# On Subscalarity of Some 2 × 2M-Hyponormal Operator Matrices

Authors: Fei Zuo; Junli Shen
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

We provide some conditions for2
×
2 operator matrices whose diagonal entries are M-hyponormal operators to be subscalar. As a consequence, we obtain that Weyl type theorem holds for such operator matrices.


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## Body

## 1. Introduction and Preliminaries

LetH be a complex separable Hilbert space and let B
(
H
) denote the algebra of all bounded linear operators on H. If T
∈
B
(
H
), we write N
(
T
), R
(
T
), σ
(
T
), and σ
a
(
T
) for the null space, the range space, the spectrum, and the approximate point spectrum of T, respectively. An operator T is called Fredholm if R
(
T
) is closed, α
(
T
)
:
=
dim
⁡
N
(
T
)
<
∞, and β
(
T
)
:
=
dim
⁡
N
(
T
*
)
<
∞. The index of a Fredholm operator T is given by i
(
T
)
=
α
(
T
)
-
β
(
T
). An operator T is called Weyl if it is Fredholm of index zero. The Weyl spectrum of T [1] is defined by w
(
T
)
:
=
{
λ
∈
ℂ
:
T
-
λ
is
not
Weyl
}.We consider the sets(1)
Φ
+
(
H
)
:
=
{
T
∈
B
(
H
)
:
R
(
T
)
is
closed
,
α
(
T
)
<
∞
}
;
Φ
+
-
(
H
)
:
=
{
T
∈
B
(
H
)
:
T
∈
Φ
+
(
H
)
,
i
(
T
)
≤
0
}
and define
(2)
σ
e
a
(
T
)
:
=
{
λ
∈
ℂ
:
T
-
λ
∉
Φ
+
-
(
H
)
}
;
π
00
(
T
)
:
=
{
λ
∈
iso
σ
(
T
)
:
0
<
α
(
T
-
λ
)
<
∞
}
;
π
00
a
(
T
)
:
=
{
λ
∈
iso
σ
a
(
T
)
:
0
<
α
(
T
-
λ
)
<
∞
}
,
where iso σ
(
T
) denotes the isolated points of σ
(
T
).Following [2], we say that Weyl’s theorem holds for T if σ
(
T
)
∖
w
(
T
)
=
π
00
(
T
) and that a-Weyl’s theorem holds for T if σ
a
(
T
)
∖
σ
e
a
(
T
)
=
π
00
a
(
T
).LetT
∈
B
(
H
). As an easy extension of normal operators, hyponormal operators have been studied by many mathematicians. Though there are many unsolved interesting problems for hyponormal operators (e.g., the invariant subspace problem), one of recent trends in operator theory is studying natural extensions of hyponormal operators. So we introduce some of these nonhyponormal operators. An operator T is said to be M-hyponormal if there exists a real positive number M such that
(3)
M
2
(
T
-
λ
)
*
(
T
-
λ
)
≥
(
T
-
λ
)
(
T
-
λ
)
*
∀
λ
∈
ℂ
.
Evidently,
(4)
T
is
hyponormal
⟹
T
is
M
-hyponormal
.There is a vast literature concerningM-hyponormal operators (see [3–5], etc.). We also note that an operator T need not be hyponormal even though T and T
* are both M-hyponormal. To see this, consider the operator
(5)
T
=
(
U
K
0
U
*
)
:
l
2
⊕
l
2
⟶
l
2
⊕
l
2
,
where U is the unilateral shift on l
2 and K
:
l
2
→
l
2 is given by K
(
x
1
,
x
2
,
x
3
,
…
)
=
(
2
x
1
,
0,0
,
…
). Then a direct calculation shows that
(6)
1
2
∥
(
T
-
z
)
x
∥
≤
∥
(
T
-
z
)
*
x
∥
≤
2
∥
(
T
-
z
)
x
∥for allz
∈
ℂ and for all x
∈
l
2
⊕
l
2, which says that T and T
* are both M-hyponormal. But since
(7)
T
*
T
=
(
I
0
0
I
+
3
2
K
)
,
while
(8)
T
T
*
=
(
I
+
3
2
K
0
0
I
)
,
T is not hyponormal.Letz be the coordinate in the complex plane ℂ and let d
μ
(
z
) denote the planar Lebesgue measure. Fix a complex (separable) Hilbert space H and a bounded (connected) open subset U of ℂ. We will denote by L
2
(
U
,
H
) the Hilbert space of measurable functions f
:
U
→
H, such that
(9)
∥
f
∥
2
,
U
=
(
∫
U
∥
f
(
z
)
∥
2
d
μ
(
z
)
)
1
/
2
<
∞
.The Bergman space forU is defined by A
2
(
U
,
H
)
=
L
2
(
U
,
H
)
∩
O
(
U
,
H
), where O
(
U
,
H
