Some criteria for the completeness of a system of root vectors of a linear operator in a Banach space

1969 ◽  
pp. 51-91 ◽  
Author(s):  
A. S. Markus
Keyword(s):  
Banach Space ◽  
Linear Operator

Projections on Tree-Like banach Spaces

1985 ◽  
Vol 37 (5) ◽  
pp. 908-920
Author(s):  
A. D. Andrew
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Banach Spaces ◽  
Bounded Linear Operator ◽  
Unconditional Basis ◽  
Reflexive Banach Space ◽  
Separable Space ◽  
Bounded Linear ◽  
Tree Space

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.

scholarly journals A note on best approximation and invertibility of operators on uniformly convex Banach spaces

1991 ◽  
Vol 14 (3) ◽  
pp. 611-614 ◽  
Author(s):  
James R. Holub
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Banach Spaces ◽  
Best Approximation ◽  
Bounded Linear Operator ◽  
Convex Banach Space ◽  
Sufficient Condition ◽  
Uniformly Convex ◽  
Bounded Linear ◽  
Uniformly Convex Banach Spaces

It is shown that ifXis a uniformly convex Banach space andSa bounded linear operator onXfor which‖I−S‖=1, thenSis invertible if and only if‖I−12S‖<1. From this it follows that ifSis invertible onXthen either (i)dist(I,[S])<1, or (ii)0is the unique best approximation toIfrom[S], a natural (partial) converse to the well-known sufficient condition for invertibility thatdist(I,[S])<1.

scholarly journals BOUNDED LINEAR OPERATORS ON SPACES IN NORMED DUALITY

2007 ◽  
Vol 49 (1) ◽  
pp. 145-154
Author(s):  
BRUCE A. BARNES
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Integral Operators ◽  
Continuous Functions ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Closed Range ◽  
Bounded Linear ◽  
Spaces Of Continuous Functions

Abstract.LetTbe a bounded linear operator on a Banach spaceW, assumeWandYare in normed duality, and assume thatThas adjointT†relative toY. In this paper, conditions are given that imply that for all λ≠0, λ−Tand λ −T†maintain important standard operator relationships. For example, under the conditions given, λ −Thas closed range if, and only if, λ −T†has closed range.These general results are shown to apply to certain classes of integral operators acting on spaces of continuous functions.

scholarly journals Iterations converging to distinct solutions of some nonlinear operator equations in Banach space

1986 ◽  
Vol 9 (3) ◽  
pp. 583-587
Author(s):  
Ioannis K. Argyros
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Nonlinear Operator ◽  
Operator Equations ◽  
Nonlinear Operator Equations

We examine the solvability of multilinear equations of the formMk(x,x,…,x)−k   times−=y,   k=2,3,…whereMkis ak-linear operator on a Banach spaceXandy∈Xis fixed.

scholarly journals Almost Surjective Epsilon-Isometry in The Reflexive Banach Spaces

CAUCHY ◽  
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

<p class="AbstractCxSpFirst">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  <em>f </em>: <em>X to</em><em> Y</em>, where <em>Y</em> is a reflexive Banach space, then there exists a bounded linear operator   <em>T</em> : <em>Y to</em><em> X</em>  with  such that</p><p class="AbstractCxSpMiddle">  </p><p class="AbstractCxSpLast">for every x in X.</p>

scholarly journals Separation problem for the Grushin differential operator in Banach spaces

2014 ◽  
Vol 30 (1) ◽  
pp. 31-37
Author(s):  
H. A. ATIA ◽  
◽  
Keyword(s):  
Banach Space ◽  
Differential Operator ◽  
Linear Operator ◽  
Banach Spaces ◽  
Bounded Linear Operator ◽  
Separation Problem ◽  
Bounded Linear

Our goal in this work is to study the separation problem for the Grushin differential operator formed by ... in the Banach space H1(R2), where the potential Q(x, y) ∈ L(1), is a bounded linear operator which transforms at 1 in value of (x, y).

