Research Statements
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Let H be a seperable infinitedimensional complex Hilbert space and let B(H) be the algebra of all bounded linear operators on H. If A,B Î B(H), define the inner derivation (or commutator map ) d_{A} and the generalized derivation d_{A}_{,B} by
d_{A}(S)= AS – SA, d_{A,B}(S) =AS – SB, SÎ B(H).
These concrete operators on B(H) occur in many settings in mathematical analysis and its applications, and their properties have been studied already during many decades.
The main emphasis on my research works is on topics linking the range Im(d_{A}) and the kernel Ker(d_{A}) (or those of d_{A,B} as well as a more general maps) in the above settings. Topics include the following:
(I) Various published results motivated by the following quite remarkable example due to Anderson (1973):
There are A Î B(H) so that the identity operator I_{H}Î cl Im(d_{A}). (1)
The related results discus sufficient conditions on A Î B(H) for I_{H}Ï cl Im(d_{A}) and more generally the size of sets such as or (and similarly for d_{A,B}). The first important contribution to the study of commutators is due to A. Wintner who in 1947 proved that the identity element 1 in a unital, normed algebra A is not a commutator, that is, there are no elements A and B such that I=ABBA. Like much good mathematics, Wintner's theorem has its roots in physics.
Indeed, it was prompted by the fact that the unbounded linear maps P and Q representing the quantummechanical momentum and position, respectively, satisfy the commutation relation PQQP = (ih/{2\pi})I, where h is the Planck's constant and I the identity operator on the underlying Hilbert space. The classical Brown Pearcy characterization of the commutators on B(H) which are not of the form lI+ K, for l ¹ 0 and K a compact operator and Wintner result are the natural motivation for Anderson result (1) and our published results.
(II) By far the largest part of my research works is devoted to results which extend another result by Anderson (1973):
If A Î B(H) is a normal that commutes with T Î B(H),
then T + AS SA³ T, SÎ B(H) (2)
It is a restatement of (2) that Ker(d_{A}) is Birkhoff orthogonal to the range Im(d_{A}) on B(H) once A Î B(H) is normal. We have obtained subsequent generalizations of Anderson result (2) in various deirections: relaxing the condition that A is normal, or considering more general operators than d_{A } , or studying the restriction of these operators to symmetrically normed ideals in B(H) . For the schatten classes C_{p}_{ }this leads to optimization problems for maps
S® T + AS SA³ T, which involve Gâteaux –type differentiation. Here we introduce a new concept of Gâteaux derivative, called j Gâteaux derivative which we have used in the case of C_{1}_{ }, C_{¥} , L_{1} which are not strictly convex , and there are many points which are not smooth. The j Gâteaux derivative can be used without care of smoothness to minimize S® , and to prove the existence and uniqueness of the best C_{1}best approximation and L_{1}best approximation.
(III) For bounded linear operators T:X®Y and S:Y®Z on Banach spaces the condition ker TÇ Im(T)={0} is equivalent to the equality ker(ST)= ker T; when X=Y=Z and T=S^{n}, this is the familiar condition that the operator S has ascent £ n. Stronger conditions would replace the range Im(T) of T by its closure, either in the norm or in some weaker topology; weaker conditions would ask that the intersection of KerS ÇIm(T) with some subspace of Y was in some sense nearly zero. Thus the
celebrate KleineckeShirokov theorem states that if X=Y=Z=A for a Banach algebra A and S=T=d_{a}:x® axxa is an inner derivation on A, then Ker(S)ÇIm(T)Ì Q, where Q=QN(A) is the quasinilpotents in A Weber showed for same S and T then when A = B(H) for separable Hilbert space H then Ker(S) Ç clw ImT Ç J Ì Q, J=K(H) is the compact operators and clw is the weak closure of ImT. We have obtained Weber's result as a consequence of a more general general result. A reasonable question is following: Does Weber's result holds when S=T=d_{A*}:B(H) ®B(H); X®A*X  XA*? We conjectured this question in one of my papers and we give a positive answer to this conjecture in an other paper. We are also intereted on the investigation of S=T=d_{A,B}:B(H) ®B(H); X®AX  XB and
S=T=d_{A*, B*}:B(H) ®B(H); X®A*X  XB*
(IV) Local spectral Theory
Weyl's theorem and Browder's theorem are related to an important property which has a leading role on local spectral theory: the single valued extension property. The study of operators satisfying Browder's theorem and Weyl's theorem is of significant interest, and is currently being done by a number of mathematicians around the world. My research works is devoted to results which extend Weyl's and Browder's theorem to generalized Weyl's theorem, or aWeyl's theorem, or generalized aWeyl's theorem and Browder's and generalized Browder's theorem.
(V) FugledePutnam's theorem
The familiar FugledePutnam Theorem is as follows
If A and B are normal operators and if X is an operator such that AX=XB, then A*X=XB*. We are interested to extend this theorem to a more general classes of operators.
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References:

 1.
 J. Anderson, On normal derivations, Proc. Amer. Math. Soc. 381(1973), 135140. MR 47:875
 2.
 B.P. Duggal, A remark on normal derivations, Proc. Amer. Math. Soc. 1267(1998), 20472052. MR 98h:47050
 3.
 F. Kittaneh, Operators that are orthogonal to the range of a derivation, J. Math. Anal. Appl. 203(1996), 863873. MR 97f:47033
 4.
 F.Kittaneh, Normal derivations in norm ideals, Proc. Amer. Math. Soc., 1236(1995)17791785. MR 95g:47054
 5.
 P.J. Maher, Commutator approximants, Proc. Amer. Math. Soc. 115(1992), 9951000. MR 92j:47059
 6.
 S. Mecheri, On minimizing Serdica Math. J. 26 (2000), no. 2, 119126. MR 2001j:47033
 7.
 S. Mecheri, On the orthogonality in von NeumannShatten classes, Int. Jour. Appl. Math, 8(2002), 441447. MR 2003b:47063
 8.
 S. Mecheri and A.Bachir, Generalized derivation modulo the ideal of all compact operators, Int.Jour.Math.Math.Sc., 32.8(2002), 501506. MR 2003i:47038 .
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