1 Introduction
Let
B
(
ℋ
)
be the algebra of all bounded linear operators acting on a complex separable Hilbert space
ℋ
. Given
A
,
B
∈
B
(
ℋ
)
, we define the generalized derivation
δ
A
,
B
:
B
(
ℋ
)
→
B
(
ℋ
)
by
δ
A
,
B
(
X
)
=
A
X
−
X
B
.
Let
X
∈
B
(
ℋ
)
be a compact operator, and let
s
1
X
≥
s
2
X
≥
…
≥
0
denote the eigenvalues of
X
=
X
∗
X
1
2
arranged in their decreasing order. The operator
X
is said to belong to the Schatten
p
-class
C
p
(
ℋ
)
, if
(1)
X
p
=
∑
∞
i
=
1
s
i
X
p
1
p
=
tr
X
p
1
p
<
+
∞
,
where
tr
denotes the trace functional. In case
p
=
∞
, we denote by
C
∞
(
ℋ
)
, the ideal of compact operators equipped with the norm
X
∞
=
s
1
X
. For
p
=
1
,
C
1
ℋ
is called the trace class, and for
p
=
2
,
C
2
ℋ
is called the Hilbert-Schmidt class and the case
p
=
∞
corresponds to the class. For more details, the reader is referred to [1]. In the sequel, we will use the following further notations and definitions. The closure of the range of an operator
T
∈
B
(
ℋ
)
will be denoted by
ran
T
¯
and
ker
T
denotes the kernel of
T
. The restriction of
T
to an invariant subspace
ℳ
will be denoted by
T
∣
ℳ
, and the commutator
A
B
−
B
A
of the operators
A
,
B
will be denoted by
[
A
,
B
]
. We recall the definition of Birkhoff-James’s orthogonality in Banach spaces [2,3].
Definition 1
If
X
is a complex Banach space, then for any elements
x
,
y
∈
X
, we say that
x
is orthogonal to
y
, noted by
x
⊥
y
, iff for all
α
,
β
∈
C
there holds
(2)
α
y
+
β
x
≥
β
x
.
for all
α
,
β
∈
C
or
R
.
If
ℳ
and
N
are linear subspaces in
X
, we say that
ℳ
is orthogonal to
N
if
x
⊥
B
−
J
y
for all
x
∈
M
and all
y
∈
N
. The orthogonality in this sense is asymmetric.
Let
∗
:
ℛ
→
B
(
X
)
be an involution defined on a linear subspace
ℛ
of
B
(
X
)
onto the algebra of all bounded linear operators acting on the Banach space
X
and
ℛ
∗
=
ℛ
. According to the definition given by Harte [4],
E
is called the Fuglede operator if
ker
E
⊆
ker
E
∗
.
The elementary operator is an operator
E
defined on Banach
A
,
ℬ
-bimodule
ℳ
with its representation
E
x
=
∑
i
=
1
n
a
i
x
b
i
, where
a
=
a
i
i
∈
A
n
,
b
=
b
i
i
∈
ℬ
n
are
n
-tuples of algebra elements. The length of
E
is defined to be the smallest number of multiplication terms required for any representation
∑
j
a
j
x
b
j
for
E
.
In this note, we consider
A
=
ℬ
=
B
(
ℋ
)
and
ℛ
=
B
(
ℋ
)
or
ℛ
=
C
p
:
(
1
≤
p
<
∞
)
and the length of
E
will be less or equal to 2, i.e., if
A
=
(
A
1
,
A
2
)
,
B
=
(
B
1
,
B
2
)
are 2-tuples of operators in
B
(
ℋ
)
2
, then the elementary operator induced by
A
and
B
is defined by
E
(
X
)
=
A
1
X
B
1
−
A
2
X
B
2
for all
X
∈
ℛ
. We will denote by
E
˜
the formal adjoint of
E
defined by
E
˜
(
X
)
=
∑
i
∈
J
A
i
∗
X
B
i
∗
for all
X
∈
ℛ
. Note also that
E
˜
∣
C
2
=
(
E
∣
C
2
)
∗
and
E
∗
(
X
)
=
∑
i
∈
J
B
i
X
A
i
on any separable ideal of compact operator, where
E
∗
is the operator adjoint of
E
in the sense of duality.
J. Anderson [5] proved that if
A
and
B
are normal operators, then
(3)
for all
X
,
S
∈
B
ℋ
:
S
∈
ker
δ
A
,
B
⇒
δ
A
,
B
X
+
S
≥
S
.
This means that the kernel of
δ
A
,
B
is orthogonal to its range.
F. Kittaneh [6] extended this result to an u.i. ideal norm
J
in
B
(
ℋ
)
, by proving that the range of
δ
A
,
B
∣
J
is orthogonal to
ker
δ
A
,
B
∩
J
.
A detailed study of range-kernel orthogonality for generalized derivation
δ
A
,
B
has received much attention in recent years and has been carried out in a large number of studies [3,5,7, 8,9,10, 11,12,13] and are based on the following result.
Theorem 2
Let
A
,
B
be operators in
B
(
ℋ
)
. If
δ
A
,
B
is Fuglede, then the range of
δ
A
,
B
(resp. the range of
δ
A
,
B
∣
C
p
) is orthogonal to the kernel of
δ
A
,
B
(resp. the kernel of
δ
A
,
B
∣
C
p
) for all
1
≤
p
≤
∞
.
D. Keckic [14] and A. Turnšek [15] extended Theorem 2 to the elementary operator
E
defined by
E
(
X
)
=
A
X
B
−
C
X
D
, where
(
A
,
C
)
and
(
B
,
D
)
are 2-tuples of commuting normal operators. Duggal [16] generalized the famous theorem to the case
(
A
,
C
)
and
(
B
,
D
)
are 2-tuples of commuting operators, where
A
,
B
are normal and
C
,
D
∗
are hyponormal.
In this paper, our goal is to extend the previous theorem to non-normal operators including quasinormal, subnormal, and
k
-quasihyponormal operators.
In the following, we recall some definitions about the range-kernel weak orthogonality.
Definition 3
[4] If
E
:
X
→
Y
and
T
:
Y
→
Z
are bounded linear operators between Banach spaces and
0
<
k
≤
1
,
(4)
s
∈
ker
T
⇒
dist
(
s
,
ran
E
)
≥
∥
s
∥
⇔
T
∠
k
E
⇔
ker
T
∠
k
ran
E
.
We say that
T
is weakly orthogonal to
E
, written
T
∠
E
, or equivalently
(5)
ker
T
∠
ran
E
⇔
∃
k
:
0
<
k
≤
1
:
T
∠
k
E
.
