Operator Theory

the branch of functional analysis that studies the properties of operators and the application of operators to the solution of various problems. The operator concept is one of the most general in mathematics.

Some examples of operators follow. (1) By assigning to each vector (ξ₁, ξ₂,ξ₃) a vector (ξʹ₁, ξʹ₂, ξʹ₃) such that ξʹ_i = a_i1ξ₁ + a_i2ξ;₂+ a_i3ξ;₃ (i = 1, 2, 3; a_i1, a_i2, and a_i3 are fixed numbers), we obtain an operator.

(2) The differentiation operation, or operator, D[f(t)] = fʹ (t) takes every differentiable function f(t) into its derivative fʹ t).

(3) The operation, or operator, of definite integration

takes every integrable function into a real number.

(4) By assigning to every function f(t) its product Φ (t) f(t) with a fixed function Φ(t), we obtain an operator.

General operator theory arose as a result of the development of such branches of classical analysis as the theory of integral equations and the solution of eigenfunction and eigenvalue problems for differential operators, for example, the Sturm-Liouville problem. Operator theory established close ties between these branches of mathematics and played an important role in their further development. Even before the general concept of an operator arose, operator methods were widely applied in the solution of various types of ordinary and partial differential equations. Operator theory provides the basic mathematical framework of quantum mechanics.

Operators in linear spaces. The most frequently encountered operators are those that act in normed linear spaces, particularly in function spaces. Such an operator is a mapping y = A(x) of a linear space R or part of it into a linear space R’, which may coincide with R. This class of operators encompasses such very important concepts as functions numerical-valued, linear transformations of a Euclidean space, and differential and integral operators (see below). Linear operators are the most widely studied class of operators and the most important from the standpoint of applications. An operator is called linear if A (α x + β y) = α A(x) + β A(y) for any elements ξ and y of the space R and any numbers o and β. If the spaces R and R’ are normed and if the ratio ǀǀA(X) ǀǀ/ǀǀX ǀǀ of the norm of A(x) and the norm of x is bounded, then the linear operator A is said to be bounded and the least upper bound of the ratio ǀǀA(x)ǀǀ/ǀǀ(x)ǀǀ is called its norm. The boundedness of a linear operator is equivalent to its continuity, that is, A(x_n → A(x) as x_n → x. The differentiation operator defined in (2) is one of the most important examples of an unbounded, and consequently discontinuous, linear operator.

The four examples given above are linear operators. The following are also examples of linear operators:

(5) Let k(s, t) be a continuous function of two variables defined on the square α ≤ s ≤ b, α ≤ t ≤ b. The equation

defines a linear integral operator called the Fredholm operator.

(6) Let f(t) be a function that is absolutely integrable on the real line. The correspondence that associates to f(t) its Fourier transform

is a linear operator. (7) The left-hand side of the linear differential equation

may be regarded as the result of the application of an operator that associates to the function x(t) the function x (t). Such an operator is termed a linear differential operator. The simplest case of a linear differential operator is the differentiation operator.

The following are examples of nonlinear operators:

(8) Let A [f(t)] =f²(t). The operator thus defined is nonlinear.

(9) Let

where F is a bounded continuous function. The correspondence g → h defined by this expression is a nonlinear integral operator.

Operations on operators. Let the operator

y = A(X)

be one-to-one. Then to each images there corresponds a unique preimage x. This correspondence is called the inverse operator and is denoted by

x = A^-1(y)

Obtaining the inverse operator is equivalent to solving the equation y= A(x) for x, that is, the finding of an unknown preimage for a given image.

If A₁ and A₂ are two operators that map R into Rʹ, then their sum A = A₁ + A₁ is (x). If the operator A₁ maps R into R’ and A₂ maps R’ into R”, then the result of the successive application of these operators is an operator that maps R into R” and is called the product A₂A₁ of the operators A₁ and A₂. If, in particular, we consider operators that map a linear space into itself, then the sum and product of two such operators are always defined. The result of η successive applications of the operator A is the nth power Aⁿ of that operator. For example, the nth power of the differentiation operator is the n-fold differentiation operator Dⁿ [f(t)] = fin)(t). The product λ A of an operator A and a number λ is defined by the formula

(λA)(x) = λA (x)

The operator E that maps every element ξ into itself is called the identity operator. The operator O that maps every element into zero is called the zero operator. The following equalities obviously hold for any A: AE = EA = A, and A + O = O + A = A. Furthermore, if A^-1 exists, then A^-1A = AA-1 = E. It should be noted, however, that in general for two arbitrary operators A and E the products AB and BA are not equal.

By means of the operations of operator addition, operator multiplication, and multiplication of an operator by a number, one may define polynomials of a linear operator and more complicated functions by using a suitably defined limit process. For example, if D is the differentiation operator, then e^D denotes the operator defined by the formula

which is valid for those f(t) for which the series on the right converges. For analytic functions the sum of this series equals f(t + 1); that is, e^D is a shift operator mapping f(t) into f(t + 1).

Linear operators in Hilbert space. Operator theory has most fully been worked out for the case of linear operators in a Hilbert space. Let A be a bounded linear operator in a Hilbert space H. The complex number λ is called an eigenvalue of A if there exists a nonzero element x in H such that A(x) = λx; x is called an eigenvector of A belonging to the eigenvalue λ. A number λ is called a regular point of A if the operator (A —λE)^-1 exists, is defined on all of H, and is bounded; the remaining values of λ are called the points of the spectrum of A. Every eigenvalue of A belongs to the spectrum; the set of eigenvalues of A is the point spectrum, and the rest of the spectrum is called the continuous spectrum. The spectrum of a linear operator on infinite-dimensional space is in general not exhausted by its eigenvalues. This characteristic feature of such linear operators distinguishes them from linear transformations on a finite-dimensional Euclidean space.

The operator A* is termed adjoint to A if the inner product (Ax, y) = (x, A*y) for all x and y in h. The operator A is called self-adjoint if A = A* and unitary if A* = A^-1. Self-adjoint and unitary operators are the most important and most thoroughly studied classes of linear operators in Hilbert space. The theory of these operators is a generalization of the theory of self-adjoint and unitary linear transformations of an n-dimensional Euclidean space.

One of the simplest classes of bounded linear operators in Hilbert space is the class of completely continuous operators. An operator A is called completely continuous if it maps every bounded set in H into a compact set. The spectrum of a completely continuous operator consists of a finite or countably infinite number of eigenvalues and has no nonzero limit points. To each λ≠ 0 there corresponds only a finite number of linearly independent eigenfunctions. The continuous spectrum is lacking.

A self-adjoint, completely continuous operator A has at least one eigenvalue. Moreover it is possible to choose in H a complete orthogonal set of elements consisting of eigenfunctions of A.

Unbounded operators. Since the concept of a bounded linear operator proved in many cases to be too restrictive, it became necessary to consider unbounded linear operators. A linear operator A in a Hilbert space H is unbounded if (1) the relation y = A(x) is defined for all ξ belonging to some linear manifold Ω, which is called the domain of definition of A,(2) A (α x + βy)) = α A(x) + β A(y), and (3) the boundedness condition does not hold.

The most important class of unbounded linear operators in Hilbert space is the class of differential operators. Many problems in mathematical physics, particularly in the theory of oscillations, reduce to finding the eigenfunctions and eigenvalues of various differential operators. For example, cylindrical functions and Legendre polynomials are simply the eigenfunctions of certain differential operators.

Nonlinear operators. The assumption of linearity plays an essential role in the study of operators, but in a number of cases it is necessary to consider nonlinear operators as well. In particular, nonlinear integral equations are very important in mechanics and physics.