scholarly journals UNA NUEVA FORMA DEL TEOREMA DE KANTOROVICH PARA EL ME´ TODO DE NEWTON

2017 ◽  
Vol 23 (1) ◽  
pp. 79
Author(s):  
Leopoldo Paredes Soria ◽  
Pedro Canales García
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Unique Solution ◽  
Newton’S Method ◽  
Convergence Theorem ◽  
Newton's Method ◽  
Convergence Condition ◽  
Palabras Clave ◽  
Lipschitz Conditions

Una nueva forma de convergencia de tipo Kantorovich para el me´todo de Newton es establecido para aproximarse localmente a una solucio´n u´nica de la ecuacio´n F (x) = 0 definido sobre un espacio de Banach. Se asume que el operador F es dos veces diferenciable Fre´chet, y que Fr, F rr satisface las condiciones de Lipschitz. Nuestra condicio´n de convergencia difiere de los me´todos conocidos y por lo tanto tiene un valor teo´rico y pra´ctico Palabras clave.-Operador lineal, Diferenciable Fre´chet, Sucesio´n convergente, Unicidad. ABSTRACTA new Kantorovich-type convergence theorem for Newton’s method is established for approximating a locally unique solution of an equation F (x) = 0 defined on a Banach space. It is assumed that the operator F is twice Fre´chet differentiable, and that Fr, F rr satisfy Lipschitz conditions. Our convergence condition differs from earlier ones and therefore it has theoretical and practical value. Keywords.-Linear operator, Differentiable Fre´chet, Convergent succession, Uniqueness.

The Range Sequence of an Operator

1974 ◽  
Vol 17 (1) ◽  
pp. 145-147
Author(s):  
F.-H. Vasilescu
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Linear Subspaces ◽  
Image Position

Let T be a linear operator on a Banach space X and consider the sequence of rangeswhere the inclusions are not necessarily proper. The linear subspaces Xn=TnX (n>0) are, in general, not closed but they have some remarkable properties [1], [2]. Let X0=X and denote by |x|0 (x∈X0) the norm of X0.

scholarly journals Semi-embeddings of Banach spaces which are hereditarily c0

1983 ◽  
Vol 26 (2) ◽  
pp. 163-167 ◽  
Author(s):  
L. Drewnowski
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Vector Space ◽  
Unit Ball ◽  
Banach Spaces ◽  
Continuous Linear Operator ◽  
Closed Unit Ball ◽  
Continuous Linear ◽  
Scattered Space ◽  
Complete Set

Following Lotz, Peck and Porta [9], a continuous linear operator from one Banach space into another is called a semi-embedding if it is one-to-one and maps the closed unit ball of the domain onto a closed (hence complete) set. (Below we shall allow the codomain to be an F-space, i.e., a complete metrisable topological vector space.) One of the main results established in [9] is that if X is a compact scattered space, then every semi-embedding of C(X) into another Banach space is an isomorphism ([9], Main Theorem, (a)⇒(b)).

scholarly journals On first-order differential operators with Bohr-Neugebauer type property

1989 ◽  
Vol 12 (3) ◽  
pp. 473-476 ◽  
Author(s):  
Aribindi Satyanarayan Rao
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Linear Operator ◽  
Real Line ◽  
Differential Operators ◽  
Bounded Solution ◽  
Almost Periodic ◽  
Bounded Linear ◽  
First Order ◽  
Type Property

We consider a differential equationddtu(t)-Bu(t)=f(t), where the functions u and f map the real line into a Banach space X and B: X→X is a bounded linear operator. Assuming that any Stepanov-bounded solution u is Stepanov almost-periodic when f is Bochner almost-periodic, we establish that any Stepanov-bounded solution u is Bochner almost-periodic when f is Stepanov almost-periodic. Some examples are given in which the operatorddt-B is shown to satisfy our assumption.

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