• Title/Summary/Keyword: Banach lattice

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NONSEPARABLE COMPLEMENTED SUBLATTICES IN THE BANACH ENVELOPE OF $WeakL_l$

  • Kang, Jeong-Heung
    • Journal of applied mathematics & informatics
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    • v.23 no.1_2
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    • pp.537-545
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    • 2007
  • We investigate complemented Banach sublattices of the Banach envelope of $Weak_L1$. In particular, the Banach envelope of $Weak_L1$ contains a complemented Banach sublattice that is isometrically isomorphic to a nonseparable Banach lattice $l_p(S),\;1{\leq}p<{\infty}\;and\;|S|{\leq}2^{{\aleph}0}$.

COMPLEMENTED SUBLATTICE OF THE BANACH ENVELOPE OF WeakL1 ISOMORPHIC TO ℓp

  • Kang, Jeong-Heung
    • Communications of the Korean Mathematical Society
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    • v.22 no.2
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    • pp.209-218
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    • 2007
  • In this paper we investigate the ${\ell}^p$ space structure of the Banach envelope of $WeakL_1$. In particular, the Banach envelope of $WeakL_1$ contains a complemented Banach sublattice that is isometrically isomorphic to the nonseparable Banach lattice ${\ell}^p$, ($1{\leq}p<\infty$) as well as the separable case.

THE CLASS OF p-DEMICOMPACT OPERATORS ON LATTICE NORMED SPACES

  • Imen Ferjani;Bilel Krichen
    • Communications of the Korean Mathematical Society
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    • v.39 no.1
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    • pp.137-147
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    • 2024
  • In the present paper, we introduce a new class of operators called p-demicompact operators between two lattice normed spaces X and Y. We study the basic properties of this class. Precisely, we give some conditions under which a p-bounded operator be p-demicompact. Also, a sufficient condition is given, under which each p-demicompact operator has a modulus which is p-demicompact. Further, we put in place some properties of this class of operators on lattice normed spaces.

On the asymptotic-norming property and the mazur intersection property

  • Cho, Sung-Jin
    • Journal of the Korean Mathematical Society
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    • v.32 no.3
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    • pp.583-591
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    • 1995
  • Unless otherwise stated, we always assume that X is a Banach space, and $1 < p, q < \infty with \frac{p}{1}+\frac{q}{1} = 1$. We use S(X) and B(X) to denote the unit sphere and the unit ball in X respectively.

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EXTREMAL STRUCTURE OF B($X^{*}$)

  • Lee, Joung-Nam
    • The Pure and Applied Mathematics
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    • v.5 no.2
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    • pp.95-100
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    • 1998
  • In this note we consider some basic facts concerning abstract M spaces and investigate extremal structure of the unit ball of bounded linear functionals on $\sigma$-complete abstract M spaces.

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Q-MEASURES ON THE DUAL UNIT BALL OF A JB-TRIPLE

  • Edwards, C. Martin;Oliveira, Lina
    • Journal of the Korean Mathematical Society
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    • v.56 no.1
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    • pp.197-224
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    • 2019
  • Let A be a $JB^*$-triple with Banach dual space $A^*$ and bi-dual the $JBW^*$-triple $A^{**}$. Elements x of $A^*$ of norm one may be regarded as normalised 'Q-measures' defined on the complete ortho-lattice ${\tilde{\mathcal{U}}}(A^{**})$ of tripotents in $A^{**}$. A Q-measure x possesses a support e(x) in ${\tilde{\mathcal{U}}}(A^{**})$ and a compact support $e_c(x)$ in the complete atomic lattice ${\tilde{\mathcal{U}}}_c(A)$ of elements of ${\tilde{\mathcal{U}}}(A^{**})$ compact relative to A. Necessary and sufficient conditions for an element v of ${\tilde{\mathcal{U}}}_c(A)$ to be a compact support tripotent $e_c(x)$ are given, one of which is related to the Q-covering numbers of v by families of elements of ${\tilde{\mathcal{U}}}_c(A)$.

Unbounded Scalar Operators on Banach Lattices

  • deLaubenfels, Ralph
    • Honam Mathematical Journal
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    • v.8 no.1
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    • pp.1-19
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    • 1986
  • We show that a (possibly unbounded) linear operator, T, is scalar on the real line (spectral operator of scalar type, with real spectrum) if and only if (iT) generates a uniformly bounded semigroup and $(1-iT)(1+iT)^{-1}$ is scalar on the unit circle. T is scalar on [0, $\infty$) if and only if T generates a uniformly bounded semigroup and $(1+T)^{-1}$ is scalar on [0,1). By analogy with these results, we define $C^0$-scalar, on the real line, or [0. $\infty$), for an unbounded operator. We show that a generator of a positive-definite group is $C^0$-scalar on the real line. and a generator of a completely monotone semigroup is $C^0$-scalar on [0, $\infty$). We give sufficient conditions for a closed operator, T, to generate a positive-definite group: the sequence < $\phi(T^{n}x)$ > $_{n=0}^{\infty}$ must equal the moments of a positive measure on the real line, for sufficiently many positive $\phi$ in $X^{*}$, x in X. If the measures are supported on [0, $\infty$), then T generates a completely monotone semigroup. On a reflexive Banach lattice, these conditions are also necessary, and are equivalent to T being scalar, with positive projection-valued measure. T generates a completely monotone semigroup if and only if T is positive and m-dispersive and generates a bounded holomorphic semigroup.

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