) denotes the Fréchet space of H-valued analytic functions on U with respect to uniform topology. Note that A
2
(
U
,
H
) is a Hilbert space. Let us define now a special Sobolev type space. Let U be again a bounded open subset of ℂ and let m be a fixed nonnegative integer. The vector valued Sobolev space W
m
(
U
,
H
) with respect to ∂
¯ and of order m will be the space of those functions f
∈
L
2
(
U
,
H
) whose derivatives ∂
¯
f
,
…
,
∂
¯
m
f in the sense of distributions still belong to L
2
(
U
,
H
). Endowed with the norm
(10)
∥
f
∥
W
m
2
=
∑
i
=
0
m
∥
∂
¯
i
f
∥
2
,
U
2
W
m
(
U
,
H
) becomes a Hilbert space contained continuously in L
2
(
U
,
H
). A bounded linear operator S on H is called scalar of order m if it possesses a spectral distribution of order m, that is, if there is a continuous unital morphism of topological algebras
(11)
Φ
:
C
0
m
(
ℂ
)
⟶
B
(
H
)
such that Φ
(
z
)
=
S, where z stands for the identity function on ℂ, and C
0
m
(
ℂ
) stands for the space of compactly supported functions on ℂ, continuously differentiable of order m
,
0
≤
m
≤
∞. An operator is subscalar if it is similar to the restriction of a scalar operator to an invariant subspace. Let U be a (connected) bounded open subset of ℂ and let m be a nonnegative integer. The linear operator M
f of multiplication by f on W
m
(
U
,
H
) is continuous and it has a spectral distribution of order m, defined by the functional calculus
(12)
Φ
M
:
C
0
m
(
ℂ
)
⟶
B
(
W
m
(
U
,
H
)
)
,
Φ
M
(
f
)
=
M
f
.
Therefore, M
z is a scalar operator of order m.An operatorT
∈
B
(
H
) is said to have the single-valued extension property (or SVEP) if for every open subset G of ℂ and any analytic function f
:
G
→
H such that (
T
-
z
)
f
(
z
)
≡
0 on G, we have f
(
z
)
≡
0 on G.An operatorT
∈
B
(
H
) is said to have Bishop’s property (
β
) if for every open subset G of ℂ and every sequence f
n
:
G
→
H of H-valued analytic functions such that (
T
-
z
)
f
n
(
z
) converges uniformly to 0 in norm on compact subsets of G, f
n
(
z
) converges uniformly to 0 in norm on compact subsets of G. It is well known that
(13)
Bishop
’
s
property
(
β
)
⟹
SVEP
.In 1984, Putinar showed in [6] that every hyponormal operator is subscalar, and then in 1987, Brown used this result to prove that a hyponormal operator with rich spectrum has a nontrivial invariant subspace (see [7]). There have been a lot of generalizations of such beautiful consequences (see [8–11]). In this paper, we provide some conditions for 2
×
2 operator matrices whose diagonal entries are M-hyponormal operators to be subscalar. As a consequence, we obtain that Weyl type theorem holds for such operator matrices.


## 2. Subscalarity

Lemma 1 (see [6, Proposition  2.1]).
For a bounded open diskD in the complex plane ℂ, there is a constant C
D such that for an arbitrary operator T
∈
B
(
H
) and f
∈
W
2
(
D
,
H
) we have
(14)
∥
(
I
-
P
)
f
∥
2
,
D
≤
C
D
(
∥
(
T
-
z
)
*
∂
¯
f
∥
2
,
D
+
∥
(
T
-
z
)
*
∂
¯
2
f
∥
2
,
D
)
,
where P denotes the orthogonal projection of L
2
(
D
,
H
) onto the Bergman space A
2
(
D
,
H
).Corollary 2.
LetD be as in Lemma 1. If T
∈
B
(
H
) is an M-hyponormal operator, then there exists a constant C
D such that for all z
∈
ℂ and f
∈
W
2
(
D
,
H
)
(15)
∥
(
I
-
P
)
f
∥
2
,
D
≤
M
C
D
c
c
×
(
∥
(
T
-
z
)
∂
¯
f
∥
2
,
D
+
∥
(
T
-
z
)
∂
¯
2
f
∥
2
,
D
)
,
where P denotes the orthogonal projection of L
2
(
D
,
H
) onto the Bergman space A
2
(
D
,
H
).Proof.
