Linear Operators

Abstract

${\mathcal{B}}$(H) denotes the set of operators in a Hilbert space H, equipped with the uniform norm. The study of elements of ${\mathcal{B}}$(H) is carried out in this Chapter. Some well-known classes of operators such as normal operators, self-adjoint operators, unitaries, isometries have been discussed. Characterisations of these classes have been accorded due importance. The adjoint of an element of ${\mathcal{B}}$(H) has been defined via bilinear forms. Several examples of different classes of operators have been included. Their adjoints have been computed. Invertibility of an element in ${\mathcal{B}}$(H) and its characterisations have been discussed. Also included in this Chapter are the square root of an operator and the polar decomposition of an operator. The Chapter ends with an application to Brownian motion. This and the following Chapter constitute the core of the book.