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Introduces linear operators, along with properties and classifications.

We begin our treatment of linear operators, also known as transformations. They can be thought as extensions of functionals that map into arbitrary vector spaces rather than a scalar space.

Definition 1 Let X and Y be be linear vector spaces and D X . A rule A : X Y which associates every element in x D with an element y = A ( x ) Y is said to be a transformation from X Y with domain D .

We have defined D because the transformation may only be defined for some subset of X .

Definition 2 A transformation A : X Y where X and Y are vector spaces over a scalar set R , is said to be linear if for every x 1 , x 2 X and all scalars α 1 , α 2 R ,

A ( α 1 x 1 + α 2 x 2 ) = α 1 A ( x 1 ) + α 2 A ( x 2 )

A common type of linear transformation is the transformation A : R n R m . In this case, A is an m × n matrix with real-valued entries (i.e., A R m × n ). There are a variety of linear transformations that arise in practice, producing equations of the form A ( x ) = y , with x X and y Y , where X and Y are linear vector spaces. For example, the equation

d x d t - a x ( t ) = y ( t )

may be written in operator notation as A ( x ( t ) ) = y ( t ) , where A : C [ T ] C [ T ] is the operator

A ( · ) = d ( · ) d t - a .

Often, we will simply write y = A x .

Definition 3 Let ( X , · X ) , ( Y , · Y ) be normed vector spaces. A linear operator A : X Y is bounded if there exists a constant M < such that A x - Y | M x X for all x X . The smallest M that satisfies this condition is the norm of A :

A X Y = max x X A x Y x X = max x X , x X = 1 A x Y = max x X , x X 1 A x Y .

Geometrically, the operator norm A measures the maximum extent A transforms the unit circle. Thus, A bounds the amplifying power of the operator A .

Operators possess many properties that are shared with functionals, with similar proofs.

Definition 4 A linear operator A : X Y is continuous on X if it is continuous at any point x X .

Theorem 1 A linear operator is bounded if and only if it is continuous.

Definition 5 The sum of two linear operators A 1 : X Y and A 2 : X Y is defined as ( A 1 + A 2 ) x = A 1 x + A 2 x . Similarly, the scaling of a linear operator A : X Y is defined as ( c A ) x = c ( A x ) . Both resulting operators are linear as well.

We can also extend the definition of the dual space to operators.

Definition 6 The normed space of all bounded linear operators from X Y is denoted B ( X , Y )

Are these spaces complete?

Theorem 2 Let X , Y be normed linear spaces. If Y is a complete space then B ( X , Y ) is complete.

Much of the terminology for operators is drawn from matrices.

Definition 7 Let X be a linear vector space. The operator I : X X given by I ( x ) = x for all x X is known as the identity operator , and I B ( X , Y ) .

Definition 8 Let A 1 : X Y and A 2 : Y Z be linear operators. The composition of these two operators ( G 2 G 1 ) x = G 2 ( G 1 x ) is called a product operator .

Definition 9 An operator A : X Y is injective (or one-to-one ) if for each y Y there exists at most one x X such that y = A x . In other words, if A x 1 = A x 2 then x 1 = x 2 .

Definition 10 An operator A : X Y is surjective (or onto ) if for all y Y there exists an x X such that y = A x .

Definition 11 An operator A : X Y is bijective if it is injective and surjective.

Lemma 1 If A 1 : X Y is a bijective operator, then there exists a transformation A 2 : Y X . such that A 2 ( A 1 x ) = x for all x X .

Note that the lemma above implies A 2 A 1 = I . Thus, we say that A 1 is invertible with inverse A 1 - 1 = A 2 .

Definition 12 An operator A : X X is non-singular if it has an inverse in B ( X , X ) ; otherwise A is singular .

In other words, if a transformation A : X X is non-singular there exists a transformation A - 1 such that A A - 1 = I . This extends the concept of singularity from matrices to arbitrary operators.

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Source:  OpenStax, Introduction to compressive sensing. OpenStax CNX. Mar 12, 2015 Download for free at http://legacy.cnx.org/content/col11355/1.4
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