scholarly journals Hilbert Space Frames Containing a Riesz Basis and Banach Spaces Which Have No Subspace Isomorphic toc0

1996 ◽  
Vol 202 (3) ◽  
pp. 940-950 ◽  
Author(s):  
Peter G. Casazza ◽  
Ole Christensen
Keyword(s):  
Hilbert Space ◽  
Banach Spaces ◽  
Riesz Basis ◽  
Space Frames

GENERALIZED p-FRAME IN SEPARABLE COMPLEX BANACH SPACES

2010 ◽  
Vol 08 (01) ◽  
pp. 133-148 ◽  
Author(s):  
XIANG-CHUN XIAO ◽  
YU-CAN ZHU ◽  
XIAO-MING ZENG
Keyword(s):  
Banach Space ◽  
Functional Analysis ◽  
Hilbert Space ◽  
Banach Spaces ◽  
Riesz Basis ◽  
Complex Banach Space ◽  
Complex Hilbert Space ◽  
The Stability

The concept of g-frame and g-Riesz basis in a complex Hilbert space was introduced by Sun.18 In this paper, we generalize the g-frame and g-Riesz basis in a complex Hilbert space to a complex Banach space. Using operators theory and methods of functional analysis, we give some characterizations of a g-frame or a g-Riesz basis in a complex Banach space. We also give a result about the stability of g-frame in a complex Banach space.

scholarly journals A characterisation of Hilbert spaces via orthogonality and proximinality

2005 ◽  
Vol 71 (1) ◽  
pp. 107-111
Author(s):  
Fathi B. Saidi
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Banach Spaces ◽  
Hilbert Spaces ◽  
Real Banach Space ◽  
Nonzero Element ◽  
Dimensional Subspace ◽  
Two Dimensional ◽  
Orthogonal Direction ◽  
Open Problems

In this paper we adopt the notion of orthogonality in Banach spaces introduced by the author in [6]. There, the author showed that in any two-dimensional subspace F of E, every nonzero element admits at most one orthogonal direction. The problem of existence of such orthogonal direction was not addressed before. Our main purpose in this paper is the investigation of this problem in the case where E is a real Banach space. As a result we obtain a characterisation of Hilbert spaces stating that, if in every two-dimensional subspace F of E every nonzero element admits an orthogonal direction, then E is isometric to a Hilbert space. We conclude by presenting some open problems.

scholarly journals Characterizing Complete Erdős Space

2009 ◽  
Vol 61 (1) ◽  
pp. 124-140 ◽  
Author(s):  
Jan J. Dijkstra ◽  
Jan van Mill
Keyword(s):  
Hilbert Space ◽  
Banach Spaces ◽  
Closed Subspace ◽  
Convergent Sequence ◽  
Cantor Set

Abstract. The space now known as complete Erdős space was introduced by Paul Erdős in 1940 as the closed subspace of the Hilbert space ℓ2 consisting of all vectors such that every coordinate is in the convergent sequence ﹛0﹜ ∪ ﹛1/n : n ∈ℕ﹜. In a solution to a problem posed by Lex G. Oversteegen we present simple and useful topological characterizations of . As an application we determine the class of factors of . In another application we determine precisely which of the spaces that can be constructed in the Banach spaces ℓp according to the ‘Erdős method’ are homeomorphic to . A novel application states that if I is a Polishable Fσ-ideal on ω, then I with the Polish topology is homeomorphic to either ℤ, the Cantor set 2ω, ℤ × 2ω, or . This last result answers a question that was asked by Stevo Todorčević.

Representation of Banach Algebras with an Involution

1957 ◽  
Vol 9 ◽  
pp. 435-442
Author(s):  
J. A. Schatz
Keyword(s):  
Hilbert Space ◽  
Banach Spaces ◽  
Banach Algebras ◽  
Bounded Operators ◽  
Uniformly Closed

In 1943 Gelfand and Neumark (3) characterized uniformly closed self-adjoint algebras of bounded operators on a Hilbert space as Banach algebras with an involution (a conjugate linear anti-isomorphism of period two) satisfying several additional conditions. The main purpose of this paper is to point out that if we consider algebras of bounded operators on complex Banach spaces more general than Hilbert space, then we can represent a larger class of algebras by essentially the same methods.

Riesz basis of exponential family for a hyperbolic system

2019 ◽  
Vol 25 (1) ◽  
pp. 13-23
Author(s):  
Abdelkader Intissar ◽  
Aref Jeribi ◽  
Ines Walha
Keyword(s):  
Hilbert Space ◽  
Linear System ◽  
Hyperbolic System ◽  
Riesz Basis ◽  
Infinitesimal Generator ◽  
Exponential Family ◽  
Analysis Method ◽  
Spectral Analysis Method ◽  
Weaker Conditions ◽  
Linear Hyperbolic System

Abstract This paper studies a linear hyperbolic system with boundary conditions that was first studied under some weaker conditions in [8, 11]. Problems on the expansion of a semigroup and a criterion for being a Riesz basis are discussed in the present paper. It is shown that the associated linear system is the infinitesimal generator of a {C_{0}} -semigroup; its spectrum consists of zeros of a sine-type function, and its exponential system {\{e^{\lambda_{n}t}\}_{n\geq 1}} constitutes a Riesz basis in {L^{2}[0,T]} . Furthermore, by the spectral analysis method, it is also shown that the linear system has a sequence of eigenvectors, which form a Riesz basis in Hilbert space, and hence the spectrum-determined growth condition is deduced.

Controllability of Abstract Systems of Fractional Order

2015 ◽  
Vol 18 (6) ◽  
Author(s):  
Therese Mur ◽  
Hernán R. Henríquez
Keyword(s):  
Differential Equation ◽  
Hilbert Space ◽  
Integral Equation ◽  
Control Systems ◽  
Banach Spaces ◽  
Fractional Differential Equation ◽  
Fractional Integral ◽  
First Order ◽  
Fractional Integral Equation ◽  
Fractional Differential

AbstractIn this paper we are concerned with the controllability of control systems governed by a fractional differential equation in Banach spaces. Using the properties of the Mittag-Leffler function we generalize to these systems a result of Korobov and Rabakh, which was established for first order systems. We apply our results to study the controllability of a system modeled by a fractional integral equation in a Hilbert space.

scholarly journals m-ISOMETRIC OPERATORS ON BANACH SPACES

2010 ◽  
Vol 03 (01) ◽  
pp. 1-19 ◽  
Author(s):  
Ould Ahmed Mahmoud Sid Ahmed
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Banach Spaces ◽  
Natural Generalization ◽  
Basic Properties

We introduce the class of m-isometric operators on Banach spaces. This generalizes to Banach space the m-isometric operators on Hilbert space introduced by Agler and Stankus. We establish some basic properties and we introduce the notion of m-invertibility as a natural generalization of the invertibility on Banach spaces.

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