This follows from Lemma1 and the definition of M-hyponormal operator.Lemma 3.
LetT
∈
B
(
H
) be an M-hyponormal operator and let D be a bounded disk in ℂ. If {
f
n
} is a sequence in W
m
(
D
,
H
)
(
m
>
2
) such that
(16)
lim
⁡
n
→
∞
⁡
∥
(
z
-
T
)
∂
¯
i
f
n
∥
2
,
D
=
0
for i
=
1,2
,
…
,
m, then lim
⁡
n
→
∞
⁡
∥
∂
¯
i
f
n
∥
2
,
D
0
=
0 for i
=
1,2
,
…
,
m
-
2, where D
0 is a disk strictly contained in D.Proof.
SinceT is an M-hyponormal operator, it follows from Corollary 2 that there exists a constant C
D such that
(17)
∥
(
I
-
P
)
∂
¯
i
f
n
∥
2
,
D
≤
M
C
D
(
∥
(
T
-
z
)
∂
¯
i
+
1
f
n
∥
2
,
D
c
c
c
c
c
c
c
c
c
c
+
∥
(
T
-
z
)
∂
¯
i
+
2
f
n
∥
2
,
D
)
for i
=
0,1
,
2
,
…
,
m
-
2. From (17), we have
(18)
lim
⁡
n
→
∞
⁡
∥
(
I
-
P
)
∂
¯
i
f
n
∥
2
,
D
=
0
for i
=
0,1
,
2
,
…
,
m
-
2. Hence,
(19)
lim
⁡
n
→
∞
⁡
∥
(
T
-
z
)
P
∂
¯
i
f
n
∥
2
,
D
=
0
for i
=
1,2
,
…
,
m
-
2. Since T has Bishop’s property (
β
) [12], we have
(20)
lim
⁡
n
→
∞
⁡
∥
P
∂
¯
i
f
n
∥
2
,
D
0
=
0
for i
=
1,2
,
…
,
m
-
2, where D
0 denotes a disk with σ
(
T
)
⫋
D
0
¯
⫋
D. From (18) and (20), we get that
(21)
lim
⁡
n
→
∞
⁡
∥
∂
¯
i
f
n
∥
2
,
D
0
=
0
for i
=
1,2
,
…
,
m
-
2.Lemma 4.
LetT
=
(
T
1
T
2
T
3
T
4
)
∈
B
(
H
⊕
H
), where T
i are mutually commuting, and both T
1 and T
4 are M-hyponormal operators. For any positive integer m and any bounded disk D in ℂ containing σ
(
T
), define the map V
m
:
H
⊕
H
→
H
(
D
) by
(22)
V
m
h
=
1
⊗
h
+
(
T
-
z
)
W
m
(
D
,
H
)
⊕
W
m
(
D
,
H
)
¯
(
=
1
⊗
h
~
)
,
where
(23)
H
(
D
)
:
=
W
m
(
D
,
H
)
⊕
W
m
(
D
,
H
)
(
T
-
z
)
W
m
(
D
,
H
)
⊕
W
m
(
D
,
H
)
¯
and 1
⊗
h denotes the constant function sending any z
∈
D to h
∈
H
⊕
H. Then the following statements hold.(i)
IfT
2
r
T
3
s
=
0 for some nonnegative integer r and s, where T
2
0
=
T
3
0
=
I, then V
4
N
+
2 is one-to-one and has closed range, where N
:
=
max
⁡
{
r
,
s
}.
(ii)
IfT
1
=
T
4, T
2
=
T
3, and T
2 is algebraic of order k, then V
4
k
+
2 is one-to-one and has closed range.
(iii)
IfT
1
+
T
4 is an M-hyponormal operator and T
1
T
4
=
T
2
T
3, then V
6 is one-to-one and has closed range.Proof.
Leth
n
=
h
n
1
⊕
h
n
2
∈
H
⊕
H and
(24)
f
n
=
f
n
1
⊕
f
n
2
∈
W
m
(
D
,
H
)
⊕
W
m
(
D
,
H
)
be sequences such that
(25)
lim
⁡
n
→
∞
⁡
∥
(
z
-
T
)
f
n
+
1
⊗
h
n
∥
W
m
⊕
W
m
=
0
.
Then (25) implies that
(26)
lim
⁡
n
→
∞
⁡
∥
(
z
-
T
1
)
f
n
1
+
T
2
f
n
2
+
1
⊗
h
n
1
∥
W
m
=
0
,
lim
⁡
n
→
∞
⁡
∥
T
3
f
n
1
+
(
z
-
T
4
)
f
n
2
+
1
⊗
h
n
2
∥
W
m
=
0
.
