<?xml version="1.0" encoding="UTF-8"?><xml><records><record><source-app name="Biblio" version="7.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Seddik, Ameur</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">Let B(H) denote the C&amp;lowast; -algebra of all bounded linear operators on a complex separable Hilbert space H , and let |||&amp;middot;||| be a unitarily invariant norm on some ideal I of B(H). In this paper, we shall show that: for f, g be two continuous non</style></title><secondary-title><style face="normal" font="default" size="100%">Mathematical Inequalities and Applications</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">2025</style></year></dates><urls><web-urls><url><style face="normal" font="default" size="100%">dx.doi.org/10.7153/mia-2025-28-35</style></url></web-urls></urls><volume><style face="normal" font="default" size="100%">28</style></volume><pages><style face="normal" font="default" size="100%">577–586 </style></pages><language><style face="normal" font="default" size="100%">eng</style></language><issue><style face="normal" font="default" size="100%">4</style></issue></record><record><source-app name="Biblio" version="7.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Seddik, Ameur</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">Some results related to the Heinz inequality in C*-algebra</style></title><secondary-title><style face="normal" font="default" size="100%">Mathematical Inequalities and Applications</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">2024</style></year></dates><volume><style face="normal" font="default" size="100%">27</style></volume><pages><style face="normal" font="default" size="100%">249–259</style></pages><language><style face="normal" font="default" size="100%">eng</style></language><issue><style face="normal" font="default" size="100%">1</style></issue></record><record><source-app name="Biblio" version="7.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Seddik, Ameur</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">Operator inequalities related to the arithmetic&amp;ndash;geometricmean inequality and characterizations</style></title><secondary-title><style face="normal" font="default" size="100%">Advances in Operator Theory</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">2023</style></year></dates><urls><web-urls><url><style face="normal" font="default" size="100%">https://doi.org/10.1007/s43036-022-00234-w</style></url></web-urls></urls><volume><style face="normal" font="default" size="100%">8</style></volume><language><style face="normal" font="default" size="100%">eng</style></language><abstract><style face="normal" font="default" size="100%">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–geometric mean inequality.</style></abstract></record><record><source-app name="Biblio" version="7.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">A. Seddik</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">Seladjoint operators, normal operators, and characterizations</style></title><secondary-title><style face="normal" font="default" size="100%">Operators and Matrices</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">2019</style></year></dates><volume><style face="normal" font="default" size="100%">835-842</style></volume><pages><style face="normal" font="default" size="100%">835-842</style></pages><language><style face="normal" font="default" size="100%">eng</style></language><abstract><style face="normal" font="default" size="100%">In this note, we present several characterizations for some distinguished&lt;br&gt;classes of bounded Hilbert space operators (self-adjoint operators, normal&lt;br&gt;operators, unitary operators, and isometry operators) in terms of operator&lt;br&gt;inequalities.</style></abstract></record><record><source-app name="Biblio" version="7.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Bouraya. C</style></author><author><style face="normal" font="default" size="100%">Seddik. A</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">On the characterization of some distinguishedclasses of Hilbert space operators</style></title><secondary-title><style face="normal" font="default" size="100%">Acta Sci. Math. (Szeged)</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">2018</style></year></dates><volume><style face="normal" font="default" size="100%">84</style></volume><pages><style face="normal" font="default" size="100%">611–627</style></pages><language><style face="normal" font="default" size="100%">eng</style></language><abstract><style face="normal" font="default" size="100%">In this note, we present several characterizations for some distinguished&lt;br&gt;classes of bounded Hilbert space operators (self-adjoint operators, normal&lt;br&gt;operators, unitary operators, and isometry operators) in terms of operator&lt;br&gt;inequalities.