Research Projects

The broad research area of my proposal is the Operator theory. In particular, this project is dedicated to analyzing the complex geometry, Nevanlinna-Pick interpolation problem, and operator theory on generalized tetrablock. These domains appear in association with structured singular value, which is a cost function on matrices that control engineers initiated in the environment of robust stabilization concerning modeling uncertainty . It is denoted by $\mu_{E}$ and defined as follows: \small{\begin{definition} Set $n\in \mathbb{Z}^{+}\,\, \text{and}\,\,n\geq2 $, and let E be a linear subspace of $\mathbb{C}^{n\times n}$. The functional \begin{equation}\label{mu} \mu_{E}(A):={\inf \{\|X\|: X \in E \text { and }(\mathbb{I}-A X) \text { is singular }\}}^{-1}, \quad A \in \mathbb{C}^{n \times n} \end{equation} is called a structured singular value. \end{definition} }. Here, $\|.\|$ denotes the operator norm with respect to the Euclidean norm on $\mathbb{C}^{n\times n}$. Usually, the subspace $E$ consists of all complex $n\times n$ matrices having a fixed block-diagonal structure. If we set $E=\mathbb{C}^{n\times n}$ then $\mu_{E}= \|.\|$. Furthermore, $\mu_{E}$ is spectral radius whenever $E$ is the space of all scalar matrices. P. Zapalowski described the domain \textit{generalized tetrablock} $\mathbb{E}(n;s;r_{1},...,r_{s})$ by considering $E$ as \small{\begin{equation}\label{E}E=E(n;s;r_{1},...,r_{s}):=\{\operatorname{diag}[z_{1}\mathbb{I}_{r_{1}},....,z_{s}\mathbb{I}_{r_{s}}]\in \mathbb{C}^{n\times n}: z_{1},...,z_{s}\in \mathbb{C},~n\geq2, \,\, s\leq n, \,\, \text{and}\,\,r_{1},...,r_{s} \,\, \text{with} \sum_{j=1}^{s}r_{j}=n.\} \end{equation} } The symmetrized polydisc $\mathbb G_n, n\geq 2$ can be obtained by considering the set $E=zI_n.$ In particular, for $n=2,$ the domain $\mathbb G_2$ is symmetrized bidisc which was inroduced by J. Agler and N. Young. When $E=w\oplus zI_{n-1}$, G. Bharli introduced the domain $\mathbb E_n$ for $n\geq 2.$ The domain $\mathbb E_2$ which is tetrablock, was addressed by Abouhajar. From the point of view of the Lempert’s theorem and the aspect of geometric function theory, the symmetrized bidisc, and the tetrablock play a crucial role in studying a long-standing open problem whether the Lempert’s theorem remains valid for $\mathbb C$-convex domain. We want to characterize $\mathbb{E}(n;s;r_{1},...,r_{s})$ for $r_j \leq 2$ and $j=1,\ldots,s$ for which the famous Lempert's theorem hold. We also want to investigate the Nevanlinna-Pick interpolation problem and operator theory for these kind of domain.