C^*

Definition

In the context of closed operators, c^* refers to the adjoint operator of a closed operator c. The adjoint operator captures important properties of the original operator and provides insights into its behavior, particularly in Hilbert spaces. Understanding c^* is crucial as it relates to the closure of operators and their spectral properties, impacting how they can be used in functional analysis.

5 Must Know Facts For Your Next Test

1. The adjoint c^* exists if and only if the operator c is closed, making it significant in characterizing closed operators.
2. For a closed operator c, the relationship between c and its adjoint c^* is defined by the inner product relation ⟨cx, y⟩ = ⟨x, c^*y⟩ for all x in the domain of c and y in the domain of c^*.
3. The operator c is self-adjoint if it equals its adjoint (c = c^*), which has implications for spectral properties and physical applications.
4. The closure of an operator c can be achieved through its adjoint by examining the densely defined operators in Hilbert spaces.
5. If a closed operator has a bounded adjoint, it leads to important results about the spectrum and resolvent of that operator.

Review Questions

• How does the concept of the adjoint operator relate to the properties of closed operators?
  • The adjoint operator is intrinsically linked to closed operators as it exists specifically when an operator is closed. This relationship is essential because it allows one to explore the symmetry and other features of the closed operator through its adjoint. The inner product relationship between an operator and its adjoint provides a framework for understanding how they interact, which is pivotal in spectral theory.
• Discuss how knowing whether an operator is self-adjoint influences our understanding of its spectrum.
  • Determining if an operator is self-adjoint (c = c^*) significantly affects our understanding of its spectrum because self-adjoint operators have real spectra. This characteristic ensures that their eigenvalues are real numbers, which is important in quantum mechanics and other applications. Moreover, self-adjoint operators lead to better stability and convergence properties when analyzing their spectral resolution.
• Evaluate the role of the adjoint operator in establishing closure properties for densely defined operators.
  • The adjoint operator plays a critical role in establishing closure properties for densely defined operators by providing a method to analyze limits of sequences within the domain. If we have a sequence that converges in norm or weakly to some limit, examining the action of both the original and adjoint operators reveals whether this limit belongs to their respective closures. This interplay helps define not only whether an operator is closed but also enhances our understanding of continuity and boundedness across different contexts in functional analysis.

Related terms

Closed Operator: An operator that maps a dense subset of a Hilbert space to another space such that its graph is closed in the product space.

Adjoint Operator: An operator associated with a given operator that represents a kind of 'transpose' operation, crucial for studying properties like symmetry and self-adjointness.

Graph of an Operator: The set of all pairs (x, Tx) where x is in the domain of an operator T, which helps in understanding the behavior and properties of T.