Abdellatif Bourhim

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Additive Maps Preserving Local Spectrum

2005-12-20

Abdellatif Bourhim , Thomas Ransford

A survey on preservers of spectra and local spectra

2015-01-01

Abdellatif Bourhim , Javad Mashreghi

MAPS PRESERVING THE LOCAL SPECTRUM OF PRODUCT OF OPERATORS

2014-12-17

Abdellatif Bourhim , Javad Mashreghi

Abstract Let X and Y be infinite-dimensional complex Banach spaces, and ${\mathcal B}$ ( X ) (resp. ${\mathcal B}$ ( Y )) be the algebra of all bounded linear operators … Abstract Let X and Y be infinite-dimensional complex Banach spaces, and ${\mathcal B}$ ( X ) (resp. ${\mathcal B}$ ( Y )) be the algebra of all bounded linear operators on X (resp. on Y ). For an operator T ∈ ${\mathcal B}$ ( X ) and a vector x ∈ X , let σ T ( x ) denote the local spectrum of T at x . For two nonzero vectors x 0 ∈ X and y 0 ∈ Y , we show that a map ϕ from ${\mathcal B}$ ( X ) onto ${\mathcal B}$ ( Y ) satisfies $ \sigma_{\varphi(T)\varphi(S)}(y_0)~=~\sigma_{TS}(x_0),~(T,~S\in{\mathcal B}(X)), $ if and only if there exists a bijective bounded linear mapping A from X into Y such that Ax 0 = y 0 and either ϕ( T ) = ATA −1 or ϕ( T ) = - ATA −1 for all T ∈ ${\mathcal B}$ ( X ).

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Local Spectral Radius Preservers

2013-02-11

Abdellatif Bourhim , Javad Mashreghi

Linear maps on Mn(C) preserving the local spectral radius

2008-01-01

Abdellatif Bourhim , V. G. Miller

Let $x_0$ be a nonzero vector in $\mathbb C^n$. We show that a linear map ${\mit\Phi}:M_n(\mathbb C)\to M_n(\mathbb C)$ preserves the local spectral radius at $x_0$ if and only if … Let $x_0$ be a nonzero vector in $\mathbb C^n$. We show that a linear map ${\mit\Phi}:M_n(\mathbb C)\to M_n(\mathbb C)$ preserves the local spectral radius at $x_0$ if and only if there is $\alpha\in\mathbb C$ of modulus one and an invertible matrix $A\in

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Maps preserving the local spectrum of triple product of operators

2014-06-11

Abdellatif Bourhim , Javad Mashreghi

Let denote the algebra of all bounded linear operators on a Banach space . Let and be infinite-dimensional complex Banach spaces, and fix two nonzero vectors and . Then we … Let denote the algebra of all bounded linear operators on a Banach space . Let and be infinite-dimensional complex Banach spaces, and fix two nonzero vectors and . Then we show that a surjective map satisfiesif and only if there exists a bijective mapping such that andwhere is a third root of unity.

Linear maps preserving the minimum and reduced minimum moduli

2009-10-28

Abdellatif Bourhim , María Burgos , Victor S. Shulman

Surjective linear maps preserving local spectra

2009-10-09

Abdellatif Bourhim

Maps preserving the local spectrum of Jordan product of matrices

2015-07-17

Abdellatif Bourhim , Mohamed Mabrouk

Maps between Banach algebras preserving the spectrum

2016-09-03

Abdellatif Bourhim , Javad Mashreghi , Anush Stepanyan

On the local spectral properties of weighted shift operators

2004-01-01

Abdellatif Bourhim

On the local spectral properties of On the local spectral properties of

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Linear maps preserving the essential spectral radius

2007-11-30

M. Bendaoud , Abdellatif Bourhim , M. Sarih

Nonlinear maps preserving the minimum and surjectivity moduli

2014-09-20

Abdellatif Bourhim , Javad Mashreghi , Anush Stepanyan

Numerical radius and product of elements in C*-algebras

2016-09-01

Abdellatif Bourhim , Mohamed Mabrouk

Let v(a) be the numerical radius of an element a in a unital -algebra . In this paper, we show that for all if and only if . We also … Let v(a) be the numerical radius of an element a in a unital -algebra . In this paper, we show that for all if and only if . We also show that for all if and only if a is a unitary element in the centre of . Furthermore, for two fixed elements a and b in , we investigate when for all , and derive a number of consequences.

a-Numerical range on $$C^*$$-algebras

2021-03-10

Abdellatif Bourhim , Mohamed Mabrouk

Bounded point evaluations for cyclic operators and local spectra

2001-07-25

Abdellatif Bourhim , C.E. Chidume , El Hassan Zerouali

In this paper we study the concept of bounded point evaluations for cyclic operators. We give a negative answer to a question of L.R. Williams, Dynamic Systems and Applications 3 … In this paper we study the concept of bounded point evaluations for cyclic operators. We give a negative answer to a question of L.R. Williams, Dynamic Systems and Applications 3 (1994), 103-112. Furthermore, we generalize some results of Williams and give a simple proof that nonnormal hyponormal weighted shifts have fat local spectra.

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Boundary behaviour of functions of Nevanlinna class

2004-01-01

Abdellatif Bourhim , Omar El-Fallah , Karim Kellay

We consider a functionf of Nevanlinna class such that its radial limit f is square summable on the unit circle T. We give a growth condition on the Taylor coeYcients … We consider a functionf of Nevanlinna class such that its radial limit f is square summable on the unit circle T. We give a growth condition on the Taylor coeYcients of f and the rate of decrease of the negatively indexed Fourier coeYcients off in order thatf be in the Hardy spaceH 2 NDO.