By the definition of the norm of Sobolev space and (26) we get
(27)
lim
⁡
n
→
∞
⁡
∥
(
z
-
T
1
)
∂
¯
i
f
n
1
+
T
2
∂
¯
i
f
n
2
∥
2
,
D
=
0
,
lim
⁡
n
→
∞
⁡
∥
T
3
∂
¯
i
f
n
1
+
(
z
-
T
4
)
∂
¯
i
f
n
2
∥
2
,
D
=
0
fori
=
1,2
,
…
,
m.(i)
Setm
=
4
N
+
2, where N
:
=
max
⁡
{
r
,
s
}. We may assume that s
≤
r. Then m
=
4
r
+
2.
We prove that for everyj
=
0,1
,
2
,
…
,
s, the following equations hold
(28)
lim
⁡
n
→
∞
⁡
∥
T
2
r
-
j
T
3
s
-
j
+
1
∂
¯
i
f
n
1
∥
2
,
D
j
=
0
,
lim
⁡
n
→
∞
⁡
∥
T
2
r
-
j
T
3
s
-
j
∂
¯
i
f
n
2
∥
2
,
D
j
=
0
for i
=
1,2
,
…
,
4
(
r
-
j
)
+
2, where σ
(
T
)
⫋
D
s
⫋
⋯
⫋
D
2
⫋
D
1
⫋
D.
To prove (28), we will use the induction on j. Since T
2
r
T
3
s
=
0, then (28) holds when j
=
0. Suppose that (28) is true for some j
<
s. From (27) and the inductions hypothesis, we have
(29)
lim
⁡
n
→
∞
⁡
∥
(
T
1
-
z
)
T
2
r
-
j
-
1
T
3
s
-
j
∂
¯
i
f
n
1
∥
2
,
D
j
=
0
,
lim
⁡
n
→
∞
⁡
∥
T
2
r
-
j
-
1
T
3
s
-
j
∂
¯
i
f
n
1
+
(
T
4
-
z
)
T
2
r
-
j
-
1
T
3
s
-
j
-
1
∂
¯
i
f
n
2
∥
2
,
D
j
=
0
for i
=
1,2
,
…
,
4
(
r
-
j
)
+
2. Since T
1 is an M-hyponormal operator, by Lemma 3 we have
(30)
lim
⁡
n
→
∞
⁡
∥
T
2
r
-
j
-
1
T
3
s
-
j
∂
¯
i
f
n
1
∥
2
,
D
j
′
=
0
fori
=
1,2
,
…
,
4
(
r
-
j
), where σ
(
T
)
⫋
D
j
′
⫋
D
j. From (30) and the second equation of (27),
(31)
lim
⁡
n
→
∞
⁡
∥
(
T
4
-
z
)
T
2
r
-
j
-
1
T
3
s
-
j
-
1
∂
¯
i
f
n
2
∥
2
,
D
j
′
=
0
for i
=
1,2
,
…
,
4
(
r
-
j
). Since T
4 is an M-hyponormal operator, by Lemma 3 we derive
(32)
lim
⁡
n
→
∞
⁡
∥
T
2
r
-
j
-
1
T
3
s
-
j
-
1
∂
¯
i
f
n
2
∥
2
,
D
j
+
1
=
0
for i
=
1,2
,
…
,
4
(
r
-
j
-
1
)
+
2, where σ
(
T
)
⫋
D
j
+
1
⫋
D
j
′. Therefore, the proof of (28) is completed. Let j
=
s in (28); we have
(33)
lim
⁡
n
→
∞
⁡
∥
T
2
r
-
s
T
3
∂
¯
i
f
n
1
∥
2
,
D
s
=
0
,
lim
⁡
n
→
∞
⁡
∥
T
2
r
-
s
∂
¯
i
f
n
2
∥
2
,
D
s
=
0
for i
=
1,2
,
…
,
4
(
r
-
s
)
+
2. From (27) and (33), it follows that
(34)
lim
⁡
n
→
∞
⁡
∥
(
T
1
-
z
)
T
2
r
-
s
-
1
∂
¯
i
f
n
1
∥
2
,
D
s
=
0
for i
=
1,2
,
…
,
4
(
r
-
s
)
+
2. Since T
1 is an M-hyponormal operator, from Lemma 3 we obtain that
(35)
lim
⁡
n
→
∞
⁡
∥
T
2
r
-
s
-
1
∂
¯
i
f
n
