Operator Theory

Overview

An operator is defined as a linear function from a vector space {% M %} to another vector space {% N %}. More often than not, an operator is a function from a vector space to itself. (an automorphism). Many times, when a function is defined to be an operator, it is implicitly assumed that it is an automorphism, but should be judged from the context.

Topics

• Definitions
  • Kernel - the Kernel of an operator is the subset of the domain, {% X %} that yields {% 0 %} after applying the operator. {% ker(T) = \{ x \in X : T(x) = 0 \} %}
  • Range - is the image the operator in {% Y %}. {% range(T) = \{ T(x) : x \in X \} %}
  • Operator Norm
  • Commutator
• Types
  • Projection Operator
  • Isometry
  • Bounded Operator
  • Adjoint Operator
  • Unitary Operator
  • Compact Linear Operator
• Spectral Theory
• Systems - systems, defined within the Signals and Systems field, are essentially just operators.