</style></abstract></record><record><source-app name="Biblio" version="7.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">A. Seddik</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">Corrigendum to Moore-Penrose Inverse and OperatorInequalities&amp;quot; Extracta Mathematicae 30 (2015), 29-39</style></title><secondary-title><style face="normal" font="default" size="100%">extracta mathematicae</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">2017</style></year></dates><volume><style face="normal" font="default" size="100%">32</style></volume><pages><style face="normal" font="default" size="100%">209 – 211</style></pages><language><style face="normal" font="default" size="100%">eng</style></language><abstract><style face="normal" font="default" size="100%">We correct a mistake which affect our main results, namely the proof of Lema 1.&lt;br&gt;The main results of the article remain unchanged.</style></abstract></record><record><source-app name="Biblio" version="7.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">A. Seddik</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">Moore-Penrose inverse and operator inequalities</style></title><secondary-title><style face="normal" font="default" size="100%">extracta mathematicae</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">2015</style></year></dates><volume><style face="normal" font="default" size="100%">30</style></volume><pages><style face="normal" font="default" size="100%">29-39</style></pages><language><style face="normal" font="default" size="100%">eng</style></language><abstract><style face="normal" font="default" size="100%">In this note, we shall give complete characterizations of the class of all normal&lt;br&gt;operators with closed range, and the class of all selfadjoint operators with closed range&lt;br&gt;multiplied by scalars in terms of some operator inequalities.</style></abstract></record><record><source-app name="Biblio" version="7.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Conde. C</style></author><author><style face="normal" font="default" size="100%">Moslehian. M.S</style></author><author><style face="normal" font="default" size="100%">Seddik. A</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">Operator inequalities related to the Corach&amp;ndash;Porta&amp;ndash;Rechtinequality</style></title><secondary-title><style face="normal" font="default" size="100%">Linear Algebra Appl.</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">2012</style></year></dates><volume><style face="normal" font="default" size="100%">436</style></volume><pages><style face="normal" font="default" size="100%">3008-3017</style></pages><language><style face="normal" font="default" size="100%">eng</style></language><abstract><style face="normal" font="default" size="100%">We prove some refinements of an inequality due to X. Zhan in an&lt;br&gt;arbitrary complex Hilbert space by using some results on the Heinz&lt;br&gt;inequality. We present several related inequalities as well as new&lt;br&gt;variants of the Corach–Porta–Recht inequality.We also characterize&lt;br&gt;the class of operators satisfying&lt;br&gt; SXS&lt;br&gt;−1 + S&lt;br&gt;−1XS + kX&lt;br&gt;&lt;br&gt; (k +&lt;br&gt;2) X under certain conditions.</style></abstract></record><record><source-app name="Biblio" version="7.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Menkad. S</style></author><author><style face="normal" font="default" size="100%">Seddik. A</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">OPERATOR INEQUALITIES AND NORMAL OPERATORS</style></title><secondary-title><style face="normal" font="default" size="100%">Banach J. Math. Anal.</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">2012</style></year></dates><volume><style face="normal" font="default" size="100%">6</style></volume><pages><style face="normal" font="default" size="100%">204-210</style></pages><language><style face="normal" font="default" size="100%">eng</style></language><abstract><style face="normal" font="default" size="100%">In the present paper, taking some advantages oered by the con-&lt;br&gt;text of nite dimensional Hilbert spaces, we shall give a complete characteriza-&lt;br&gt;tions of certain distinguished classes of operators (self-adjoint, unitary re ec-&lt;br&gt;tion, normal) in terms of operator inequalities. These results extend previous&lt;br&gt;characterizations obtained by the second author.</style></abstract></record><record><source-app name="Biblio" version="7.