1
∥
2
,
D
s
′
=
0
for i
=
1,2
,
…
,
4
(
r
-
s
), where σ
(
T
)
⫋
D
s
′
⫋
D
s, and hence
(36)
lim
⁡
n
→
∞
⁡
∥
(
T
4
-
z
)
T
2
r
-
s
-
1
∂
¯
i
f
n
2
∥
2
,
D
s
′
=
0
for i
=
1,2
,
…
,
4
(
r
-
s
). Since T
4 is an M-hyponormal operator, Lemma 3 implies that
(37)
lim
⁡
n
→
∞
⁡
∥
T
2
r
-
s
-
1
∂
¯
i
f
n
2
∥
2
,
D
s
+
1
=
0
for i
=
1,2
,
…
,
4
(
r
-
s
-
1
)
+
2, where σ
(
T
)
⫋
D
s
+
1
⫋
D
s
′. By repeating the process from (33) to (37), it holds for all j
=
0,1
,
2
,
…
,
r
-
s that
(38)
lim
⁡
n
→
∞
⁡
∥
T
2
r
-
s
-
j
∂
¯
i
f
n
2
∥
2
,
D
s
+
j
=
0
for i
=
1,2
,
…
,
4
(
r
-
s
-
j
)
+
2, where σ
(
T
)
⫋
D
r
+
1
⫋
D
r
⫋
D
r
-
1
⫋
⋯
⫋
D
s
+
1
⫋
D
s. In particular, let r
=
s
+
j,
(39)
lim
⁡
n
→
∞
⁡
∥
∂
¯
i
f
n
2
∥
2
,
D
r
=
0
for i
=
1,2. Hence, from the first equation of (27), we have
(40)
lim
⁡
n
→
∞
⁡
∥
(
T
1
-
z
)
∂
¯
i
f
n
1
∥
2
,
D
r
=
0
for i
=
1,2. Applying Corollary 2, we have
(41)
lim
⁡
n
→
∞
⁡
∥
(
I
-
P
)
f
n
1
∥
2
,
D
r
′
=
0
,
where P denotes the orthogonal projection of L
2
(
D
r
′
,
H
) onto A
2
(
D
r
′
,
H
) and σ
(
T
)
⫋
D
r
′
⫋
D
r. By combining (26) with (39) and (41), we obtain that
(42)
lim
⁡
n
→
∞
⁡
∥
(
z
-
T
)
P
f
n
+
1
⊗
h
n
∥
2
,
D
r
′
=
0
,
where P
f
n
:
=
(
P
f
n
1
P
f
n
2
).
LetΓ be a curve in D
r
′ surrounding σ
(
T
). Then for z
∈
Γ
(43)
lim
⁡
n
→
∞
⁡
∥
P
f
n
(
z
)
+
(
z
-
T
)
-
1
(
1
⊗
h
n
)
(
z
)
∥
=
0
uniformly. Hence, by the Riesz functional calculus,
(44)
lim
⁡
n
→
∞
⁡
∥
1
2
π
i
∫
Γ
P
f
n
(
z
)
d
z
+
h
n
∥
=
0
.
But (
1
/
2
π
i
)
∫
Γ
‍
P
f
n
(
z
)
d
z
=
0 by Cauchy’s theorem. Hence, lim
⁡
n
→
∞
⁡
h
n
=
0, and so V
4
r
+
2 is one-to-one and has closed range.(ii)
Setm
=
4
k
+
2. By the hypothesis and (27), we have
(45)
lim
⁡
n
→
∞
⁡
∥
(
T
1
-
z
)
∂
¯
i
f
n
1
+
T
2
∂
¯
i
f
n
2
∥
2
,
D
=
0
,
lim
⁡
n
→
∞
⁡
∥
T
2
∂
¯
i
f
n
1
+
(
T
1
-
z
)
∂
¯
i
f
n
2
∥
2
,
D
=
0for i
=
1,2
,
…
,
4
k
+
2. Since T
2 is algebraic with order k, there exists a nonconstant polynomial p
(
z
)
=
(
z
-
z
1
)
(
z
-
z
2
)
⋯
(
z
-
z
k
) such that p
(
T
2
)
=
0. Set q
j
(
z
)
=
(
z
-
z
j
+
1
)
⋯
(
z
-
z
k
) for j
=
0,1
,
2
,
…
,
k
-
1 and q
k
(
z
)
=
1.
