Expert answer:By definition, 〈 퐴 ∗ 푥 , 푦 〉 = 〈 푥
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Homework 3
1. Let {en : n 2 N} be the standard othonormal basis for `2 (N), that is en : N ! C where
(
1 i=n
en (i) =
0 i 6= n.
Let {↵n }1
n=1 be a sequence of complex numbers. Show that there is an operator A 2
B(`2 (N)) such that Aen = ↵n en for all n 2 N if and only if {↵n }1
n=1 is a bounded sequence.
In this case, show that kAk = sup{|↵n | : n 2 N}.
2. Prove Proposition 2.2 (b), (c), and (d).
3. Show that the product of two self-adjoint operators is self-adjoint if and only if they
commute.
4. Find the adjoint of the operator A in Problem 1.
5. Prove Proposition 2.15 from the text. (Hint: Use the real and imaginary parts of A.)
6. Let P and Q be projections. Show that P + Q is a projection if and only if ran P ? ran Q.
Show that P Q is a projection if and only if P Q = QP .
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