.

UGD property of Musielak-Orlicz sequence spaces

By the properties of the Musielak-Orlicz function's sequence, the necessary and sufficient condition for uniform Gateaux differential (UGD) property of Musielak-Orlicz sequence spaces equipped with the Luxemburg norm and a criterion for weakly uniform rotundity of Musielak-Orlicz sequence space with Orlicz norm are given.