Additive maps preserving the reduced minimum modulus of Banach space operators

2009-01-01

Abdellatif Bourhim

Let ${\mathcal B}(X)$ be the algebra of all bounded linear operators on an infinite dimensional complex Banach space $X$. We prove that an additive surjective map $ϕ$ on ${\mathcal B}(X)$ … Let ${\mathcal B}(X)$ be the algebra of all bounded linear operators on an infinite dimensional complex Banach space $X$. We prove that an additive surjective map $ϕ$ on ${\mathcal B}(X)$ preserves the reduced minimum modulus if and only if either there are bijective isometries $U:X\to X$ and $V:X\to X$ both linear or both conjugate linear such that $ϕ(T)=UTV$ for all $T\in{\mathcal B}(X)$, or $X$ is reflexive and there are bijective isometries $U:X^*\to X$ and $V:X\to X^*$ both linear or both conjugate linear such that $ϕ(T)=UT^*V$ for all $T\in{\mathcal B}(X)$. As immediate consequences of the ingredients used in the proof of this result, we get the complete description of surjective additive maps preserving the minimum, the surjectivity and the maximum moduli of Banach space operators.

Linear maps preserving the minimum modulus

2010-01-01

Abdellatif Bourhim , María Burgos

We characterize surjective linear maps that preserve the minimum modulus between unital semisimple Banach algebras, one of them is a unital C * -algebra having either real rank zero or … We characterize surjective linear maps that preserve the minimum modulus between unital semisimple Banach algebras, one of them is a unital C * -algebra having either real rank zero or essential socle.We also describe surjective linear maps on L (H) , with H an infinitedimensional Hilbert space, preserving the essential minimum modulus.Results concerning surjectivity and maximum modulus are also obtained.

Jordan product and local spectrum preservers

2016-01-01

Abdellatif Bourhim , Mohamed Mabrouk

Let $X$ and $Y$ be two infinite-dimensional complex Banach spaces, and fix two nonzero vectors $x_0\in X$ and $y_0\in Y$. Let ${\mathscr B}(X)$ (resp. ${\mathscr B}(Y)$) denote the algebra of … Let $X$ and $Y$ be two infinite-dimensional complex Banach spaces, and fix two nonzero vectors $x_0\in X$ and $y_0\in Y$. Let ${\mathscr B}(X)$ (resp. ${\mathscr B}(Y)$) denote the algebra of all bounded linear operators on $X$ (resp. on $Y$). We show tha

Linear maps preserving Fredholm and Atkinson elements ofC*-algebras

2008-12-19

M. Bendaoud , Abdellatif Bourhim , María Burgos +1 more

Let A be a unital C*-algebra of real rank zero and B be a unital semisimple complex Banach algebra. We characterize linear maps from A onto B preserving different essential … Let A be a unital C*-algebra of real rank zero and B be a unital semisimple complex Banach algebra. We characterize linear maps from A onto B preserving different essential spectral sets and quantities such as the essential spectrum, the (left, right) essential spectrum, the Weyl spectrum, the index and the essential spectral radius.

Multiplicatively local spectrum-preserving maps

2018-03-23

Abdellatif Bourhim , Ji Eun Lee

Linear maps preserving regularity in C∗-algebras

2009-01-01

Abdellatif Bourhim , María Burgos

Let $A$ and $B$ be unital $C^*$-algebras such that at least one of them is of real rank zero. We investigate surjective linear maps from $A$ to $B$ preserving the … Let $A$ and $B$ be unital $C^*$-algebras such that at least one of them is of real rank zero. We investigate surjective linear maps from $A$ to $B$ preserving the conorm, the (von Neumann) regularity, the generalized spectrum, and their essential versions. As a consequence, we recover results of Mbekhta, and Mbekhta and Šemrl for $\mathcal L(H)$ when $H$ is an infinite-dimensional complex Hilbert space.

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The single-valued extension property for bilateral operator weighted shifts

2004-09-08

Abdellatif Bourhim , C.E. Chidume

In this paper, we give necessary and sufficient conditions for a bilateral operator weighted shift to enjoy the single-valued extension property. In this paper, we give necessary and sufficient conditions for a bilateral operator weighted shift to enjoy the single-valued extension property.

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THE SINGLE-VALUED EXTENSION PROPERTY IS NOT PRESERVED UNDER SUMS AND PRODUCTS OF COMMUTING OPERATORS

2007-01-01

Abdellatif Bourhim , V. G. Miller

Abstract. We show that the sum and the product of two commuting operators with the single-valued extension property need not inherit this property. Abstract. We show that the sum and the product of two commuting operators with the single-valued extension property need not inherit this property.

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Bounded Point Evaluations and Local Spectral Theory

2000-01-01

Abdellatif Bourhim

We study in this paper the concept of bounded point evaluations for cyclic operators. We give a negative answer to a question of L.R. Williams {\it Dynamic Systems and Apllications} … We study in this paper the concept of bounded point evaluations for cyclic operators. We give a negative answer to a question of L.R. Williams {\it Dynamic Systems and Apllications} 3(1994) 103-112. Furthermore, we generalize some results of Williams and give a simple proof of theorem 2.5 of L.R. Williams (The Local Spectra of Pure Quasinormal Operators J. Math. anal. Appl. 187(1994) 842-850) that non normal hyponormal weighted shifts have fat local spectra.

Essentially spectrally bounded linear maps

2009-06-18

M. Bendaoud , Abdellatif Bourhim

Let ${\mathcal L}({\mathcal {H}})$ be the algebra of all bounded linear operators on an infinite dimensional complex Hilbert space ${\mathcal {H}}$. We characterize essentially spectrally bounded linear maps from ${\mathcal … Let ${\mathcal L}({\mathcal {H}})$ be the algebra of all bounded linear operators on an infinite dimensional complex Hilbert space ${\mathcal {H}}$. We characterize essentially spectrally bounded linear maps from ${\mathcal L}( {\mathcal {H}})$ onto ${\mathcal L}({\mathcal {H}})$ itself. As a consequence, we characterize linear maps from ${\mathcal L}( {\mathcal {H}})$ onto ${\mathcal L}({\mathcal {H}})$ itself that compress different essential spectral sets such as the the essential spectrum, the (left, right) essential spectrum, and the semi-Fredholm spectrum.

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Peripheral local spectrum preservers and maps increasing the local spectral radius

2016-01-01

Abdellatif Bourhim , Tarik Jari , Javad Mashreghi

We address two long standing problems in the context of local spectral radius preservers.First, we completely describe the form of maps preserving the peripheral local spectrum of product or triple … We address two long standing problems in the context of local spectral radius preservers.First, we completely describe the form of maps preserving the peripheral local spectrum of product or triple product of operators.Second, we establish the automatic continuity of linear maps increasing the local spectral radius of operators at a fixed nonzero vector.

