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  B(H), define the inner
derivation (or commutator map ) A and the generalzed
derivation A,B by
A(S)= AS – SA, A,B(S) =AS – SB, S 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 •
A) (or  A) and the kernel Ker(linking the range Im(
A,B as well as a more general maps) in the those of
above settings.
.
(I) Various published results motivated by the following quite remarkable
example due to Anderson (1973):
There are A  B(H) so that the identity operator IH. (1)
The related results discus sufficient conditions on A  B(H) for IH, and more
generally the size of sets such as or (and similarly for 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
0 and K a compact operator and Wintner result  I+ K, for not of the form
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  B(H) is a normal that commutes with T  B(H), then
S B(H).
(2)
It is a restatement of (2) that Ker(A) is Birkhoff orthogonal to the range
Im(A) on B(H) once A  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 A , or studying the restriction of these operators to symmetrically
normed ideals in B(H) . For the schatten classes Cp this leads to
optimization problems for maps S , which involve Gâteaux –type
differentiation. Here we introduce a new concept of Gâteaux derivative,
called - 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 - 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 T 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 Im(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=a:x ax-xa is an inner derivation on A,
then Ker(S)Im(T) 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)   J  Q, J=K(H) is the compact
operators and 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=A*:B(H)
B(H); XA*X - XA*? We have conjectured this question in one of
my papers and we give a positive answer to this conjecture in an other
paper.
(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 see 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.