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Seddik. A</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">CLOSED OPERATOR INEQUALITIES AND OPEN PROBLEMS</style></title><secondary-title><style face="normal" font="default" size="100%">Mathematical Inequalities and Applications</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">2011</style></year></dates><volume><style face="normal" font="default" size="100%">14</style></volume><pages><style face="normal" font="default" size="100%">147-154</style></pages><language><style face="normal" font="default" size="100%">eng</style></language><abstract><style face="normal" font="default" size="100%">In [3, 4] , we have given some characterizations of some subclasses of complex Hilbert&lt;br&gt;space normal operators by inequalities. In this note, we shall reformulate these results using a&lt;br&gt;concept of closed inequalities. Also, we shall give some new characterizations. Some open&lt;br&gt;problems concerning closed inequalities are posed at the end of this note.</style></abstract></record><record><source-app name="Biblio" version="7.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Seddik. A</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">On the injective norm and characterization of some subclassesof normal operators by inequalities or equalities</style></title><secondary-title><style face="normal" font="default" size="100%">J. Math. Anal. Appl.</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">2009</style></year></dates><volume><style face="normal" font="default" size="100%">35</style></volume><pages><style face="normal" font="default" size="100%">277-284</style></pages><language><style face="normal" font="default" size="100%">eng</style></language></record><record><source-app name="Biblio" version="7.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Seddik. A</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">ON THE INJECTIVE NORM OF sigma(Ai &amp;otimes; Bi) AND CHARACTERIZATION OF NORMALOID OPERATORS</style></title><secondary-title><style face="normal" font="default" size="100%">Operators and Matrices</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">2008</style></year></dates><volume><style face="normal" font="default" size="100%">2</style></volume><pages><style face="normal" font="default" size="100%">67-77</style></pages><language><style face="normal" font="default" size="100%">eng</style></language><abstract><style face="normal" font="default" size="100%">Let B(H) denotes the C∗-algebra of all bounded linear operators acting on the&lt;br&gt;complex Hilbert space H . In this note, we shall give some lower estimates for the injective norm&lt;br&gt;of the element&lt;br&gt;n&lt;br&gt;i=1&lt;br&gt;Ai⊗Bi in the tensor product B(H)⊗B(H), where A = (A1, ..., An) and B =&lt;br&gt;(B1, ..., Bn) are two n-tuples of elements in B(H) ; and we shall characterize the normaloid&lt;br&gt;operators in B(H) using the injective norm.&lt;br&gt;1.</style></abstract></record><record><source-app name="Biblio" version="7.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Seddik. A</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">Rank one operators and norm of elementary operators</style></title><secondary-title><style face="normal" font="default" size="100%">Linear Algebra Appl.</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">2007</style></year></dates><volume><style face="normal" font="default" size="100%">424</style></volume><pages><style face="normal" font="default" size="100%">177-183</style></pages><language><style face="normal" font="default" size="100%">eng</style></language><abstract><style face="normal" font="default" size="100%">Let A be a standard operator algebra acting on a (real or complex) normed space E. For two n-tuples&lt;br&gt;A = (A1, . . . , An) and B = (B1, . . . , Bn) of elements in A, we define the elementary operator RA,B on&lt;br&gt;A by the relation RA,B(X) = n&lt;br&gt;i=1 AiXBi for all X in A. For a single operator A ∈ A, we define the&lt;br&gt;two particular elementary operators LA and RA onAby LA(X) = AX and RA(X) = XA, for every X in&lt;br&gt;A. We denote by d(RA,B) the supremum of the norm of RA,B(X) over all unit rank one operators on E.&lt;br&gt;In this note, we shall characterize: (i) the supremun d(RA,B), (ii) the relation d(RA,B) = n&lt;br&gt;i=1&lt;br&gt;Ai&lt;br&gt;Bi&lt;br&gt;,&lt;br&gt;(iii) the relation d(LA&lt;br&gt;− RB) = A + B, (iv) the relation d(LARB&lt;br&gt;+ LBRA) = 2AB. Moreover,&lt;br&gt;we shall showthe lower estimate d(LA&lt;br&gt;− RB)  max{supλ∈V (B)&lt;br&gt;A − λI, supλ∈V (A)&lt;br&gt;B − λI} (where&lt;br&gt;V (X) is the algebraic numerical range of X inA).</style></abstract></record><record><source-app name="Biblio" version="7.