408
7 Variational Principles
of (X, Y ), and both are nonempty, since at least one zero operator θ ∈ B(X, Y ). When
X = Y , B(X, X ) is denoted by B(X ).
Let X be a normed linear space, All of the continuous linear functionals in X are
denoted by X
∗ , and X
∗ by the usual linear operation and the norm of functional form
a normed linear space, then X
∗ is called the dual space, conjugate space or adjoint
space of X .
Let X and Y be both the Hilbert space, T is a bounded linear operator from X
to Y , if there exists the bounded linear operator T
∗ from Y to X , and for arbitrary
x ∈ X and y ∈ Y , such that
(T x, y) = (x, T
∗ y) + B(x, y)
(7.4.6)
holds, where B(x, y) is the boundary term, then T
∗ is called the conjugate operator
or Hilbert adjoint operator of T , the latter is called the adjoint operator for short.
If T and its conjugate operator T
∗ can be exchanged, namely T T
∗
= T
∗ T , then
T is called the normal operator. If T
∗ T = I X , T T
∗
= I Y , where I X , I Y are the
unit operator on X and Y respectively, then T is called the unitary operator. The
unitary operator is usually denoted by U . If the boundary term B(x, y) vanishes,
then the conjugate operator is reduced to (T x, y) = (x, T
∗ y). If X and Y are the
same Hilbert space, then T and T
∗ are the operators of the same space, at this point,
if T = T
∗ , then T is called the self-conjugate operator or self-adjoint operator,
it is also called the Hermitian operator or Hermite operator. As can be seen from
the definitions, the unitary operator and the conjugate operator are the special cases
of the normal operator.
Let X be a Hilbert space, for all x ∈ X , an obvious property of unitary operator is
U x =
U
∗ x
= x
It is that the following relations hold
U x
2
= (U x, U x) = (x, U
∗ U x) = (x, x) = x
2
U
∗ x
2
= (U
∗ x, U
∗ x) = (x, UU
∗ x) = (x, x) = x
2
In the definition of the above mentioned adjoint operator, the operators T and
T
∗ just act on the left sides of x and y respectively, T
∗ is called the left conjugate
operator or left adjoint operator of the left operator T . If T = T
∗ , the T is called
the left self-conjugate operator or left self-adjoint operator. Because the boundary
term B(x, y) may be zero in a functional, it can be neglected. From which the other
three definitions of adjoint operator can be elicited.
Let X and Y be both the Hilbert space, T is a bounded linear operator from X
to Y , if there exists the bounded linear operator T
∗ from Y to X , and for arbitrary
x ∈ X and y ∈ Y , such that the equations
(xT, y) = (x, yT
∗
)
(7.4.7)
(T x, y) = (x, yT
∗
)
(7.4.8)
7 Variational Principles
of (X, Y ), and both are nonempty, since at least one zero operator θ ∈ B(X, Y ). When
X = Y , B(X, X ) is denoted by B(X ).
Let X be a normed linear space, All of the continuous linear functionals in X are
denoted by X
∗ , and X
∗ by the usual linear operation and the norm of functional form
a normed linear space, then X
∗ is called the dual space, conjugate space or adjoint
space of X .
Let X and Y be both the Hilbert space, T is a bounded linear operator from X
to Y , if there exists the bounded linear operator T
∗ from Y to X , and for arbitrary
x ∈ X and y ∈ Y , such that
(T x, y) = (x, T
∗ y) + B(x, y)
(7.4.6)
holds, where B(x, y) is the boundary term, then T
∗ is called the conjugate operator
or Hilbert adjoint operator of T , the latter is called the adjoint operator for short.
If T and its conjugate operator T
∗ can be exchanged, namely T T
∗
= T
∗ T , then
T is called the normal operator. If T
∗ T = I X , T T
∗
= I Y , where I X , I Y are the
unit operator on X and Y respectively, then T is called the unitary operator. The
unitary operator is usually denoted by U . If the boundary term B(x, y) vanishes,
then the conjugate operator is reduced to (T x, y) = (x, T
∗ y). If X and Y are the
same Hilbert space, then T and T
∗ are the operators of the same space, at this point,
if T = T
∗ , then T is called the self-conjugate operator or self-adjoint operator,
it is also called the Hermitian operator or Hermite operator. As can be seen from
the definitions, the unitary operator and the conjugate operator are the special cases
of the normal operator.
Let X be a Hilbert space, for all x ∈ X , an obvious property of unitary operator is
U x =
U
∗ x
= x
It is that the following relations hold
U x
2
= (U x, U x) = (x, U
∗ U x) = (x, x) = x
2
U
∗ x
2
= (U
∗ x, U
∗ x) = (x, UU
∗ x) = (x, x) = x
2
In the definition of the above mentioned adjoint operator, the operators T and
T
∗ just act on the left sides of x and y respectively, T
∗ is called the left conjugate
operator or left adjoint operator of the left operator T . If T = T
∗ , the T is called
the left self-conjugate operator or left self-adjoint operator. Because the boundary
term B(x, y) may be zero in a functional, it can be neglected. From which the other
three definitions of adjoint operator can be elicited.
Let X and Y be both the Hilbert space, T is a bounded linear operator from X
to Y , if there exists the bounded linear operator T
∗ from Y to X , and for arbitrary
x ∈ X and y ∈ Y , such that the equations
(xT, y) = (x, yT
∗
)
(7.4.7)
(T x, y) = (x, yT
∗
)
(7.4.8)