My research interests lie in the fields of functional analysis and operator theory. More precisely, I have been working on the classification of the commutators on various Banach spaces as well as the structure of the commutator ideals. More generally, I am interested in the structure of the norm closed ideals in Banach spaces. This research activity is motivated by the fact that the structure of commutator ideals is essential for investigating traces which in turn is relevant for the calculation of the cyclic homology and the algebraic K-theory of operator ideals. During my graduate study at Texas A&M University, I have made contributions to the classification of the commutators on $\ell_1$, $\ell_{\infty}$ and $\ell_{p_1}\oplus \ell_{p_2}\oplus\cdots\oplus\ell_{p_n}$ where I was able to obtain a complete classification. As a visiting assistant professor at Texas A&M University and later as a postdoctoral fellow at Weizmann Institute of science, together with W.B. Johnson and G. Schechtman we were able obtain a complete classification of the commutators on $L_p$ for $1\leq p<\infty$.

### Overview

Commutators, linear operators of the form $AB - BA$, first appeared in physics, for instance in a mathematical formulation of the Heisenberg Uncertainty Principle. Later, they became an important tool for investigating the derivations of a general Lie algebra, which in turn has numerous applications in many areas of mathematics. A natural problem that arises in the study of derivations on a Banach algebra is to classify the commutators in the algebra. The first major contribution in this direction was due to Wintner, who proved that the identity in a unital Banach algebra is not a commutator. This immediately implies that no operator of the form $\lambda I +K$, where $K$ belongs to a norm closed proper ideal $I(X)$ of $L(X)$ and $\lambda\neq 0$, is a commutator in the Banach algebra $L(X)$ of all bounded linear operators on the Banach space $X$. In 1965 Brown and Pearcy made a breakthrough by classifying the commutators on a Hilbert space :

Theorem. An operator $T\in L(\ell_2)$ is a commutator if and only if $T -\lambda I$ is not compact for any $\lambda\neq 0$.

The theorem above suggests what the classification might be in the other classical sequence spaces, and in 1972, Apostol proved that every non-commutator on the space $\ell_p$ for $1 < p < \infty$ is of the form $\lambda I +K$, where $K$ is compact and $\lambda\neq 0$. A year later he also showed that the same classification holds for the space $c_0$. It should be also noted that in 1971, Schneeberger proved that the compact operators on $L_p$ for $1 < p < \infty$ are commutators.

### The conjecture

Note that if $X = \ell_p, 1 < p <\infty$, or $X = c_0$, the ideal of compact operators $K(X)$ is the largest proper ideal in $L(X)$. The results of Brown and Pearcy and Apostol suggest the following.

Conjecture. Let $X$ be a Banach space for which there exists a largest proper ideal in $L(X)$. Then the only operators on $X$ that are not commutators are the ones of the form $\lambda I + K$, where $K$ belongs to the largest ideal in $L(X)$ and $\lambda\neq 0$.

We were able to show that the conjecture remains valid for the spase $\ell_\infty$, where the largest ideal is the ideal of strictly singular operators, and it is also valid for $L_p$ for $1 < p < \infty$, where the largest ideal is the ideal of non-E ($p=1$) and non-A ($1 < p < \infty$) operators.