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Lie product and local spectrum preservers

2018-05-17

Zine El Abidine Abdelali , Abdellatif Bourhim , Mohamed Mabrouk

Spectrum of bilateral shifts with operator-valued weights

2006-01-31

Abdellatif Bourhim

We describe the spectrum of bilateral operator-weighted shifts. We describe the spectrum of bilateral operator-weighted shifts.

On the largest analytic set for cyclic operators

2003-01-01

Abdellatif Bourhim

We describe the set of analytic bounded point evaluations for an arbitrary cyclic bounded linear operator T on a Hilbert space ℋ ; some related consequences are discussed. Furthermore, we … We describe the set of analytic bounded point evaluations for an arbitrary cyclic bounded linear operator T on a Hilbert space ℋ ; some related consequences are discussed. Furthermore, we show that two densely similar cyclic Banach‐space operators possessing Bishop′s property ( β ) have equal approximate point spectra.

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Linear Maps Preserving Matrices of Local Spectral Radius Zero at a Fixed Vector

2018-04-25

Abdellatif Bourhim , Constantin Costara

Abstract In this paper, we characterize linear maps on matrix spaces that preserve matrices of local spectral radius zero at some fixed nonzero vector. Abstract In this paper, we characterize linear maps on matrix spaces that preserve matrices of local spectral radius zero at some fixed nonzero vector.

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Maps Preserving the Numerical Radius Distance Between $$C^*$$ C ∗ -Algebras

2019-02-04

Abdellatif Bourhim , Mohamed Mabrouk

Maps preserving the local reduced minimum modulus

2023-06-10

Abdellatif Bourhim , Mostafa Mbekhta

Comportement radial des fonctions de la classe de Nevanlinna

2001-09-01

Abdellatif Bourhim , Omar El-Fallah , Karim Kellay

Maps preserving the local spectrum of quadratic products of matrices

2018-06-01

Zine El Abidine Abdelali , Abdellatif Bourhim

Maps preserving matrices of local reduced minimum modulus zero at a fixed vector

2023-09-29

Abdellatif Bourhim , Mohamed Mabrouk , Mostafa Mbekhta

Unitary equivalence and reduced minimum modulus preservers

2024-08-25

Abdellatif Bourhim , Mostafa Mbekhta

Bounded point evaluations for cyclic Hilbert space operators

2003-10-01

Abdellatif Bourhim

In this talk, to be given at a conference at Seconda Università degli Studi di Napoli in September 2001, we shall describe the set of analytic bounded point evaluations for … In this talk, to be given at a conference at Seconda Università degli Studi di Napoli in September 2001, we shall describe the set of analytic bounded point evaluations for an arbitrary cyclic bounded linear operator T on a Hilbert space H and shall answer some questions due to L. R. Williams.

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LOCAL SPECTRA OF UNILATERAL OPERATOR WEIGHTED SHIFTS

2004-07-01

Abdellatif Bourhim

Numerical radius characterizations of elements in -algebras

2017-09-22

Abdellatif Bourhim , Mohamed Mabrouk

For an element a in a unital -algebra , let V(a) and v(a) denote the numerical range and the numerical radius of a, respectively. First, we show that a is … For an element a in a unital -algebra , let V(a) and v(a) denote the numerical range and the numerical radius of a, respectively. First, we show that a is a unitary central element in if and only if for all normal elements . We then use this result to show that if a and b are two elements in with , then for all normal elements if and only if ba is a unitary central element in and for some positive scalar . If, however, the numerical radius is replaced by the numerical range, we show that such a condition is redundant and prove that for all normal elements if and only if a is invertible such that is a central element in and . Furthermore, we show that for all invertible elements if and only if |a| is in the centre of and is a unitary element.

On the local spectral properties of weighted shift operators

2003-01-01

Abdellatif Bourhim

In this paper, we study the local spectral properties for both unilateral and bilateral weighted shift operators. In this paper, we study the local spectral properties for both unilateral and bilateral weighted shift operators.

Recent progress on local spectrum-preserving maps

2020-01-01

Abdellatif Bourhim , Javad Mashreghi

Maps preserving the Aluthge transform of unitarily similar operators

2025-01-01

Abdellatif Bourhim , Mostafa Mbekhta

Unitary similarity and the numerical radius preservers

2025-03-01

Abdellatif Bourhim , Mohamed Mabrouk

Local spectra of operator weighted shifts

2003-01-01

Abdellatif Bourhim

In this paper, we study the local spectral properties of unilateral operator weighted shifts. In this paper, we study the local spectral properties of unilateral operator weighted shifts.

Spectrum and local spectrum preservers of skew Lie products of matrices

2020-01-01

Zine El Abidine Abdelali , Abdellatif Bourhim , Mohamed Mabrouk

Composition operators on the Hardy–Hilbert space 𝐻² and model spaces 𝐾_{Θ}

2016-01-01

Abdellatif Bourhim , Javad Mashreghi

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Nonsurjective Numerical Radius Isometries Between Matrix Algebras