For
0
<
k
≤
1
, we say that
(
E
,
T
)
has a
1
k
-gap if
T
∠
k
E
.
If
T
=
E
and
k
≠
1
, we shall call
E
w
-orthogonal
(
E
∠
E
)
, consequently we get a
1
k
-gap between the subspaces
ker
E
and
ran
E
, which corresponds to the “range-kernel weak-orthogonality” for an operator
E
. If
k
=
1
, we shall say that
T
is orthogonal to
E
, written
T
⊥
E
, also if
X
=
(
Y
)
=
Z
and
T
=
E
we get a 1-gap between the subspaces
ker
E
and
ran
E
.
T
is said to be a quasi-normal if
[
T
,
T
∗
T
]
=
0
, subnormal if there exist a Hilbert space
K
⊇
ℋ
and a normal operator
N
∈
B
(
K
)
such that
N
∣
ℋ
=
T
. Also,
T
is called hyponormal if
[
T
∗
,
T
]
is a positive operator. Furthermore, we have the following proper inclusion
T
normal
⇒
T
quasi-normal
⇒
T
subnormal
⇒
T
hyponormal
.
A
∈
B
(
X
)
is said the Fuglede operator [13] if
ker
A
⊆
ker
A
∗
.
Recall that an
n
-tuple
A
=
(
A
1
,
A
2
,
…
,
A
n
)
∈
B
(
ℋ
)
n
is said commuting (resp. doubly commuting) if
[
A
i
,
A
j
]
=
0
(resp.
[
A
i
,
A
j
]
=
0
and
[
A
i
∗
,
A
j
]
=
0
) for all
i
,
j
=
1
,
…
,
n
,
i
≠
j
. The
n
-tuple
A
is said to be normal if
A
is commuting and each
A
i
(
i
=
1
,
…
,
n
)
is normal, and
A
is subnormal if
A
is the restriction of a normal
n
-tuple to a common invariant subspace. Clearly, every normal
n
-tuple is subnormal
n
-tuple. Any other notation or definition will be explained as and when required.
3 Main results
Proposition 7
Let
ℋ
,
K
be Hilbert spaces and
A
∈
B
(
ℋ
)
,
B
∈
B
(
K
)
, and
E
∈
B
(
B
(
K
,
ℋ
)
)
such that
E
(
X
)
=
A
X
B
. If
E
is the Fuglede operator, then
E
is
w
-orthogonal,
E
∠
E
˜
, and the inequality
(6)
min
{
∥
E
(
X
)
+
S
∥
J
,
∥
E
˜
(
X
)
+
S
∥
J
}
≥
∥
S
∥
J
holds if
J
=
B
(
K
,
ℋ
)
with
k
=
1
/
2
or
J
=
C
p
(
K
,
ℋ
)
with
k
=
1
2
2
−
p
p
, if
1
≤
p
≤
2
and
k
=
1
2
p
−
2
p
, if
2
≤
p
<
∞
.
Proof
If
A
and
B
∗
are injective operators, then
E
is injective. So there is nothing to prove.
Suppose that
A
or
B
∗
is the non-injective operator. From Lemma 5(2) and with respect to the decompositions:
ℋ
=
(
ker
A
)
⊥
⊕
ker
A
,
K
=
(
ker
B
∗
)
⊥
⊕
ker
B
∗
we obtain
A
=
A
1
⊕
A
;
B
=
B
1
⊕
0
. Let
X
=
(
X
i
,
j
)
i
,
j
=
1
,
2
:
K
→
ℋ
. Then
E
(
X
)
=
A
1
X
11
B
1
⊕
0
.
From the injectivity of
A
1
and
B
1
∗
it yields that any
S
=
(
S
i
j
)
i
,
j
=
1
,
2
in
ker
E
can be written as
S
=
0
S
12
S
21
S
22
,
where the operators
S
12
,
S
21
, and
S
22
are arbitrary.
Hence, for all
S
∈
ker
E
and all
X
∈
B
(
K
,
ℋ
)
, by Lemma 5(2), we have
min
{
∥
E
(
X
)
+
S
∥
,
∥
E
˜
(
X
)
+
S
∥
}
=
A
1
X
11
B
1
S
12
S
21
S
22
≥
1
2
(
∥
S
12
∥
2
+
∥
S
21
∥
2
+
∥
S
22
∥
2
)
1
/
2
≥
1
2
∥
S
∥
.
Also, for all
S
∈
ker
E
∩
C
p
(
K
,
ℋ
)
and all
X
∈
B
(
K
,
ℋ
)
such that
E
(
X
)
∈
C
p
(
K
,
ℋ
)
, we get
if
1
≤
p
≤
2
, then
min
{
∥
E
(
X
)
+
S
∥
p
,
∥
E
˜
(
X
)
+
S
∥
p
}
=
A
1
X
11
B
1
S
12
S
21
S
22
p
≥
1
2
2
−
p
p
∥
S
12
∥
p
p
+
∥
S
21
∥
p
p
+
∥
S
22
∥
p
p
1
/
p
≥
1
2
2
−
p
p
∥
S
∥
p
.
if
2
≤
p
<
∞
, then
min
{
∥
E
(
X
)
+
S
∥
p
,
∥
E
˜
(
X
)
+
S
∥
p
}
=
A
1
X
11
B
1
S
12
S
21
S
22
p
≥
1
2
p
−
2
p
∥
S
12
∥
p
p
+
∥
S
21
∥
p
p
+
∥
S
22
∥
p
p
1
/
p
≥
1
2
p
−
2
p
∥
S
∥
p
.
□
Using Lemma 5(2), we get a simple form of the previous result as follows.
Corollary 8
If
ker
A
reduces
A
and
ker
B
∗
reduces
B
∗
, then
E
is
w
-orthogonal,
E
E
˜
and satisfies the relation (6).
In the sequel
ξ
denotes the elementary operator defined by
ξ
(
X
)
=
A
X
A
∗
−
B
X
B
∗
from
B
(
ℋ
)
to
J
, where
A
and
B
are operators in
B
(
ℋ
)
and
J
=
B
(
ℋ
)
or
J
=
C
p
(
ℋ
)
;
1
≤
p
<
∞
.
Lemma 9
Let
Δ
be the elementary operator defined on
B
(
ℋ
)
by
Δ
(
X
)
=
A
X
B
−
X
, where
A
,
B
∈
B
(
ℋ
)
. If
Δ
is a Fuglede operator, then
Δ
is orthogonal and
Δ
⊥
Δ
˜
.
Proof
The proof is the same as the one in Theorem 4.□
Proposition 10
Let
A
,
B
be doubly commuting operators in
B
(
ℋ
)
and
ξ
:
B
(
ℋ
)
→
C
p
(
ℋ
)
;
(
1
≤
p
≤
∞
,
p
≠
2
)
.
