Operator (math)

Operator ( late late. Operator - employee, performer, from operor - I work, I act) - a mathematical mapping between sets in which each of them is endowed with some additional structure (order, topology, algebraic operations). The concept of an operator is used in various branches of mathematics to distinguish it from other kinds of mappings (mainly numerical functions ); the exact value depends on the context, for example, in functional analysis, operators are understood as mappings that map a function to another function (“operator on the space of functions” instead of “function of function”).

Some types of operators:

operators on function spaces ( differentiation , integration , convolution with the kernel , Fourier transform ) in functional analysis;
mappings (especially linear) between vector spaces (projectors, coordinate rotations, homotheties, vector multiplication by a matrix) in linear algebra ;
sequence conversion (convolution of discrete signals, median filter) in discrete mathematics .

Content

1 Basic terminology
2 Simple Examples
3 Linear Operators
4 Zero operator
5 Unit (identity) operator
6 Record
7 Linear differential operator symbol
8 See also
9 notes
10 Literature

Basic terminology

About the operator $A:X\to Y$ ${\ displaystyle A: X \ to Y}$ they say that he acts out of many $X$ ${\ displaystyle X}$ in many $Y$ ${\ displaystyle Y}$ . An operator may not be everywhere defined on $X$ ${\ displaystyle X}$ ; then talk about his domain $D_{A}=D(A)\subset X$ ${\ displaystyle D_ {A} = D (A) \ subset X}$ . For $x\in X$ ${\ displaystyle x \ in X}$ result of operator application $A$ ${\ displaystyle A}$ to $x$ ${\ displaystyle x}$ denote $A(x)$ ${\ displaystyle A (x)}$ or $A x$ ${\ displaystyle Ax}$ .

If $X$ ${\ displaystyle X}$ and $Y$ ${\ displaystyle Y}$ Are vector spaces , then in the set of all operators from $X$ ${\ displaystyle X}$ at $Y$ ${\ displaystyle Y}$ we can distinguish a class of linear operators .

If $X$ ${\ displaystyle X}$ and $Y$ ${\ displaystyle Y}$ Are vector topological spaces , then in the set of operators from $X$ ${\ displaystyle X}$ at $Y$ ${\ displaystyle Y}$ the class of continuous operators is naturally distinguished, as well as the class of linear bounded operators and the class of linear compact operators (also called completely continuous).

Simple examples

An operator acting on function spaces is a rule according to which one function is transformed into another. Function conversion $x(t)$ ${\ displaystyle x (t)}$ according to the rule $A$ ${\ displaystyle A}$ to another function $y(t)$ ${\ displaystyle y (t)}$ has the form $y(t)=A\{x(t)\}$ ${\ displaystyle y (t) = A \ {x (t) \}}$ or, more simply, $y=Ax$ ${\ displaystyle y = Ax}$ .

Examples of such transformations are multiplication by a number: $y(t)=cx(t)$ ${\ displaystyle y (t) = cx (t)}$ and differentiation: $\scriptstyle y(t)={\frac {dx(t)}{dt}}$ ${\ displaystyle \ scriptstyle y (t) = {\ frac {dx (t)} {dt}}}$ . The corresponding operators are called the operators of multiplication by number, differentiation, integration, solution of a differential equation, etc.

Operators that modify a function argument are called transform operators or transforms . The transformation replaces the coordinate axes, maps the function to another space. For example, the Fourier transform from time to frequency domain:

F(\omega )={\frac {1}{\sqrt {2\pi }}}\int \limits _{-\infty }^{\infty }f(t)e^{-it\omega }\,dt={\mathcal {F}}\{f(t)\}.

{\ displaystyle F (\ omega) = {\ frac {1} {\ sqrt {2 \ pi}}} \ int \ limits _ {- \ infty} ^ {\ infty} f (t) e ^ {- it \ omega} \, dt = {\ mathcal {F}} \ {f (t) \}.}

The difference between an operator and a simple superposition of functions in this case is that the value of the function $y$ ${\ displaystyle y}$ generally speaking at every point $t$ ${\ displaystyle t}$ depends not only on $x(t)$ ${\ displaystyle x (t)}$ , and from the values of the function $x$ ${\ displaystyle x}$ at all points $t$ ${\ displaystyle t}$ . Let us illustrate with the example of the Fourier transform. The value of this transformation (function spectrum) at the point $\omega$ ${\ displaystyle \ omega}$ changes with continuous change of the original function in the neighborhood of any point $t$ ${\ displaystyle t}$ .