2023-05-29

Abdellatif Bourhim , Mohamed Mabrouk

The M-reduced minimum modulus of operators

2024-12-01

Abdellatif Bourhim , Mostafa Mbekhta

Linear maps compressing the set of local reduced minimum moduli

2025-05-08

Abdellatif Bourhim , Mostafa Mbekhta

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper B left-parenthesis upper H right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">B</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>H</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {B}(H)</mml:annotation> </mml:semantics> … Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper B left-parenthesis upper H right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">B</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>H</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {B}(H)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the algebra of all bounded linear operators on a complex Hilbert space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. For any unit vector <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x element-of upper H"> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>H</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">x\in H</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P Subscript x"> <mml:semantics> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>x</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">P_x</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the orthogonal projection onto the span of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x"> <mml:semantics> <mml:mi>x</mml:mi> <mml:annotation encoding="application/x-tex">x</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. For any <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T element-of script upper B left-parenthesis upper H right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo>∈</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">B</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>H</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">T\in \mathcal {B}(H)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="gamma left-parenthesis upper T comma x right-parenthesis colon-equal sup left-brace right-brace colon greater-than-or-equal-to greater-than-or-equal-to t 0 colon less-than-or-equal-to less-than-or-equal-to times times tPx vertical-bar vertical-bar upper T"> <mml:semantics> <mml:mrow> <mml:mi>γ</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>T</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>≔</mml:mo> <mml:mo movablelimits="true" form="prefix">sup</mml:mo> <mml:mrow> <mml:mo>{</mml:mo> <mml:mi>t</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>0</mml:mn> <mml:mspace width="thickmathspace"/> <mml:mo>:</mml:mo> <mml:mspace width="thickmathspace"/> <mml:mi>t</mml:mi> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>x</mml:mi> </mml:msub> <mml:mo>≤</mml:mo> <mml:mo fence="false" stretchy="false">|</mml:mo> <mml:mi>T</mml:mi> <mml:mo fence="false" stretchy="false">|</mml:mo> <mml:mo>}</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\gamma (T,x)≔\sup \left \{ t\geq 0 \; : \; t P_x \leq \vert T\vert \right \}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the local reduced minimum modulus of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding="application/x-tex">T</mml:annotation> </mml:semantics> </mml:math> </inline-formula> at <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x"> <mml:semantics> <mml:mi>x</mml:mi> <mml:annotation encoding="application/x-tex">x</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and set <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Lamda left-parenthesis upper T right-parenthesis colon-equal left-brace gamma left-parenthesis upper T comma x right-parenthesis colon x element-of upper H comma double-vertical-bar x double-vertical-bar equals 1 right-brace"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal">Λ</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>T</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>≔</mml:mo> <mml:mrow> <mml:mo>{</mml:mo> <mml:mi>γ</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>T</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>:</mml:mo> <mml:mspace width="1em"/> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>H</mml:mi> <mml:mo>,</mml:mo> <mml:mspace width="thickmathspace"/> <mml:mo fence="false" stretchy="false">‖</mml:mo> <mml:mi>x</mml:mi> <mml:mo fence="false" stretchy="false">‖</mml:mo> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>}</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\Lambda (T)≔\left \{ \gamma (T, x): \quad x\in H,\;\|x\|=1\right \}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In this paper, we investigate surjective linear maps on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper B left-parenthesis upper H right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">B</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>H</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {B}(H)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> compressing this set.

Maps preserving the mean transforms of pairs of unitarily similar operators

2025-05-06

Abdellatif Bourhim , Mostafa Mbekhta

Unitary similarity and the numerical radius preservers

2025-03-01

Abdellatif Bourhim , Mohamed Mabrouk

Joint numerical radius distance preservers on C *-algebras

2025-02-12

Abdellatif Bourhim , Mohamed Mabrouk

Multiplicatively maximal numerical range-preserving maps

2025-01-31

Abdellatif Bourhim , Mohamed Mabrouk

Maps preserving the Aluthge transform of unitarily similar operators

2025-01-01

Abdellatif Bourhim , Mostafa Mbekhta

The M-reduced minimum modulus of operators

2024-12-01

Abdellatif Bourhim , Mostafa Mbekhta

Unitary equivalence and reduced minimum modulus preservers

2024-08-25

Abdellatif Bourhim , Mostafa Mbekhta

Maps preserving matrices of local reduced minimum modulus zero at a fixed vector

2023-09-29

Abdellatif Bourhim , Mohamed Mabrouk , Mostafa Mbekhta

Maps preserving the local reduced minimum modulus

2023-06-10

Abdellatif Bourhim , Mostafa Mbekhta

Nonsurjective Numerical Radius Isometries Between Matrix Algebras

2023-05-29

Abdellatif Bourhim , Mohamed Mabrouk

a-Numerical range on $$C^*$$-algebras

2021-03-10

Abdellatif Bourhim , Mohamed Mabrouk

Spectrum and local spectrum preservers of skew Lie products of matrices

2020-01-01

Zine El Abidine Abdelali , Abdellatif Bourhim , Mohamed Mabrouk

Recent progress on local spectrum-preserving maps

2020-01-01

Abdellatif Bourhim , Javad Mashreghi

Maps Preserving the Numerical Radius Distance Between $$C^*$$ C ∗ -Algebras

2019-02-04

Abdellatif Bourhim , Mohamed Mabrouk

Maps preserving the local spectrum of quadratic products of matrices

2018-06-01

Zine El Abidine Abdelali , Abdellatif Bourhim

Lie product and local spectrum preservers

2018-05-17

Zine El Abidine Abdelali , Abdellatif Bourhim , Mohamed Mabrouk

Linear Maps Preserving Matrices of Local Spectral Radius Zero at a Fixed Vector

2018-04-25

Abdellatif Bourhim , Constantin Costara

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Multiplicatively local spectrum-preserving maps

2018-03-23

Abdellatif Bourhim , Ji Eun Lee

Numerical radius characterizations of elements in -algebras

2017-09-22

Abdellatif Bourhim , Mohamed Mabrouk

Maps between Banach algebras preserving the spectrum

2016-09-03

Abdellatif Bourhim , Javad Mashreghi , Anush Stepanyan

Numerical radius and product of elements in C*-algebras

2016-09-01

Abdellatif Bourhim , Mohamed Mabrouk

Peripheral local spectrum preservers and maps increasing the local spectral radius

2016-01-01

Abdellatif Bourhim , Tarik Jari , Javad Mashreghi

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Jordan product and local spectrum preservers

2016-01-01

Abdellatif Bourhim , Mohamed Mabrouk

Composition operators on the Hardy–Hilbert space 𝐻² and model spaces 𝐾_{Θ}

2016-01-01

Abdellatif Bourhim , Javad Mashreghi

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Maps preserving the local spectrum of Jordan product of matrices

2015-07-17

Abdellatif Bourhim , Mohamed Mabrouk

A survey on preservers of spectra and local spectra

2015-01-01

Abdellatif Bourhim , Javad Mashreghi

MAPS PRESERVING THE LOCAL SPECTRUM OF PRODUCT OF OPERATORS

2014-12-17

Abdellatif Bourhim , Javad Mashreghi

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Nonlinear maps preserving the minimum and surjectivity moduli