If
ker
A
reduces
A
,
ker
B
reduces
B
and
ξ
orthogonal, then
ker
A
∩
ker
B
=
{
0
}
.
Proof
We consider the following three cases:
Let us suppose that
N
=
ker
A
∩
ker
B
≠
{
0
}
.
If
ker
A
≠
ker
B
and
ker
B
⊈
ker
A
, then with respect to the decomposition
ℋ
=
(
ker
B
)
⊥
⊕
(
ker
B
⊖
N
)
⊕
N
and from the hypothesis it yields
A
=
A
1
⊕
A
2
⊕
0
,
B
=
B
1
⊕
0
,
and
S
=
(
S
i
j
)
i
,
j
=
1
,
…
,
3
,
where
A
2
is an injective operator. Hence,
S
∈
ker
ξ
⇒
A
1
S
11
A
1
∗
=
B
1
S
11
B
1
∗
;
A
1
S
12
A
2
∗
=
A
2
S
21
A
1
∗
=
0
;
S
22
=
0
and the other entries are arbitrary. Choosing
X
and
S
as follows:
X
=
0
⊕
(
e
⊗
e
)
⊕
0
,
S
=
0
⊕
0
R
R
∗
C
,
where
e
is a non-zero vector in
ℋ
,
R
is an operator of rank one, and
C
is a self-adjoint operator of rank one. Then
ξ
(
X
)
+
S
=
0
⊕
A
2
e
⊕
A
2
e
R
R
∗
C
.
Applying Lemma (2.4) [15], we get
∥
ξ
(
X
)
+
S
∥
p
<
0
R
R
∗
C
p
=
∥
S
∥
p
(
p
≠
2
)
.
If
ker
A
≠
ker
B
and
ker
A
⊈
ker
B
, we proceed similarly as in the first case, it suffices to replace
A
by
B
and
B
by
A
in the preceding argument.
If
ker
A
=
ker
B
, then with respect to the decomposition
ℋ
=
(
ker
B
)
⊥
⊕
ker
B
,
we get
A
=
A
1
⊕
0
,
B
=
B
1
⊕
0
.
Letting
S
=
(
S
i
j
)
i
,
j
=
1
,
…
,
3
. Then
S
∈
ker
ξ
⇒
A
1
S
11
A
1
∗
=
B
1
S
11
B
1
∗
,
and the other entries are arbitrary. Choosing
X
and
S
as
X
=
(
e
⊗
e
)
⊕
0
,
S
=
0
R
R
∗
C
,
where
e
and
R
are as in (i). Then
ξ
(
X
)
+
S
=
A
1
e
⊗
A
1
e
−
B
1
e
⊗
B
1
e
R
R
∗
C
.
If
A
1
e
⊗
A
1
e
=
B
1
e
⊗
B
1
e
for all
e
∈
ℋ
, then it follows from this fact and the injectivity of
A
1
and
B
1
that
A
1
=
α
B
1
with
∣
α
∣
=
1
and hence
ξ
=
0
, which is a contradiction with the assumption
ξ
≠
0
. We use Lemma (2.4) [15] to complete the proof for
p
∉
{
1
,
2
}
.
In the case
p
=
1
, let us assume that
A
1
e
⊗
A
1
e
−
B
1
e
⊗
B
1
e
is an operator of rank 2 and has eigenvalues
λ
1
,
λ
2
with
∣
λ
1
∣
=
∣
λ
2
∣
for all
e
∈
ℋ
, then we can check that
∣
λ
1
∣
=
∣
λ
2
∣
⇔
∥
A
1
e
∥
=
∥
B
1
e
∥
and
⟨
A
1
e
,
B
1
e
⟩
=
0
∀
e
∈
ℋ
.
If
ker
B
∗
⊆
ker
B
, then by the injectivity of
B
1
∗
, it follows that
A
1
=
0
, which is a contradiction since
A
≠
0
.
If
ker
B
∗
⊈
ker
B
, with respect to the decomposition
ℋ
=
(
ker
B
1
∗
)
⊥
⊕
ker
B
1
∗
⊕
ker
B
we get
A
=
A
1
⊕
A
2
⊕
0
,
B
=
B
1
B
2
0
0
⊕
0
,
and
S
=
(
S
i
j
)
i
,
j
=
1
,
…
,
3
.
Since
A
1
is injective and
S
∈
ker
ξ
, we obtain
A
1
S
11
A
1
∗
=
B
1
S
11
B
1
∗
;
S
12
=
S
21
=
S
22
=
0
and the other entries are arbitrary.
We rewrite
S
on the following decomposition
ℋ
=
(
ker
B
1
∗
)
⊥
⊕
ker
B
⊕
ker
B
1
∗
and choose
S
11
=
S
23
=
S
32
=
0
,
S
13
=
R
,
S
31
=
R
∗
,
S
34
=
C
, and
X
=
e
⊗
e
⊕
0
(
R
,
C
,
e
are as in (i)). Then
ξ
(
X
)
+
S
=
A
1
e
⊗
A
1
e
−
B
1
e
⊗
B
1
e
R
R
∗
C
⊕
0
.
If
A
1
e
⊗
A
1
e
−
B
1
e
⊗
B
1
e
is an operator of rank two and has eigenvalues
λ
1
,
λ
2
with
∣
λ
1
∣
=
∣
λ
2
∣
for all
e
∈
ℋ
, then from the previous argument we conclude that
A
1
=
0
.
On the other hand, if
B
2
≠
0
, then from
[
A
,
B
]
=
0
it follows that
A
2
=
0
, also a contradiction with the fact that
A
≠
0
.
If
B
2
=
0
(
ker
B
∗
reduces
B
∗
), then
B
=
B
1
⊕
0
,
A
=
0
⊕
A
2
⊕
0
(
A
2
is injective), and
S
=
(
S
i
j
)
i
,
j
=
1
,
…
,
3
. Hence, it follows from
A
S
A
∗
=
B
S
B
∗
that
S
=
0
S
12
S
13
S
21
0
S
23
S
31
S
32
S
33
and the other entries are arbitrary.
To conclude the proof we can argue similarly as in the first case (i).□
Corollary 11
Let
(
A
1
,
A
2
)
,
(
B
1
,
B
2
)
be 2-tuples of doubly commuting operators in
B
(
ℋ
)
and
E
:
B
(
ℋ
)
→
C
p
(
ℋ
)
;
(
1
≤
p
≤
∞
,
p
≠
2
)
be the elementary operator defined by
E
(
X
)
=
A
1
X
B
1
−
A
2
X
B
2
such that
ker
A
1
,
ker
A
2
,
ker
B
1
∗
, and
ker
B
2
∗
reduce
A
1
,
A
2
,
B
1
∗
, and
B
2
∗
, respectively. If
E
is orthogonal, then
ker
A
1
∩
ker
A
2
=
{
0
}
=
ker
B
1
∗
∩
ker
B
2
∗
.
