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<∞.

Complex strict and uniform convexity and hyponormal operators

1984 ◽  
Vol 96 (3) ◽  
pp. 483-493 ◽  
Author(s):  
Kirsti Mattila
Keyword(s):  
Banach Space ◽  
Dual Space ◽  
Numerical Range ◽  
Complex Banach Space ◽  
Linear Operators ◽  
Uniform Convexity ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Hyponormal Operators ◽  
Image Position

Let X be a complex Banach space. We denote by X* the dual space of X and by B(X) the space of all bounded linear operators on X. The (spatial) numerical range of an operator TεB(X) is defined as the setIf V(T) ⊂ ℝ, then T is called hermitian. More about numerical ranges may be found in [8] and [9].

scholarly journals A result on hermitian operators

1989 ◽  
Vol 31 (1) ◽  
pp. 71-72
Author(s):  
J. E. Jamison ◽  
Pei-Kee Lin
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Numerical Range ◽  
Complex Banach Space ◽  
Bounded Linear ◽  
Image Position

Let X be a complex Banach space. For any bounded linear operator T on X, the (spatial) numerical range of T is denned as the setIf V(T) ⊆ R, then T is called hermitian. Vidav and Palmer (see Theorem 6 of [3, p. 78] proved that if the set {H + iK:H and K are hermitian} contains all operators, then X is a Hilbert space. It is natural to ask the following question.

Growth conditions on subharmonic functions and resolvents of operators

1985 ◽  
Vol 97 (2) ◽  
pp. 321-324
Author(s):  
J. R. Partington
Keyword(s):  
Banach Space ◽  
Numerical Range ◽  
Bounded Operator ◽  
Complex Banach Space ◽  
Growth Conditions ◽  
Subharmonic Functions ◽  
Image Position

Let X be a complex Banach space and T a bounded operator on X. The numerical range of T is defined by

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 Biconcave-function characterisations of UMD and Hilbert spaces

1993 ◽  
Vol 47 (2) ◽  
pp. 297-306 ◽  
Author(s):  
Jinsik Mok Lee
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Hilbert Spaces ◽  
Complex Banach Space ◽  
Unit Interval ◽  
Martingale Differences ◽  
Almost Everywhere ◽  
Integrable Functions ◽  
Image Position ◽  
Umd Banach Spaces

Suppose that X is a real or complex Banach space with norm |·|. Then X is a Hilbert space if and only iffor all x in X and all X-valued Bochner integrable functions Y on the Lebesgue unit interval satisfying EY = 0 and |x − Y| ≤ 2 almost everywhere. This leads to the following biconcave-function characterisation: A Banach space X is a Hilbert space if and only if there is a biconcave function η: {(x, y) ∈ X × X: |x − y| ≤ 2} → R such that η(0, 0) = 2 andIf the condition η(0, 0) = 2 is eliminated, then the existence of such a function η characterises the class UMD (Banach spaces with the unconditionally property for martingale differences).

scholarly journals Numerical ranges of conjugations and antilinear operators on a Banach space

Filomat ◽  
2021 ◽  
Vol 35 (8) ◽  
pp. 2715-2720
Author(s):  
Muneo Chō ◽  
Injo Hur ◽  
Ji Lee
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Unit Circle ◽  
Banach Spaces ◽  
Unit Disc ◽  
Numerical Range ◽  
Space Setting ◽  
Path Connectedness ◽  
The Given

In this paper, we prove that the numerical range of a conjugation on Banach spaces, using the connected property, is either the unit circle or the unit disc depending the dimension of the given Banach space. When a Banach space is reflexive, we have the same result for the numerical range of a conjugation by applying path-connectedness which is applicable to the Hilbert space setting. In addition, we show that the numerical ranges of antilinear operators on Banach spaces are contained in annuli.

On martingales with values in a complex Banach space

1988 ◽  
Vol 104 (2) ◽  
pp. 399-406 ◽  
Author(s):  
D. J. H. Garling
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Complex Banach Space ◽  
Uniform Convexity ◽  
Martingale Inequalities ◽  
Nikodym Property

In recent years it has become clear that there are several ways in which complex Banach spaces can differ quite markedly from their real counterparts, and many of these concern martingales. Thus, in [6] complex uniform convexity was related to martingale inequalities, in [3] and [7] the convergence of L1-bounded analytic martingales was considered and in [8] this property was related to the analytic Radon–Nikodym property.

Orthonormal systems in Banach spaces and their applications

1976 ◽  
Vol 79 (3) ◽  
pp. 493-510 ◽  
Author(s):  
N. J. Kalton ◽  
G. V. Wood
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Orthonormal System ◽  
Numerical Range ◽  
Complex Banach Space ◽  
General Complex

By an orthonormal system in a general complex Banach space, we mean a collection {eα: α ∈ } it vectors such that, for each α, there is an hermitian (in the numerical range sense, see (4)) projection Pα whose range is lin (eα) and such that PαPβ = 0, if α ≠ β. This paper is devoted to the study of orthonormal systems in general Banach spaces, and their applications to problems of characterizing isometries and hermitian operators.

scholarly journals Modulus of nearly uniform smoothness and Lindenstrauss formulae

1995 ◽  
Vol 37 (2) ◽  
pp. 143-153 ◽  
Author(s):  
Tomás Domínguez Benavides
Keyword(s):  
Banach Space ◽  
Strong Relationship ◽  
Uniform Convexity ◽  
Measures Of Noncompactness ◽  
Conjugate Space ◽  
Uniform Smoothness ◽  
Image Position ◽  
Smooth Space

AbstractThe Lindenstrauss formulawhich states a strong relationship between the (Clarkson) modulus of uniform convexity δx of a Banach space X and the modulus of uniform smoothness px* of the conjugate space X*, is well known. Following the idea of the definitions of nearly uniform smooth space by S. Prus and modulus of uniform smoothness we define a modulus of nearly uniform smoothness and prove some Lindenstrauss type formulae concerning this modulus and the modulus of nearly uniform convexity for some measures of noncompactness.

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