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 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.

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 AN INTERNAL CHARACTERIZATION OF COMPLETE POSITIVITY FOR ELEMENTARY OPERATORS

2002 ◽  
Vol 45 (2) ◽  
pp. 285-300
Author(s):  
Richard M. Timoney
Keyword(s):  
Linear Operators ◽  
Type I ◽  
Complete Positivity ◽  
Elementary Operator ◽  
Bounded Linear ◽  
C Algebra ◽  
C Algebras ◽  
Metric Operator ◽  
Elementary Operators

AbstractComplete positivity of ‘atomically extensible’ bounded linear operators between $C^*$-algebras is characterized in terms of positivity of a bilinear form on certain finite-rank operators. In the case of an elementary operator on a $C^*$-algebra, the approach leads us to characterize k-positivity of the operator in terms of positivity of a quadratic form on a subset of the dual space of the algebra and in terms of a certain inequality involving factorial states of finite type I.As an application we characterize those $C^*$-algebras where every k-positive elementary operator on the algebra is completely positive. They are either k-subhomogeneous or k-subhomogeneous by antiliminal. We also give a dual approach to the metric operator space introduced by Arveson.AMS 2000 Mathematics subject classification: Primary 46L05. Secondary 47B47; 47B65

MONOTONE OPERATOR FUNCTIONS ON C*-ALGEBRAS

2005 ◽  
Vol 16 (02) ◽  
pp. 181-196 ◽  
Author(s):  
HIROYUKI OSAKA ◽  
SERGEI SILVESTROV ◽  
JUN TOMIYAMA
Keyword(s):  
Classical Case ◽  
Linear Operators ◽  
Monotone Functions ◽  
Bounded Linear ◽  
C Algebra ◽  
C Algebras ◽  
Operator Monotone ◽  
Operator Functions ◽  
Operator Monotone Functions

The article is devoted to investigation of classes of functions monotone as functions on general C*-algebras that are not necessarily the C*-algebra of all bounded linear operators on a Hilbert space as in classical case of matrix and operator monotone functions. We show that for general C*-algebras the classes of monotone functions coincide with the standard classes of matrix and operator monotone functions. For every class we give exact characterization of C*-algebras with this class of monotone functions, providing at the same time a monotonicity characterization of subhomogeneous C*-algebras. We use this result to generalize characterizations of commutativity of a C*-algebra based on monotonicity conditions for a single function to characterizations of subhomogeneity. As a C*-algebraic counterpart of standard matrix and operator monotone scaling, we investigate, by means of projective C*-algebras and relation lifting, the existence of C*-subalgebras of a given monotonicity class.

scholarly journals Chain-finite operators and locally chain-finite operators

2001 ◽  
Vol 43 (1) ◽  
pp. 113-121
Author(s):  
Teresa Bermúdez ◽  
Antonio Martinón
Keyword(s):  
Positive Integer ◽  
Linear Operator ◽  
Banach Spaces ◽  
Bounded Linear Operator ◽  
Generalized Inverse ◽  
Algebraic Characterization ◽  
Bounded Linear ◽  
A Chain

We give algebraic conditions characterizing chain-finite operators and locally chain-finite operators on Banach spaces. For instance, it is shown that T is a chain-finite operator if and only if some power of T is relatively regular and commutes with some generalized inverse; that is there are a bounded linear operator B and a positive integer k such that TkBTk =Tk and TkB=BTk. Moreover, we obtain an algebraic characterization of locally chain-finite operators similar to (1).

scholarly journals The n-inverses of a matrix

Filomat ◽  
2017 ◽  
Vol 31 (12) ◽  
pp. 3801-3813
Author(s):  
Caixing Gu ◽  
Heidi Keas ◽  
Robert Lee
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Tensor Products ◽  
Complex Banach Space ◽  
Spectral Characterization ◽  
Bounded Linear ◽  
Nilpotent Operators ◽  
Nilpotent Matrices

The concept of a left n-inverse of a bounded linear operator on a complex Banach space was introduced recently. Previously, there have been results on products and tensor products of left n-inverses, and the representation of left n-inverses as the sum of left inverses and nilpotent operators was being discussed. In this paper, we give a spectral characterization of the left n-inverses of a finite (square) matrix. We also show that a left n-inverse of a matrix T is the sum of the inverse of T and two nilpotent matrices.

scholarly journals Classification of Integrodifferential C-Algebras*

Symmetry ◽  
2021 ◽  
Vol 13 (10) ◽  
pp. 1900
Author(s):  
Anton A. Kutsenko
Keyword(s):  
Infinite Product ◽  
Complete Characterization ◽  
Difference Operators ◽  
C Algebra ◽  
C Algebras ◽  
Finite Dimensional ◽  
Differential Algebras ◽  
Af Algebras ◽  
Product Of Matrices

