Lecture 22: The Spectral Theorem for a Compact Self-Adjoint Operator

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Spectral Theory for Self-Adjoint Compact Operators

Spectrum and Resolving Set Basics (00:00:19 - 00:01:17)

Resolving Set Definition:

Complex numbers such that the operator is invertible and bounded.

This means the inverse must also be continuous, again by the open mapping theorem.

Spectrum Definition:

Complement of the resolving set.

Values such that the operator is not invertible.

Eigenvalue Properties in Finite Dimensional Spaces (00:01:17 - 00:03:08)

Hermitian/Self-Adjoint Matrices:

• Real eigenvalues
• Diagonalization via an orthonormal basis
• Diagonal entries are the eigenvalues

Compact Operator Spectral Properties (00:03:08 - 00:05:46)

• Compact operators are limits of finite-rank matrices.
• In infinite-dimensional spaces, spectrum is more complicated.
• Zero can be in the spectrum.
• Non-degenerate cases have countably infinite eigenvalues converging to zero.

Spectral Characteristics Table

Dimension	Spectrum Behavior	Eigenvalue Properties
Finite	Exactly eigenvalues	Real, diagonalizable
Infinite	More complicated	Potentially countably infinite

Fredholm Alternative (00:06:53 - 00:10:45)

Theorem Conditions:

• The operator is a self-adjoint compact operator.
• The number is a nonzero real number.

Key Conclusions:

• Range of the operator is closed.
• Eigenspace corresponding to the operator is finite-dimensional.
• Eigenvectors for different eigenvalues are orthogonal.

Maximum Principle for Eigenvalues (00:41:08 - 00:47:36)

Eigenvalue Ordering Process:

• Eigenvalues can be systematically found.
• Nonzero eigenvalues ordered.
• Eigenvalues converge to zero if infinite.

Eigenvalue Computation Diagram

Spectral Theorem for Compact Self-Adjoint Operators (01:07:52 - 01:23:43)

Main Conclusions:

• Can construct orthonormal basis of eigenvectors.
• Basis includes vectors from null space and non-zero eigenvalue spaces.
• Works for separable Hilbert spaces.

Mathematical Nuances

Spectral Decomposition:

• Eigenvalues potentially infinite but converging to zero.

Practical Implications

• Provides systematic method for understanding linear operators.
• Generalizes matrix diagonalization to infinite-dimensional spaces.
• Is critical for the analysis of differential equations and functional analysis.

Key Takeaways

• The spectrum is more complex in infinite dimensions.
• Eigenvalues have predictable convergence properties.
• An orthonormal eigenvector basis can always be constructed.