Claim. For everyj
=
0,1
,
2
,
…
,
k, the following equations hold:
(46)
lim
⁡
n
→
∞
⁡
∥
q
j
(
T
2
)
∂
¯
i
f
n
1
∥
2
,
D
j
=
0
,
lim
⁡
n
→
∞
⁡
∥
q
j
(
T
2
)
∂
¯
i
f
n
2
∥
2
,
D
j
=
0
for i
=
1,2
,
…
,
4
(
k
-
j
)
+
2, where σ
(
T
)
⫋
D
k
⫋
⋯
⫋
D
2
⫋
D
1
⫋
D.
To prove the claim, we use the induction onj. Since q
0
(
T
2
)
=
p
(
T
2
)
=
0, then when j
=
0 the claim holds. Suppose that the claim is true for some j
=
r, where 0
≤
r
<
k. Multiplying (45) by q
r
+
1
(
T
2
), we obtain
(47)
lim
⁡
n
→
∞
⁡
∥
(
T
1
-
z
)
q
r
+
1
(
T
2
)
∂
¯
i
f
n
1
+
z
r
+
1
q
r
+
1
(
T
2
)
∂
¯
i
f
n
2
∥
2
,
D
r
=
0
,
lim
⁡
n
→
∞
⁡
∥
z
r
+
1
q
r
+
1
(
T
2
)
∂
¯
i
f
n
1
+
(
T
1
-
z
)
q
r
+
1
(
T
2
)
∂
¯
i
f
n
2
∥
2
,
D
r
=
0
for i
=
1,2
,
…
,
4
(
k
-
r
)
+
2. From (47), we derive
(48)
lim
⁡
n
→
∞
⁡
∥
(
T
1
-
(
z
-
z
r
+
1
)
)
q
r
+
1
(
T
2
)
(
∂
¯
i
f
n
1
+
∂
¯
i
f
n
2
)
∥
2
,
D
r
=
0
for i
=
1,2
,
…
,
4
(
k
-
r
)
+
2. Since T
1 is an M-hyponormal operator, from (48) and Lemma 3 we obtain
(49)
lim
⁡
n
→
∞
⁡
∥
q
r
+
1
(
T
2
)
(
∂
¯
i
f
n
1
+
∂
¯
i
f
n
2
)
∥
2
,
D
r
′
=
0
for i
=
1,2
,
…
,
4
(
k
-
r
), where σ
(
T
)
⫋
D
r
′
⫋
D
r. Combining (49) with the first equation of (47), we have
(50)
lim
⁡
n
→
∞
⁡
∥
(
T
1
-
(
z
+
z
r
+
1
)
)
q
r
+
1
(
T
2
)
∂
¯
i
f
n
1
∥
2
,
D
r
′
=
0
for i
=
1,2
,
…
,
4
(
k
-
r
). Since T
1 is an M-hyponormal operator, we obtain from Lemma 3 and (50) that
(51)
lim
⁡
n
→
∞
⁡
∥
q
r
+
1
(
T
2
)
∂
¯
i
f
n
1
∥
2
,
D
r
+
1
=
0
for i
=
1,2
,
…
,
4
(
k
-
r
-
1
)
+
2, where σ
(
T
)
⫋
D
r
+
1
⫋
D
r
′. From (49) and (51), we obtain
(52)
lim
⁡
n
→
∞
⁡
∥
q
r
+
1
(
T
2
)
∂
¯
i
f
n
2
∥
2
,
D
r
+
1
=
0
for i
=
1,2
,
…
,
4
(
k
-
r
-
1
)
