Title: Linear isometries on the Fréchet space of holomorphic functions on the open unit disc

Abstract: Let X be a topological space of holomorphic functions on the open unit disc $D$. The study of the geometry of a space X is centered on the identification of the linear isometries on X, and there is an obvious connection between weighted composition operators and isometries. This connection can be traced back to Banach himself and emphasized by Forelli, El-Gebeily, Wolfe, Kolaski, Cima, Wogen, Colonna and many others. A characterisation is given of all the linear isometries of Hol (D), the Fréchet space of all holomorphic functions on the unit disc D, when it is given one of the two standard metrics. Operators similar to an isometry are also classified. Further, the larger class of operators isometric when restricted to one of the defining seminorms is identified.