Range

Definition

The range of a linear operator is the set of all possible outputs that can be produced by applying the operator to all inputs in its domain. This concept is crucial for understanding how operators behave and interact with functions, as it reflects the extent to which an operator can transform elements from one space to another. The range provides insights into properties such as injectivity and surjectivity, which are important in defining bounded operators, analyzing polar decompositions, and understanding symmetric and self-adjoint unbounded operators.

5 Must Know Facts For Your Next Test

1. The range of a linear operator is a vector space itself, formed by linear combinations of outputs corresponding to inputs in the domain.
2. In the context of bounded linear operators, if an operator is surjective, its range will be equal to its codomain.
3. For unbounded operators, determining the range can be more complex due to issues related to convergence and domains.
4. The polar decomposition illustrates how any bounded operator can be expressed in terms of its range and an orthogonal projection onto that range.
5. An operator's range plays a key role in understanding the spectral properties of self-adjoint operators, especially concerning eigenvalues and eigenvectors.

Review Questions

• How does the concept of range relate to determining if a linear operator is injective or surjective?
  • The range of a linear operator provides crucial information for assessing its injectivity and surjectivity. If the range covers the entire codomain, then the operator is surjective. Conversely, if different inputs yield different outputs, it indicates that the kernel only contains the zero vector, thus demonstrating injectivity. Therefore, by analyzing the range, one can determine both properties of the operator.
• Discuss how polar decomposition utilizes the concept of range and why this is important for understanding bounded operators.
  • Polar decomposition breaks down any bounded linear operator into a product involving an orthogonal projection onto its range and a partial isometry. This relationship highlights how the structure of the range influences the behavior of the operator. Understanding this decomposition helps in analyzing how bounded operators transform input vectors while maintaining certain geometric properties, essential for applications in functional analysis.
• Evaluate the implications of an unbounded self-adjoint operator's range on its spectral characteristics.
  • The range of an unbounded self-adjoint operator has significant implications for its spectral characteristics. Specifically, it determines whether certain spectral values (eigenvalues) exist or are part of the continuous spectrum. If the range is not dense in a particular subspace, it may lead to gaps in possible eigenvalues or affect convergence properties of associated eigenfunctions. Thus, examining the range aids in understanding how these operators behave in relation to their spectra.

Related terms

Kernel: The kernel of an operator consists of all inputs that map to the zero vector, indicating the operator's nullity and providing insights into its injective properties.

Isomorphism: An isomorphism is a mapping between two structures that preserves their operations and properties, often implying a bijective correspondence between the range and the codomain.

Closed Operator:

A closed operator is one where the graph of the operator is a closed set in the product space, relating to its domain and range and affecting its boundedness.