On the Poincaré-Hopf Theorem for Functionals Defined on Banach Spaces

1. In this paper, we investigate the ranges of projections on certain Banach spaces of functions defined on a diadic tree. The notion of a “tree-like” Banach space is due to James 4], who used it to construct the separable space JT which has nonseparable dual and yet does not contain l1. This idea has proved useful. In [3], Hagler constructed a hereditarily c0 tree space, HT, and Schechtman [6] constructed, for each 1 ≦ p ≦ ∞, a reflexive Banach space, STp with a 1-unconditional basis which does not contain lp yet is uniformly isomorphic to for each n.In [1] we showed that if U is a bounded linear operator on JT, then there exists a subspace W ⊂ JT, isomorphic to JT such that either U or (1 — U) acts as an isomorphism on W and UW or (1 — U)W is complemented in JT. In this paper, we establish this result for the Hagler and Schechtman tree spaces.

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Almost Surjective Epsilon-Isometry in The Reflexive Banach Spaces

CAUCHY ◽

10.18860/ca.v4i4.4100 ◽

2017 ◽

Vol 4 (4) ◽

pp. 167

Author(s):

Minanur Rohman

Keyword(s):

Banach Space ◽

Linear Operator ◽

Banach Spaces ◽

Lorentz Space ◽

Bounded Linear Operator ◽

Closed Subspace ◽

Reflexive Banach Space ◽

Reflexive Banach Spaces ◽

Bounded Linear ◽

Star Topology

In this paper, we will discuss some applications of almost surjective epsilon-isometry mapping, one of them is in Lorentz space ( L_(p,q)-space). Furthermore, using some classical theorems of w star-topology and concept of closed subspace -complemented, for every almost surjective epsilon-isometry mapping f : X to Y, where Y is a reflexive Banach space, then there exists a bounded linear operator T : Y to X with such that for every x in X.

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Hermitian operators and isometries on sums of Banach spaces

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500028583 ◽

1989 ◽

Vol 32 (2) ◽

pp. 169-191 ◽

Cited By ~ 9

Author(s):

R. J. Fleming ◽

J. E. Jamison

Keyword(s):

Banach Space ◽

Banach Spaces ◽

Sequence Space ◽

Reflexive Banach Space ◽

Geometric Theory ◽

Orlicz Sequence Space ◽

Vector Sequence ◽

Banach Sequence Space ◽

Banach Sequence

Let E be a Banach sequence space with the property that if (αi) ∈ E and |βi|≦|αi| for all i then (βi) ∈ E and ‖(βi)‖E≦‖(αi)‖E. For example E could be co, lp or some Orlicz sequence space. If (Xn) is a sequence of real or complex Banach spaces, then E can be used to construct a vector sequence space which we will call the E sum of the Xn's and symbolize by ⊕EXn. Specifically, ⊕EXn = {(xn)|(xn)∈Xn and (‖xn‖)∈E}. The E sum is a Banach space with norm defined by: ‖(xn)‖ = ‖(‖xn‖)‖E. This type of space has long been the source of examples and counter-examples in the geometric theory of Banach spaces. For instance, Day [7] used E=lp and Xk=lqk, with appropriate choice of qk, to give an example of a reflexive Banach space not isomorphic to any uniformly conves Banach space. Recently VanDulst and Devalk [33] have considered Orlicz sums of Banach spaces in their studies of Kadec-Klee property.

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Interpolating sequence on certain Banach spaces of analytic functions

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700020219 ◽

2002 ◽

Vol 65 (2) ◽

pp. 177-182 ◽

Cited By ~ 4

Author(s):

B. Yousefi

Keyword(s):

Banach Space ◽

Banach Spaces ◽

Connected Domain ◽

Analytic Functions ◽

Reflexive Banach Space ◽

Bounded Operator ◽

Interpolating Sequence ◽

Spaces Of Analytic Functions

Let G be a finitely connected domain and let X be a reflexive Banach space of functions analytic on G which admits the multiplication Mz as a polynomially bounded operator. We give some conditions that a sequence in G has an interpolating subsequence for X.

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Duals of Banach spaces which admit nontrivial smooth functions

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700043768 ◽

1974 ◽

Vol 11 (2) ◽

pp. 161-166 ◽

Cited By ~ 3

Author(s):

K. John ◽

V. Zizler

Keyword(s):

Banach Space ◽

Banach Spaces ◽

Differentiable Function ◽

Continuous Linear ◽

Smooth Functions ◽

One To One ◽

Fréchet Differentiable

If a Banach space X admits a continuously Fréchet differentiable function with bounded nonempty support, then X* admits a projectional resolution of identity and a continuous linear one-to-one map into co(Γ).

