MAZUR–ULAM PROPERTY OF THE SUM OF TWO STRICTLY CONVEX BANACH SPACES

2015 ◽  
Vol 93 (3) ◽  
pp. 473-485 ◽  
Author(s):  
JIAN-ZE LI
Keyword(s):  
Banach Spaces ◽  
Equivalent Form ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Extension Problem ◽  
Isometric Extension ◽  
Strictly Convex ◽  
Necessary And Sufficient ◽  
Strictly Convex Banach Spaces ◽  
Equivalent Conditions

In this article, we study the Mazur–Ulam property of the sum of two strictly convex Banach spaces. We give an equivalent form of the isometric extension problem and two equivalent conditions to decide whether all strictly convex Banach spaces admit the Mazur–Ulam property. We also find necessary and sufficient conditions under which the $\ell ^{1}$-sum and the $\ell ^{\infty }$-sum of two strictly convex Banach spaces admit the Mazur–Ulam property.

scholarly journals A NEW CONSTRUCTION FOR REGULAR SEMIGROUPS WITH QUASI-IDEAL ORTHODOX TRANSVERSALS

2009 ◽  
Vol 86 (2) ◽  
pp. 177-187 ◽  
Author(s):  
XIANGJUN KONG ◽  
XIANZHONG ZHAO
Keyword(s):  
Regular Semigroup ◽  
Structure Theorem ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Green’S Relations ◽  
Regular Semigroups ◽  
New Construction ◽  
Necessary And Sufficient ◽  
Equivalent Conditions ◽  
Orthodox Semigroups

AbstractIn any regular semigroup with an orthodox transversal, we define two sets R and L using Green’s relations and give necessary and sufficient conditions for them to be subsemigroups. By using R and L, some equivalent conditions for an orthodox transversal to be a quasi-ideal are obtained. Finally, we give a structure theorem for regular semigroups with quasi-ideal orthodox transversals by two orthodox semigroups R and L.

scholarly journals Checking of the positivity of descriptor linear systems by the use of the shuffle algorithm

2011 ◽  
Vol 21 (3) ◽  
pp. 287-298 ◽  
Author(s):  
Tadeusz Kaczorek
Keyword(s):  
Linear Systems ◽  
Continuous Time ◽  
Equivalent Form ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Descriptor Systems ◽  
State Equations ◽  
Necessary And Sufficient ◽  
Descriptor Linear Systems ◽  
Time Linear

Checking of the positivity of descriptor linear systems by the use of the shuffle algorithmNecessary and sufficient conditions for the positivity of descriptor continuous-time and discrete-time linear systems are established. The shuffle algorithm is applied to transform the state equations of the descriptor systems to their equivalent form for which necessary and sufficient conditions for their positivity have been derived. A procedure for checking the positivity of the descriptor systems is proposed and illustrated by numerical examples.

scholarly journals Balls intersection properties of Banach spaces

1992 ◽  
Vol 45 (2) ◽  
pp. 333-342 ◽  
Author(s):  
Dongjian Chen ◽  
Zhibao Hu ◽  
Bor-Luh Lin
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Sufficient Conditions ◽  
Intersection Property ◽  
Necessary And Sufficient Conditions ◽  
Asplund Space ◽  
Necessary And Sufficient

Necessary and sufficient conditions for a Banach space with the Mazur intersection property to be an Asplund space are given. It is proved that Mazur intersection property is determined by the separable subspaces of the space. Corresponding problems for a space to have the ball-generated property are considered. Some comments on possible renorming so that a space having the Mazur intersection property are given.

scholarly journals Datko criteria for uniform instability in Banach spaces

2021 ◽  
Vol 66 (1) ◽  
pp. 115-122
Author(s):  
Rovana Boruga Toma ◽  
Mihail Megan
Keyword(s):  
Banach Spaces ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Evolution Operators ◽  
Necessary And Sufficient

The main objective of this paper is to present some necessary and sufficient conditions of Datko type for the uniform exponential and uniform polynomial instability concepts for evolution operators in Banach spaces.

scholarly journals A unifying approach for some nonexpansiveness conditions on modular vector spaces

2020 ◽  
Vol 25 (5) ◽  
Author(s):  
Andrea Bejenaru ◽  
Mihai Postolache
Keyword(s):  
Fixed Points ◽  
Banach Spaces ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Vector Spaces ◽  
Minimizing Sequences ◽  
Ishikawa Iteration ◽  
New Class ◽  
Condition C ◽  
Necessary And Sufficient

This paper provides a new, symmetric, nonexpansiveness condition to extend the classical Suzuki mappings. The newly introduced property is proved to be equivalent to condition (E) on Banach spaces, while it leads to an entirely new class of mappings when going to modular vector spaces; anyhow, it still provides an extension for the modular version of condition (C). In connection with the newly defined nonexpansiveness, some necessary and sufficient conditions for the existence of fixed points are stated and proved. They are based on Mann and Ishikawa iteration procedures, convenient uniform convexities and properly selected minimizing sequences.

