7.4 Operators and Functionals
409
(xT, y) = (x, T
∗ y)
(7.4.9)
hold, then T
∗ in Eq. (7.2.3) is called the right conjugate operator or right adjoint
operator of the right operator T ; T
∗ in Eq. (7.2.4) is called the right conjugate
operator or right adjoint operator of the left operator T ; T
∗ in Eq. (7.2.5) is
called the left conjugate operator or left adjoint operator of right operator T . T
∗
in Eqs. (7.2.2)–(7.2.5) are all called the conjugate operator or adjoint operator
of T . If X and Y are the same Hilbert space, then T and T
∗ are the operators in
the same space, at this point, if T = T
∗ , then T in Eq. (7.2.3) is called the right
self-conjugate operator or right self-adjoint operator. T in Eq. (7.2.4) is called
the left and right mixed self-conjugate operator or left and right mixed selfadjoint operator. T in Eq. (7.2.5) is called the right and left mixed self-conjugate
operator or right and left mixed self-adjoint operator. T in Eqs. (7.2.4)–(7.2.5)
are commonly called the mixed self-conjugate operator or mixed self-adjoint
operator. When T in Eqs. (7.2.2)–(7.2.5) corresponds with the different operators,
they are all called self-conjugate operator or self-adjoint operator.
Theorem 7.4.3 The conjugate operator of a bounded operator is also a bounded
operator. The norms of the conjugate operators equal each other.
Proof According to Schwarz inequality, it is deduced from (T x, y) = (x, T
∗ y)
(x, T
∗ y)
T x · y T · x · y
(1)
Putting x = T
∗ y, then (x, T
∗ y) = (T
∗ y, T
∗ y) = T
∗ y
2 . Eliminating T
∗ y
on both sides of the inequality (1), we get
T
∗ y
T · y
(2)
It can be proved from this that T
∗
T . Transposing T
∗ and T , it can also
be proved that T T
∗
. Comparing the two inequalities, T
∗
= T can be
obtained. Quod erat demonstrandum.
Let H be the Hilbert space, T is a linear operator in H , if its domain D(T ) is
dense in H , then T is called the densely defined operator or operator with dense
domain in H .
Let H be the Hilbert space, X is the subset in H , T is the linear operator of X to
H , if for arbitrary x, y ∈ X , such that (T x, y) = (x, T y) holds, then T is called the
symmetric operator.
Theorem 7.4.4 The necessary and the sufficient conditions that operator T is the
symmetry operator is: The inner product (T x, x) is a real numbers.
Proof The necessity. Let T be a symmetric operator, According to the definition of
the symmetric operator, there is (T x, y) = (x, T y), putting x = y, then (T x, x) =
(x, T x), or according to the symmetric of the inner product, there is (T x, x) =
409
(xT, y) = (x, T
∗ y)
(7.4.9)
hold, then T
∗ in Eq. (7.2.3) is called the right conjugate operator or right adjoint
operator of the right operator T ; T
∗ in Eq. (7.2.4) is called the right conjugate
operator or right adjoint operator of the left operator T ; T
∗ in Eq. (7.2.5) is
called the left conjugate operator or left adjoint operator of right operator T . T
∗
in Eqs. (7.2.2)–(7.2.5) are all called the conjugate operator or adjoint operator
of T . If X and Y are the same Hilbert space, then T and T
∗ are the operators in
the same space, at this point, if T = T
∗ , then T in Eq. (7.2.3) is called the right
self-conjugate operator or right self-adjoint operator. T in Eq. (7.2.4) is called
the left and right mixed self-conjugate operator or left and right mixed selfadjoint operator. T in Eq. (7.2.5) is called the right and left mixed self-conjugate
operator or right and left mixed self-adjoint operator. T in Eqs. (7.2.4)–(7.2.5)
are commonly called the mixed self-conjugate operator or mixed self-adjoint
operator. When T in Eqs. (7.2.2)–(7.2.5) corresponds with the different operators,
they are all called self-conjugate operator or self-adjoint operator.
Theorem 7.4.3 The conjugate operator of a bounded operator is also a bounded
operator. The norms of the conjugate operators equal each other.
Proof According to Schwarz inequality, it is deduced from (T x, y) = (x, T
∗ y)
(x, T
∗ y)
T x · y T · x · y
(1)
Putting x = T
∗ y, then (x, T
∗ y) = (T
∗ y, T
∗ y) = T
∗ y
2 . Eliminating T
∗ y
on both sides of the inequality (1), we get
T
∗ y
T · y
(2)
It can be proved from this that T
∗
T . Transposing T
∗ and T , it can also
be proved that T T
∗
. Comparing the two inequalities, T
∗
= T can be
obtained. Quod erat demonstrandum.
Let H be the Hilbert space, T is a linear operator in H , if its domain D(T ) is
dense in H , then T is called the densely defined operator or operator with dense
domain in H .
Let H be the Hilbert space, X is the subset in H , T is the linear operator of X to
H , if for arbitrary x, y ∈ X , such that (T x, y) = (x, T y) holds, then T is called the
symmetric operator.
Theorem 7.4.4 The necessary and the sufficient conditions that operator T is the
symmetry operator is: The inner product (T x, x) is a real numbers.
Proof The necessity. Let T be a symmetric operator, According to the definition of
the symmetric operator, there is (T x, y) = (x, T y), putting x = y, then (T x, x) =
(x, T x), or according to the symmetric of the inner product, there is (T x, x) =