2014-09-20

Abdellatif Bourhim , Javad Mashreghi , Anush Stepanyan

Maps preserving the local spectrum of triple product of operators

2014-06-11

Abdellatif Bourhim , Javad Mashreghi

Local Spectral Radius Preservers

2013-02-11

Abdellatif Bourhim , Javad Mashreghi

Linear maps preserving the minimum modulus

2010-01-01

Abdellatif Bourhim , María Burgos

Linear maps preserving the minimum and reduced minimum moduli

2009-10-28

Abdellatif Bourhim , María Burgos , Victor S. Shulman

Surjective linear maps preserving local spectra

2009-10-09

Abdellatif Bourhim

Essentially spectrally bounded linear maps

2009-06-18

M. Bendaoud , Abdellatif Bourhim

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Linear maps preserving regularity in C∗-algebras

2009-01-01

Abdellatif Bourhim , María Burgos

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Additive maps preserving the reduced minimum modulus of Banach space operators

2009-01-01

Abdellatif Bourhim

Linear maps preserving Fredholm and Atkinson elements ofC*-algebras

2008-12-19

M. Bendaoud , Abdellatif Bourhim , María Burgos +1 more

Linear maps on Mn(C) preserving the local spectral radius

2008-01-01

Abdellatif Bourhim , V. G. Miller

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Linear maps preserving the essential spectral radius

2007-11-30

M. Bendaoud , Abdellatif Bourhim , M. Sarih

THE SINGLE-VALUED EXTENSION PROPERTY IS NOT PRESERVED UNDER SUMS AND PRODUCTS OF COMMUTING OPERATORS

2007-01-01

Abdellatif Bourhim , V. G. Miller

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Spectrum of bilateral shifts with operator-valued weights

2006-01-31

Abdellatif Bourhim

We describe the spectrum of bilateral operator-weighted shifts. We describe the spectrum of bilateral operator-weighted shifts.

Additive Maps Preserving Local Spectrum

2005-12-20

Abdellatif Bourhim , Thomas Ransford

The single-valued extension property for bilateral operator weighted shifts

2004-09-08

Abdellatif Bourhim , C.E. Chidume

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LOCAL SPECTRA OF UNILATERAL OPERATOR WEIGHTED SHIFTS

2004-07-01

Abdellatif Bourhim

On the local spectral properties of weighted shift operators

2004-01-01

Abdellatif Bourhim

On the local spectral properties of On the local spectral properties of

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Boundary behaviour of functions of Nevanlinna class

2004-01-01

Abdellatif Bourhim , Omar El-Fallah , Karim Kellay

Bounded point evaluations for cyclic Hilbert space operators

2003-10-01

Abdellatif Bourhim

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Local spectra of operator weighted shifts

2003-01-01

Abdellatif Bourhim

In this paper, we study the local spectral properties of unilateral operator weighted shifts. In this paper, we study the local spectral properties of unilateral operator weighted shifts.

On the largest analytic set for cyclic operators

2003-01-01

Abdellatif Bourhim

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Coauthor	Papers Together
Mohamed Mabrouk	13
Javad Mashreghi	9
Mostafa Mbekhta	7
María Burgos	4
M. Bendaoud	3
Zine El Abidine Abdelali	3
M. Sarih	2
Karim Kellay	2
V. G. Miller	2
C.E. Chidume	2
Omar El-Fallah	2
Anush Stepanyan	2
El Hassan Zerouali	1
Tarik Jari	1
Victor S. Shulman	1
Ji Eun Lee	1
Thomas Ransford	1
Constantin Costara	1

An Introduction to Local Spectral Theory

2000-03-30

Kjeld Laursen , Michael Neumann

Abstract Modern local spectral theory is built on the classical spectral theorem, a fundamental result in single-operator theory and Hilbert spaces. This book provides an in-depth introduction to the natural … Abstract Modern local spectral theory is built on the classical spectral theorem, a fundamental result in single-operator theory and Hilbert spaces. This book provides an in-depth introduction to the natural expansion of this fascinating topic of Banach space operator theory, whose pioneers include Dunford, Bishop, Foias, and others. Assuming only modest prerequisites of its readership, it gives complete coverage of the field, including the fundamental recent work by Albrecht and Eschmeier which provides the full duality theory for Banach space operators. It is highlighted by many characterizations of decomposable operators, and of other related, important classes of operators, as well as an in-depth study of their spectral properties, including identifications of distinguished parts, and results on permanence properties of spectra with respect to several types of similarity. Also found is a thorough and quite elementary treatment of the modern single- operator duality theory; this theory has many applications, both to general issues of classification and to such celebrated problems as the invariant subspace problems. A long chapter - almost a book in itself - is devoted to the use of local spectral theory in the study of spectral properties of multipliers and convolution operators. Another one describes its connections to automatic continuity theory. Written in a careful and detailed style, it contains numerous examples, many simplified proofs of classical results, and extensive references. It concludes with a list of interesting open problems, suitable for continued research.

A Primer on Spectral Theory

1991-01-01

Bernard Aupetit

Fredholm and Local Spectral Theory, with Applications to Multipliers

2004-01-01

Pietro Aiena

Linear Maps Preserving the Spectral Radius

1996-12-01

Matej Brešar , Peter Šemrl

Additive Maps Preserving Local Spectrum

2005-12-20

Abdellatif Bourhim , Thomas Ransford

Weighted shift operators and analytic function theory

1974-12-31

Allen Shields

Spectrum-preserving linear maps

1986-04-01

Ali Jafarian , A. R. Sourour

Maps on matrices that preserve the spectral radius distance

1999-01-01

Rajendra Bhatia , Peter Šemrl , A. R. Sourour

Let ϕ be a surjective map on the space of n×n complex matrices such that r(ϕ(A)-ϕ(B))=r(A-B) for all A,B, where r(X) is the spectral radius of X. We show that … Let ϕ be a surjective map on the space of n×n complex matrices such that r(ϕ(A)-ϕ(B))=r(A-B) for all A,B, where r(X) is the spectral radius of X. We show that ϕ must be a composition of five types of maps: translation, multiplication by a scalar of modulu

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MAPS PRESERVING THE LOCAL SPECTRUM OF PRODUCT OF OPERATORS

2014-12-17

Abdellatif Bourhim , Javad Mashreghi

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Spectral Theory of Linear Operators and Spectral Systems in Banach Algebras