+
2. Therefore, the proof of the claim is completed.
From the claim withj
=
k, we have
(53)
lim
⁡
n
→
∞
⁡
∥
∂
¯
i
f
n
1
∥
2
,
D
k
=
lim
⁡
n
⟶
∞
⁡
∥
∂
¯
i
f
n
2
∥
2
,
D
k
=
0
for i
=
1,2. From Lemma 1 we derive that
(54)
lim
⁡
n
→
∞
⁡
∥
(
I
-
P
)
f
n
1
∥
2
,
D
k
=
lim
⁡
n
⟶
∞
⁡
∥
(
I
-
P
)
f
n
2
∥
2
,
D
k
=
0
,
where P denotes the orthogonal projection of L
2
(
D
k
,
H
) onto A
2
(
D
k
,
H
). By applying the proof of (i), we obtain that V
4
k
+
2 is one-to-one and has closed range.(iii)  Setm
=
6. By (27), we have
(55)
lim
⁡
n
→
∞
⁡
∥
(
T
1
T
3
-
z
T
3
)
∂
¯
i
f
n
1
+
T
2
T
3
∂
¯
i
f
n
2
∥
2
,
D
=
0
,
lim
⁡
n
→
∞
⁡
∥
T
1
T
3
∂
¯
i
f
n
1
+
(
T
1
T
4
-
z
T
1
)
∂
¯
i
f
n
2
∥
2
,
D
=
0i
=
1,2
,
…
,
6
T
1
T
4
=
T
2
T
3(56)
lim
⁡
n
→
∞
⁡
∥
z
(
T
1
∂
¯
i
f
n
2
-
T
3
∂
¯
i
f
n
1
)
∥
2
,
D
=
0i
=
1,2
,
…
,
6(57)
lim
⁡
n
→
∞
⁡
∥
T
1
∂
¯
i
f
n
2
-
T
3
∂
¯
i
f
n
1
∥
2
,
D
1
=
0i
=
1,2
,
…
,
4
σ
(
T
)
⫋
D
1
⫋
D(58)
lim
⁡
n
→
∞
⁡
∥
(
T
1
+
T
4
-
z
)
∂
¯
i
f
n
2
∥
2
,
D
1
=
0i
=
1,2
,
3,4
T
1
+
T
4
M(59)
lim
⁡
n
→
∞
⁡
∥
∂
¯
i
f
n
2
∥
2
,
D
2
=
0i
=
1,2
σ
(
T
)
⫋
D
2
⫋
D
1(60)
lim
⁡
n
→
∞
⁡
∥
(
T
1
-
z
)
∂
¯
i
f
n
1
∥
2
,
D
2
=
0i
=
1,2
T
1
M(61)
lim
⁡
n
→
∞
⁡
∥
(
I
-
P
)
f
n
1
∥
2
,
D
3
=
0
,P
L
2
(
D
3
,
H
)
A
2
(
D
3
,
H
)
σ
(
T
)
⫋
D
3
⫋
D
2
V
6Theorem 5.
LetT
=
(
T
1
T
2
T
3
T
4
)
∈
B
(
H
⊕
H
), where T
i are mutually commuting, and both T
1 and T
4 are M-hyponormal operators. If {
T
i
}
i
=
1
4 satisfy one of the conditions in Lemma 4, then T is a subscalar operator of order m, where m is the appropriately chosen integer as in Lemma 4.Proof.
LetD be a bounded disk in ℂ containing σ
(
T
) and consider the quotient space
(62)
H
(
D
)
:
=
W
m
(
D
,
H
)
⊕
W
m
(
D
,
H
)
(
T
-
z
)
W
m
(
D
,
H
)
⊕
W
m
(
D
,
H
)
¯
endowed with the Hilbert space norm, where m
=
4
N
+
2, N
:
=
max
⁡
{
r
,
s
} for (i), m
=
4
k
+
2 for (ii), and m
=
6 for (iii) in Lemma 4. The class of a vector f or an operator S on H
(
D
) will be denoted by f
~, S
~, respectively. Let M be the operator of multiplication by z on W
m
(
D
,
H
)
⊕
W
m
(
D
,
H
). Then M is a scalar operator of order m and has a spectral distribution Φ. Since R
(
T
-
z
) is invariant under M, M
~ can be well defined. Moreover, consider the spectral distribution Φ
:
C
0
m
(
ℂ
)
→
B
(
W
m
(
D
,
H
)
⊕
W
m
(
D
,
H
)
) defined by the following relation: for φ
∈
C
0
m
(
ℂ
) and f
∈
W
m
(
D
,
H
)
⊕
W
m
(
D
,
H
), Φ
(
ϕ
)
f
=
ϕ
f. Then the spectral distribution Φ of M commutes with T
-
z, and so M
~ is still a scalar operator of order m with Φ
~ as a spectral distribution. As in Lemma 4, if we define the map V
m
:
H
1
⊕
H
2
→
H
(
D
) by
(63)
V
m
h
=
1
⊗
h
cc
+
(
T
-
z
)
W
m
(
D
,
H
)
⊕
W
m
(
D
,
H
)
¯
(
=
1
⊗
h
~
)
,
then V
m is one-to-one and has closed range. Since
(64)
V
m
T
h
=
1
⊗
T
h
~
=
z
⊗
h
~
=
M
~
(
1
⊗
h
~
)
=
M
~
V
m
h
for all h
∈
H
⊕
H
,
V
m
T
=
M
~
V
m. In particular, R
(
V
m
) is invariant under M
~ and R
(
V
m
) is closed; it is a closed invariant subspace of the scalar operator M
~. Since T is similar to the restriction M
~
∣
R
(
V
m
) and M
~ is scalar of order m, T is a subscalar operator of order m.Corollary 6.
