1. Let T be a bounded linear operator on a Banach space X. Prove that σ(T ∗ ) = σ(T ).
2. Let (·, ·) be a form on a complex vector space X satisfying all properties of a scalar
product except nondegeneracy (that is, (x, x) may be zero for some nonzero x’s). Then
(i) |(x, y)| ≤ (x, x)1/2 (y, y)1/2 for any x and y;
(ii) the set N = {x | (x, x) = 0} is a subspace of X, and the form defines a scalar product on
X/N .
3. Let a be a real continuous function on [0, 1]. Consider the operator A of multiplication
by a on L2 [0, 1], so (Af )(t) = a(t)f (t). Show that A is not compact unless a is identically
zero.