The theory of operators is studying the general properties of operators and applying them to solving various problems. For example, it turns out that the operator of multiplying a vector by a matrix and the operator of convolution of a function with a weight have many common properties.

Fundamental to practice is the class of so-called linear operators . He is also the most researched. An example of a linear operator is the multiplication operation $n$ ${\ displaystyle n}$ -dimensional vector per matrix size $n\times m$ ${\ displaystyle n \ times m}$ . This operator displays $n$ ${\ displaystyle n}$ -dimensional space of vectors in $m$ ${\ displaystyle m}$ -dimensional.

Linear Operators

Operator $L$ ${\ displaystyle L}$ (acting from a vector space to a vector one) is called linear homogeneous (or simply linear ) if it has the following properties:

can be applied term by term to the sum of arguments:
$L(x_{1}+x_{2})=L(x_{1})+L(x_{2})$ ${\ displaystyle L (x_ {1} + x_ {2}) = L (x_ {1}) + L (x_ {2})}$ ;
scalar (constant value) $c$ ${\ displaystyle c}$ can be taken out of the operator sign:
$L(cx)=cL(x)$ ${\ displaystyle L (cx) = cL (x)}$ ;

From the second property it follows that for a linear homogeneous operator the property $L(0)=0$ ${\ displaystyle L (0) = 0}$ .

Operator $L$ ${\ displaystyle L}$ is called linear inhomogeneous if it consists of a linear homogeneous operator with the addition of some fixed element:

L\{x\}=L_{0}\{x\}+\varphi

{\ displaystyle L \ {x \} = L_ {0} \ {x \} + \ varphi}

,

Where $L_{0}$ ${\ displaystyle L_ {0}}$ Is a linear homogeneous operator.

In the case of a linear transformation of discrete functions (sequences, vectors), the new values of the functions $y_{k}$ ${\ displaystyle y_ {k}}$ are linear functions of old values $x_{k}$ ${\ displaystyle x_ {k}}$ :

y_{k}=\sum _{l=1}^{n}T_{kl}\,x_{l}

{\ displaystyle y_ {k} = \ sum _ {l = 1} ^ {n} T_ {kl} \, x_ {l}}

.

In the more general case of continuous functions, the two-dimensional weight matrix takes the form of a function of two variables $K(t,\;\omega )$ ${\ displaystyle K (t, \; \ omega)}$ , and is called the kernel of the linear integral transformation:

\varphi (t)=\int \limits _{V}\!K(t,\omega )f(\omega )\,d\omega =K\{f(\omega )\}.

{\ displaystyle \ varphi (t) = \ int \ limits _ {V} \! K (t, \ omega) f (\ omega) \, d \ omega = K \ {f (\ omega) \}.}

Operand function $f(\omega )$ ${\ displaystyle f (\ omega)}$ in this case is called the spectral function . The spectrum can be discrete, then $f(\omega )$ ${\ displaystyle f (\ omega)}$ replaced by vector $W$ ${\ displaystyle W}$ . In this case $\varphi (t)$ ${\ displaystyle \ varphi (t)}$ representable by a finite or infinite series of functions:

\varphi (t)=\sum _{i=1}^{n}T_{i}(t)w_{i}.

{\ displaystyle \ varphi (t) = \ sum _ {i = 1} ^ {n} T_ {i} (t) w_ {i}.}

Zero operator

Operator $O$ ${\ displaystyle O}$ matching each vector $\mathbf {a}$ ${\ displaystyle \ mathbf {a}}$ zero vector $\mathbf {0}$ ${\ displaystyle \ mathbf {0}}$ obviously linear; it is called the null operator ^[1] .

Single (identical) operator

Operator $E$ ${\ displaystyle E}$ matching each vector $\mathbf {a}$ ${\ displaystyle \ mathbf {a}}$ the vector itself $\mathbf {a}$ ${\ displaystyle \ mathbf {a}}$ obviously linear; it is called the identity or identity operator.