The infinite product of matrices with integer entries, known as a modified Glimm–Bratteli symbol n, is a new, sufficiently simple, and very powerful tool for the characterization of approximately finite-dimensional (AF) algebras. This symbol provides a convenient algebraic representation of the Bratteli diagram for AF algebras in the same way as was previously performed by J. Glimm for more simple uniformly hyperfinite (UHF) algebras. We apply this symbol to characterize integrodifferential algebras. The integrodifferential algebra FN,M is the C*-algebra generated by the following operators acting on L2([0,1)N→CM): (1) operators of multiplication by bounded matrix-valued functions, (2) finite-difference operators, and (3) integral operators. Most of the operators and their approximations studying in physics belong to these algebras. We give a complete characterization of FN,M. In particular, we show that FN,M does not depend on M, but depends on N. At the same time, it is known that differential algebras HN,M, generated by the operators (1) and (2) only, do not depend on both dimensions N and M; they are all *-isomorphic to the universal UHF algebra. We explicitly compute the Glimm–Bratteli symbols (for HN,M, it was already computed earlier) which completely characterize the corresponding AF algebras. This symbol n is an infinite product of matrices with nonnegative integer entries. Roughly speaking, all the symmetries appearing in the approximation of complex infinite-dimensional integrodifferential and differential algebras by finite-dimensional ones are coded by a product of integer matrices.

scholarly journals An observation about pseudospectra

Filomat ◽  
2021 ◽  
Vol 35 (3) ◽  
pp. 995-1000
Author(s):  
Boting Jia ◽  
Youling Feng
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Bounded Linear ◽  
Finite Dimensional

For ? > 0 and a bounded linear operator T acting on some Hilbert space, the ?-pseudospectrum of T is ??(T) = {z ? C : ||(zI-T)-1|| > ?-1}. This note provides a characterization of those operators T satisfying ??(T) = ?(T) + B(0,?) for all ? > 0. Here B(0,?) = {z ? C : |z| < ?}. In particular, such operators on finite dimensional spaces must be normal.

Projections on Tree-Like banach Spaces

1985 ◽  
Vol 37 (5) ◽  
pp. 908-920
Author(s):  
A. D. Andrew
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Banach Spaces ◽  
Bounded Linear Operator ◽  
Unconditional Basis ◽  
Reflexive Banach Space ◽  
Separable Space ◽  
Bounded Linear ◽  
Tree Space

1. In this paper, we investigate the ranges of projections on certain Banach spaces of functions defined on a diadic tree. The notion of a “tree-like” Banach space is due to James 4], who used it to construct the separable space JT which has nonseparable dual and yet does not contain l1. This idea has proved useful. In [3], Hagler constructed a hereditarily c0 tree space, HT, and Schechtman [6] constructed, for each 1 ≦ p ≦ ∞, a reflexive Banach space, STp with a 1-unconditional basis which does not contain lp yet is uniformly isomorphic to for each n.In [1] we showed that if U is a bounded linear operator on JT, then there exists a subspace W ⊂ JT, isomorphic to JT such that either U or (1 — U) acts as an isomorphism on W and UW or (1 — U)W is complemented in JT. In this paper, we establish this result for the Hagler and Schechtman tree spaces.

scholarly journals EXPLICIT INTERPOLATION BOUNDS BETWEEN HARDY SPACE AND

2013 ◽  
Vol 95 (2) ◽  
pp. 158-168
Author(s):  
H.-Q. BUI ◽  
R. S. LAUGESEN
Keyword(s):  
Growth Rate ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Bounded Mean Oscillation ◽  
Complex Interpolation ◽  
Bounded Linear ◽  
Exponential Bound ◽  
Mean Oscillation ◽  
Good Lambda Inequality ◽  
Interpolation Result

AbstractEvery bounded linear operator that maps ${H}^{1} $ to ${L}^{1} $ and ${L}^{2} $ to ${L}^{2} $ is bounded from ${L}^{p} $ to ${L}^{p} $ for each $p\in (1, 2)$, by a famous interpolation result of Fefferman and Stein. We prove ${L}^{p} $-norm bounds that grow like $O(1/ (p- 1))$ as $p\downarrow 1$. This growth rate is optimal, and improves significantly on the previously known exponential bound $O({2}^{1/ (p- 1)} )$. For $p\in (2, \infty )$, we prove explicit ${L}^{p} $ estimates on each bounded linear operator mapping ${L}^{\infty } $ to bounded mean oscillation ($\mathit{BMO}$) and ${L}^{2} $ to ${L}^{2} $. This $\mathit{BMO}$ interpolation result implies the ${H}^{1} $ result above, by duality. In addition, we obtain stronger results by working with dyadic ${H}^{1} $ and dyadic $\mathit{BMO}$. The proofs proceed by complex interpolation, after we develop an optimal dyadic ‘good lambda’ inequality for the dyadic $\sharp $-maximal operator.

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