Operator (mathematics)

An operator is a mathematical rule ( a calculus ), through which you can make new objects of mathematical objects. It can be a standard function or a rule of functions. Application, see the operators for arithmetic operations, ie in manual or mechanical calculations.

Operator

Standardized operators are usually then defined in mathematics, if it is a common, recurring requirement, usually a one-or two -digit shortcut. The arguments of this linkage called operands. The operators are represented by a special, distinctive mathematical symbol (a special character of the formula notation).

Examples:

• The operators used for the basic arithmetic operations of addition, which is written as the plus sign " ", the minus sign "-" for subtraction, the mark of "·", " x " or "* " for multiplication, and the division's Divided characters " ÷ ", " :", " /" and the fraction bar
• The unary operator for the negative number, which is also with a minus - is written " "
• The pipe symbol " " for the composition of functions
• The operator class education

Operand

The arguments on which to apply an operator are called operands. When printing, so the numbers and the addresses that are associated with the two-sided operator.

Operators in the Functional Analysis

In the functional analysis one has to do with vector spaces, whose elements are themselves functions. To better distinguish the elements of this vector spaces by the maps between such vector spaces, the latter also called operators.

Examples

Examples of well-known operators that assign a number to a function or any other function, is:

• The differential operator in the formation of differentials.
• The Volterraoperator to form the definite integral. Operators like this, assign the function of a number, called functional.
• The nabla operator to determine the gradient of a multidimensional function.

Linear operators between vector spaces

In the ( linear ) functional analysis we consider properties of linear mappings between ( infinite-dimensional ) Banach spaces. Such maps are called operators. So do and Banach spaces, then they are called with pictures

Operators. Sometimes, in order to emphasize the nonlinearity, one also speaks of a linear operator. Corresponds to the space is called as the space of real or complex numbers, the functional imaging.

The operators described in the previous section are examples of operators in the sense of (linear) functional analysis. Special classes of linear operators are compact operators or about Fredholm operators.

Nonlinear Operators

In the ( non-linear ) functional analysis is sometimes called mappings between normed vector spaces and operators. Be so and normed spaces and a subset, then that means the picture also ( non-linear ) operator. Is the field of real or complex numbers, you sometimes also called ( nonlinear ) functional. Just as in the field of linear operators the compact operators form an important subset even in the non-linear operators.

Operators of Physics

Operators are also defined in the mathematical calculus of Physics:

• Density operator,
• Position operator,
• Momentum operator,
• Energy operator,
• Hamiltonian of quantum mechanics