# Operator (math)

An operator is a mathematical rule (a calculus ) that can be used to create new objects from mathematical objects . It can be a standardized function or a rule about functions. The operators are used in arithmetic operations , i.e. in manual or machine calculations.

Some operators for the four basic arithmetic operations : plus , minus , times and divided .

## operator

Standardized operators are usually defined in mathematics when it is a question of a frequent, recurring rule, usually a one- or two-digit link . The arguments of this link are called operands. The operators are represented by a special, identifying mathematical symbol (a special character of the formula notation ).

Examples:

## operand

The arguments to which an operator is applied are called operands. In the expression , the numbers and the operands are linked with the two-sided operator . ${\ displaystyle 1 + 2}$${\ displaystyle 1}$${\ displaystyle 2}$${\ displaystyle +}$

## Operators in functional analysis

In functional analysis one has to do with vector spaces , the elements of which are themselves functions. In order to better distinguish the elements of these vector spaces from the mappings between such vector spaces, the latter are also called operators . Mapping of function spaces in the field of real or complex numbers is also called functional. Special classes of operators are compact operators or Fredholm operators .

### Examples

Well-known examples of operators that assign a number or another function to a function are:

• The differential operator for forming differentials .${\ displaystyle \ textstyle {\ frac {\ mathrm {d}} {\ mathrm {d} x}}}$
• The Volterra operator for the formation of the definite integral . Operators like these, which assign a number to a function, are called functional .${\ displaystyle \ textstyle \ int _ {0} ^ {t}}$
• The Nabla operator for determining the gradient of a multi-dimensional function.${\ displaystyle \ nabla}$

### Linear and non-linear operators

In functional analysis one considers the properties of mappings between (infinite-dimensional) Banach spaces . Linear maps are called linear operators, nonlinear maps are called nonlinear operators.

## Operators of physics

Observables in quantum mechanics are operators. They are usually named after the quantity to be measured: the operator for measuring the position is then called the position operator . Similarly, there are the momentum operator , the spin operator , etc. ${\ displaystyle {\ hat {\ mathbf {x}}}}$ ${\ displaystyle {\ hat {\ mathbf {p}}}}$ ${\ displaystyle {\ hat {\ vec {\ mathbf {s}}}}}$

The operator for energy is called the Hamilton operator and denoted by. It occurs in particular in the Schrödinger equation . ${\ displaystyle {\ hat {H}}}$ ${\ displaystyle \ mathrm {i} \ hbar {\ tfrac {\ partial} {\ partial t}} | \, \ psi (t) \ rangle = {\ hat {H}} | \, \ psi (t) \ rangle}$

The density operator is a ensemble the probability is chosen at a system is with in a particular state. ${\ displaystyle \ rho}$