By Daniel T. Gillespie

Booklet by means of Gillespie, Daniel T

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**Example text**

Such a vector is called an eigenvector corresponding to the eigenvalue λ. These eigenvectors and 0 form the eigenspace corresponding to the eigenvalue λ. This eigenspace is thus the kernel ker(Tλ ) of the operator Tλ , hence closed. The dimension of the eigenspace corresponding to an eigenvalue λ is called the multiplicity or degeneracy of λ. We usually denote ker Tλ = Nλ when T is understood, even if ker Tλ = {0}. In the general case any one of the three parts of the spectrum may be void (although the spectrum of a bounded operator is always nonempty).

If ϕ = ∞ n=1 ϕn converges if and only if n=1 ϕn n=1 ϕn , ∞ 2 2 then ϕ = n=1 ϕn . (b) If (ϕn ) is orthonormal, then for every ϕ ∈ H the series ∞ k=1 ϕk |ϕ ϕk converges, and if ψ is its sum, then (ϕ − ψ) ⊥ ϕn for all n ∈ N. Proof (a) If n > m, then by orthogonality n m ϕk − k=1 ϕk 2 n = k=1 ϕk 2 k=m+1 n = ϕk k=1 n = ϕk k=m+1 m 2 − ϕk 2 . 7 Orthonormal Sets 29 This implies that the sequence of the partial sums sn = nk=1 ϕk is a Cauchy sequence in H if and only if the sequence of the partial sums nk=1 ϕk 2 is a Cauchy sequence in R.

B) Let T be as in the previous exercise. Does the sequence (T n )n∈N converge strongly? Does it converge weakly? 17. Define the mapping T : 2 → 2 via the formula (T f )(n) = f (n + 1). Show that the adjoint of T satisfies the formula (T ∗ f )(n) = f (n − 1) when n ≥ 2, and (T ∗ f )(1) = 0. 18. We saw in Exercise 15 that the operator T defined there is isometric. Is T ∗ isometric? Is any one of the formulas T ∗ T = I , T T ∗ = I true (when I is the identity operator of 2 )? Is T normal? 19. e. P and Q are projections).