2.2 Linear operators

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 $D\subseteq X$ . A rule $A:X\to Y$ which associates every element in $x\in D$ with an element $y=A\left(x\right)\in Y$ is said to be a transformation from $X\to Y$ with domain $D$ .

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

Definition 2 A transformation $A:$ $X\to Y$ where X and Y are vector spaces over a scalar set $R$ , is said to be linear if for every $x_1,x_2\in X$ and all scalars ${\alpha }_1,{\alpha }_2\in R$ ,

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

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

Often, we will simply write $y=Ax$ .

Definition 3 Let $(X,\parallel ·{\parallel }_X)$ , $(Y,\parallel ·{\parallel }_Y)$ be normed vector spaces. A linear operator $A:X\to Y$ is bounded if there exists a constant $M<\infty$ such that $\parallel Ax-Y|\le {M\parallel x\parallel }_X$ for all $x\in X$ . The smallest M that satisfies this condition is the norm of $A$ :

Geometrically, the operator norm $\parallel A\parallel$ measures the maximum extent $A$ transforms the unit circle. Thus, $\parallel A\parallel$ 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 $A:X\to Y$ is continuous on $X$ if it is continuous at any point $x\in X$ .

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

Definition 5 The sum of two linear operators $A_1:X\to Y$ and $A_2:X\to Y$ is defined as $(A_1+A_2)x=A_{1}x+A_{2}x$ . Similarly, the scaling of a linear operator $A:X\to Y$ is defined as $\left(cA\right)x=c\left(Ax\right)$ . 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 $X\to Y$ 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 $I:X\to X$ given by $I\left(x\right)=x$ for all $x\in X$ is known as the identity operator , and $I\in B(X,Y)$ .

Definition 8 Let $A_1:X\to Y$ and $A_2:Y\to Z$ be linear operators. The composition of these two operators $\left(G_2{G}_1\right)x=G_2\left(G_{1}x\right)$ is called a product operator .

Definition 9 An operator $A:X\to Y$ is injective (or one-to-one ) if for each $y\in Y$ there exists at most one $x\in X$ such that $y=Ax$ . In other words, if $A{x}_1=A{x}_2$ then $x_1=x_2$ .

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

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

Lemma 1 If $A_1:X\to Y$ is a bijective operator, then there exists a transformation $A_2:Y\to X$ . such that $A_2\left(A_{1}x\right)=x$ for all $x\in X$ .

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 $A:X\to X$ is non-singular if it has an inverse in $B(X,X)$ ; otherwise $A$ is singular .

In other words, if a transformation $A:X\to X$ 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.