Lecture 22: The Spectral Theorem for a Compact Self-Adjoint Operator | Summary and Q&A

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November 17, 2022
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Lecture 22: The Spectral Theorem for a Compact Self-Adjoint Operator

TL;DR

This content explains the spectral theory for self-adjoint compact operators, including the spectrum, eigenvalues, and eigenvectors.

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Key Insights

  • 👾 The spectrum of a compact self-adjoint operator consists of eigenvalues and can be characterized in terms of the null space and invertibility of the operator.
  • 🤳 Self-adjoint compact operators can be diagonalized, just like self-adjoint matrices, in an orthonormal basis of eigenvectors.
  • 🥺 The maximum principle theorem allows for the construction of a sequence of eigenvalues and eigenvectors for self-adjoint compact operators, ultimately leading to an orthonormal basis for the Hilbert space.

Transcript

[SQUEAKING] [RUSTLING] [CLICKING] CASEY RODRIGUEZ: OK, so let's continue our discussion about spectral theory for self adjoint compact operators. So let me just briefly recall the spectrum of a bounded operator, which was supposed to be a generalization of the eigenvalues of a matrix. So we defined the resolvent set of A to be those complex numbers... Read More

Questions & Answers

Q: What is the spectrum of a bounded linear operator?

The spectrum of a bounded linear operator is the set of complex numbers for which the operator minus that number times the identity is not invertible. It consists of the numbers that make the operator non-invertible.

Q: Are the eigenvalues of a self-adjoint matrix real?

Yes, in the case of self-adjoint matrices, the eigenvalues are always real.

Q: Can self-adjoint compact operators be diagonalized?

Yes, similar to self-adjoint matrices, self-adjoint compact operators can be diagonalized in an orthonormal basis of eigenvectors.

Q: What does the Fredholm alternative theorem state for self-adjoint compact operators?

The Fredholm alternative for self-adjoint compact operators states that either the operator minus a given eigenvalue is bijective, or the null space of the operator corresponds to the eigenvalue and is nontrivial and finite dimensional.

Summary & Key Takeaways

  • Spectral theory is a generalization of eigenvalues for matrices to bounded linear operators. It involves the spectrum, which is the set of numbers for which the operator is not invertible.

  • In the case of self-adjoint operators, the spectrum consists of the eigenvalues, and the operator can be diagonalized in an orthonormal basis of eigenvectors.

  • Compact operators, which are limits of finite rank operators, also have similar properties, with countably infinite eigenvalues that converge to zero.

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