The other identities are proven the exact same way

The other identities are proven the exact same way.
Comment. If we work with one Hilbert space H , then the above result gives
the fact that B ( H ) is a unital involutive Banach algebra . (See Section 5 for the
terminology.) This will be crucial for the development of the theory.
Another important set of results deals with the kernel and the range.

304 CHAPTER II: ELEMENTS OF FUNCTIONAL ANALYSIS
Proposition 7.2 (Kernel-Range Identities) . Let H
1
and H
2
be Hilbert spaces.
For any operator T ∈ B ( H
1
, H
2
) , one has the equalities
(i) Ker T = (Ran T )
⊥
;
(ii) Ran T = (Ker T )
⊥
.
Proof. (i). If we start with some vector η ∈ Ker T , then for every ξ ∈ H
1
,
we have
( η | Tξ )
H
2
= ( T η | ξ )
H
1
= 0 ,
thus proving that η ⊥ Tξ , ∀ ξ ∈ H
1
, i.e. η ∈ (Ran T ) ⊥ . This proves the inclusion
Ker T = (Ran T )
⊥
.
To prove the inclusion in the other direction, we start with some vector η ∈
Ker T = (Ran T )
⊥
, and we prove that T η = 0. This is however pretty clear
since we have η ⊥ ( TT η ), i.e.
0 = ( η | TT η )
H
2
= ( T η | T η )
H
1
= T η
2
,
which forces T η = 0.
(ii). This follows immediately from part (i) applied to T :
Ran T =
(
[Ran T ]
⊥
)
⊥
= (Ker T )
⊥
.
Example 7.1. Let n ≥ 1 be some integer, and consider the Hilbert space C n ,
whose inner product is the standard one:
( ξ | η ) =
n
k =1
¯
ξ
k
η
k
, ∀ ξ = ( ξ
1
, . . . , ξ
n
) , η = ( η
1
, . . . , η
n
) ∈ C
n
.
The Banach algebra B ( C n ) is obviously identified with the algebra Mat
n × n
( C ) of
n × n matrices with complex coefficients. The adjoint operation then corresponds
to a (familiar) operation in linear algebra. For A ∈ Mat
n × n
( C ), say A = [ a
jk
] n
j,k =1
,
one takes A = [ b
jk
]
n
j,k =1
to be the conjugate transpose of A , i.e. the b ’s are
defined as b
jk
= ¯ a
kj
. A similar identification works with B ( C
m
, C
n
), identified
with Mat
n × m
( C ).
The adjoint operation is used for defining certain types of operators.
Definitions. Let H be a Hilbert space.
A. We say that an operator T ∈ B ( H ) is normal , if T T = TT .
B. We say that an operator T ∈ B ( H ) is self-adjoint , if T = T .
C. We say that an operator T ∈ B ( H ) is positive , if
( Tξ | ξ )
H
≥ 0 , ∀ ξ ∈ H .
Remarks 7.2. A. Every self-adjoint operator T ∈ B ( H ) is normal.
B. The set { T ∈ B ( H ) : T normal } is closed in B ( H ), in the norm topology.
Indeed, if we start with a sequence ( T
n
) ∞
n =1
of normal operators, which converges
(in norm) to some T ∈ B ( H ), then ( T
n
)
∞
n =1
converges to T , and since the multi-
plication map
B ( H ) × B ( H ) ( X, Y ) -→ XY ∈ B ( H )
is continuous, have T T = lim
n →∞
T
n
T
n
and TT = lim
n →∞
T
n
T
n
, so we imme-
diately get T T = TT .
C. For T ∈ B ( H ), the following are equivalent (see Remark 3.1):
• T is self-adjoint;

§ 7. Operator Theory on Hilbert spaces 305
• the sesquilinear map
φ
T
: H × H ( ξ, η ) -→ ( Tξ | η )
H
∈ C
is sesqui-symmetric, i.e. ( Tξ | η ) = ( Tη | ξ ), ∀ ξ, η ∈ H ;
• ( Tξ | ξ ) ∈ R , ∀ ξ ∈ H .
In particular, we see that every positive operator T is self-adjoint.