scholarly journals The evaluation functionals associated with an algebra of bounded operators

1969 ◽  
Vol 10 (1) ◽  
pp. 73-76 ◽  
Author(s):  
J. Duncan
Keyword(s):  
Banach Space ◽  
Normed Space ◽  
Finite Rank ◽  
Bounded Operators ◽  
Image Position

In this note we shall employ the notation of [1] without further mention. Thus X denotes a normed space and P the subset of X × X′ given byGiven a subalgebra of B(X), the set {Φ(X,f):(x,f) ∈ P} of evaluation functional on is denoted by II. We shall prove that if X is a Banach space and if contains all the bounded operators of finite rank, then Π is norm closed in ′. We give an example to show that Π need not be weak* closed in ″. We show also that FT need not be norm closed in ″ if X is not complete.

scholarly journals On well-bounded operators of class Г

1983 ◽  
Vol 24 (1) ◽  
pp. 1-5
Author(s):  
Adnan A. S. Jibril
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Bounded Operators ◽  
Strong Limit ◽  
Riemann Sums ◽  
Image Position

Let T be a linear operator acting in a Banach space X. It has been shown by Smart [5] and Ringrose [3] that, if X is reflexive, then T is well-bounded if and only if it may be expressed in the formwhere {E(λ)} is a suitable family of projections in X and the integral exists as the strong limit of Riemann sums.

scholarly journals Some inequalities for norm unitaries in Banach algebras

1978 ◽  
Vol 21 (1) ◽  
pp. 17-23 ◽  
Author(s):  
M. J. Crabb ◽  
J. Duncan
Keyword(s):  
Banach Space ◽  
Banach Algebra ◽  
Unit Circle ◽  
Banach Algebras ◽  
Bounded Operators ◽  
Image Position ◽  
Unital Banach Algebra

Let A be a complex unital Banach algebra. An element u∈A is a norm unitary if(For the algebra of all bounded operators on a Banach space, the norm unitaries arethe invertible isometries.) Given a norm unitary u∈A, we have Sp(u)⊃Γ, where Sp(u) denotes the spectrum of u and Γ denotes the unit circle in C. If Sp(u)≠Γ we may suppose, by replacing eiθu, that . Then there exists h ∈ A such that

TWO NEW GENERALISED HYPERSTABILITY RESULTS FOR THE DRYGAS FUNCTIONAL EQUATION

2017 ◽  
Vol 95 (2) ◽  
pp. 269-280 ◽  
Author(s):  
LADDAWAN AIEMSOMBOON ◽  
WUTIPHOL SINTUNAVARAT
Keyword(s):  
Banach Space ◽  
Functional Equation ◽  
Normed Space ◽  
Nonempty Subset ◽  
Image Position

Let $X$ be a nonempty subset of a normed space such that $0\notin X$ and $X$ is symmetric with respect to $0$ and let $Y$ be a Banach space. We study the generalised hyperstability of the Drygas functional equation $$\begin{eqnarray}f(x+y)+f(x-y)=2f(x)+f(y)+f(-y),\end{eqnarray}$$ where $f$ maps $X$ into $Y$ and $x,y\in X$ with $x+y,x-y\in X$. Our first main result improves the results of Piszczek and Szczawińska [‘Hyperstability of the Drygas functional equation’, J. Funct. Space Appl.2013 (2013), Article ID 912718, 4 pages]. Hyperstability results for the inhomogeneous Drygas functional equation can be derived from our results.

On the Operator Identity ∑ AkXBk ≡ 0

1978 ◽  
Vol 31 (4) ◽  
pp. 845-857 ◽  
Author(s):  
C. K. Fong ◽  
A. R. Sourour
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Sufficient Conditions ◽  
Bounded Operators ◽  
Operator Identity ◽  
Image Position ◽  
Space Case ◽  
Necessary And Sufficient ◽  
Induced Mapping ◽  
Compact Map

Let Aj and Bj (1 ≦ j ≦ m) be bounded operators on a Banach space ᚕ and let Φ be the mapping on , the algebra of bounded operators on ᚕ, defined by(1)We give necessary and sufficient conditions for Φ to be identically zero or to be a compact map or (in the Hilbert space case) for the induced mapping on the Calkin algebra to be identically zero. These results are then used to obtain some results about inner derivations and, more generally, about mappings of the formFor example, it is shown that the commutant of the range of C(S, T) is “small” unless S and T are scalars.

Spectral characterization of the socle in Jordan–Banach algebras

1995 ◽  
Vol 117 (3) ◽  
pp. 479-489 ◽  
Author(s):  
Bernard Aupetit
Keyword(s):  
Banach Space ◽  
Banach Algebra ◽  
Finite Rank ◽  
Banach Algebras ◽  
Spectral Characterization ◽  
Bounded Operators ◽  
Finite Rank Operators ◽  
Finite Dimensional ◽  
Minimal Idempotent

If A is a complex Banach algebra the socle, denoted by Soc A, is by definition the sum of all minimal left (resp. right) ideals of A. Equivalently the socle is the sum of all left ideals (resp. right ideals) of the form Ap (resp. pA) where p is a minimal idempotent, that is p2 = p and pAp = ℂp. If A is finite-dimensional then A coincides with its socle. If A = B(X), the algebra of bounded operators on a Banach space X, the socle of A consists of finite-rank operators. For more details about the socle see [1], pp. 78–87 and [3], pp. 110–113.

