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5.6. LINEAR OPERATORS 207

Properties of a unitary operator U:
U = U –1 or U U = UU = I,
∗
∗
∗
Ux = x for all x in V.
A linear operator A in L(V, V)is saidtobe normal if
∗
A A = AA .
∗
THEOREM. A bounded linear operator A is normal if and only if Ax = A x .
Remark. Any unitary or Hermitian operator is normal.

5.6.1-7. Transpose, symmetric, and orthogonal operators.

The transpose operator of a bounded linear operator A L(V, V) in a real Hilbert space V
is the operator A T L(V, V) such that for all x, y in V, the following relation holds:
T
(Ax) ⋅ y = x ⋅ (A y).
THEOREM. Any bounded linear operator A in a real Hilbert space has a unique transpose
operator.
The properties of transpose operators in a real Hilbert space are similar to the properties
T
∗
of adjoint operators considered in Paragraph 5.6.1-5 if one takes A instead of A .
A linear operator A L(V, V) in a real Hilbert space V is said to be symmetric if
T
A = A or (Ax) ⋅ y = x ⋅ (Ay).
A linear operator A L(V, V) in a real Hilbert space V is said to be skew-symmetric if
T
A =–A or (Ax) ⋅ y =–x ⋅ (Ay).
The properties of symmetric linear operators in a real Hilbert space are similar to the
T
properties of Hermitian operators considered in Paragraph 5.6.1-5 if one takes A instead
of A .
∗
A linear operator P L(V, V) in a real Hilbert space V is said to be orthogonal if for
any x and y in V, the following relations hold:
(Px) ⋅ (Py)= x ⋅ y.
This relation is called the orthogonality condition.
Properties of orthogonal operator P:
T
T
T
P = P –1 or P P = PP = I,
Px = x for all x in V.
5.6.1-8. Positive operators. Roots of an operator.
A Hermitian (symmetric, in the case of a real space) operator A is said to be
a) nonnegative (resp., nonpositive), and one writes A ≥ 0 (resp., A ≤ 0)if (Ax) ⋅ x ≥ 0
(resp., (Ax) ⋅ x ≤ 0)for any x in V.
b) positive or positive definite (resp., negative or negative definite) and one writes A > 0
(A < 0)if (Ax) ⋅ x > 0 (resp., (Ax) ⋅ x < 0)for any x ≠ 0.
An mth root of an operator A is an operator B such that B m = A.
THEOREM. If A is a nonnegative Hermitian (symmetric) operator, then for any positive
integer m there exists a unique nonnegative Hermitian (symmetric) operator A 1/m .