Proof
Consider the space
ℋ
⊕
ℋ
and the following decompositions
A
=
A
1
⊕
B
1
∗
,
B
=
A
2
⊕
B
2
∗
,
and
Y
=
0
X
0
0
.
Then, for all
X
∈
B
(
ℋ
)
, we have
A
Y
A
∗
−
B
Y
B
∗
=
A
1
X
B
1
−
A
2
X
B
2
,
ker
A
=
ker
A
1
∩
ker
B
1
∗
, and
ker
B
=
ker
A
2
∩
ker
B
2
∗
. Hence,
ker
A
1
∩
ker
A
2
=
{
0
}
=
ker
B
1
∗
∩
ker
B
2
∗
⇔
ker
A
∩
ker
B
=
{
0
}
.
So to achieve the proof, it suffices to apply the previous proposition.□
Theorem 12
Let
A
1
,
A
2
,
N
1
,
and
N
2
be operators in
B
(
ℋ
)
such that
(
N
1
,
N
2
)
is 2-tuple normal and
[
A
1
,
N
1
]
=
[
A
2
,
N
2
]
=
0
. Let
E
(
X
)
=
A
1
X
A
2
−
N
1
X
N
2
such that
E
(
X
)
,
E
˜
(
X
)
∈
J
. If
E
is the Fuglede operator, then
E
is
w
-orthogonal and
E
∠
E
˜
. Furthermore,
If
J
=
C
p
(
ℋ
)
:
(
1
≤
p
<
∞
,
p
≠
2
)
, then
E
is orthogonal if and only if
ker
A
1
∩
ker
N
1
=
{
0
}
=
ker
A
2
∗
∩
ker
N
2
∗
;
If
ker
A
1
∩
ker
N
1
=
{
0
}
=
ker
A
2
∗
∩
ker
N
2
∗
, then
E
is orthogonal and
E
⊥
E
˜
;
If
ker
A
1
∩
ker
N
1
≠
{
0
}
or
ker
A
2
∗
∩
ker
N
2
∗
≠
0
, then for all
X
∈
B
(
ℋ
)
and all
S
∈
ker
E
∩
J
,
min
{
∥
E
(
X
)
+
S
∥
J
,
∥
E
˜
(
X
)
+
S
∥
J
}
≥
k
∥
S
∥
J
,
where
J
=
B
(
ℋ
)
:
k
=
1
2
n
;
J
=
C
p
(
ℋ
)
:
k
=
1
2
4
−
2
p
p
, if
1
≤
p
≤
2
and
k
=
1
2
2
p
−
4
p
, if
2
≤
p
<
∞
.
Proof
The implication
(
⇒
)
follows from Corollary 8.
Let
ξ
:
B
(
ℋ
)
→
J
be the elementary operator defined by
ξ
(
X
)
=
A
X
A
∗
−
N
X
N
∗
, where
A
,
N
∈
B
(
ℋ
)
,
N
is normal with
[
A
,
N
]
=
0
. Assume that
ξ
is the Fuglede operator.
Suppose that
N
is invertible and set
D
=
N
−
1
A
.
ξ
is Fuglede implies that
ξ
D
is Fuglede, where
ξ
D
is the elementary operator induced by
D
. By Lemma 2.4 [16], we have
∥
ξ
(
X
)
+
S
∥
J
=
∥
A
X
A
∗
−
N
X
N
∗
+
S
∥
J
=
∥
D
(
N
X
N
∗
)
D
∗
−
N
X
N
∗
+
S
∥
J
≥
∥
S
∥
J
.
Similarly, it can be shown that
∥
ξ
˜
(
X
)
+
S
∥
J
≥
∥
S
∥
J
for any operator
S
∈
(
ker
ξ
)
∩
J
.
Suppose that
N
is injective, set
Δ
n
=
{
λ
∈
C
:
∣
λ
∣
≤
1
n
;
n
∈
N
}
and
μ
(
Δ
n
)
denotes the corresponding spectral projection, where
P
n
=
I
−
μ
(
Δ
n
)
converges strongly to
I
.
From
[
A
,
N
]
=
0
and by Fuglede-Putnam’s theorem it follows that
[
A
,
N
∗
]
=
0
and therefore
P
n
ℋ
reduces both
A
and
N
. Let
ℋ
=
P
n
ℋ
⊕
(
I
−
P
n
)
ℋ
,
then
A
=
A
1
n
⊕
A
2
n
;
N
=
N
1
n
⊕
N
2
n
and
P
n
=
I
⊕
0
, where
N
1
n
is invertible. Hence,
P
n
(
ξ
(
X
)
+
S
)
P
n
=
[
A
1
n
X
11
A
1
n
∗
−
N
1
n
X
11
N
1
n
∗
+
S
11
]
⊕
0
for all
X
=
(
X
i
j
)
i
,
j
=
1
,
2
∈
B
(
ℋ
)
and
S
=
(
S
i
j
)
i
,
j
=
1
,
2
∈
ker
ξ
implying
∥
ξ
(
X
)
+
S
∥
J
≥
∥
P
n
(
ξ
(
X
)
+
S
)
P
n
∥
J
=
∣
A
1
n
X
11
A
1
n
∗
−
N
1
n
X
11
N
1
n
∗
+
S
11
∥
J
≥
∥
S
11
∣
J
.
Since
P
n
(
ξ
∗
(
X
)
+
S
)
P
n
=
[
A
1
n
∗
X
11
A
1
n
−
N
1
n
∗
X
11
N
1
n
+
S
11
]
⊕
0
, then
∥
ξ
∗
(
X
)
+
S
∥
J
≥
∥
P
n
(
ξ
∗
(
X
)
+
S
)
P
n
∥
J
=
∥
A
1
n
∗
X
11
A
1
n
−
N
1
n
∗
X
11
N
1
n
+
S
11
∥
J
≥
∥
S
11
∥
J
.
On the other hand, for all positive integer
n
, we have
∥
S
11
∥
J
=
∥
P
n
S
P
n
∥
≤
min
{
∥
ξ
(
X
)
+
S
∥
J
,
∥
ξ
∗
(
X
)
+
S
∥
J
}
<
∞
and thus
sup
n
∥
P
n
S
P
n
∥
<
∞
for all
S
∈
ker
ξ
∩
J
and
X
∈
B
(
ℋ
)
. It follows by Lemma 3 [14] that
min
{
∥
ξ
(
X
)
+
S
∥
J
,
∥
ξ
˜
(
X
)
+
S
∥
J
}
≥
∥
S
∥
J
.