On the Ranges of Certain Fractional Integrals

1972 ◽  
Vol 24 (6) ◽  
pp. 1198-1216 ◽  
Author(s):  
P. G. Rooney
Keyword(s):  
Banach Space ◽  
Fractional Integrals ◽  
Complex Numbers ◽  
Bounded Operators ◽  
Image Position

Suppose 1 ≦ P < ∞, μ is real, and denote by Lμ,p the collection of functions f, measurable on (0, ∞ ), and which satisfy1.1Also denote by [X] the collection of bounded operators from a Banach space X to itself. For v > 0, Re α > 0, Re β > 0, let1.2and1.3where ξ and η are complex numbers. Iv,α,ξ and Jv,β,η, are generalizations of the Riemann-Liouville and Weyl fractional integrals respectively, and consequently we shall refer to them as fractional integrals.

An analytical study of bifurcation problems for equations involving Fredholm mappings

1988 ◽  
Vol 110 (3-4) ◽  
pp. 199-225 ◽  
Author(s):  
N.X. Tan
Keyword(s):  
Banach Space ◽  
Open Subset ◽  
Bifurcation Point ◽  
Normed Space ◽  
Analytical Study ◽  
Fredholm Mapping ◽  
Characteristic Value ◽  
Bifurcation Problems ◽  
Image Position

Let us consider equations in the formwhere Λ is an open subset of a normed space. For any fixed λ ∊ Λ, T, L(λ,.) and M(λ,.) are mappings from the closure D0 of a neighbourhood D0 of the origin in a Banach space X into another Banach space Y with T(0) = L(λ, 0) = M(λ, 0) = 0. Let λ be a characteristic value of the pair (T, L) such that T − L(λ,.) is a Fredholm mapping with nullity p and index s, p> s≧ 0. Under sufficient hypotheses on T, L and M, (λ, 0) is a bifurcation point of the above equations. Some well-known results obtained by Crandall and Rabinowitz [2], McLeod and Sattinger [5] and others will be generalised. The results in this paper are extensions of the results obtained by the author in [7].

scholarly journals The dual of the space of bounded operators on a Banach space

2021 ◽  
Vol 8 (1) ◽  
pp. 48-59
Author(s):  
Fernanda Botelho ◽  
Richard J. Fleming
Keyword(s):  
Banach Space ◽  
Tensor Product ◽  
Banach Spaces ◽  
Dual Space ◽  
Tensor Products ◽  
Bounded Operators ◽  
Projective Tensor Product ◽  
Projective Tensor

Abstract Given Banach spaces X and Y, we ask about the dual space of the 𝒧(X, Y). This paper surveys results on tensor products of Banach spaces with the main objective of describing the dual of spaces of bounded operators. In several cases and under a variety of assumptions on X and Y, the answer can best be given as the projective tensor product of X ** and Y *.

The wigner property for CL-spaces and finite-dimensional polyhedral banach spaces

2021 ◽  
pp. 1-17
Author(s):  
Dongni Tan ◽  
Xujian Huang
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Phase Function ◽  
Linear Isometry ◽  
Basic Properties ◽  
Finite Dimensional ◽  
Image Position

Abstract We say that a map $f$ from a Banach space $X$ to another Banach space $Y$ is a phase-isometry if the equality \[ \{\|f(x)+f(y)\|, \|f(x)-f(y)\|\}=\{\|x+y\|, \|x-y\|\} \] holds for all $x,\,y\in X$ . A Banach space $X$ is said to have the Wigner property if for any Banach space $Y$ and every surjective phase-isometry $f : X\rightarrow Y$ , there exists a phase function $\varepsilon : X \rightarrow \{-1,\,1\}$ such that $\varepsilon \cdot f$ is a linear isometry. We present some basic properties of phase-isometries between two real Banach spaces. These enable us to show that all finite-dimensional polyhedral Banach spaces and CL-spaces possess the Wigner property.

On the centres of hereditary JBW-subalgebras of a JBW-algebra

1979 ◽  
Vol 85 (2) ◽  
pp. 317-324 ◽  
Author(s):  
C. M. Edwards
Keyword(s):  
Banach Space ◽  
General Theory ◽  
Jordan Algebra ◽  
Identity Element ◽  
Image Position

A JB-algebra A is a real Jordan algebra, which is also a Banach space, the norm in which satisfies the conditions thatandfor all elements a and b in A. It follows from (1.1) and (l.2) thatfor all elements a and b in A. When the JB-algebra A possesses an identity element then A is said to be a unital JB-algebra and (1.2) is equivalent to the condition thatfor all elements a and b in A. For the general theory of JB-algebras the reader is referred to (2), (3), (7) and (10).

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