صالح احمد محمد مشرى

 

Research Statements

 If you want working in this area of operator theory, you will find many papers as references. You can download a pdf.files.

Let H be a seperable infinite-dimensional 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_HÎ cl Im(d_A). (1)

The related results discus sufficient conditions on A Î B(H) for I_HÏ 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=AB-BA. 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 quantum-mechanical momentum and position, respectively, satisfy the commutation relation PQ-QP = (-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 -SA||³ ||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 -SA||³ ||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 C₁ , C_¥ , L₁ 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 C₁-best approximation and L₁-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=Sⁿ, 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 Kleinecke-Shirokov theorem states that if X=Y=Z=A for a Banach algebra A and S=T=d_a:x® ax-xa 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 a-Weyl's theorem, or generalized a-Weyl's theorem and Browder's and generalized Browder's theorem.

 (V) Fuglede-Putnam's theorem

The familiar Fuglede-Putnam 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.

To know more about Inner Derivations, Commutator, Generalized Derivations. Search here .http://scholar.google.com/.

 

References:

1.

J. Anderson, On normal derivations, Proc. Amer. Math. Soc. 38-1(1973), 135-140. MR 47:875

2.

B.P. Duggal, A remark on normal derivations, Proc. Amer. Math. Soc. 126-7(1998), 2047-2052. MR 98h:47050

3.

F. Kittaneh, Operators that are orthogonal to the range of a derivation, J. Math. Anal. Appl. 203(1996), 863-873. MR 97f:47033

4.

F.Kittaneh, Normal derivations in norm ideals, Proc. Amer. Math. Soc., 123-6(1995)1779-1785. MR 95g:47054

5.

P.J. Maher, Commutator approximants, Proc. Amer. Math. Soc. 115(1992), 995-1000. MR 92j:47059

6.

S. Mecheri, On minimizing Serdica Math. J. 26 (2000), no. 2, 119-126. MR 2001j:47033

7.

S. Mecheri, On the orthogonality in von Neumann-Shatten classes, Int. Jour. Appl. Math, 8(2002), 441-447. 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), 501-506. MR 2003i:47038 .

 

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