Suppose that
N
is an arbitrary normal operator.
If
ker
A
∩
ker
N
=
{
0
}
, then
ℋ
may be decomposed as
ℋ
=
[
(
ker
N
)
⊥
⊖
ker
A
]
⊕
ker
A
⊕
ker
N
.
Since
ξ
is Fuglede and by Lemma 5(1.(i)), we have
ker
A
reduces
A
,
A
=
A
11
⊕
0
⊕
A
22
and
N
=
N
11
⊕
N
22
⊕
0
.
For
X
=
(
X
i
j
)
i
,
j
=
1
,
2
,
3
∈
B
(
ℋ
)
, set
ξ
1
(
X
11
)
=
A
11
X
11
A
11
∗
−
N
11
X
11
N
11
∗
and
S
=
(
S
i
j
)
i
,
j
=
1
,
2
,
3
∈
ker
ξ
, then
ξ
1
(
S
11
)
=
0
and
A
11
S
13
A
22
∗
=
A
22
S
31
A
11
∗
=
A
22
S
33
A
22
∗
=
0
,
N
11
S
12
N
22
∗
=
N
22
S
21
N
11
∗
=
N
22
S
22
N
22
∗
=
0
.
Since
ξ
is Fuglede, then
S
∈
ker
ξ
˜
and
(7)
ξ
1
∗
(
S
11
)
=
0
and
A
11
∗
S
13
A
22
=
A
22
∗
S
31
A
11
=
A
22
∗
S
33
A
22
=
0
,
N
11
∗
S
12
N
22
=
N
22
∗
S
21
N
11
=
N
22
∗
S
22
N
22
=
0
.
Since
N
11
,
N
22
,
A
11
, and
A
22
are injective, we get from (ii) that
ξ
1
is orthogonal,
ξ
1
⊥
ξ
1
˜
and
S
13
=
S
31
=
S
33
=
S
12
=
S
22
=
S
21
=
0
and any operator
S
∈
ker
ξ
has the form
S
=
S
11
0
0
0
0
S
23
0
S
32
0
,
where
S
23
and
S
32
are arbitrary.
Let
S
23
=
U
23
∣
S
23
∣
,
S
32
=
U
32
∣
S
32
∣
be the polar decompositions of
S
23
and
S
32
, respectively, let
V
be the operator defined by
V
=
I
⊕
0
U
32
∗
U
23
∗
0
.
Then
∥
ξ
(
X
)
+
S
∥
J
≥
∥
V
(
ξ
(
X
)
+
S
)
∥
J
=
ξ
1
(
X
11
)
+
S
11
∗
∗
∗
∣
S
32
∣
∗
∗
∗
∣
S
23
∣
.
Applying Lemma 6, we get
if
J
=
B
(
ℋ
)
:
∥
ξ
(
X
)
+
S
∥
≥
max
{
∥
ξ
1
(
X
11
)
+
S
11
∥
,
∥
S
32
∥
,
∥
S
23
∥
}
≥
max
{
∥
S
11
∥
,
∥
S
32
∥
,
∥
S
23
∥
}
=
∥
S
∥
.
if
J
=
C
p
(
ℋ
)
;
(
1
≤
p
<
∞
)
:
∥
ξ
(
X
)
+
S
∥
p
≥
(
∥
ξ
1
(
X
11
)
+
S
11
∥
p
p
+
∥
S
32
∥
p
p
+
∥
S
23
∥
p
p
)
1
/
p
≥
(
∥
S
11
∥
p
p
+
∥
S
32
∥
p
p
+
∥
S
23
∥
p
p
)
1
/
p
=
∥
S
∣
p
.
By the same method, we have that
ξ
⊥
ξ
˜
.
If
M
=
ker
A
∩
ker
N
≠
{
0
}
and
ℋ
is decomposed as
ℋ
=
(
ker
N
)
⊥
⊕
[
ker
N
⊖
M
]
⊕
M
,
then by Lemma 5(ii) and the fact that
ξ
is Fuglede, it follows that
M
reduces
A
, and hence
A
=
A
11
⊕
A
22
⊕
0
,
N
=
N
11
⊕
0
.
For
X
=
(
X
i
j
)
i
,
j
=
1
,
2
,
3
, we set
ξ
1
(
X
)
=
A
11
X
11
A
11
∗
−
N
11
X
11
N
11
∗
and let
S
=
(
S
i
j
)
i
,
j
=
1
,
2
,
3
∈
ker
ξ
. From the injectivity of
A
11
and
A
22
, we obtain
ξ
1
(
S
11
)
=
0
and
S
12
=
S
21
=
S
22
=
0
.
By simple computation, we get
∥
ξ
(
X
)
+
S
∥
J
=
ξ
1
(
X
11
)
+
S
11
∗
S
13
∗
∗
S
23
S
31
S
32
S
33
J
.
Applying Lemma
6 yields
for
J
=
B
(
ℋ
)
:
∥
ξ
(
X
)
+
S
∥
2
≥
1
2
2
(
∥
ξ
1
(
X
11
+
S
11
)
2
+
∥
S
13
∥
2
+
∥
S
23
∥
2
+
∥
S
32
∥
2
+
∥
S
13
∥
2
+
∥
S
33
∥
2
)
≥
1
2
4
∥
S
∥
2
.
for
J
=
C
p
(
ℋ
)
;
1
≤
p
≤
2
:
∥
ξ
(
X
)
+
S
∥
p
p
≥
1
2
2
−
p
∥
ξ
1
(
X
11
+
S
11
)
p
p
+
∥
S
13
∥
p
p
+
∥
S
23
∥
p
p
+
∥
S
32
∥
p
p
+
∥
S
13
∥
p
p
+
∥
S
33
∥
p
p
≥
1
2
4
−
2
p
∥
S
∥
p
p
.
for
J
=
C
p
(
ℋ
)
;
2
≤
p
<
∞
:
∥
ξ
(
X
)
+
S
∥
p
p
≥
1
2
2
−
p
∥
ξ
1
(
X
11
+
S
11
)
p
p
+
∥
S
13
∥
p
p
+
∥
S
23
∥
p
p
+
∥
S
32
∥
p
p
+
∥
S
13
∥
p
p
+
∥
S
33
∥
p
p
≥
1
2
2
p
−
4
∥
S
∥
p
p
.