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Bregmanf-Projection Operator with Applications to Variational Inequalities in Banach Spaces

Abstract and Applied Analysis ◽

10.1155/2014/594285 ◽

2014 ◽

Vol 2014 ◽

pp. 1-10 ◽

Cited By ~ 1

Author(s):

Chin-Tzong Pang ◽

Eskandar Naraghirad ◽

Ching-Feng Wen

Keyword(s):

Banach Space ◽

Variational Inequalities ◽

Banach Spaces ◽

Projection Operator ◽

Dual Space ◽

Reflexive Banach Space ◽

Lower Semicontinuous ◽

Existence Of A Solution ◽

Bregman Functions ◽

Proper Convex

Using Bregman functions, we introduce the new concept of Bregman generalizedf-projection operatorProjCf, g:E*→C, whereEis a reflexive Banach space with dual spaceE*; f: E→ℝ∪+∞is a proper, convex, lower semicontinuous and bounded from below function;g: E→ℝis a strictly convex and Gâteaux differentiable function; andCis a nonempty, closed, and convex subset ofE. The existence of a solution for a class of variational inequalities in Banach spaces is presented.

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A General Iterative Method for a Nonexpansive Semigroup in Banach Spaces with Gauge Functions

Journal of Applied Mathematics ◽

10.1155/2012/506976 ◽

2012 ◽

Vol 2012 ◽

pp. 1-14

Author(s):

Kamonrat Nammanee ◽

Suthep Suantai ◽

Prasit Cholamjiak

Keyword(s):

Banach Space ◽

Iterative Method ◽

Strong Convergence ◽

Banach Spaces ◽

Reflexive Banach Space ◽

Duality Mapping ◽

Nonexpansive Semigroup ◽

Gauge Functions ◽

Implicit And Explicit ◽

General Iterative Method

We study strong convergence of the sequence generated by implicit and explicit general iterative methods for a one-parameter nonexpansive semigroup in a reflexive Banach space which admits the duality mappingJφ, whereφis a gauge function on[0,∞). Our results improve and extend those announced by G. Marino and H.-K. Xu (2006) and many authors.

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On projectional resolution of identity on the duals of certain Banach spaces

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700013344 ◽

1987 ◽

Vol 35 (3) ◽

pp. 363-371 ◽

Cited By ~ 5

Author(s):

M. Fabian

Keyword(s):

Banach Space ◽

Banach Spaces ◽

Differentiable Function ◽

Continuous Derivative ◽

Continuous Linear ◽

Resolution Of The Identity ◽

One To One ◽

Rotund Norm

A consequence of the main proposition includes results of Tacon, and John and Zizler and says: If a Banach space X possesses a continuous Gâteaux differentiable function with bounded nonempty support and with norm-weak continuous derivative, then its dual X* admits a projectional resolution of the identity and a continuous linear one-to-one mapping into c0 (Γ). The proof is easy and selfcontained and does not use any complicated geometrical lemma. If the space X is in addition weakly countably determined, then X* has an equivalent dual locally uniformly rotund norm. It is also shown that l∞ admits no continuous Gâteaux differentiable function with bounded nonempty support.

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Ideals of compact operators

Journal of the Australian Mathematical Society ◽

10.1017/s144678870001017x ◽

2004 ◽

Vol 77 (1) ◽

pp. 91-110 ◽

Cited By ~ 8

Author(s):

Åsvald Lima ◽

Eve Oja

Keyword(s):

Banach Space ◽

Unit Ball ◽

Banach Spaces ◽

Approximation Property ◽

Closed Subspace ◽

Reflexive Banach Space ◽

Compact Operators ◽

Compact Approximation

AbstractWe give an example of a Banach space X such that K (X, X) is not an ideal in K (X, X**). We prove that if z* is a weak* denting point in the unit ball of Z* and if X is a closed subspace of a Banach space Y, then the set of norm-preserving extensions H B(x* ⊗ z*) ⊆ (Z*, Y)* of a functional x* ⊗ Z* ∈ (Z ⊗ X)* is equal to the set H B(x*) ⊗ {z*}. Using this result, we show that if X is an M-ideal in Y and Z is a reflexive Banach space, then K (Z, X) is an M-ideal in K(Z, Y) whenever K (Z, X) is an ideal in K (Z, Y). We also show that K (Z, X) is an ideal (respectively, an M-ideal) in K (Z, Y) for all Banach spaces Z whenever X is an ideal (respectively, an M-ideal) in Y and X * has the compact approximation property with conjugate operators.

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Strictly Cyclic Functionals, Reflexivity, and Hereditary Reflexivity of Operator Algebras

Abstract and Applied Analysis ◽

10.1155/2012/434308 ◽

2012 ◽

Vol 2012 ◽

pp. 1-12

Author(s):

Quanyuan Chen ◽

Xiaochun Fang

Keyword(s):

Banach Space ◽

Banach Spaces ◽

Operator Algebra ◽

Operator Algebras ◽

Reflexive Banach Space ◽

Cyclic Operator ◽

Abelian Subalgebra ◽

Cyclic Vectors ◽

Reflexive Algebra

This paper is concerned with strictly cyclic functionals of operator algebras on Banach spaces. It is shown that ifXis a reflexive Banach space andAis a norm-closed semisimple abelian subalgebra ofB(X)with a strictly cyclic functionalf∈X∗, thenAis reflexive and hereditarily reflexive. Moreover, we construct a semisimple abelian operator algebra having a strictly cyclic functional but having no strictly cyclic vectors. The hereditary reflexivity of an algbra of this type can follow from theorems in this paper, but does not follow directly from the known theorems that, if a strictly cyclic operator algebra on Banach spaces is semisimple and abelian, then it is a hereditarily reflexive algebra.

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