Key Concepts in Functional Calculus to Know for Operator Theory

Functional calculus is a powerful tool in operator theory, allowing functions to be applied to operators. It connects numerical functions to linear operators, enabling the creation of new operators from existing ones through various types of functions, including continuous and holomorphic.

1. Definition of functional calculus

   • A mathematical framework that allows the application of functions to operators.
   • Extends the concept of functions of numbers to functions of linear operators on Hilbert spaces.
   • Provides a way to define new operators based on existing ones using continuous, holomorphic, or Borel functions.

2. Continuous functional calculus for normal operators

   • Applicable to normal operators, which commute with their adjoint.
   • Utilizes continuous functions defined on the spectrum of the operator.
   • Ensures that the resulting operator is also normal and retains spectral properties.

3. Holomorphic functional calculus

   • Extends functional calculus to holomorphic functions on a neighborhood of the spectrum.
   • Allows for the definition of operators using complex analytic functions.
   • Provides powerful tools for studying spectral properties and functional equations.

4. Borel functional calculus

   • Deals with Borel measurable functions and their application to self-adjoint operators.
   • Useful for constructing operators from more general classes of functions.
   • Bridges the gap between continuous and more general functional calculi.

5. Spectral theorem and its relation to functional calculus

   • States that normal operators can be diagonalized via a unitary transformation.
   • Provides the foundation for defining functional calculus through the spectral measure.
   • Connects the operator's spectrum directly to the functional calculus framework.

6. Riesz-Dunford functional calculus

   • A method for applying functions to bounded operators using contour integrals.
   • Generalizes the continuous functional calculus to a broader class of functions.
   • Essential for dealing with operators that may not be normal.

7. Functional calculus for unbounded operators

   • Extends the concepts of functional calculus to unbounded operators, which are common in quantum mechanics.
   • Requires careful handling of domains and closures to ensure well-defined operations.
   • Important for applications in differential operators and quantum mechanics.

8. Applications of functional calculus in quantum mechanics

   • Provides a framework for defining observables as self-adjoint operators.
   • Facilitates the study of quantum states and their evolution through operator functions.
   • Essential for understanding the mathematical foundation of quantum mechanics.

9. Functional calculus for C-algebras*

   • Generalizes functional calculus to the setting of C*-algebras, which are algebraic structures that include operators.
   • Allows for the application of continuous functions to elements of C*-algebras.
   • Plays a crucial role in the study of non-commutative geometry and quantum physics.

10. Weyl-von Neumann theorem and its connection to functional calculus

    • Addresses the classification of self-adjoint operators based on their spectral properties.
    • Establishes conditions under which certain operators can be approximated by simpler ones.
    • Connects to functional calculus by providing insights into the behavior of operators under perturbations.