Similarly,
for
J
=
B
(
ℋ
)
:
∥
ξ
˜
(
X
)
+
S
∥
≥
1
2
2
∥
S
∥
.
for
J
=
C
p
(
ℋ
)
;
1
≤
p
≤
2
:
∥
ξ
˜
(
X
)
+
S
∥
p
≥
1
2
4
−
2
p
p
∥
S
∥
p
.
for
J
=
C
p
(
ℋ
)
;
2
≤
p
<
∞
:
∥
ξ
˜
(
X
)
+
S
∥
p
≥
1
2
2
p
−
4
p
∥
S
∥
p
.
Let us now finish the proof for the elementary operator
E
(
X
)
=
A
1
X
A
2
−
N
1
X
N
2
. Consider the space
ℋ
⊕
ℋ
and define the following operators on
B
(
ℋ
⊕
ℋ
)
as
N
=
N
1
⊕
N
2
∗
;
A
=
A
1
⊕
A
2
∗
,
and
Z
=
0
X
0
0
.
It is clear that
N
is normal operator and
[
A
1
,
N
1
]
=
[
A
2
,
N
2
]
=
0
imply
[
A
,
N
]
=
0
and
E
(
Z
)
=
A
Z
A
∗
−
N
Z
N
∗
.
Applying the preceding result, the proof is complete.□
Let
A
,
B
∈
B
(
ℋ
)
and
Ω
be a set in
B
(
ℋ
)
2
defined by
(
A
,
B
)
∈
Ω
if and only if
Δ
(
X
)
=
A
N
X
B
M
−
X
is the Fuglede operator for any normal
N
,
M
satisfying
[
N
,
A
]
=
[
M
,
B
]
=
0
.
It follows from the definition that
Δ
is Fuglede and
Ω
≠
∅
since
(
I
,
I
)
∈
Ω
.
It is shown in [13] that if
(
A
,
B
)
∈
Ω
, then the elementary operator
E
defined by
E
(
X
)
=
N
X
M
−
A
X
B
is Fuglede for any normal operators
N
and
M
in
B
(
ℋ
)
. So as a consequence of the previous theorem, we get the following corollaries.
Corollary 13
Let
A
,
B
∈
B
(
ℋ
)
and
N
,
M
be arbitrary normal operators in
B
(
ℋ
)
such that
[
N
,
A
]
=
[
M
,
B
]
=
0
and
E
be the elementary operator defined by
E
(
X
)
=
N
X
M
−
A
X
B
for all
X
∈
B
(
ℋ
)
.
If
(
A
,
B
)
∈
Ω
, then
E
is
w
-orthogonal and
E
∠
E
˜
. Furthermore,
E
and
E
˜
verify assertions (i), (ii), and (iii) in Theorem 12.
Lemma 14
[13] Let
A
,
B
∈
B
(
ℋ
)
and
N
,
M
be normal operators in
B
(
ℋ
)
such that
[
N
,
A
]
=
[
M
,
B
]
=
0
, then
(
A
,
B
)
∈
Ω
in each of the following cases:
A
and
B
∗
are hyponormal operators;
A
is
k
-quasihyponormal and
B
∗
is injective
k
-quasihyponormal operator.
Corollary 15
If
(
A
1
,
A
2
)
,
(
B
1
,
B
2
)
are 2-tuples of commuting operators in
B
(
ℋ
)
such that
A
1
,
B
1
are normal operators and
E
(
X
)
=
A
1
X
B
1
−
A
2
X
B
2
, then
E
is
w
-orthogonal and
E
∠
E
˜
. Furthermore,
E
and
E
˜
satisfy assertions (i), (ii), and (iii) cited in Theorem 12, in each of the following cases:
A
1
and
B
1
∗
are hyponormal operators;
A
1
is
k
-quasihyponormal and
B
1
∗
is injective
k
-quasihyponormal operator.
In the next theorem, we give a positive answer to a question raised by P. B. Duggal [16]: Is Theorem 2 still true if the hypothesis is related to
A
and
B
∗
being subnormal?
Theorem 16
If
A
and
B
∗
are 2-tuples of commuting subnormal operators in
[
B
(
ℋ
)
]
2
such that
A
=
(
A
1
,
A
2
)
,
B
=
(
B
1
,
B
2
)
, and
E
(
X
)
=
A
1
X
B
1
−
A
2
X
B
2
, then
E
is
w
-orthogonal and
E
∠
E
˜
.
Proof
From the definition of subnormality of 2-tuple operator, we have
A
is the restriction of a 2-tuple normal
N
=
(
N
1
,
N
2
)
to a common invariant subspace
ℋ
and
B
∗
is the restriction of a 2-tuple normal
M
=
(
M
1
,
M
2
)
to a common invariant subspace
ℋ
equivalently to
A
i
=
N
i
∣
ℋ
;
B
i
∗
=
M
i
∣
ℋ
,
i
=
1
,
2
with
N
i
,
M
i
are normal operators on a Hilbert space
K
⊇
ℋ
and
[
N
1
,
N
2
]
=
[
M
1
,
M
2
]
=
0
. If
S
∈
ker
E
, then for all
X
∈
B
(
ℋ
)
, we have
N
1
X
˜
M
1
∗
−
N
2
X
˜
M
2
∗
+
S
˜
=
E
(
X
)
+
S
0
0
0
,
where
X
˜
=
X
⊕
0
and
S
˜
=
S
⊕
0
. Hence,
∥
N
1
X
˜
M
1
∗
−
N
2
X
˜
M
2
∗
+
S
˜
∥
J
=
∥
E
(
X
)
+
S
∥
J
.
Since
N
i
,
M
i
:
i
=
1
,
2
are normal, we get the
w
-orthogonality of
E
. With similar argument,
E
∠
E
˜
follows.□
Corollary 17
If
(
A
1
,
A
2
)
and
B
=
(
B
1
,
B
2
)
are 2-tuples of commuting operators in
[
B
(
ℋ
)
]
2
such that
[
A
i
,
A
1
∗
A
1
+
A
2
∗
A
2
]
=
[
B
i
∗
,
B
1
B
1
∗
+
B
2
B
2
∗
]
=
0
;
i
=
1
,
2
and
E
(
X
)
=
A
1
X
B
1
−
A
2
X
B
2
, then
E
is
w
-orthogonal and
E
∠
E
˜
.
