scholarly journals Linear Maps Which are Anti-derivable at Zero

2020 ◽  
Vol 43 (6) ◽  
pp. 4315-4334
Author(s):  
Doha Adel Abulhamil ◽  
Fatmah B. Jamjoom ◽  
Antonio M. Peralta
Keyword(s):  
Linear Operator ◽  
Bounded Linear Operator ◽  
Complete Characterization ◽  
Linear Operators ◽  
Continuous Linear ◽  
Bounded Linear ◽  
Linear Maps ◽  
Continuous Linear Operators

Abstract Let $$T:A\rightarrow X$$ T : A → X be a bounded linear operator, where A is a $$\hbox {C}^*$$ C ∗ -algebra, and X denotes an essential Banach A-bimodule. We prove that the following statements are equivalent: (a) T is anti-derivable at zero (i.e., $$ab =0$$ a b = 0 in A implies $$T(b) a + b T(a)=0$$ T ( b ) a + b T ( a ) = 0 ); (b) There exist an anti-derivation $$d:A\rightarrow X^{**}$$ d : A → X ∗ ∗ and an element $$\xi \in X^{**}$$ ξ ∈ X ∗ ∗ satisfying $$\xi a = a \xi ,$$ ξ a = a ξ , $$\xi [a,b]=0,$$ ξ [ a , b ] = 0 , $$T(a b) = b T(a) + T(b) a - b \xi a,$$ T ( a b ) = b T ( a ) + T ( b ) a - b ξ a , and $$T(a) = d(a) + \xi a,$$ T ( a ) = d ( a ) + ξ a , for all $$a,b\in A$$ a , b ∈ A . We also prove a similar equivalence when X is replaced with $$A^{**}$$ A ∗ ∗ . This provides a complete characterization of those bounded linear maps from A into X or into $$A^{**}$$ A ∗ ∗ which are anti-derivable at zero. We also present a complete characterization of those continuous linear operators which are $$^*$$ ∗ -anti-derivable at zero.

scholarly journals Antinormal composition operators on $ \mbf{\ell^2}$

2008 ◽  
Vol 39 (4) ◽  
pp. 347-352 ◽  
Author(s):  
Gyan Prakash Tripathi ◽  
Nand Lal
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Composition Operators ◽  
Complete Characterization ◽  
Inner Product ◽  
Complex Numbers ◽  
Bounded Linear ◽  
Normal Operators

A bounded linear operator $ T $ on a Hilbert space $ H $ is called antinormal if the distance of $ T $ from the set of all normal operators is equal to norm of $ T $. In this paper, we give a complete characterization of antinormal composition operators on $ \ell^2 $, where $ \ell^2 $ is the Hilbert space of all square summable sequences of complex numbers under standard inner product on it.

ORTHOGONALITY PRESERVERS REVISITED

2009 ◽  
Vol 02 (03) ◽  
pp. 387-405 ◽  
Author(s):  
María Burgos ◽  
Francisco J. Fernández-Polo ◽  
Jorge J. Garcés ◽  
Antonio M. Peralta
Keyword(s):  
Linear Operator ◽  
Natural Product ◽  
Bounded Linear Operator ◽  
Complete Characterization ◽  
Bounded Linear ◽  
C Algebra ◽  
Jordan Homomorphism ◽  
C Algebras ◽  
Orthogonality Preservers

We obtain a complete characterization of all orthogonality preserving operators from a JB *-algebra to a JB *-triple. If T : J → E is a bounded linear operator from a JB *-algebra (respectively, a C *-algebra) to a JB *-triple and h denotes the element T**(1), then T is orthogonality preserving, if and only if, T preserves zero-triple-products, if and only if, there exists a Jordan *-homomorphism [Formula: see text] such that S(x) and h operator commute and T(x) = h•r(h) S(x), for every x ∈ J, where r(h) is the range tripotent of h, [Formula: see text] is the Peirce-2 subspace associated to r(h) and •r(h) is the natural product making [Formula: see text] a JB *-algebra. This characterization culminates the description of all orthogonality preserving operators between C *-algebras and JB *-algebras and generalizes all the previously known results in this line of study.

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.

A Note on the Caradus Class of Bounded Linear Operators on a Complex Banach Space

1969 ◽  
Vol 21 ◽  
pp. 592-594 ◽  
Author(s):  
A. F. Ruston
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Finite Number ◽  
Bounded Linear Operator ◽  
Present Note ◽  
Complex Banach Space ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
The Subject

1. In a recent paper (1) on meromorphic operators, Caradus introduced the class of bounded linear operators on a complex Banach space X. A bounded linear operator T is put in the class if and only if its spectrum consists of a finite number of poles of the resolvent of T. Equivalently, T is in if and only if it has a rational resolvent (8, p. 314).Some ten years ago (in May, 1957), I discovered a property of the class g which may be of interest in connection with Caradus' work, and is the subject of the present note.2. THEOREM. Let X be a complex Banach space. If T belongs to the class, and the linear operator S commutes with every bounded linear operator which commutes with T, then there is a polynomial p such that S = p(T).

