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206 ALGEBRA

The rank of a linear operator A is the dimension of its range: rank (A)= dim (im A).
Properties of the rank of a linear operator:

rank (AB) ≤ min{rank (A), rank (B)},
rank (A)+ rank (B)– n ≤ rank (AB),

where A and B are linear operators in L(V, V)and n =dim V.
Remark. If rank (A)= n then rank (AB)=rank (BA)=rank (B).

THEOREM. Let A : V→ V be a linear operator. Then the following statements are
equivalent:
–1
1. A is invertible (i.e., there exists A ).
2. ker A = 0.
3. im A = V.
4. rank (A)= dim V.

5.6.1-5. Notion of a adjoint operator. Hermitian operators.

Let A L(V, V) be a bounded linear operator in a Hilbert space V. The operator A in
∗
L(V, V) is called its adjoint operator if
(Ax) ⋅ y = x ⋅ (A y)
∗
for all x and y in V.
THEOREM. Any bounded linear operator A in a Hilbert space has a unique adjoint
operator.
Properties of adjoint operators:
¯
∗ ∗
∗
∗
∗
∗
∗
(A + B) = A + B , (λA) = λA ,(A ) = A,
∗
∗
∗
(AB) = B A , O = O, I = I,
∗
∗
2
∗ –1
(A ) =(A ) , A = A , A A = A ,
∗
∗
–1 ∗
(Ax) ⋅ (By) ≡ x ⋅ (A By) ≡ (B Ax) ⋅ y for all x and y in V,
∗
∗
¯
where A and B are bounded linear operators in a Hilbert space V, λ is the complex conjugate
of a number λ.
A linear operator A L(V, V) in a Hilbert space V is said to be Hermitian (self-adjoint) if
A = A or (Ax) ⋅ y = x ⋅ (Ay).
∗
A linear operator A (V, V) in a Hilbert space V is said to be skew-Hermitian if
A =–A or (Ax) ⋅ y =–x ⋅ (Ay).
∗
5.6.1-6. Unitary and normal operators.
A linear operator U L(V, V) in a Hilbert space V is called a unitary operator if for all x
and y in V, the following relation holds:
(Ux) ⋅ (Uy)= x ⋅ y.
This relation is called the unitarity condition.