The conception of the ideaIdeas of operators has played a fruitful role in the evolution of mathematics and physicsPhysics. The implementation of this idea made it possible to explore properties of solution spaces, the topology of certain spaces and to discover some underlying dualities, which made it possible to understand deep aspects of mathematical and physicalPhysical theories. For instance, the equivalence between integral equations and differential equations has allowed the resolution of many essential problems in mathematics and physics. This is the case of what is called the Fredholm operator in mathematics and the formulation of wave mechanics by Schrödinger. Our aim in this paper is to study and discuss the origin and development of the ideaIdeas of operators in mathematics and physics and the entanglement between the two, thanks to the properties of certain operators, which play a prominent role in old and recent developments in mathematics and physics.