@article {89884,
	title = {Some results related to the Heinz inequality in C*-algebra},
	journal = {Mathematical Inequalities and Applications},
	volume = {27},
	number = {1},
	year = {2024},
	pages = {249{\textendash}259},
	author = {Seddik, Ameur}
}
@article {89885,
	title = {Operator inequalities related to the arithmetic\&ndash;geometricmean inequality and characterizations},
	journal = {Advances in Operator Theory},
	volume = {8},
	year = {2023},
	abstract = {In this survey, we shall present the characterizations of some distinguished classes of bounded linear operators acting on a complex separable Hilbert space in terms of operator inequalities related to the arithmetic{\textendash}geometric mean inequality.},
	url = {https://doi.org/10.1007/s43036-022-00234-w},
	author = {Seddik, Ameur}
}
@article {14895,
	title = {Seladjoint operators, normal operators, and characterizations},
	journal = {Operators and Matrices},
	volume = {835-842},
	year = {2019},
	pages = {835-842},
	abstract = {In this note, we present several characterizations for some distinguishedclasses of bounded Hilbert space operators (self-adjoint operators, normaloperators, unitary operators, and isometry operators) in terms of operatorinequalities.},
	author = {A. Seddik}
}
@article {14901,
	title = {On the characterization of some distinguishedclasses of Hilbert space operators},
	journal = {Acta Sci. Math. (Szeged)},
	volume = {84},
	year = {2018},
	pages = {611{\textendash}627},
	abstract = {In this note, we present several characterizations for some distinguishedclasses of bounded Hilbert space operators (self-adjoint operators, normaloperators, unitary operators, and isometry operators) in terms of operatorinequalities.},
	author = {Bouraya. C and Seddik. A}
}
@article {14908,
	title = {Corrigendum to Moore-Penrose Inverse and OperatorInequalities\&quot; Extracta Mathematicae 30 (2015), 29-39},
	journal = {extracta mathematicae},
	volume = {32},
	year = {2017},
	pages = {209 {\textendash} 211},
	abstract = {We correct a mistake which affect our main results, namely the proof of Lema 1.The main results of the article remain unchanged.},
	author = {A. Seddik}
}
@article {14912,
	title = {Moore-Penrose inverse and operator inequalities},
	journal = {extracta mathematicae},
	volume = {30},
	year = {2015},
	pages = {29-39},
	abstract = {In this note, we shall give complete characterizations of the class of all normaloperators with closed range, and the class of all selfadjoint operators with closed rangemultiplied by scalars in terms of some operator inequalities.},
	author = {A. Seddik}
}
@article {15206,
	title = {Operator inequalities related to the Corach\&ndash;Porta\&ndash;Rechtinequality},
	journal = {Linear Algebra Appl.},
	volume = {436},
	year = {2012},
	pages = {3008-3017},
	abstract = {We prove some refinements of an inequality due to X. Zhan in anarbitrary complex Hilbert space by using some results on the Heinzinequality. We present several related inequalities as well as newvariants of the Corach{\textendash}Porta{\textendash}Recht inequality.We also characterizethe class of operators satisfying SXS-1 + S-1XS + kX (k +2) X under certain conditions.},
	author = {Conde. C and Moslehian. M.S and Seddik. A}
}
@article {14920,
	title = {OPERATOR INEQUALITIES AND NORMAL OPERATORS},
	journal = {Banach J. Math. Anal.},
	volume = {6},
	year = {2012},
	pages = {204-210},
	abstract = {In the present paper, taking some advantages oered by the con-text of nite dimensional Hilbert spaces, we shall give a complete characteriza-tions of certain distinguished classes of operators (self-adjoint, unitary re ec-tion, normal) in terms of operator inequalities. These results extend previouscharacterizations obtained by the second author.},
	author = {Menkad. S and Seddik. A}
}
@article {15207,
	title = {CLOSED OPERATOR INEQUALITIES AND OPEN PROBLEMS},
	journal = {Mathematical Inequalities and Applications},
	volume = {14},
	year = {2011},
	pages = {147-154},