LetT
=
(
T
1
T
2
T
3
T
4
)
∈
B
(
H
⨁
H
), where T
i are mutually commuting, both T
1 and T
4 are M-hyponormal operators, and {
T
i
}
i
=
1
4 satisfy one of the conditions in Lemma 4. Then T has property (
β
) and the single-valued extension property.Proof.
From section one, we need only to prove thatT has property (
β
). Since property (
β
) is transmitted from an operator to its restrictions to closed invariant subspaces, we are reduced by Theorem 5 to the case of a scalar operator. Since every scalar operator has property (
β
) (see [6]), T has property (
β
).Define the quasi-nilpotent part ofλ
I
-
T
(65)
H
0
(
λ
I
-
T
)
:
=
{
x
∈
H
:
lim
⁡
n
→
∞
⁡
∥
(
λ
I
-
T
)
n
x
∥
1
/
n
=
0
}
.Definition 7.
An operatorT
∈
B
(
H
) is said to belong to the class H
(
p
) if there exists a natural number p
:
=
p
(
λ
) such that
(66)
H
0
(
λ
I
-
T
)
=
N
(
λ
I
-
T
)
p
∀
λ
∈
ℂ
.Theorem 8 (see [13]).
Every subscalar operatorT
∈
B
(
H
) is H
(
p
).Definition 9.
An operatorT
∈
B
(
H
) is said to be polaroid if every λ
∈
iso
σ
(
T
) is a pole of the resolvent of T.Note that(67)
T
is
polaroid
⟺
T
*
is
polaroid
.
The condition of being polaroid may be characterized by means of the quasi-nilpotent part.Theorem 10 (see [14]).
An operatorT
∈
B
(
H
) is polaroid if and only if there exists a natural number p
:
=
p
(
λ
) such that
(68)
H
0
(
λ
I
-
T
)
=
N
(
λ
I
-
T
)
p
∀
λ
∈
i
s
o
σ
(
T
)
.Corollary 11.
EveryH
(
p
) operator is polaroid.Since a subscalar operator isH
(
p
), we have the following.Corollary 12.
Every subscalar operator is polaroid.Corollary 13.
LetT
=
(
T
1
T
2
T
3
T
4
)
∈
B
(
H
⊕
H
), where T
i are mutually commuting, both T
1 and T
4 are M-hyponormal operators, and {
T
i
}
i
=
1
4 satisfy one of the conditions in Lemma 4. Then T is polaroid.IfT
∈
B
(
H
) has SVEP, then T and T
* satisfy Browder’s theorem. A sufficient condition for an operator T satisfying Browder’s theorem to satisfy Weyl’s theorem is that T is polaroid. Then we have the following result.Corollary 14.
LetT
=
(
T
1
T
2
T
3
T
4
)
∈
B
(
H
⊕
H
), where T
i are mutually commuting, both T
1 and T
4 are M-hyponormal operators, and {
T
i
}
i
=
1
4 satisfy one of the conditions in Lemma 4. Then Weyl’s theorem holds for T and T
*.Observe that ifT
∈
B
(
H
) has SVEP, then σ
(
T
)
=
σ
a
(
T
*
)
¯. Hence, if T has SVEP and is polaroid, then T
* satisfies a-Weyl’s theorem [15, Theorem  3.10].Corollary 15.
LetT
=
(
T
1
T
2
T
3
T
4
)
∈
B
(
H
⊕
H
), where T
i are mutually commuting, both T
1 and T
4 are M-hyponormal operators, and {
T
i
}
i
=
1
4 satisfy one of the conditions in Lemma 4. Then a-Weyl’s theorem holds for T
*.Proof.
SinceT is polaroid and has SVEP, then a-Weyl’s theorem holds for T
*.In the following,f is an analytic function on σ
(
T
) and f is not constant on each connected component of the open set U containing σ
(
T
).Corollary 16.
LetT
=
(
T
1
T
2
T
3
T
4
)
∈
B
(
H
⊕
H
), where T
i are mutually commuting, both T
1 and T
4 are M-hyponormal operators, and {
T
i
}
i
=
1
4 satisfy one of the conditions in Lemma 4. Then the following assertions hold:(i)
Weyl’s theorem holds forf
(
T
);
(ii)
a-Weyl’s theorem holds for f
(
T
*
).Proof.
(
i
) Since T is polaroid and has SVEP, then f
(
T
) satisfies Weyl’s theorem by [15, Theorem  3.14].
(
ii) Since T is polaroid and has SVEP, then f
(
T
*
) satisfies a-Weyl’s theorem by [15, Theorem  3.12].



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*Source: 461567-2014-02-25.xml*