Adjoint operators

Definition

Adjoint operators are linear operators associated with a given linear operator on a Hilbert space, which generalize the concept of transpose matrices to infinite-dimensional spaces. The adjoint operator is denoted as $A^*$ for an operator $A$, and it has the property that for any two vectors $x$ and $y$, the inner product satisfies $ig\<Ax, yig\> = ig\<x, A^*yig\ ext{ for all } x, y$. This connection to inner products is crucial for understanding properties like self-adjointness and the spectral theorem, which relate to both the resolvent set and dual spaces.

5 Must Know Facts For Your Next Test

1. The adjoint of a bounded linear operator exists in a Hilbert space, ensuring that many important properties can be studied through this framework.
2. An operator is normal if it commutes with its adjoint, which has implications for its spectral properties and the structure of the spectrum.
3. The concept of adjoints extends beyond Hilbert spaces to other inner product spaces, demonstrating its versatility in functional analysis.
4. In quantum mechanics, observables are represented by self-adjoint operators, emphasizing the importance of adjoint operators in physical contexts.
5. The relationship between an operator and its adjoint is fundamental in deriving results such as the spectral theorem, which provides insight into the structure of operators.

Review Questions

• How does the concept of adjoint operators relate to self-adjointness and what implications does this have for eigenvalues?
  • Self-adjoint operators satisfy the property $A = A^*$, meaning that the operator is equal to its adjoint. This characteristic ensures that all eigenvalues of self-adjoint operators are real numbers and their corresponding eigenvectors can be chosen to be orthogonal. Therefore, understanding adjoint operators is crucial for establishing these fundamental properties in spectral theory.
• Discuss how adjoint operators play a role in understanding the resolvent set and its significance in spectral theory.
  • The resolvent set consists of those complex numbers $ ho$ for which the resolvent operator $(A - ho I)^{-1}$ exists. The relationship between an operator and its adjoint influences the behavior of this resolvent operator, particularly when analyzing spectra. Understanding how adjoints interact with the resolvent set helps in determining important properties such as point spectrum and continuous spectrum.
• Evaluate the impact of adjoint operators on the dual spaces and how this connection enriches the study of functional analysis.
  • The interplay between adjoint operators and dual spaces reveals deep insights into functional analysis. Since every bounded linear operator on a Hilbert space has a corresponding adjoint, this relationship allows us to explore how linear functionals operate within dual spaces. By analyzing how an operator acts on elements in a dual space through its adjoint, we gain a richer understanding of linear mappings and their properties across various mathematical contexts.

Related terms

Self-adjoint operator: An operator $A$ is self-adjoint if $A = A^*$, meaning it is equal to its adjoint, which implies real eigenvalues and orthogonal eigenvectors.

Resolvent operator: The resolvent operator $(A - ho I)^{-1}$ plays a key role in spectral theory, relating to the adjoint operator in analyzing the spectrum of $A$.

Dual space: The dual space of a vector space consists of all linear functionals defined on that space, connecting to adjoint operators through the relationship of operators on these spaces.