A special case of a linear operator that returns an operand unchanged:

E\mathbf {a} =\mathbf {a} ,

{\ displaystyle E \ mathbf {a} = \ mathbf {a},}

that is, how the matrix operator is determined by the equality

\sum _{k}E_{ik}\,a_{k}=a_{i}

{\ displaystyle \ sum _ {k} E_ {ik} \, a_ {k} = a_ {i}}

and as an integral operator - by the equality

\int \limits _{\alpha }^{\beta }\!E(x,t)a(t)\,dt=a(x)

{\ displaystyle \ int \ limits _ {\ alpha} ^ {\ beta} \! E (x, t) a (t) \, dt = a (x)}

.

Unit matrix $E_{ik}$ ${\ displaystyle E_ {ik}}$ written mostly with the symbol $\delta _{ik}=\delta _{ki}$ ${\ displaystyle \ delta _ {ik} = \ delta _ {ki}}$ ( Kronecker symbol ). We have: $\delta _{ik}=1$ ${\ displaystyle \ delta _ {ik} = 1}$ at $i=k$ ${\ displaystyle i = k}$ and $\delta _{ik}=0$ ${\ displaystyle \ delta _ {ik} = 0}$ at $i\neq k$ ${\ displaystyle i \ neq k}$ .

Single core $E(x,t)$ ${\ displaystyle E (x, t)}$ written as $E(x,t)=\delta (t-x)$ ${\ displaystyle E (x, t) = \ delta (tx)}$ ( delta function ). $\delta (x-t)=0$ ${\ displaystyle \ delta (xt) = 0}$ everywhere except $x=t$ ${\ displaystyle x = t}$ , where the function becomes infinite and, moreover, such that

\int \limits _{\alpha }^{\beta }\!\delta (x-t)\,dt=1

{\ displaystyle \ int \ limits _ {\ alpha} ^ {\ beta} \! \ delta (xt) \, dt = 1}

.

Record

In mathematics and technology, the conventional form of operator notation, similar to algebraic symbolism, is widely used. Such symbolism in some cases avoids complex transformations and writes formulas in a simple and convenient form. The arguments of the operator are called operands , the number of operands is called the arity of the operator (for example, single, binary). Writing statements can be systematized as follows:

prefix : where operator and operands come next, for example:

Q(x_{1},\;x_{2},\;\ldots ,\;x_{n});

{\ displaystyle Q (x_ {1}, \; x_ {2}, \; \ ldots, \; x_ {n});}

postfix : if the operator character follows the operands, for example:

(x_{1},\;x_{2},\;\ldots ,\;x_{n})\;Q;

{\ displaystyle (x_ {1}, \; x_ {2}, \; \ ldots, \; x_ {n}) \; Q;}

infix : the operator is inserted between operands, it is mainly used with binary operators:

x_{1}\;Q\;x_{2};

{\ displaystyle x_ {1} \; Q \; x_ {2};}

positional : the operator sign is omitted, the operator is present implicitly. Most often, the product operator is not written (variables, numerical values per physical unit, matrices, composition of functions), for example, 3 kg. This ability of one operator to act on heterogeneous entities is achieved by overloading operators ;
subscript or superscript left or right; It is mainly used for exponentiation and element selection of a vector by index.

As you can see, the operator record often takes an abbreviated form from the generally accepted function record. When using prefix or postfix notation, the brackets are omitted in most cases if the arity of the operator is known. So, a single operator $Q$ ${\ displaystyle Q}$ over function $f$ ${\ displaystyle f}$ usually written for brevity $Q f$ ${\ displaystyle Qf}$ instead $Q(f)$ ${\ displaystyle Q (f)}$ ; brackets are used for clarity, for example, operation on a work $Q(fg)$ ${\ displaystyle Q (fg)}$ . $Q$ ${\ displaystyle Q}$ acting on $f(x)$ ${\ displaystyle f (x)}$ also record $(Qf)(x)$ ${\ displaystyle (Qf) (x)}$ . To indicate some operators, special characters are introduced, for example, unary $n!$ ${\ displaystyle n!}$ (factorial “!”, to the right of the operand), $-n$ ${\ displaystyle -n}$ (negation, left) or calligraphic characters, as in the case of the Fourier transform of a function ${\mathcal {F}}\{f(t)\}$ ${\ displaystyle {\ mathcal {F}} \ {f (t) \}}$ . Exponentiation $n^{x}$ ${\ displaystyle n ^ {x}}$ can be considered a binary operator of two arguments, or a power or exponential function of one argument.