	abstract = {In [3, 4] , we have given some characterizations of some subclasses of complex Hilbertspace normal operators by inequalities. In this note, we shall reformulate these results using aconcept of closed inequalities. Also, we shall give some new characterizations. Some openproblems concerning closed inequalities are posed at the end of this note.},
	author = {Seddik. A}
}
@article {15209,
	title = {On the injective norm and characterization of some subclassesof normal operators by inequalities or equalities},
	journal = {J. Math. Anal. Appl.},
	volume = {35},
	year = {2009},
	pages = {277-284},
	author = {Seddik. A}
}
@article {15210,
	title = {ON THE INJECTIVE NORM OF sigma(Ai \&otimes; Bi) AND CHARACTERIZATION OF NORMALOID OPERATORS},
	journal = {Operators and Matrices},
	volume = {2},
	year = {2008},
	pages = {67-77},
	abstract = {Let B(H) denotes the C*-algebra of all bounded linear operators acting on thecomplex Hilbert space H . In this note, we shall give some lower estimates for the injective normof the elementni=1Ai(x)Bi in the tensor product B(H)(x)B(H), where A = (A1, ..., An) and B =(B1, ..., Bn) are two n-tuples of elements in B(H) ; and we shall characterize the normaloidoperators in B(H) using the injective norm.1.},
	author = {Seddik. A}
}
@article {15211,
	title = {Rank one operators and norm of elementary operators},
	journal = {Linear Algebra Appl.},
	volume = {424},
	year = {2007},
	pages = {177-183},
	abstract = {Let A be a standard operator algebra acting on a (real or complex) normed space E. For two n-tuplesA = (A1, . . . , An) and B = (B1, . . . , Bn) of elements in A, we define the elementary operator RA,B onA by the relation RA,B(X) = ni=1 AiXBi for all X in A. For a single operator A ∈ A, we define thetwo particular elementary operators LA and RA onAby LA(X) = AX and RA(X) = XA, for every X inA. We denote by d(RA,B) the supremum of the norm of RA,B(X) over all unit rank one operators on E.In this note, we shall characterize: (i) the supremun d(RA,B), (ii) the relation d(RA,B) = ni=1AiBi,(iii) the relation d(LA- RB) = A + B, (iv) the relation d(LARB+ LBRA) = 2AB. Moreover,we shall showthe lower estimate d(LA- RB)  max{supλ∈V (B)A - λI, supλ∈V (A)B - λI} (whereV (X) is the algebraic numerical range of X inA).},
	author = {Seddik. A}
}
@article {15230,
	title = {Onthe Numerical Range and NormofElementary Operators},
	journal = {Linear and Multilinear Algebra},
	volume = {52},
	year = {2004},
	pages = {293-302},
	abstract = {Let B{\dh}E{\TH} be the complex Banach algebra of all bounded linear operators on a complex Banach space E: Forn-tuples A {\textonequarter} {\dh}A1, . . . ,An{\TH} and B {\textonequarter} {\dh}B1, . . . ,Bn{\TH} of operators on E, let RA, B denote the operator on B{\dh}E{\TH}defined by RA, B{\dh}X{\TH} {\textonequarter}Pni{\textonequarter}1AiXBi :For A, B 2 B{\dh}E{\TH}, we put UA, B {\textonequarter} R{\dh}A, B{\TH}, {\dh}B,A{\TH}:In this note, we prove thatcoXni{\textonequarter}1ii : {\dh}1, . . . , n{\TH} 2 V{\dh}A{\TH}, {\dh}1, . . . , n{\TH} 2 V{\dh}B{\TH}( ) W0{\dh}RA, BjJ{\TH}where V{\dh}{\TH} is the joint spatial numerical range, W0{\dh}{\TH} is the algebraic numerical range and J is a norm ideal ofB{\dh}E{\TH}: We shall show that this inclusion becomes an equality when RA, B is taken to be a derivation. Also, wededuce that w{\dh}UA, BjJ{\TH}  2{\dh}ffiffiffi2p 1{\TH}w{\dh}A{\TH}w{\dh}B{\TH}, for A,B 2 B{\dh}E{\TH} and J is a norm ideal of B{\dh}E{\TH}, where w{\dh}{\TH} is thenumerical radius.On the other hand, in the particular case when E is a Hilbert space, we shall prove that the lower estimatebound kUA, BjJk  2{\dh}ffiffiffi2p 1{\TH}kAkkBk holds, if one of the following two conditions is satisfied:(i) J is a standard operator algebra of B{\dh}E{\TH} and A,B 2 J:(ii) J is a norm ideal of B{\dh}E{\TH} and A, B 2 B{\dh}E{\TH}:},
	author = {Seddik. A}
}
@article {15214,
	title = {On the norm of elementary operatorsin standard operator algebras},
	journal = {Acta Sci. Math. (Szeged)},
