Lorentz Sequence Spaces Research Articles

INTERPOLATION BY ANALYTIC FUNCTIONS ON PREDUALS OF LORENTZ SEQUENCE SPACES

Let (en) be the canonical basis of the predual of the Lorentz sequence space d∗(w, 1). We consider the restriction operator R associated to the basis (ei )f rom some Banach space of analytic functions into the complex sequence space and we characterize the ranges of R.

Weighted Hardy inequalities for decreasing sequences and functions

We obtain a complete characterization of the weights for which Hardy's inequality holds on the cone of non-increasing sequences. Our proofs translate immediately to the analogous inequality for non-increasing functions, thereby also completing the investigation in that direction. As an application of our results we characterize the boundedness of the Hardy-Littlewood maximal operator on Lorentz sequence spaces.

EXTENSIONS OF POLYNOMIALS ON PREDUALS OF LORENTZ SEQUENCE SPACES

We show that there is a unique norm-preserving extension for norm-attaining 2-homogeneous polynomials on the predual $d_*(w,1)$ of a complex Lorentz sequence space $d(w,1)$ to $d^*(w,1)$ , but there is no unique norm-preserving extension from $\mathcal{P}(^nd_*(w,1))$ to $\mathcal{P}(^nd^*(w,1))$ for $n\geq3$ .

On holomorphic functions attaining their norms

We show that on a complex Banach space X, the functions uniformly continuous on the closed unit ball and holomorphic on the open unit ball that attain their norms are dense provided that X has the Radon–Nikodym property. We also show that the same result holds for Banach spaces having a strengthened version of the approximation property but considering just functions which are also weakly uniformly continuous on the unit ball. We prove that there exists a polynomial such that for any fixed positive integer k, it cannot be approximated by norm attaining polynomials with degree less than k. For X=d ∗(ω,1) , a predual of a Lorentz sequence space, we prove that the product of two polynomials with degree less than or equal two attains its norm if, and only if, each polynomial attains its norm.

Sequence Spaces lp,qin Probabilistic Characterizations of Weak Type Operators

We study operators \(T:X \mapsto L_ \circ ([0,1],{\mathcal{M}},m)\) (not necessarily linear) defined on a quasi-Bahach space X and taking values in the space of real-valued Lebesgue-measurable functions. Factorization theorems for linear and superlinear operators with values in the space \(L_ \circ \) are proved with the help of the Lorentz sequence spaces \(l_{p,q} \). Sequences of functions belonging to fixed bounded sets in the spaces \(L_{p,\infty } \) are characterized for \(0 < p < \infty \) and \(0 < q \leqslant p\). The possibility of distinguishing weak type operators (bounded in the space \(L_{p,\infty } \)) from operators factorizable through \(L_{p,\infty } \) is obtained in terms of sequences of independent random variables. A criterion under which an operator is symmetrically bounded in order in \(L_{p,r} ,{\text{ }}0 < r \leqslant \infty \), is established. Some refinements of the above-mentioned results are obtained for translation shift-invariant sets and operators. Bibliography: 30 titles.

BOUNDARIES FOR AN ALGEBRA OF BOUNDED HOLOMORPHIC FUNCTIONS

Let Ab(BE) be the Banach algebra of all complex val- ued bounded continuous functions on the closed unit ball BE of a complex Banach space E, and holomorphic in the interior of BE, endowed with the sup norm. We present some sucient conditions for a set to be a boundary for Ab(BE) in case E belongs to a class of Banach spaces that includes the pre-dual of a Lorentz sequence space studied by Gowers in (6). We also prove the non-existence of the Shilov boundary for Ab(BE) and give some examples of bound- aries.

Boundaries for algebras of holomorphic functions

Let A u ( B G ) be the Banach algebra of all complex valued functions defined on the closed unit ball B G of a complex Banach space G which are uniformly continuous on B G and holomorphic in the interior of B G , endowed with the sup norm. A characterization of the boundaries for A u ( B G ) is given in case G belongs to a class of Banach spaces that includes the pre-dual of a Lorentz sequence space studied by Gowers in Israel J. Math. 69 (1990) 129–151. The non-existence of the Shilov boundary for A u ( B G ) is also proved.

Uniform convexity of ψ-direct sums of Banach spaces

Let X and Y be Banach spaces and ψ a continuous convex function on the unit interval [0,1] satisfying certain conditions. Let X⊕ψY be the direct sum of X and Y equipped with the associated norm with ψ. We show that X⊕ψY is uniformly convex if and only if X,Y are uniformly convex and ψ is strictly convex. As a corollary we obtain that the ℓp,q-direct sum X⊕p,qY,1⩽q⩽p⩽∞ (not p=q=1 nor ∞), is uniformly convex if and only if X,Y are, where ℓp,q is the Lorentz sequence space. These results extend the well-known fact for the ℓp-sum X⊕pY,1<p<∞. Some other examples are also presented.

