Compact Self-Adjoint Operators and the Spectral Theorem · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Spectra that shrink to zero

Mathematics · Calculus & Analysis · ages 23-24
Name ______________________   Date ____________

  1. Every eigenvalue of a self-adjoint operator is real.

    Circle one:   True   False

  2. For the right shift on square-summable sequences, which number is NOT in the spectrum?

    • 0
    • 0.5
    • 2
    • 1

  3. A student says every eigenvalue of a self-adjoint operator is real. Is that right?

    Circle one:   True   False

  4. A diagonal operator has diagonal entries 2 and 5. What is the larger eigenvalue?

    Answer: ______________

  5. For a compact self-adjoint operator, a student claims every nonzero point of the spectrum is an eigenvalue. Is that right?

    Circle one:   True   False

  6. Where can the eigenvalues of a compact self-adjoint operator accumulate?

    • Only at zero
    • At every real number
    • At 1
    • Nowhere

  7. A compact diagonal operator has eigenvalues 1/n for n = 1, 2, 3, and on. What is the largest eigenvalue?

    Answer: ______________

  8. Where can the eigenvalues of a compact self-adjoint operator accumulate?

    • At every real number
    • Only at zero
    • Nowhere at all

  9. Nina claims every bounded operator has an orthonormal eigenbasis. What is wrong?

    • She dropped both hypotheses: compactness and self-adjointness are needed
    • She used the wrong diagonal entries
    • She tested invertibility instead of eigenvalues

  10. Eigenvectors of a self-adjoint operator for distinct eigenvalues are orthogonal. A student says this always holds. Is that right?

    Circle one:   True   False

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Answer key

For grown-ups. Fold this page away before handing over the rest.

Spectra that shrink to zero W1-mt_P_Jr9LWFQj-s1

  1. True · Self-adjointness forces each eigenvalue to equal its own conjugate.
  2. 2 · The spectrum is the closed unit disc, so 2 lies outside while 0, 0.5, and 1 sit inside.
  3. True · Self-adjointness forces every eigenvalue to equal its own conjugate, hence real.
  4. 5 · Eigenvalues sit on the diagonal, so the larger is 5.
  5. True · Fredholm theory at nonzero points makes every nonzero spectral value an eigenvalue of finite multiplicity.
  6. Only at zero · Compactness forces finite-rank approximation, squeezing all but finitely many eigenvalues near zero, so they accumulate only at zero.
  7. 1 · The eigenvalues 1, 1/2, 1/3, and on decrease to 0, so the largest is 1.
  8. Only at zero · Only zero can be a cluster point of the eigenvalues.
  9. She dropped both hypotheses: compactness and self-adjointness are needed · The shift alone refutes the general claim.
  10. True · Orthogonality of eigenvectors for distinct eigenvalues is the finite-dimensional shadow of the spectral theorem.
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