Proof
By assumption,
A
and
B
∗
are spherically quasi-normal commuting 2-tuples (see definition [17]) and also by [17],
A
and
B
∗
are subnormal 2-tuples. So the desired result follows from Theorem 16.□
Theorem 18
Let
(
A
1
,
A
2
)
,
(
B
1
,
B
2
)
be 2-tuples of doubly commuting operators in
B
(
ℋ
)
and
E
be the elementary operator defined by
E
(
X
)
=
A
1
X
B
1
−
A
2
X
B
2
such that
E
(
X
)
,
E
˜
(
X
)
∈
J
,
A
1
,
B
1
∗
are quasi-normal operators and
A
2
,
B
2
∗
are
k
-quasihyponormal operators
(
k
≥
1
)
with
ker
A
2
⊆
ker
A
2
∗
and
ker
B
2
∗
⊆
ker
B
2
. Then
E
is
w
-orthogonal and
E
∠
E
˜
. Furthermore,
If
J
=
C
p
(
ℋ
)
;
(
1
≤
p
<
∞
,
p
≠
2
)
, then
E
is
orthogonal
if
and
only
if
ker
A
1
∩
ker
N
1
=
ker
A
2
∗
∩
ker
N
2
∗
=
{
0
}
;
If
ker
A
1
ker
N
1
=
{
0
}
=
ker
A
2
∗
∩
ker
N
2
∗
, then
E
is orthogonal and
E
⊥
E
˜
;
If
ker
A
1
∩
ker
A
2
≠
0
or
ker
B
1
∗
∩
ker
B
2
∗
≠
{
0
}
, then for all
X
∈
B
(
ℋ
)
and all
S
∈
ker
E
∩
J
,
min
{
∥
E
(
X
)
+
S
∥
J
,
∥
E
˜
(
X
)
+
S
∥
J
}
≥
k
∥
S
∥
J
.
For
J
=
B
(
ℋ
)
,
k
=
1
6
;
For
J
=
C
p
(
ℋ
)
,
k
=
1
6
2
−
p
p
, if
1
≤
p
≤
2
;
For
J
=
C
p
(
ℋ
)
,
k
=
1
6
p
−
2
p
, if
2
≤
p
<
∞
.
Proof
Consider the following decompositions
ℋ
=
ℋ
1
=
(
ker
A
2
)
⊥
⊕
ker
A
2
,
ℋ
=
ℋ
2
=
(
ker
B
2
∗
)
⊥
⊕
ker
B
2
∗
.
Then
A
2
=
C
1
⊕
0
,
B
2
=
C
2
⊕
0
,
where
C
1
,
C
2
∗
are injective
k
-quasihyponormal and by hypothesis, we get
A
1
=
T
1
⊕
T
2
,
B
1
=
R
1
⊕
R
2
with
T
1
,
T
2
,
R
1
∗
, and
R
2
∗
are quasinormal operators and
(
T
1
,
C
1
)
,
(
R
1
,
C
2
)
are 2-tuples of doubly commuting operators.
Since
ker
T
1
reduces
T
1
(resp.
ker
R
1
∗
reduces
R
1
∗
) and by commutativity, we have that
T
1
=
A
11
⊕
0
,
R
1
=
B
11
⊕
0
,
C
1
=
A
21
⊕
A
22
,
C
2
=
B
21
⊕
B
22
with respect to the following decompositions
(
ker
A
2
)
⊥
=
(
ker
T
1
)
⊥
⊕
ker
T
1
,
(
ker
B
2
∗
)
⊥
=
(
ker
R
1
∗
)
⊥
⊕
ker
R
1
∗
,
where
A
11
and
B
11
∗
are injective quasinormal operators,
A
21
,
A
22
,
B
21
∗
, and
B
22
∗
are injective
k
-quasihyponormal. We set
A
11
=
U
∣
A
11
∣
and
B
11
=
V
∣
B
11
∣
=
∣
B
11
∗
∣
V
.
Then it follows from the injectivity of
A
11
and
B
11
∗
that
U
and
V
∗
are isometry operators and by
[
T
1
,
C
1
]
=
[
T
1
∗
,
C
1
]
=
0
, we get that
(8)
A
11
,
A
21
=
[
A
11
∗
,
A
21
]
=
0
.
Then
[
A
11
∗
A
11
,
A
21
]
=
[
A
11
A
11
∗
,
A
21
]
=
0
and
[
∣
A
11
∣
,
A
21
]
=
[
∣
A
11
∗
∣
,
A
21
]
=
0
. Hence,
[
U
,
A
21
]
=
[
U
∗
,
A
21
]
=
0
. Similarly, we obtain that
[
V
,
B
21
]
=
[
V
∗
,
B
21
]
=
0
,
and
(
U
A
21
∗
)
∗
is
k
-quasihyponormal. Indeed,
(
A
21
∗
U
)
∗
k
[
U
∗
A
21
,
A
21
∗
U
]
A
21
∗
k
U
k
=
(
A
21
)
k
+
1
U
∗
k
[
U
∗
,
U
]
U
k
A
21
∗
k
+
1
+
U
∗
k
U
A
21
k
[
A
21
,
A
21
∗
]
A
21
∗
k
U
∗
U
k
=
U
∗
U
∗
k
A
21
k
[
A
21
,
A
21
∗
]
A
21
∗
k
U
∗
U
k
≤
0
.
Similarly, we get
(
V
∗
B
21
)
∗
is an injective
k
-quasihyponormal.
Let
X
,
S
∈
B
(
ℋ
)
:
X
=
(
X
i
j
)
i
,
j
=
1
,
2
,
3
and
S
=
(
S
i
j
)
i
,
j
=
1
,
2
,
3
. If
S
∈
ker
E
, then
(9)
A
11
S
11
B
11
=
A
21
S
11
B
21
,
(10)
A
11
S
13
R
2
=
T
2
S
31
B
11
=
T
2
S
33
R
2
=
0
,
(11)
A
21
S
12
B
22
=
A
22
S
21
B
21
=
A
22
S
22
B
22
=
0
.
And
U
∗
(
A
11
X
11
B
11
−
A
21
X
11
B
21
+
S
11
)
V
∗
=
∣
A
11
∣
X
11
∣
B
11
∗
∣
−
U
∗
A
21
X
11
B
21
V
∗
+
U
∗
S
11
V
∗
.
We derive from 8 that
∣
A
11
∣
U
S
11
V
∗
∣
B
11
∣
=
U
∗
A
21
U
∗
S
11
V
∗
B
21
V
∗
.
Applying Corollary 8,
∥
A
11
X
11
B
11
−
A
21
X
11
B
21
+
S
11
∥
≥
∥
U
∗
(
A
11
X
11
B
11
−
A
21
X
11
B
21
+
S
11
)
V
∗
∥
≥
∥
U
∗
S
11
V
∗
∥
.
from the injectivity of
A
11
and its polar decomposition, we have
(
ker
T
1
)
⊥
=
(
ker
A
11
)
⊥
=
(
ker
U
)
⊥
;
ker
A
11
¯
=
ran
U
and
U
:
(
ker
U
)
⊥
⟶
ran
U
is unitary. Taking the following decompositions yields
(
ker
U
)
⊥
=
ran
U
⊕
(
ran
U
)
⊥
;
(
ker
V
∗
)
⊥
=
ran
V
∗
¯
⊕
(
ran
V
∗
)
⊥
.