Right and Left Weak Approximation Properties in Banach Spaces

2009 ◽  
Vol 52 (1) ◽  
pp. 28-38 ◽  
Author(s):  
Changsun Choi ◽  
Ju Myung Kim ◽  
Keun Young Lee
Keyword(s):  
Banach Spaces ◽  
Approximation Property ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Weak Approximation ◽  
Weak Version ◽  
Approximation Properties ◽  
Necessary And Sufficient

AbstractNew necessary and sufficient conditions are established for Banach spaces to have the approximation property; these conditions are easier to check than the known ones. A shorter proof of a result of Grothendieck is presented, and some properties of a weak version of the approximation property are addressed.

scholarly journals FRÉCHET FRAMES, GENERAL DEFINITION AND EXPANSIONS

2014 ◽  
Vol 12 (02) ◽  
pp. 195-208 ◽  
Author(s):  
STEVAN PILIPOVIĆ ◽  
DIANA T. STOEVA
Keyword(s):  
Banach Spaces ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Sequence Spaces ◽  
General Definition ◽  
Fréchet Space ◽  
Frechet Space ◽  
Bk Space ◽  
Frame Expansions ◽  
Necessary And Sufficient

We define an (X1, Θ, X2)-frame with Banach spaces X2 ⊆ X1, ‖ ⋅ ‖1 ≤ ‖ ⋅ ‖2, and a BK-space [Formula: see text]. Then by the use of decreasing sequences of Banach spaces [Formula: see text] and of sequence spaces [Formula: see text], we define a General Fréchet frame on the Fréchet space [Formula: see text]. We obtain frame expansions of elements of XF and its dual [Formula: see text], as well of some of the generating spaces of XF with convergence in appropriate norms. Moreover, we determine necessary and sufficient conditions for a General pre-Fréchet frame to be a General Fréchet frame, as well as for the complementedness of the range of the analysis operator U : XF → ΘF. Several examples illustrate our investigations.

The Influence of Generalized Frattini Subgroups on the Solvability of a Finite Group

1970 ◽  
Vol 22 (1) ◽  
pp. 41-46 ◽  
Author(s):  
James C. Beidleman
Keyword(s):  
Finite Group ◽  
Solvable Group ◽  
Proper Subgroup ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Fitting Subgroup ◽  
Frattini Subgroup ◽  
A Finite Group ◽  
Necessary And Sufficient ◽  
Equivalent Conditions

1. The Frattini and Fitting subgroups of a finite group G have been useful subgroups in establishing necessary and sufficient conditions for G to be solvable. In [1, pp. 657-658, Theorem 1], Baer used these subgroups to establish several very interesting equivalent conditions for G to be solvable. One of Baer's conditions is that ϕ(S), the Frattini subgroup of S, is a proper subgroup of F(S), the Fitting subgroup of S, for each subgroup S ≠ 1 of G. Using the Fitting subgroup and generalized Frattini subgroups of certain subgroups of G we provide certain equivalent conditions for G to be a solvable group. One such condition is that F(S) is not a generalized Frattini subgroup of S for each subgroup S ≠ 1 of G. Our results are given in Theorem 1.

scholarly journals Integral Characterizations for Uniform Stability with Growth Rates in Banach Spaces

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 235
Author(s):  
Rovana Boruga(Toma) ◽  
Mihail Megan ◽  
Daniela Maria-Magdalena Toth
Keyword(s):  
Exponential Stability ◽  
Banach Spaces ◽  
Growth Rates ◽  
Evolution Operator ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Uniform Stability ◽  
Polynomial Stability ◽  
Uniform Exponential Stability ◽  
Necessary And Sufficient

The aim of this paper is to present some integral characterizations for the concept of uniform stability with growth rates in Banach spaces. In this sense, we prove necessary and sufficient conditions (of Barbashin and Datko type) for an evolution operator to be uniform h- stable. As particular cases of this notion, we obtain four characterizations for uniform exponential stability and two characterizations for uniform polynomial stability.

scholarly journals On uniform exponential trisplitting for cocycles of linear operators in Banach spaces

2018 ◽  
Vol 56 (2) ◽  
pp. 81-103
Author(s):  
Larisa Elena Biriş ◽  
Claudia Luminiţa Mihiţ ◽  
Traian Ceauşu ◽  
Ioan-Lucian Popa
Keyword(s):  
Banach Spaces ◽  
Lyapunov Functions ◽  
Sufficient Conditions ◽  
Type A ◽  
Point Of View ◽  
Necessary And Sufficient Conditions ◽  
Linear Operators ◽  
Skew Product ◽  
Exponential Trichotomy ◽  
Necessary And Sufficient

AbstractThe aim of this paper is to study the concept of uniform exponential trisplitting for skew-product semiflow in Banach spaces. This concept is a generalisation of the well-known concept of uniform exponential trichotomy. We obtain necessary and sufficient conditions for this concept of Datko’s type. a character-isation in terms of Lyapunov functions is provided. The results are obtained from the point of view of the projector families, i.e. invariant and strongly invariant.

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