¿Cómo se construye un operador compacto con un espectro y vectores propios dados?

To construct a compact operator with a given spectrum and eigenvectors, one approach is to use a Schauder basis for the underlying Hilbert or Banach space. Start by selecting a sequence of eigenvalues corresponding to the desired spectrum, ensuring the sequence converges to zero, as compact operators have this property. Then, choose a Schauder basis consisting of eigenvectors associated with these eigenvalues. Define the operator by mapping each basis vector to its corresponding eigenvalue times the same basis vector. This construction ensures the operator is compact, with the specified spectrum and eigenvectors, leveraging the properties of the Schauder basis to maintain the desired structure.

The diagonalization theorem states that a linear operator on a finite-dimensional vector space can be represented as a diagonal matrix if and only if there is a basis consisting of its eigenvectors. This means that the operator can be "diagonalized," simplifying its representation and making it easier to analyze and compute its powers. In this basis, the operator acts by scaling each basis vector by the corresponding eigenvalue. Diagonalization is a powerful tool in linear algebra, providing insights into the structure of operators and simplifying many problems involving linear transformations and systems of linear equations.

To construct a diagonalizable operator, start with a finite-dimensional vector space and identify a basis of eigenvectors. Ensure the operator has a full set of linearly independent eigenvectors, each associated with a unique eigenvalue. Construct the operator's matrix representation in this eigenvector basis, where the matrix is diagonal, with eigenvalues on the diagonal. For a given operator, this involves solving the characteristic equation to find eigenvalues, and then finding corresponding eigenvectors. If successful, reframe the operator in this eigenvector basis to achieve a diagonal matrix, simplifying analysis and computation by leveraging its diagonal structure.

To verify the diagonalization of an operator, follow these steps: Find Eigenvalues: Solve the characteristic equation of the operator to determine its eigenvalues. Find Eigenvectors: For each eigenvalue, solve the system (A−λI)v=0 to find the corresponding eigenvectors. Check Linear Independence: Ensure the eigenvectors are linearly independent. For an n×n matrix, there should be n independent eigenvectors. Construct the Diagonal Matrix: Form the diagonal matrix with the eigenvalues on the diagonal. Verify Basis Transformation: Confirm that the original operator can be expressed as A=PDP(−1) where P is the matrix of eigenvectors and D is the diagonal matrix.