Representation of certain Linear Operators in Hilbert Space

1971 ◽  
Vol 23 (1) ◽  
pp. 132-150 ◽  
Author(s):  
Bernard Niel Harvey
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Separable Hilbert Space ◽  
Finite Dimension ◽  
Linear Operators ◽  
Inner Product ◽  
Indefinite Metric ◽  
Bounded Linear ◽  
Bilinear Functional

In this paper we represent certain linear operators in a space with indefinite metric. Such a space may be a pair (H, B), where H is a separable Hilbert space, B is a bilinear functional on H given by B(x, y) = [Jx, y], [, ] is the Hilbert inner product in H, and J is a bounded linear operator such that J = J* and J2 = I. If T is a linear operator in H, then ‖T‖ is the usual operator norm. The operator J above has two eigenspaces corresponding to the eigenvalues + 1 and –1.In case the eigenspace in which J induces a positive operator has finite dimension k, a general spectral theory is known and has been developed principally by Pontrjagin [25], Iohvidov and Kreĭn [13], Naĭmark [20], and others.

scholarly journals The generalized condition numbers of bounded linear operators in Banach spaces

2004 ◽  
Vol 76 (2) ◽  
pp. 281-290 ◽  
Author(s):  
Guoliang Chen ◽  
Yimin Wei ◽  
Yifeng Xue
Keyword(s):  
Linear Operator ◽  
Banach Spaces ◽  
Perturbation Analysis ◽  
Bounded Linear Operator ◽  
Linear Operators ◽  
Condition Numbers ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Ill Posed ◽  
Linear Operator Equations

AbstractFor any bounded linear operator A in a Banach space, two generalized condition numbers, k(A) and k(A) are defined in this paper. These condition numbers may be applied to the perturbation analysis for the solution of ill-posed differential equations and bounded linear operator equations in infinite dimensional Banach spaces. Different expressions for the two generalized condition numbers are discussed in this paper and applied to the perturbation analysis of the operator equation.

scholarly journals On the sublinear operators factoring throughLq

2004 ◽  
Vol 2004 (50) ◽  
pp. 2695-2704
Author(s):  
Lahcène Mezrag ◽  
Abdelmoumene Tiaiba
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Positive Constant ◽  
Bounded Linear Operator ◽  
Bounded Operator ◽  
Linear Operators ◽  
Sublinear Operator ◽  
Bounded Linear ◽  
Sublinear Operators

Let0<p≤q≤+∞. LetTbe a bounded sublinear operator from a Banach spaceXinto anLp(Ω,μ)and let∇Tbe the set of all linear operators≤T. In the present paper, we will show the following. LetCbe a positive constant. For alluin∇T,Cpq(u)≤C(i.e.,uadmits a factorization of the formX→u˜Lq(Ω,μ)→MguLq(Ω,μ), whereu˜is a bounded linear operator with‖u˜‖≤C,Mguis the bounded operator of multiplication byguwhich is inBLr+(Ω,μ)(1/p=1/q+1/r),u=Mgu∘u˜andCpq(u)is the constant ofq-convexity ofu) if and only ifTadmits the same factorization; This is under the supposition that{gu}u∈∇Tis latticially bounded. Without this condition this equivalence is not true in general.

scholarly journals Asymptotically quasi-compact products of bounded linear operators

1973 ◽  
Vol 16 (3) ◽  
pp. 286-289 ◽  
Author(s):  
Anthony F. Ruston
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Present Note ◽  
Fredholm Determinant ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear

It is known (see, for instance, [1] p. 64, [6] p. 264) that, if A and B are bounded linear operators on a Banach space into itself (or, more generally, if A is a bounded linear operator on into a Banach space and B is a bounded linear operator on into), then AB and BA have the same spectrum except (possibly) for zero. In the present note, it is shown that AB is asymptotically quasi-compact if and only if BA is asymptotically quasi-compact, and that then any Fredholm determinant for AB is a Fredholm determinant for BA and vice versa.

scholarly journals Bounded operators on topological vector spaces and their spectral radii

Filomat ◽  
2012 ◽  
Vol 26 (6) ◽  
pp. 1283-1290
Author(s):  
Shirin Hejazian ◽  
Madjid Mirzavaziri ◽  
Omid Zabeti
Keyword(s):  
Linear Operator ◽  
Vector Space ◽  
Topological Vector Space ◽  
Bounded Linear Operator ◽  
Sufficient Conditions ◽  
Linear Operators ◽  
Vector Spaces ◽  
Bounded Operators ◽  
Bounded Linear ◽  
Topological Vector Spaces

In this paper, we consider three classes of bounded linear operators on a topological vector space with respect to three different topologies which are introduced by Troitsky. We obtain some properties for the spectral radii of a linear operator on a topological vector space. We find some sufficient conditions for the completeness of these classes of operators. Finally, as a special application, we deduce some sufficient conditions for invertibility of a bounded linear operator.

On Certain Classes of Bounded Linear Operators

1970 ◽  
Vol 13 (4) ◽  
pp. 469-473
Author(s):  
C-S Lin
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Banach Algebra ◽  
Fredholm Operator ◽  
Bounded Linear Operator ◽  
Complex Banach Space ◽  
Linear Operators ◽  
Complex Numbers ◽  
Bounded Linear Operators ◽  
Bounded Linear

Let T—c be a Fredholm operator, where T is a bounded linear operator on a complex Banach space and c is a scalar, the set of all such scalars is called the Φ-set of T [2] and was studied by many authors. In this connection, the purpose of the present paper is to investigate some classes Φ(V) of all such operators for any subset V of the complex plane.Let X be a Banach space over the field C of complex numbers with dim Z = ∞, unless otherwise stated, B(X) the Banach algebra of all bounded linear operators and K(X) the closed two-sided ideal of all compact operators on X.

Sign in / Sign up

Export Citation Format

Share Document