2.1 Bounded linear operators and their properties

Bounded linear operators are crucial in functional analysis, mapping between normed spaces while preserving structure. They're defined by their ability to transform bounded sets into bounded sets, with the operator norm quantifying this property.

Understanding bounded linear operators is key to grasping how functions behave in infinite-dimensional spaces. Their properties, like linearity and boundedness, form the foundation for studying more complex operators and functional analysis concepts.

Bounded Linear Operators

Definition of bounded linear operators

• A linear operator $T: X \to Y$ between normed spaces $(X, \|\cdot\|_X)$ and $(Y, \|\cdot\|_Y)$ is bounded if there exists a constant $M \geq 0$ such that $\|Tx\|_Y \leq M\|x\|_X$ for all $x \in X$
  • The smallest such $M$ is called the operator norm of $T$, denoted as $\|T\|$
  • Intuitively, a bounded linear operator maps bounded sets in $X$ to bounded sets in $Y$
• Examples of bounded linear operators:
  • Identity operator $I: X \to X$ defined by $Ix = x$ for all $x \in X$ is bounded with $\|I\| = 1$
  • Differentiation operator $D: C^1[a,b] \to C[a,b]$ defined by $(Df)(x) = f'(x)$ for all $f \in C^1[a,b]$ is bounded
  • Integration operator $J: C[a,b] \to C[a,b]$ defined by $(Jf)(x) = \int_a^x f(t) dt$ for all $f \in C[a,b]$ is bounded

Properties of bounded linear operators

• Linearity: Let $T: X \to Y$ be a bounded linear operator between normed spaces $X$ and $Y$. For any $x_1, x_2 \in X$ and scalars $\alpha, \beta$, we have:
  • $T(\alpha x_1 + \beta x_2) = \alpha T(x_1) + \beta T(x_2)$
  • This property ensures that bounded linear operators preserve the vector space structure
• Boundedness: Let $T: X \to Y$ be a bounded linear operator with operator norm $\|T\|$. For any $x \in X$, we have:
  • $\|Tx\|_Y \leq \|T\| \|x\|_X$
  • Proof: By definition of the operator norm, $\|Tx\|_Y \leq \|T\| \|x\|_X$ for all $x \in X$
  • This property ensures that the image of a bounded set under a bounded linear operator remains bounded

Boundedness and operator norm

• Boundedness test: A linear operator $T: X \to Y$ is bounded if and only if there exists a constant $M \geq 0$ such that $\|Tx\|_Y \leq M\|x\|_X$ for all $x \in X$
  • This test provides a necessary and sufficient condition for a linear operator to be bounded
• Finding the operator norm:
  • The operator norm of a bounded linear operator $T: X \to Y$ is defined as:
    • $\|T\| = \sup\{\|Tx\|_Y : \|x\|_X \leq 1\} = \sup\{\|Tx\|_Y / \|x\|_X : x \neq 0\}$
    • Intuitively, the operator norm is the smallest upper bound for the "stretching" of vectors under the operator
  • Equivalently, the operator norm can be found using:
    • $\|T\| = \inf\{M \geq 0 : \|Tx\|_Y \leq M\|x\|_X \text{ for all } x \in X\}$
    • This definition provides an alternative way to compute the operator norm

Space of bounded linear operators

• The space of bounded linear operators from $X$ to $Y$, denoted as $\mathcal{B}(X,Y)$, is a normed space with the operator norm
  • This space contains all bounded linear operators between $X$ and $Y$
• Linearity of $\mathcal{B}(X,Y)$: For any $T_1, T_2 \in \mathcal{B}(X,Y)$ and scalars $\alpha, \beta$, we have:
  • $(\alpha T_1 + \beta T_2)(x) = \alpha T_1(x) + \beta T_2(x)$ for all $x \in X$
  • $\|\alpha T_1 + \beta T_2\| \leq |\alpha| \|T_1\| + |\beta| \|T_2\|$
  • These properties ensure that $\mathcal{B}(X,Y)$ is a vector space and the operator norm is a norm on this space
• Completeness of $\mathcal{B}(X,Y)$: If $Y$ is a Banach space, then $\mathcal{B}(X,Y)$ is also a Banach space
  • Proof: Let $(T_n)$ be a Cauchy sequence in $\mathcal{B}(X,Y)$. Define $T: X \to Y$ by $Tx = \lim_{n \to \infty} T_n x$ for each $x \in X$. Then $T \in \mathcal{B}(X,Y)$ and $\lim_{n \to \infty} \|T_n - T\| = 0$
  • This property ensures that $\mathcal{B}(X,Y)$ is complete with respect to the operator norm, making it a suitable space for analysis