2003-01-01

Veronika Müller

Preface .- I. Banach Algebras .- II. Operators .- III. Essential Spectrum .- IV. Taylor Spectrum .- V. Orbits and Capacity .- Appendix .- Bibliography Preface .- I. Banach Algebras .- II. Operators .- III. Essential Spectrum .- IV. Taylor Spectrum .- V. Orbits and Capacity .- Appendix .- Bibliography

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Approximate point spectrum of a weighted shift

1970-01-01

William C. Ridge

Notation. If T is a Hilbert space operator, let A(T) denote its spectrum, fl(T) its approximate point spectrum, flo(T) its point spectrum, r(T) its compression spectrum, m(T) its lower bound … Notation. If T is a Hilbert space operator, let A(T) denote its spectrum, fl(T) its approximate point spectrum, flo(T) its point spectrum, r(T) its compression spectrum, m(T) its lower bound (i.e., inf{ lTxll/llxll: x7&0}), and r(T) its spectral radius. Let i(T) denote supn m(Tn)ln, which equals limn m(Tn)1n. Let R denote a weighted right shift on 11, defined by Ren=snen +I , where (en) is an orthonormal basis of 12, n = 1, 2, .... Let L denote its adjoint, a weighted left shift. Let B denote a weighted two-sided shift on 12, defined by Ben=sSnen+1 n=0, ? 1, ? 2, .. ., (en) here being an orthonormal basis of 12. If B has purely nonzero weights (sn), then let

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Mappings preserving spectra of products of matrices

2006-11-21

Jor-Ting Chan , Chi-Kwong Li , Nung-Sing Sze

Let $M_n$ be the set of $n\times n$ complex matrices, and for every $A\in M_n$, let $\operatorname {Sp}(A)$ denote the spectrum of $A$. For various types of products $A_1* \cdots … Let $M_n$ be the set of $n\times n$ complex matrices, and for every $A\in M_n$, let $\operatorname {Sp}(A)$ denote the spectrum of $A$. For various types of products $A_1* \cdots *A_k$ on $M_n$, it is shown that a mapping $\phi : M_n \rightarrow M_n$ satisfying $\operatorname {Sp}(A_1*\cdots *A_k) = \operatorname {Sp}(\phi (A_1)* \cdots *\phi (A_k))$ for all $A_1, \dots , A_k \in M_n$ has the form \[ X \mapsto \xi S^{-1}XS \quad \mathrm { or } \quad A \mapsto \xi S^{-1}X^tS\] for some invertible $S \in M_n$ and scalar $\xi$. The result covers the special cases of the usual product $A_1* \cdots * A_k = A_1 \cdots A_k$, the Jordan triple product $A_1*A_2 = A_1*A_2*A_1$, and the Jordan product $A_1*A_2 = (A_1A_2+A_2A_1)/2$. Similar results are obtained for Hermitian matrices.

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A survey on preservers of spectra and local spectra

2015-01-01

Abdellatif Bourhim , Javad Mashreghi

Maps preserving the nilpotency of products of operators

2007-01-18

Chi-Kwong Li , Peter Šemrl , Nung-Sing Sze

Local spectrum and local spectral radius of an operator at a fixed vector

2009-01-01

Janko Bračič , Veronika Müller

Let $\mathscr X$ be a complex Banach space and $e\in\mathscr X$ a nonzero vector. Then the set of all operators $T\in{\cal L}(\mathscr X)$ with $\sigma_T(e)=\sigma_\delta(T)$, respectively $r_T(e)=r(T)$, is residual. This … Let $\mathscr X$ be a complex Banach space and $e\in\mathscr X$ a nonzero vector. Then the set of all operators $T\in{\cal L}(\mathscr X)$ with $\sigma_T(e)=\sigma_\delta(T)$, respectively $r_T(e)=r(T)$, is residual. This is an analogy to the well known r

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Maps preserving the local spectrum of triple product of operators

2014-06-11

Abdellatif Bourhim , Javad Mashreghi

n-JORDAN HOMOMORPHISMS

2009-06-19

M‎. ‎Eshaghi Gordji

Abstract Let n ∈ℕ and let A and B be rings. An additive map h : A → B is called an n -Jordan homomorphism if h ( a n … Abstract Let n ∈ℕ and let A and B be rings. An additive map h : A → B is called an n -Jordan homomorphism if h ( a n )=( h ( a )) n for all a ∈ A . Every Jordan homomorphism is an n -Jordan homomorphism, for all n ≥2, but the converse is false in general. In this paper we investigate the n -Jordan homomorphisms on Banach algebras. Some results related to continuity are given as well.

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Some characterizations of the automorphisms of $B(H)$ and $C(X)$

2001-06-08

Lajos Molnár

We present some nonlinear characterizations of the automorphisms of the operator algebra $B(H)$ and the function algebra $C(X)$ by means of their spectrum preserving properties. We present some nonlinear characterizations of the automorphisms of the operator algebra $B(H)$ and the function algebra $C(X)$ by means of their spectrum preserving properties.

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Linear Maps Preserving Operators of Local Spectral Radius Zero

2012-03-01

Constantin Costara

Operators Which Do Not Have the Single Valued Extension Property

2000-10-01

Pietro Aiena , Osmin Monsalve

Maps preserving numerical ranges of operator products

2005-10-13

Jinchuan Hou , Qinghui Di

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a complex Hilbert space, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B left-parenthesis upper H right-parenthesis"> <mml:semantics> … Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a complex Hilbert space, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B left-parenthesis upper H right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>B</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>H</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">B(H)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> the algebra of all bounded linear operators on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S Superscript a Baseline left-parenthesis upper H right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>S</mml:mi> <mml:mi>a</mml:mi> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>H</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">S^a(H)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> the real linear space of all self-adjoint operators on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We characterize the surjective maps on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B left-parenthesis upper H right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>B</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>H</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">B(H)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> or <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S Superscript a Baseline left-parenthesis upper H right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>S</mml:mi> <mml:mi>a</mml:mi> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>H</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">S^a(H)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> that preserve the numerical ranges of products or Jordan triple-products of operators.

