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.

A representation theorem for a complete Boolean algebra of projections

1979 ◽  
Vol 83 (3-4) ◽  
pp. 225-237 ◽  
Author(s):  
H. R. Dowson ◽  
T. A. Gillespie
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Boolean Algebra ◽  
Complex Banach Space ◽  
Complex Hilbert Space ◽  
Complete Boolean Algebra ◽  
Norm Topology ◽  
Weak Operator ◽  
Bounded Sets ◽  
Algebra Of Operators

SynopsisLet B be a complete Boolean algebra of projections on a complex Banach space X and let (B) denote the closed algebra of operators generated by B in the norm topology. It is shown that there is a complex Hilbert space H, a complete Boolean algebra B0 of self-adjoint projections on H, and an algebraic isomorphism of B onto B. This isomorphism is bicontinuous when B and B are endowed with the norm topologies, the weak operator topologies or the ultraweak operator topologies. It is also bicontinuous on bounded sets with respect to the strong operator topologies on B and B. As an application, it is shown that the weak and ultraweak operator topologies in fact coincide on B.

scholarly journals Isometries on extremely non-complex Banach spaces

2010 ◽  
Vol 10 (2) ◽  
pp. 325-348 ◽  
Author(s):  
Piotr Koszmider ◽  
Miguel Martín ◽  
Javier Merí
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Linear Operator ◽  
Banach Spaces ◽  
Real Banach Space ◽  
Complex Banach Space ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Group Of Isometries

AbstractGiven a separable Banach space E, we construct an extremely non-complex Banach space (i.e. a space satisfying that ‖ Id + T2 ‖ = 1 + ‖ T2 ‖ for every bounded linear operator T on it) whose dual contains E* as an L-summand. We also study surjective isometries on extremely non-complex Banach spaces and construct an example of a real Banach space whose group of surjective isometries reduces to ±Id, but the group of surjective isometries of its dual contains the group of isometries of a separable infinite-dimensional Hilbert space as a subgroup.

scholarly journals Support projections on Banach spaces

1977 ◽  
Vol 18 (1) ◽  
pp. 13-15 ◽  
Author(s):  
P. G. Spain
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Linear Operator ◽  
Banach Algebra ◽  
Banach Spaces ◽  
Bounded Linear Operator ◽  
Complex Banach Space ◽  
Linear Operators ◽  
Bounded Linear ◽  
Support Projection

Each bounded linear operator a on a Hilbert space K has a hermitian left-support projection p such that and (1 – p)K = ker α* = ker αα*. I demonstrate here that certain operators on Banach spaces also have left supports.Throughout this paper X will be a complex Banach space with norm-dual X', and L(X) will be the Banach algebra of bounded linear operators on X. Two linear subspaces Y and Z of X are orthogonal (in the sense of G. Birkhoff) if ∥ y ∥ ≦ ∥ y + z ∥ (y ∈Y, z ∈ Z); this orthogonality relation is not, in general, symmetric. It is easy to see that pX is orthogonal to (1 – p)X if and only if the norm of p is 0 or 1, when p is a projection on X.

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 On the M-hypercyclicity of cosine function on Banach spaces

2019 ◽  
Vol 38 (3) ◽  
pp. 133-140
Author(s):  
Abdelaziz Tajmouati ◽  
Abdeslam El Bakkali ◽  
Ahmed Toukmati
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Complex Banach Space ◽  
Cosine Function ◽  
Infinite Dimensional ◽  
Uniformly Continuous

In this paper we introduce and study the M-hypercyclicity of strongly continuous cosine function on separable complex Banach space, and we give the criteria for cosine function to be M-hypercyclic. We also prove that every separable infinite dimensional complex Banach space admits a uniformly continuous cosine function.

scholarly journals Multipliers on Vector Valued Bergman Spaces

2002 ◽  
Vol 54 (6) ◽  
pp. 1165-1186 ◽  
Author(s):  
Oscar Blasco ◽  
José Luis Arregui
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Bergman Space ◽  
Unit Disc ◽  
Bergman Spaces ◽  
Complex Banach Space ◽  
Bounded Operators ◽  
Taylor Coefficients ◽  
Vector Valued

AbstractLet X be a complex Banach space and let Bp(X) denote the vector-valued Bergman space on the unit disc for 1 ≤ p < ∞. A sequence (Tn)n of bounded operators between two Banach spaces X and Y defines a multiplier between Bp(X) and Bq(Y) (resp. Bp(X) and lq(Y)) if for any function we have that belongs to Bq(Y) (resp. (Tn(xn))n ∈ lq(Y)). Several results on these multipliers are obtained, some of them depending upon the Fourier or Rademacher type of the spaces X and Y. New properties defined by the vector-valued version of certain inequalities for Taylor coefficients of functions in Bp(X) are introduced.

Weyl's theorem for the square of operator and perturbations

2015 ◽  
Vol 17 (05) ◽  
pp. 1450042
Author(s):  
Weijuan Shi ◽  
Xiaohong Cao
Keyword(s):  
Hilbert Space ◽  
Compact Operator ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Weyl’S Theorem ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Weyl Spectrum ◽  
The Stability

Let H be an infinite-dimensional separable complex Hilbert space and B(H) the algebra of all bounded linear operators on H. T ∈ B(H) satisfies Weyl's theorem if σ(T)\σw(T) = π00(T), where σ(T) and σw(T) denote the spectrum and the Weyl spectrum of T, respectively, π00(T) = {λ ∈ iso σ(T) : 0 < dim N(T - λI) < ∞}. T ∈ B(H) is said to have the stability of Weyl's theorem if T + K satisfies Weyl's theorem for all compact operator K ∈ B(H). In this paper, we characterize the operator T on H satisfying the stability of Weyl's theorem holds for T2.

scholarly journals On a Numerical Radius Preserving Onto Isometry onL(X)

2016 ◽  
Vol 2016 ◽  
pp. 1-3
Author(s):  
Sun Kwang Kim
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Numerical Range ◽  
Complex Banach Space ◽  
Uniform Convexity ◽  
Numerical Radius ◽  
Uniform Smoothness

We study a numerical radius preserving onto isometry onL(X). As a main result, whenXis a complex Banach space having both uniform smoothness and uniform convexity, we show that an onto isometryTonL(X)is numerical radius preserving if and only if there exists a scalarcTof modulus 1 such thatcTTis numerical range preserving. The examples of such spaces are Hilbert space andLpspaces for1<p<∞.

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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