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Seddik. A</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">Onthe Numerical Range and NormofElementary Operators</style></title><secondary-title><style face="normal" font="default" size="100%">Linear and Multilinear Algebra</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">2004</style></year></dates><volume><style face="normal" font="default" size="100%">52</style></volume><pages><style face="normal" font="default" size="100%">293-302</style></pages><language><style face="normal" font="default" size="100%">eng</style></language><abstract><style face="normal" font="default" size="100%">Let BðEÞ be the complex Banach algebra of all bounded linear operators on a complex Banach space E: For&lt;br&gt;n-tuples A ¼ ðA1, . . . ,AnÞ and B ¼ ðB1, . . . ,BnÞ of operators on E, let RA, B denote the operator on BðEÞ&lt;br&gt;defined by RA, BðXÞ ¼&lt;br&gt;Pn&lt;br&gt;i¼1AiXBi :&lt;br&gt;For A, B 2 BðEÞ, we put UA, B ¼ RðA, BÞ, ðB,AÞ:&lt;br&gt;In this note, we prove that&lt;br&gt;co&lt;br&gt;Xn&lt;br&gt;i¼1&lt;br&gt;ii : ð1, . . . , nÞ 2 VðAÞ, ð1, . . . , nÞ 2 VðBÞ&lt;br&gt;( )&lt;br&gt; W0ðRA, BjJÞ&lt;br&gt;where VðÞ is the joint spatial numerical range, W0ðÞ is the algebraic numerical range and J is a norm ideal of&lt;br&gt;BðEÞ: We shall show that this inclusion becomes an equality when RA, B is taken to be a derivation. Also, we&lt;br&gt;deduce that wðUA, BjJÞ  2ð&lt;br&gt;ffiffiffi2&lt;br&gt;p&lt;br&gt; 1ÞwðAÞwðBÞ, for A,B 2 BðEÞ and J is a norm ideal of BðEÞ, where wðÞ is the&lt;br&gt;numerical radius.&lt;br&gt;On the other hand, in the particular case when E is a Hilbert space, we shall prove that the lower estimate&lt;br&gt;bound kUA, BjJk  2ð&lt;br&gt;ffiffiffi2&lt;br&gt;p&lt;br&gt; 1ÞkAkkBk holds, if one of the following two conditions is satisfied:&lt;br&gt;(i) J is a standard operator algebra of BðEÞ and A,B 2 J:&lt;br&gt;(ii) J is a norm ideal of BðEÞ and A, B 2 BðEÞ:</style></abstract></record><record><source-app name="Biblio" version="7.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Seddik. A</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">On the norm of elementary operatorsin standard operator algebras</style></title><secondary-title><style face="normal" font="default" size="100%">Acta Sci. Math. (Szeged)</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">2004</style></year></dates><volume><style face="normal" font="default" size="100%">70</style></volume><pages><style face="normal" font="default" size="100%">229-236</style></pages><language><style face="normal" font="default" size="100%">eng</style></language><abstract><style face="normal" font="default" size="100%">Let A be a complex normed algebra. For A,B ∈ A, define a basic&lt;br&gt;elementary operator MA,B : A → A by MA,B(X) = AXB.&lt;br&gt;Given a standard operator algebra A acting on a complex normed space&lt;br&gt;and A,B ∈ A we have:&lt;br&gt;(i) The lower estimate kMA,B +MB,Ak ≥ 2(√2 − 1)kAkkBk holds.&lt;br&gt;(ii) The lower estimate kMA,B +MB,Ak ≥ kAkkBk holds if&lt;br&gt;inf&lt;br&gt;∈C kA + Bk = kAk or inf&lt;br&gt;∈C kB + Ak = kBk.&lt;br&gt;(iii) The equality kMA,B +MB,Ak = 2kAkkBk holds if&lt;br&gt;kA + Bk = kAk + kBk for some unit scalar .&lt;br&gt;These results extend analogous estimates established earlier for standard&lt;br&gt;operator subalgebras of Hilbert space operators.</style></abstract></record><record><source-app name="Biblio" version="7.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Seddik. A</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">On some operator norm inequalities</style></title><secondary-title><style face="normal" font="default" size="100%">Linear Algebra Appl.</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">2004</style></year></dates><volume><style face="normal" font="default" size="100%">389</style></volume><pages><style face="normal" font="default" size="100%">183-187</style></pages><language><style face="normal" font="default" size="100%">eng</style></language><abstract><style face="normal" font="default" size="100%">Let B(H) be the C&lt;br&gt;∗-algebra of all bounded linear operators on a complex Hilbert space&lt;br&gt;H, S be an invertible and selfadjoint operator in B(H) and let (I, .&lt;br&gt;I ) denote a norm ideal&lt;br&gt;of B(H). In this note, we shall show the following inequality:&lt;br&gt;∀X ∈ I : SXS&lt;br&gt;−1 − S&lt;br&gt;−1XS&lt;br&gt;I  (SS&lt;br&gt;−1 − 1)SXS&lt;br&gt;−1 + S&lt;br&gt;−1XS&lt;br&gt;I .</style></abstract></record><record><source-app name="Biblio" version="7.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Seddik. A</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">THE NUMERICAL RANGE OF ELEMENTARY OPERATORS</style></title><secondary-title><style face="normal" font="default" size="100%">Integr. equ. oper. theory</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">2002</style></year></dates><volume><style face="normal" font="default" size="100%">43</style></volume><pages><style face="normal" font="default" size="100%">248-252</style></pages><language><style face="normal" font="default" size="100%">eng</style></language><abstract><style face="normal" font="default" size="100%">For n-tuples A = (At, ..., AN) and B = (BI, ..., Bn) of operators on aHilber! space&lt;br&gt;H, let RA,B denote the operator on L(H) defined by RA,B(X) = ~i=1AiX Bi. In&lt;br&gt;this paper we prove that&lt;br&gt;co c~i13i : (a,,..., an) C W(A), (131, ..., 1~) e W(B) C Wo(RA,B)&lt;br&gt;where W is the joint spatial numerical range and W0 is the numerical range. We&lt;br&gt;will show also that this inclusion becomes an equality when I~A,B is taken to be&lt;br&gt;a generalized derivation, and it is strict when RA,B is taken to be an elementary&lt;br&gt;multiplication operator induced by non scalar self-adjoints operators.