Basic Properties of Operators

- The
**sum**and**difference**of two operators and are given by

(33) (34)

- The
**product**of two operators is defined by

(35) - Two operators are
**equal**if

(36) - The
**identity operator**does nothing (or multiplies by 1)

(37)

(38) - The
**associative law**holds for operators

(39) - The
**commutative law**does*not*generally hold for operators. In general, . It is convenient to define the quantity

(40) **commutator**of and . Note that the order matters, so that . If and happen to commute, then . - The
**n-th power**of an operator is defined as successive applications of the operator, e.g.

(41) - The
**exponential**of an operator is defined via the power series

(42)