Cotype 2 estimates for spaces of polynomials on sequence spaces

We give asymptotically correct estimations for the cotype 2 constant C2(P(mXn)) ofthe spaceP(mXn) of allm-homogenous polynomials onXn, the span of the firstn sequencesek=(\gdkj)j in a Banach sequence spaceX. Applications to Minkowski, Orlicz and Lorentz sequence spaces are given.

Pitt's theorem for operators between general Lorentz sequence spaces

We characterize the pairs of general Lorentz sequence spaces $\ell^{u,v}(\nu),$ $\ell^{p,q}(\mu), 0 &lt; u, v, p, q &lt; \infty$ such that all continuous linear maps from the first space into the second one are compact.

On M-Ideals of Compact Operators in Lorentz Sequence Spaces

Let X=d(v,p) and Y=d(w,q) be Lorentz sequence spaces. We investigate when the space K(X,Y) of compact linear operators acting from X to Y forms or does not form an M-ideal (in the space of bounded linear operators). We show that K(X,Y) fails to be a non-trivial M-ideal whenever p=1 or p>q. In the case when 1>p≤q, we establish a general (essential) condition guaranteeing that K(X,Y) is not an M-ideal. In contrast, we prove that non-trivial M-ideals K(X,Y) do exist whenever 1<p<q, and we give a description of them.

On James and Jordan–von Neumann Constants of Lorentz Sequence Spaces

The nonsquare or James constant J(X) and the Jordan–von Neumann constant CNJ(X) are computed for two-dimensional Lorentz sequence spaces d(2)(w,q) in the case where 2≤q<∞. The Jordan–von Neumann constant is also calculated in the case where 1≤q<2.

Lorentz-Schatten Classes and Pointwise Domination of Matrices

AbstractWe investigate pointwise domination property in operator spaces generated by Lorentz sequence spaces.

Norms and lower bounds of operators on the Lorentz sequence space $d(w,1)$

Norms and lower bounds of operators on the Lorentz sequence space $d(w,1)$

Compactness of operators acting from a Lorentz sequence space to an Orlicz sequence space

LetX andY be closed subspaces of the Lorentz sequence spaced(v, p) and the Orlicz sequence spacel M , respectively. It is proved that every bounded linear operator fromX toY is compact whenever $$p > \beta _M : = \inf \{ q > 0:\inf \{ M(\lambda t)/M(\lambda )t^q :0 0.$$ As an application, the reflexivity of the space of bounded linear operators acting fromd(v, p) tol M is characterized.

Norm attaining multilinear forms and polynomials on preduals of Lorentz sequence spaces

For each natural number N, we give an example of a Banach space X such that the set of norm attaining N{linear forms is dense in the space of all continuous N{linear forms on X, but there are continuous (N +1){linear forms on X which cannot be approximated by norm attaining (N+1){linear forms. Actually, X is the canonical predual of a suitable Lorentz sequence space. We also get the analogous result for homogeneous polynomials.

The $q$ -concavity constants of Lorentz sequence spaces and related inequalitites

The $q$ -concavity constants of Lorentz sequence spaces and related inequalitites

Geometric Properties of Symmetric Spaces with Applications to Orlicz–Lorentz Spaces

We deal with the basic convexity properties –rotundity, and uniform, local uniform and full rotundity –- for symmetric spaces. A characterization of Orlicz–Lorentz spaces with the Kadec–Klee property for pointwise convergence is given. These results are applied to obtain criteria of convexity properties for Orlicz–Lorentz sequence spaces, and some new proofs of the sufficiency part of criteria for rotundity and uniform rotundity for Orlicz–Lorentz function spaces.

1-Complemented Subspaces of Spaces With 1-Unconditional Bases

AbstractWe prove that if X is a complex strictly monotone sequence space with 1-unconditional basis, Y ⊆ X has no bands isometric to ℓ22 and Y is the range of norm-one projection from X, then Y is a closed linear span a family of mutually disjoint vectors in X.We completely characterize 1-complemented subspaces and norm-one projections in complex spaces ℓp(ℓq) for 1 ≤ p,q &gt; ∞.Finally we give a full description of the subspaces that are spanned by a family of disjointly supported vectors and which are 1-complemented in (real or complex) Orlicz or Lorentz sequence spaces. In particular if an Orlicz or Lorentz space X is not isomorphic to ℓp for some 1 ≤ p,q &gt; ∞ then the only subspaces of X which are 1-complemented and disjointly supported are the closed linear spans of block bases with constant coefficients.

Characterization of weak type by the entropy distribution of r-nuclear operators

The dual of a Banach space X is of weak type p if and only if the entropy numbers of an r-nuclear operator with values in a Banach space of weak type q belong to the Lorentz sequence space $ℓ_{s,r}$ with 1/s + 1/p + 1/q = 1 + 1/r (0 < r < 1, 1 ≤ p, q ≤ 2)