Symbol of a linear differential operator

The symbol of a linear differential operator maps a differential operator to a polynomial, roughly speaking, replacing the composition of partial derivatives with the product of the variables associated with them. The highest monomials of the operator symbol (the main operator symbol) reflect the qualitative behavior of the solution of the partial differential equation corresponding to this operator. Partial linear elliptic equations are characterized by the fact that their main symbol does not turn into 0 anywhere.

Let be $x=(x_{1},\ldots ,x_{n})$ ${\ displaystyle x = (x_ {1}, \ ldots, x_ {n})}$ and there are multi-indices $\alpha =(\alpha _{1},\ldots ,\alpha _{n})$ ${\ displaystyle \ alpha = (\ alpha _ {1}, \ ldots, \ alpha _ {n})}$ and $\beta =(\beta _{1},\ldots ,\beta _{n})$ ${\ displaystyle \ beta = (\ beta _ {1}, \ ldots, \ beta _ {n})}$ . Then we put

{\begin{aligned}D^{\alpha }x^{\beta }&={\frac {\partial ^{\vert \alpha \vert }}{\partial x_{1}^{\alpha _{1}}\cdots \partial x_{n}^{\alpha _{n}}}}x_{1}^{\beta _{1}}\cdots x_{n}^{\beta _{n}}\\&={\frac {\partial ^{\alpha _{1}}}{\partial x_{1}^{\alpha _{1}}}}x_{1}^{\beta _{1}}\cdots {\frac {\partial ^{\alpha _{n}}}{\partial x_{n}^{\alpha _{n}}}}x_{n}^{\beta _{n}}.\end{aligned}}

{\ displaystyle {\ begin {aligned} D ^ {\ alpha} x ^ {\ beta} & = {\ frac {\ partial ^ {\ vert \ alpha \ vert}} {\ partial x_ {1} ^ {\ alpha _ {1}} \ cdots \ partial x_ {n} ^ {\ alpha _ {n}}}} x_ {1} ^ {\ beta _ {1}} \ cdots x_ {n} ^ {\ beta _ {n }} \\ & = {\ frac {\ partial ^ {\ alpha _ {1}}} {\ partial x_ {1} ^ {\ alpha _ {1}}}} x_ {1} ^ {\ beta _ { 1}} \ cdots {\ frac {\ partial ^ {\ alpha _ {n}}} {\ partial x_ {n} ^ {\ alpha _ {n}}}} x_ {n} ^ {\ beta _ {n }}. \ end {aligned}}}

Let be $P$ ${\ displaystyle P}$ Is a linear differential order operator $k$ ${\ displaystyle k}$ in Euclidean space $\mathbb {R} ^{d}$ ${\ displaystyle \ mathbb {R} ^ {d}}$ . Then $P$ ${\ displaystyle P}$ is a polynomial from the derivative $D$ ${\ displaystyle D}$ , in a multi-index record it will be written like this

P=p(x,D)=\sum _{|\alpha |\leq k}a_{\alpha }(x)D^{\alpha }.

{\ displaystyle P = p (x, D) = \ sum _ {| \ alpha | \ leq k} a _ {\ alpha} (x) D ^ {\ alpha}.}

Polynomial $p$ ${\ displaystyle p}$ , by definition, is a complete symbol $P$ ${\ displaystyle P}$ :

\sigma P(\xi )=p(x,\xi )=\sum _{|\alpha |\leq k}a_{\alpha }\xi ^{\alpha }.

{\ displaystyle \ sigma P (\ xi) = p (x, \ xi) = \ sum _ {| \ alpha | \ leq k} a _ {\ alpha} \ xi ^ {\ alpha}.}

The main symbol of the operator consists of maximal monomials $\sigma _{P}$ ${\ displaystyle \ sigma _ {P}}$ :

\sigma _{P}(\xi )=\sum _{|\alpha |=k}a_{\alpha }\xi ^{\alpha }

{\ displaystyle \ sigma _ {P} (\ xi) = \ sum _ {| \ alpha | = k} a _ {\ alpha} \ xi ^ {\ alpha}}

and is part of the complete symbol of the operator, which is transformed as a tensor when changing coordinates.

Notes

↑ Shilov G.E. Mathematical analysis. Special course. - M .: Fizmatlit, 1961. - C. 203

Literature

(1995) Operator. Mathematical Encyclopedic Dictionary. Ch. ed. Yu. V. Prokhorov. M .: "Big Russian Encyclopedia."

[1] Shilov G.E. Mathematical analysis. Special course. - M .: Fizmatlit, 1961. - C. 203