The Theory of Subnormal Operators

1991-06-26

John B. Conway

Preliminaries Subnormal operators: The elementary theory Function theory on the unit circle Hyponormal operators Uniform rational approximation Weak-star rational approximation Some structure theory for subnormal operators Bounded point evaluations. Preliminaries Subnormal operators: The elementary theory Function theory on the unit circle Hyponormal operators Uniform rational approximation Weak-star rational approximation Some structure theory for subnormal operators Bounded point evaluations.

Invertibility preserving linear maps on ℒ(𝒳)

1996-01-01

A. R. Sourour

For Banach spaces <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Y"> <mml:semantics> <mml:mi>Y</mml:mi> <mml:annotation encoding="application/x-tex">Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, … For Banach spaces <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Y"> <mml:semantics> <mml:mi>Y</mml:mi> <mml:annotation encoding="application/x-tex">Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, we show that every unital bijective invertibility preserving linear map between <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper L left-parenthesis upper X right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">L</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal L(X)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper L left-parenthesis upper Y right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">L</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>Y</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal L(Y)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a Jordan isomorphism. The same conclusion holds for maps between <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper C upper I plus script upper K left-parenthesis upper X right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:mi>I</mml:mi> <mml:mo>+</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">K</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb CI+ \mathcal K(X)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper C upper I plus script upper K left-parenthesis upper Y right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:mi>I</mml:mi> <mml:mo>+</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">K</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>Y</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb CI+\mathcal K(Y)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.

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Maps preserving the spectrum of generalized Jordan product of operators

2009-11-18

Jinchuan Hou , Chi-Kwong Li , Ngai‐Ching Wong

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Linear maps preserving the minimum and surjectivity moduli of operators

2010-01-01

Mostafa Mbekhta

Let B(H) be the algebra of all bounded linear operators on a complex Hilbert space H , and denote by m(T) and q(T) respectively the minimum modulus and the surjectivity … Let B(H) be the algebra of all bounded linear operators on a complex Hilbert space H , and denote by m(T) and q(T) respectively the minimum modulus and the surjectivity modulus for every T ∈ B(H) .In this paper, we prove that if φ is a surjective unital linear map on B(H) , then m(φ (T)) = m(T) for every T ∈ B(H) if and only if q(φ (T)) = q(T) for every T ∈ B(H) if and only if there exists an unitary operator U ∈ B(H) such that φ (T ) = UTU * for all T ∈ B(H) .

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Linear maps preserving the minimum modulus

2010-01-01

Abdellatif Bourhim , María Burgos

Linear maps preserving the minimum and reduced minimum moduli

2009-10-28

Abdellatif Bourhim , María Burgos , Victor S. Shulman

Spectrum-Preserving Linear Mappings between Banach Algebras or Jordan-Banach Algebras

2000-12-01

Bernard Aupetit

Spectrum-preserving linear mappings were studied for the first time by G. Frobenius [18]. He proved that a linear mapping Φ from Mn(C) onto Mn(C) which preserves the spectrum has one … Spectrum-preserving linear mappings were studied for the first time by G. Frobenius [18]. He proved that a linear mapping Φ from Mn(C) onto Mn(C) which preserves the spectrum has one of the forms Φ(x) = axa−1 or Φ(x) = atxa−1, for some invertible matrix a. (Incidentally the hypothesis that Φ is onto is superfluous; see Proposition 2.1(i).) This result was extended by J. Dieudonné [17] supposing Φ onto and satisfying SpΦ(x) ⊂ Sp x, for every n × n matrix x. Several results of M. Nagasawa, S. Banach and M. Stone, R. V. Kadison, A. Gleason and J. P. Kahane and W. Żelazko led I. Kaplansky in [22] to the following problem: given two Banach algebras with unit and Φ a linear mapping from A into B such that Φ(1) = 1 and SpΦ(x) ⊂ Sp x, for every x ∈ A, is it true that Φ is a Jordan morphism? With this general formulation, this question cannot be true (see [2], p. 28).

Surjective maps on matrices preserving the local spectral radius distance

2013-07-16

Constantin Costara

Let be a non-zero vector in . We prove that if is a surjective map from the space of complex matrices into itself such that and the local spectral radius … Let be a non-zero vector in . We prove that if is a surjective map from the space of complex matrices into itself such that and the local spectral radius of at equals the local spectral radius of at for all and , there exists then an invertible matrix and of modulus one such that either and for all , or and for all . We arrive at the same conclusion by supposing that the local spectral radius of at equals the local spectral radius of at for all and .

Spectrum preserving linear mappings in Banach algebras

1994-01-01

Bernard Aupetit , Toit Mouton

Let A and B be two unitary Banach algebras. We study linear mappings from A into B which preserve the polynomially convex hull of the spectrum. In particular, we give … Let A and B be two unitary Banach algebras. We study linear mappings from A into B which preserve the polynomially convex hull of the spectrum. In particular, we give conditions under which such surjective linear mappings are Jordan morphisms.

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Linear maps on Mn(C) preserving the local spectral radius

2008-01-01

Abdellatif Bourhim , V. G. Miller

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Local Spectral Radius Preservers

2013-02-11

Abdellatif Bourhim , Javad Mashreghi

The surjectivity radius, packing numbers and boundedness below of linear operators

1983-12-01

Endre Makai , Jaroslav Zemánek

Invariant Subspaces of the Shift Operator

2015-04-20

Javad Mashreghi , Emmanuel Fricain , William T. Ross

A note on invertibility preservers on Banach algebras

2003-07-09

Matej Brešar , Ajda Fošner , Peter Šemrl

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper A"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">{\mathcal {A}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script … Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper A"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">{\mathcal {A}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper B"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">B</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">{\mathcal {B}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be semisimple Banach algebras and let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi colon script upper A right-arrow script upper B"> <mml:semantics> <mml:mrow> <mml:mi>ϕ</mml:mi> <mml:mo>:</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> <mml:mo stretchy="false">→</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">B</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\phi :\mathcal {A}\to \mathcal {B}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a unital bijective linear operator that preserves invertibility. If the socle of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper A"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">{\mathcal A}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an essential ideal of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper A"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">{\mathcal A}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi"> <mml:semantics> <mml:mi>ϕ</mml:mi> <mml:annotation encoding="application/x-tex">\phi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a Jordan isomorphism.

