Spectral Theory in Hilbert Space

CHAPTER 4
Operators
A bounded linear operator from a Banach space B1 to another Banach
space B2 is a linear mapping T : B1 → B2 such that for some
constant C we have T u2 ≤ Cu1 for every u ∈ B1. The smallest
such constant C is called the norm of the operator T and denoted by
T . Like in the discussion of linear forms in the last chapter it follows
that the boundedness of T is equivalent to continuity, in the sense that
T uj − T u2 → 0 if uj − u1 → 0 (Exercise 4.1). If B1 = B2 = B one
says that T is an operator on B. The operator-norm defined above has
the following properties (Here T : B1 → B2 and S are bounded linear
operators, and B1, B2 and B3 Banach spaces).
(1) T ≥ 0, equality only if T = 0,
(2) λT = |λ|T for any λ ∈ C,
(3) S + T ≤ S + T if S : B1 → B2,
(4) ST ≤ ST if S : B2 → B3.
We leave the proof to the reader (Exercise 4.1). Thus we have made the
set of bounded operators from B1 to B2 into a normed space B(B1, B2).
In fact, B(B1, B2) is a Banach space (Exercise 4.2). We write B(B)
for the bounded operators on B. Because of the property (4) B(B) is
called a Banach algebra.
Now let H1 and H2 be Hilbert spaces. Then every bounded operator
T : H1 → H2 has an adjoint 1 T ∗ : H2 → H1 defined as follows. Consider
a fixed element v ∈ H2 and the linear form H1 ∋ u ↦→ 〈T u, v〉2
which is obviously bounded by T v2. By the Riesz’ representation
theorem there is therefore a unique element v ∗ ∈ H1, such that
〈T u, v〉2 = 〈u, v ∗ 〉1. By the uniqueness, and since 〈T u, v〉2 depends
anti-linearly on v, it follows that T ∗ : v ↦→ v ∗ is a linear operator from
H2 to H1. It is also bounded, since v ∗ 2 1 = 〈T v ∗ , v〉2 ≤ T v ∗ 1v2,
so that T ∗ ≤ T . The adjoint has the following properties.
Proposition 4.1. The adjoint operation B(H1, H2) ∋ T ↦→ T ∗ ∈
B(H2, H1) has the properties:
(1) (T1 + T2) ∗ = T ∗ 1 + T ∗ 2 ,
(2) (λT ) ∗ = λT ∗ for any complex number λ,
(3) (T2T1) ∗ = T ∗ 1 T ∗ 2 if T2 : H2 → H3,
(4) T ∗∗ = T ,
1 Also operators between general Banach spaces, or even more general topological
vector spaces, have adjoints, but they will not concern us here.
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