On the Commutativity of Unbounded Unclosed Symmetric Operators

This thesis consists of three chapters. The first chapter establishes the necessary background, covering linear operators both bounded and unbounded in Hilbert spaces, as well as the spectral theorem and the Fuglede-Putnam theorems. The second chapter looks at past research, showing how results improved over time by using weaker conditions on the operators. The third chapter builds on the work of S. Dehimi, M. H. Mortad, and Bachir [11]. It explores what happens when we remove the closedness assumption on the unbounded operator. Even without this condition, we show that the main results still hold and present additional generalizations.