</style></abstract></record><record><source-app name="Biblio" version="7.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Seddik. A</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">SOME RESULTS RELATEDTO THE CORACH-PORTA-RECHT INEQUALITY</style></title><secondary-title><style face="normal" font="default" size="100%">Proc. Amer. Math. Soc.</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">2001</style></year></dates><volume><style face="normal" font="default" size="100%">129</style></volume><pages><style face="normal" font="default" size="100%">3009-3015</style></pages><language><style face="normal" font="default" size="100%">eng</style></language></record><record><source-app name="Biblio" version="7.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Seddik. A</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">The numerical range of elementary operators II</style></title><secondary-title><style face="normal" font="default" size="100%">Linear Algebra Appl.</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">2001</style></year></dates><volume><style face="normal" font="default" size="100%">338</style></volume><pages><style face="normal" font="default" size="100%">239-244</style></pages><language><style face="normal" font="default" size="100%">eng</style></language><abstract><style face="normal" font="default" size="100%">For A,B ∈ L(H) (the algebra of all bounded linear operators on the Hilbert space H),&lt;br&gt;it is proved that: (i) the generalized derivation δA,B is convexoid if and only if A and B are&lt;br&gt;convexoid; (ii) the operators δA,B and δA,B&lt;br&gt;|Cp (where p  1) have the same numerical range&lt;br&gt;and are equal to W0(A) − W0(B) (where Cp is the Banach space of the p-Schatten class&lt;br&gt;operators on H).</style></abstract></record><record><source-app name="Biblio" version="7.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Seddik. A</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">Sur I&amp;rsquo;lmage et le Noyau d&amp;rsquo;une Dbrivation G6n6ralis6e</style></title><secondary-title><style face="normal" font="default" size="100%">Linear Algebra Appl.</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">1998</style></year></dates><volume><style face="normal" font="default" size="100%">274</style></volume><pages><style face="normal" font="default" size="100%">77~83</style></pages><language><style face="normal" font="default" size="100%">eng</style></language><abstract><style face="normal" font="default" size="100%">Let A E_Y(H~), B E_F(H.J ( w h ere Hi, H, are Hilbert spaces), and let S,, H&lt;br&gt;denote the operator on _Y( H,, H,) given by&lt;br&gt;6,,,(X) = AX - XB, X =.Y( H,, H,)&lt;br&gt;J. P. Williams asked: For which A is R(i%,)-n {A*) = {O}?(w here S,, A = 6,) We&lt;br&gt;obtain some operators in this class. The case of 6, s, A f B, is interesting in itself;&lt;br&gt;moreover it is useful if we have to use a decomposition of the Hilbert space in a direct&lt;br&gt;some for the consideration of 6,. In this note we describe some classes of operators&lt;br&gt;A, B for which we have R(6,, H) - f? ker S,,, se = (0). 0 1998 Elsevier Science</style></abstract></record><record><source-app name="Biblio" version="7.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Seddik. A</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">DERIVATION AND JORDAN OPERATORS</style></title><secondary-title><style face="normal" font="default" size="100%">Integr. equ. oper. theory</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">1997</style></year></dates><volume><style face="normal" font="default" size="100%">28</style></volume><pages><style face="normal" font="default" size="100%">120 - 124</style></pages><language><style face="normal" font="default" size="100%">eng</style></language><abstract><style face="normal" font="default" size="100%">For A E/:(H) ( the algebra of all operators on the complex Hilbert space H)~ let 5A&lt;br&gt;denote the operator on/~(H) defined by : (~A(X) = AX - XA.&lt;br&gt;We show here that for all Jordan operators A : R(SA) N {A*)' = {0}, where R(6A) is&lt;br&gt;the range of 6A and (A*}; is the commntant of the adjoint of A.</style></abstract></record></records></xml>