Additive mappings preserving operators of rank one

1993-03-01

Matjaž Omladič , Peter Šemrl

Isometries of Operator Algebras

1951-09-01

Richard V. Kadison

Well known results of Banach [1]2 and M. H. Stone [8] determine all linear isometric maps of one C(X) onto another (where 'C(X)' denotes, throughout this paper, the set of … Well known results of Banach [1]2 and M. H. Stone [8] determine all linear isometric maps of one C(X) onto another (where 'C(X)' denotes, throughout this paper, the set of all real-valued, continuous functions on the compact Hausdorff space X). Such isometries are the maps induced by homeomorphisms of the spaces involved followed by possible changes of sign in the function values on the various closed and open sets. An internal characterization of these isometries would classify them as an algebra isomorphism of the C(X)'s followed by a real unitary multiplication, i.e., multiplication by a real continuous function whose absolute value is 1. The situation in the case of the ring of complex continuous functions (which we denote by 'C'(X)' throughout) is exactly the same; the real unitary multiplication being replaced, of course, by a complex unitary multiplication. It is the purpose of this paper to present the non-commutative extension of the results stated above. A comment as to why this noncommutative extension takes form in a statement about algebras of operators on a Hilbert space seems to be in order. The work of Gelfand-Neumark [2T has as a very particular consequence the fact that each C'(X) is faithfully representable as a self-adjoint, uniformly closed algebra of operators (C*algebra) on a Hilbert space. The representing algebra of operators is, of course, commutative. A statement about the norm and algebraic structure of C' (X) finds then its natural non-commutative extension in the corresponding statement about not necessarily commutative C*algebras. A cursory examination shows that one cannot hope for a word for word transference of the C'(X) result to the non-commutative situation. An isometry between operator algebras is as likely to be an anti-isomorphism as an isomorphism. The direct sum of two C* algebras, which is again a C* algebra, by [2], with an automorphism in one component and an anti-automorphism in the other shows that isomorphisms and anti-isomorphisms together do not encompass all isometries. It is slightly surprising, in view of these facts, that any orderly classification of the isometries of a C* algebra is at all possible. It turns out, in fact, that all isometric maps are composites of a unitary multiplication and a map preserving the C*or quantum mechanical structure (see Segal [7])of the operator algebra in question. More specifically, such maps are linear isomorphisms which commute with the * operation and are multiplicative on powers, composed with a multiplication by a unitary operator in the algebra.

A generalization of peripherally-multiplicative surjections between standard operator algebras

2009-08-11

Takeshi Miura , Dai Honma

Abstract Let A and B be standard operator algebras on Banach spaces X and Y, respectively. The peripheral spectrum σπ (T) of T is defined by σπ (T) = z … Abstract Let A and B be standard operator algebras on Banach spaces X and Y, respectively. The peripheral spectrum σπ (T) of T is defined by σπ (T) = z ∈ σ(T): |z| = maxw∈σ(T) |w|. If surjective (not necessarily linear nor continuous) maps φ, ϕ: A → B satisfy σπ (φ(S)ϕ(T)) = σπ (ST) for all S; T ∈ A, then φ and ϕ are either of the form φ(T) = A 1 TA 2 −1 and ϕ(T) = A 2 TA 1 −1 for some bijective bounded linear operators A 1; A 2 of X onto Y, or of the form φ(T) = B 1 T*B 2 −1 and ϕ(T) = B 2 T*B −1 for some bijective bounded linear operators B 1;B 2 of X* onto Y.

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Jordan homomorphisms

1956-01-01

I. N. Herstein

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On generalized inverses in C*-algebras

1992-01-01

Robin Harte , Mostafa Mbekhta

We investigate when a C*-algebra element generates a closed ideal, and discuss Moore-Penrose and commuting generalized inverses. We investigate when a C*-algebra element generates a closed ideal, and discuss Moore-Penrose and commuting generalized inverses.

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Automatic continuity for linear surjective mappings decreasing the local spectral radius at some fixed vector

2010-11-05

Constantin Costara

Linear Maps Preserving Generalized Invertibility

2006-03-27

Mostafa Mbekhta , Leiba Rodman , Peter Šemrl

Bounded point evaluations for cyclic operators and local spectra

2001-07-25

Abdellatif Bourhim , C.E. Chidume , El Hassan Zerouali

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$p$-hyponormal operators are subscalar

2003-04-07

Lin Chen , Yingbin Ruan , Yan Zikun

We prove that if $R, S\in B(\mathbf {X }),\ R, S$ are injective, then $RS$ is subscalar if and only if $SR$ is subscalar. As corollaries, it is shown that … We prove that if $R, S\in B(\mathbf {X }),\ R, S$ are injective, then $RS$ is subscalar if and only if $SR$ is subscalar. As corollaries, it is shown that $p$-hyponormal operators $(0<p\le 1)$ and log-hyponormal operators are subscalar; also w-hyponormal operators $T$ with Ker$T\subset$ Ker$T^{*}$ and their generalized Aluthge transformations $T(r, 1-r) \ (0<r<1)$ are subscalar.

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JORDAN ISOMORPHISM OF PURELY INFINITE C*-ALGEBRAS

2007-02-09

Ying‐Fen Lin , Martin Mathieu

We prove that every unital bounded linear mapping from a unital purely infinite C*-algebra of real rank zero into a unital Banach algebra which preserves elements of square zero is … We prove that every unital bounded linear mapping from a unital purely infinite C*-algebra of real rank zero into a unital Banach algebra which preserves elements of square zero is a Jordan homomorphism. This entails a description of unital surjective spectral isometries as the Jordan isomorphisms in this setting.

The single valued extension property on a Banach space

1975-05-01

J.A. Finch

An operator T which maps a Banach space X into itself has the single valued extension property if the only analytic function / which satisfies (λl -Γ)/(λ) = 0 is … An operator T which maps a Banach space X into itself has the single valued extension property if the only analytic function / which satisfies (λl -Γ)/(λ) = 0 is / = 0. Clearly the point spectrum of any operator which does not have the single valued extension property must have nonempty interior.The converse does not hold.However, it is shown below that if λ o l -T is semi-Fredholm and λ 0 is an interior point of the point spectrum of Γ, then T does not have the single valued extension property.

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