Operator

In mathematics, logic and physics the term operator can be used with various meanings.

In some version, an operator is a mathematical symbol that indicates that a specified operation is to be performed on a certain number of operands (number, function, vector, etc.).

Operators are usually interpreted as functions, better yet as applications, for example addition, multiplication, etc., can be understood as functions of two arguments. Or an application of SxS in S, or simply of D in F, case of indefinite integral or derivative which are linear operators.

Operators in a vector space

The term operator is a map between two vector spaces. It is used more frequently when one of them has infinite dimension. This is usually the case for a vector space whose elements are functions. If it is a linear application, we can call it a linear operator.

The study of linear operators is of particular interest in a Banach space. In this space, there is a norm and we can define a sphere of unit radius. The linear operator that is bounded in this sphere is called a bounded linear operator. The bounded linear operators between two Banach spaces in turn form a Banach space, the study of which is quite interesting. An extension of the real derivative to operators is the Frechet derivative which is a bounded linear operator.

Not all interesting linear operators are bounded: there are many examples of important operators in quantum mechanics that are not bounded.

The most typical example of non-accused linear operator is the derivative - considered as an application between two real functions spaces-. The differential operator, ${displaystyle {frac {d}{dx}}}}$, acts on the function ${displaystyle f(x),}$ that is written to your right, producing a new derivative function: ${displaystyle f'(x),}$

If an operator is defined between two vector spaces of functions, it acts by transforming some functions into others.

Bilinear or bivariate operators

(For stricter definitions of linearity and bilinearity, see related topics)

Their name depends on the author; they are operators that act on two objects (generally written on both sides of the operator) producing a single result. See the following cases.

General types of operators

Condition operators

They relate a term A with another B establishing their equality, hierarchy or any other possible relationship, as examples we have:

• A = B states that A is equal to B.

In this case we must distinguish between operator = and the operator = comparison. The first takes the value of B and assigned to it A; the second only compares the values A and B without modifying them and returns a logical or true value true if both values are equal or false if such values are not equal.

• B or inequality.

This case is precisely the opposite of the previous one, although here we cannot speak of assignment, but if of comparison. Now the result of this operation will be F if the values A and B are equal and V if they're different.

Order operators

Order operators establish or verify classifications between numbers (A < B, A > B, etc.) or other types of values (characters, strings,...).

Any type of data that can be ordered by any criterion can be compared to these operators; as the former return a true value based on the result that the comparison has in each case.

• A  B Return V Yeah. A is strictly greater than B and F otherwise
• A Δ B Return V Yeah. A is strictly lower than B and F otherwise
• A ≥ B Return V Yeah. A is greater or equal than B and F otherwise
• A ≤ B Return V Yeah. A is less or equal than B and F otherwise

• Other less common relational operators are the so-called geometric operators: parallelism (A ATAB), perpendicularity and others

Logical operators

Widely used in Computer Science, propositional logic and Boolean algebra, among other disciplines. Logical operators provide us with a result based on whether or not a certain condition is met. This generates a series of values that, in the simplest cases, can be parameterized with the numerical values 0 and 1, as can be seen in the examples below. The combination of two or more logical operators forms a logical function.

• The simplest are (note your relationship with the relational operators):
  • Non-logical Operator: '¬a' means everything that is not A'
  • Operator Y-logic: 'A ∧ B' means 'A and B at once'; resulting FALSO (0) if it is not fulfilled and TRUE (1) if it does.
  • O-logic Operator: 'A or B' if A is true and B is true, the result is true, if A is false and B is true or veceversa is still true and if A and B are false no matter what false means; 'Or A, or B, or both'; resulting FALSO (0) if they do not give A or B and TRUE (1) if one of the two or both is given at the same time.
  • Operator =: 'A = B' means 'A must be equal to B'; resulting FALSO (0) if this is not so and REAL (1) otherwise.
  • Operator ”: 'A θ B' means 'A must be less than B'; resulting FALSO (0) if not satisfied and TRUE (1) otherwise.
  • Operator : 'A  B' means 'A must be greater than B'; resulting FALSO (0) if not satisfied and TRUE (1) otherwise.

• The most complex operators are built from the previous ones (some more could be included) and already enter into what would be a logical function. A very used example would be 'SI(condition;A;B)' ('IF condition THEN A ELSE B' in most programming languages) whose result would be A if the 'condition' or B is satisfied otherwise.

Arithmetic operations

Arithmetic operations can be understood, from an operational point of view, as bivariate operators or as right-hand operators. For example, '2 × 3' It can be the bivariate multiplication operator acting on the numbers 2 and 3, or the '2 ×' which acts on 3. In this group are addition, subtraction, multiplication and division.

Other operations, derived from the usual arithmetic operations, are potentiation, radicalization and logarithmation.

Other operators

Related topics

• Linear application
• Defined bilineal shape
• Calculation
• Logical calculation