M. A. Aouichaoui

This paper delves into the investigation of compact sets within the complex plane under a boundary constraint. Specifically, we focus on scenarios where a compact set $A$ is enclosed by a curve $\partial A$ that lies within the boundary of the augmented unit disk $\partial \mathbb{D}$, including the origin. The main goal is to establish a theorem that characterizes the possible configurations of such sets. By interweaving the principles of topology and operator theory, this study not only enhances our comprehension of compact sets under specialized boundary conditions but also underscores a practical implication in the realm of operator theory. This connection is particularly evident in the examination of the spectrum of operators that meet specific isometric conditions.