Then
A
11
=
A
11
1
γ
0
0
,
B
11
=
B
11
1
0
ζ
0
,
S
11
=
S
11
1
S
11
2
S
11
3
S
11
4
,
U
=
U
1
0
:
(
ker
U
)
⊥
→
ran
U
⊕
(
ran
U
)
⊥
,
V
=
V
1
0
:
ran
V
∗
¯
⊕
(
ran
V
∗
)
⊥
→
(
ker
V
∗
)
⊥
.
From the commutativity, we obtain
A
21
=
A
21
1
⊕
A
21
2
;
B
21
=
B
21
1
⊕
B
21
2
.
By simple computation, we get
U
∗
S
11
V
∗
=
U
1
∗
S
11
1
V
1
∗
,
A
21
1
S
11
2
B
21
2
=
A
21
2
S
11
3
B
21
1
=
A
21
2
S
11
4
B
21
2
=
0
.
From the injectivity of
A
21
i
and
B
21
i
:
i
=
1
,
2
, we derive that
S
11
2
=
S
11
3
=
S
11
4
=
0
.
The injectivity of
A
21
,
B
22
,
A
22
,
B
21
,
A
11
, and
B
11
∗
in the equalities 10 and 11 implies that
S
12
=
S
21
=
S
22
=
0
,
S
13
R
2
=
T
2
S
31
=
T
2
S
33
R
2
=
0
.
Setting
E
11
(
X
11
)
=
A
11
X
11
B
11
−
A
21
X
11
B
21
.
If
ker
A
1
∩
ker
A
2
=
{
0
}
=
ker
B
1
∗
∩
ker
B
2
∗
, then
S
13
=
S
33
=
S
31
=
0
and therefore any operator
S
∈
ker
E
has the form
S
=
S
11
0
0
0
0
S
23
0
S
32
0
,
where
S
23
and
S
32
are arbitrary with
∥
S
11
∥
J
=
∥
S
11
1
∥
J
=
∥
U
1
∗
S
11
1
V
1
∗
∥
J
=
∥
U
∗
S
11
V
∗
∥
J
and
∥
E
(
X
)
+
S
∥
J
=
E
11
(
X
11
)
+
S
11
∗
∗
∗
∗
S
23
∗
S
32
∗
J
.
Let
S
23
=
U
23
∣
S
23
∣
,
S
32
=
S
32
∣
S
32
∣
be the polar decomposition of
S
23
and
S
32
, respectively, and set the operator
V
=
I
⊕
0
U
32
∗
U
23
∗
0
. Then
∥
E
(
X
)
+
S
∥
J
≥
∥
∥
V
(
E
(
X
)
+
S
)
∥
J
≥
E
11
(
X
11
)
+
S
11
∗
∗
∗
∣
S
32
∣
∗
∗
∗
∣
S
23
∣
J
.
Applying Lemma 5, we get
J
=
B
(
ℋ
)
:
∥
E
(
X
)
+
S
∥
≥
max
{
∥
E
(
X
11
)
+
S
11
∥
,
∥
S
32
∥
,
∥
S
23
∥
}
≥
max
{
∥
S
11
∥
,
∥
S
32
∥
,
∥
S
23
∥
}
=
∥
S
∥
.
J
=
C
p
(
ℋ
)
:
(
1
≤
p
<
∞
)
∥
E
(
X
)
+
S
∥
p
≥
(
∥
E
(
X
11
)
+
S
11
∥
p
p
+
∥
S
32
∥
p
p
+
∥
S
23
∥
p
p
)
1
/
p
≥
(
∥
S
11
∥
p
p
+
∥
S
32
∥
p
p
+
∥
S
23
∥
p
p
)
1
/
p
=
∥
S
∥
p
p
.
If
ker
A
1
∩
ker
A
2
≠
{
0
}
or
ker
B
1
∗
∩
ker
B
2
∗
≠
{
0
}
, then any operator
S
∈
ker
E
has the form
S
=
S
11
0
S
13
0
0
S
23
S
13
S
32
S
33
,
where
S
23
and
S
32
are arbitrary. By simple calculation, we have
∥
E
(
X
)
+
S
∥
J
=
E
11
(
X
11
)
+
S
11
∗
A
11
X
13
R
2
+
S
13
∗
∗
S
23
T
2
X
31
B
11
+
S
31
S
32
T
2
X
33
R
2
+
S
33
J
.
It is well known that the kernel of a quasinormal operator is a reduced subspace, then by application of Corollary 8, we obtain
∥
A
11
X
13
R
2
+
S
13
∥
J
≥
k
∥
S
13
∥
J
∥
T
2
X
31
B
11
+
S
31
∥
J
≥
k
∥
S
31
∥
J
∥
T
2
X
33
R
2
+
S
33
∥
J
≥
k
∥
S
33
∥
J
.
Therefore, by Lemma 6, we get
J
=
B
(
ℋ
)
;
∥
E
(
X
)
+
S
∥
2
≥
1
3
2
∥
E
11
(
X
11
)
+
S
11
∥
2
+
1
2
2
[
∥
S
32
∥
2
+
∥
S
23
∥
2
+
∥
S
13
∥
2
+
∥
S
31
∥
2
+
∥
S
33
∥
2
]
≥
1
6
2
∥
S
∥
2
.
J
=
C
p
(
ℋ
)
:
(
2
≤
p
<
∞
)
∥
E
(
X
)
+
S
∥
p
p
≥
1
3
p
−
2
∥
S
11
∥
p
p
+
1
2
p
−
2
[
∥
S
32
∥
p
p
+
∥
S
23
∥
p
p
+
∥
S
13
∥
p
p
+
∥
S
31
∥
p
p
+
∥
S
33
∥
p
p
]
≥
1
2
p
−
2
1
3
p
−
2
∥
S
∥
p
p
=
1
6
p
−
2
∥
S
∥
p
p
.
J
=
C
p
(
ℋ
)
:
(
1
≤
p
≤
2
)
:
∥
E
(
X
)
+
S
∥
p
p
≥
1
3
2
−
p
∥
S
11
∥
p
p
+
1
2
2
−
p
[
∥
S
32
∥
p
p
+
∥
S
23
∥
p
p
+
∥
S
13
∥
p
p
+
∥
S
31
∥
p
p
+
∥
S
33
∥
p
p
]
≥
1
2
2
−
p
1
3
2
−
p
∥
S
∥
p
p
=
1
6
p
−
2
∥
S
∥
p
p
.
□