	volume = {70},
	year = {2004},
	pages = {229-236},
	abstract = {Let A be a complex normed algebra. For A,B ∈ A, define a basicelementary operator MA,B : A {\textrightarrow} A by MA,B(X) = AXB.Given a standard operator algebra A acting on a complex normed spaceand A,B ∈ A we have:(i) The lower estimate kMA,B +MB,Ak >= 2(√2 - 1)kAkkBk holds.(ii) The lower estimate kMA,B +MB,Ak >= kAkkBk holds ifinf∈C kA + Bk = kAk or inf∈C kB + Ak = kBk.(iii) The equality kMA,B +MB,Ak = 2kAkkBk holds ifkA + Bk = kAk + kBk for some unit scalar .These results extend analogous estimates established earlier for standardoperator subalgebras of Hilbert space operators.},
	author = {Seddik. A}
}
@article {15212,
	title = {On some operator norm inequalities},
	journal = {Linear Algebra Appl.},
	volume = {389},
	year = {2004},
	pages = {183-187},
	abstract = {Let B(H) be the C*-algebra of all bounded linear operators on a complex Hilbert spaceH, S be an invertible and selfadjoint operator in B(H) and let (I, .I ) denote a norm idealof B(H). In this note, we shall show the following inequality:∀X ∈ I : SXS-1 - S-1XSI  (SS-1 - 1)SXS-1 + S-1XSI .},
	author = {Seddik. A}
}
@article {15232,
	title = {THE NUMERICAL RANGE OF ELEMENTARY OPERATORS},
	journal = {Integr. equ. oper. theory},
	volume = {43},
	year = {2002},
	pages = {248-252},
	abstract = {For n-tuples A = (At, ..., AN) and B = (BI, ..., Bn) of operators on aHilber! spaceH, let RA,B denote the operator on L(H) defined by RA,B(X) = ~i=1AiX Bi. Inthis paper we prove thatco c~i13i : (a,,..., an) C W(A), (131, ..., 1~) e W(B) C Wo(RA,B)where W is the joint spatial numerical range and W0 is the numerical range. Wewill show also that this inclusion becomes an equality when I~A,B is taken to bea generalized derivation, and it is strict when RA,B is taken to be an elementarymultiplication operator induced by non scalar self-adjoints operators.},
	author = {Seddik. A}
}
@article {16334,
	title = {SOME RESULTS RELATEDTO THE CORACH-PORTA-RECHT INEQUALITY},
	journal = {Proc. Amer. Math. Soc.},
	volume = {129},
	year = {2001},
	pages = {3009-3015},
	author = {Seddik. A}
}
@article {15234,
	title = {The numerical range of elementary operators II},
	journal = {Linear Algebra Appl.},
	volume = {338},
	year = {2001},
	pages = {239-244},
	abstract = {For A,B ∈ L(H) (the algebra of all bounded linear operators on the Hilbert space H),it is proved that: (i) the generalized derivation δA,B is convexoid if and only if A and B areconvexoid; (ii) the operators δA,B and δA,B|Cp (where p  1) have the same numerical rangeand are equal to W0(A) - W0(B) (where Cp is the Banach space of the p-Schatten classoperators on H).},
	author = {Seddik. A}
}
@article {16336,
	title = {Sur I\&rsquo;lmage et le Noyau d\&rsquo;une Dbrivation G6n6ralis6e},
	journal = {Linear Algebra Appl.},
	volume = {274},
	year = {1998},
	pages = {77~83},
	abstract = {Let A E_Y(H~), B E_F(H.J ( w h ere Hi, H, are Hilbert spaces), and let S,, Hdenote the operator on _Y( H,, H,) given by6,,,(X) = AX - XB, X =.Y( H,, H,)J. P. Williams asked: For which A is R(i\%,)-n {A*) = {O}?(w here S,, A = 6,) Weobtain some operators in this class. The case of 6, s, A f B, is interesting in itself;moreover it is useful if we have to use a decomposition of the Hilbert space in a directsome for the consideration of 6,. In this note we describe some classes of operatorsA, B for which we have R(6,, H) - f? ker S,,, se = (0). 0 1998 Elsevier Science},
	author = {Seddik. A}
}
@article {16337,
	title = {DERIVATION AND JORDAN OPERATORS},
	journal = {Integr. equ. oper. theory},
	volume = {28},
	year = {1997},
	pages = {120 - 124},
	abstract = {For A E/:(H) ( the algebra of all operators on the complex Hilbert space H)~ let 5Adenote the operator on/~(H) defined by : (~A(X) = AX - XA.We show here that for all Jordan operators A : R(SA) N {A*){\textquoteright} = {0}, where R(6A) isthe range of 6A and (A*}; is the commntant of the adjoint of A.},
	author = {Seddik. A}
}
