Volume 8 (2007), Issue 3, Article 90, 7 pp.
ON CLASS wF (p, r, q) OPERATORS AND QUASISIMILARITY
CHANGSEN YANG AND YULIANG ZHAO
C OLLEGE OF M ATHEMATICS AND I NFORMATION S CIENCE
H ENAN N ORMAL U NIVERSITY,
X INXIANG 453007, P EOPLE ’ S R EPUBLIC OF C HINA
yangchangsen117@yahoo.com.cn
D EPARTMENT OF M ATHEMATICS
A NYANG I NSTITUTE OF T ECHNOLOGY
A NYANG C ITY, H ENAN P ROVINCE 455000
P EOPLE ’ S R EPUBLIC OF C HINA
zhaoyuliang512@163.com
Received 17 October, 2006; accepted 15 June, 2007
Communicated by C.K. Li
A BSTRACT. Let T be a bounded linear operator on a complex Hilbert space H. In this paper, we
show that if T belongs to class wF (p, r, q) operators, then we have (i) T ∗ X = XN ∗ whenever
T X = XN for some X ∈ B(H), where N is normal and X is injective with dense range.
(ii) T satisfies the property (β)ε , i.e., T is subscalar, moreover, T is subdecomposable. (iii)
Quasisimilar class wF (p, r, q) operators have the same spectra and essential spectra.
Key words and phrases: Class wF (p, r, q) operators, Fuglede-Putnam’s theorem, Property (β)ε , Subscalar, Subdecomposable.
2000 Mathematics Subject Classification. 47B20, 47A30.
1. I NTRODUCTION
Let X denote a Banach space, T ∈ B(X) is said to be generalized scalar ([3]) if there exists
a continuous algebra homomorphism (called a spectral distribution of T ) Φ : ε(C) → B(X)
with Φ(1) = I and Φ(z) = T , where ε(C) denotes the algebra of all infinitely differentiable
functions on the complex plane C with the topology defined by uniform convergence of such
functions and their derivatives ([2]). An operator similar to the restriction of a generalized
scalar (decomposable) operator to one of its closed invariant subspaces is said to be subscalar
(subdecomposable). Subscalar operators are subdecomposable operators ([3]). Let H, K be
complex Hilbert spaces and B(H), B(K) be the algebra of all bounded linear operators in H
and K respectively, B(H, K) denotes the algebra of all bounded linear operators from H to K.
A capital letter (such as T ) means an element of B(H). An operator T is said to be positive
(denoted by T ≥ 0) if (T x, x) ≥ 0 for any x ∈ H. An operator T is said to be p−hyponormal
if (T ∗ T )p ≥ (T T ∗ )p , 0 < p ≤ 1.
The authors are grateful to the referee for comments which improved the paper.
266-06
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C HANGSEN YANG AND Y ULIANG Z HAO
Definition 1.1 ([10]). For p > 0, r ≥ 0, and q ≥ 1, an operator T belongs to class wF (p, r, q)
if
2(p+r)
1
(|T ∗ |r |T |2p |T ∗ |r ) q ≥ |T ∗ | q
and
1
1
|T |2(p+r)(1− q ) ≥ (|T |p |T ∗ |2r |T |p )1− q .
Let T = U |T | be the polar decomposition of T . We define
Tep,r = |T |p U |T |r (p + r = 1).
The operator Tep,r is known as the generalized Aluthge transform of T . We define (Tep,r )(1) =
^
Tep,r , (Tep,r )(n) = [(Tep,r )(n−1) ]p,r , where n ≥ 2.
The following Fuglede-Putnam’s theorem is famous. We extend this theorem for class wF (p, r, q)
operators.
Theorem 1.1 (Fuglede-Putnam’s Theorem [7]). Let A and B be normal operators and X be
an operator on a Hilbert space. Then the following hold and follow from each other:
(i) (Fuglede) If AX = XA, then A∗ X = XA∗ .
(ii) (Putnam) If AX = XB, then A∗ X = XB ∗ .
2. P RELIMINARIES
Lemma 2.1 ([9]). If N is a normal operator on H, then we have
\
(N − λ)H = {0}.
λ∈C
Lemma 2.2 ([5]). Let T = U |T | be the polar decomposition of a p-hyponormal operator for
p > 0. Then the following assertions hold:
(i) Tes,t = |T |s U |T |t is p+min(s,t)
-hyponormal for any s > 0 and t > 0 such that max{s, t} ≥
s+t
p.
(ii) Tes,t = |T |s U |T |t is hyponormal for any s > 0 and t > 0 such that max{s, t} ≤ p.
Lemma 2.3 ([8]). Let T ∈ B(H), D ∈ B(H) with 0 ≤ D ≤ M (T − λ)(T − λ)∗ for all λ in C,
1
where M is a positive real number. Then for every x ∈ D 2 H there exists a bounded function
f : C → H such that (T − λ)f (λ) ≡ x.
2m
Lemma 2.4 ([10]). If T ∈ wF (p, r, q), then Tep,r ≥ |T |2m ≥ (Tep,r )∗ |2m , where m =
n
n
oo
p
min 1q , max p+r
, 1 − 1q , i.e., Tep,r = |T |p U |T |r is m-hyponormal operator.
Lemma 2.5 ([11]). Let A, B ≥ 0, α0 , β0 > 0 and −β0 ≤ δ ≤ α0 , −β0 ≤ δ̄ ≤ α0 , if 0 ≤ δ ≤ α0
+δ
β
αβ0+β
β0
0
0
0
α
0
≥ B β0 +δ , then
and B 2 A B 2
β
2
α
B A B
β
2
β+δ
α+β
≥ B β+δ ,
and
α
α
Aα−δ̄ ≥ A 2 B β A 2
hold for each α ≥ α0 , β ≥ β0 and 0 ≤ δ̄ ≤ α.
1
1
α−δ̄
α+β
1
1
Lemma 2.6 ([6]). Let A ≥ 0, B ≥ 0, if B 2 AB 2 ≥ B 2 and A 2 BA 2 ≥ A2 then A = B.
Lemma 2.7. Let A, B ≥ 0, s, t ≥ 0, if B s A2t B s = B 2s+2t , At B 2s At = A2s+2t then A = B.
J. Inequal. Pure and Appl. Math., 8(3) (2007), Art. 90, 7 pp.
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O N C LASS wF (p, r, q) O PERATORS A ND Q UASISIMILARITY
3
Proof. We choose k > max{s, t. Since B s A2t B s = B 2s+2t , At B 2s At = A2s+2t it follows from
Lemma 2.5 that:
2k+2t
(B k A2k B k ) 4k ≥ B 2k+2t ,
A2k−2t ≥ (Ak B 2k Ak )
2k−2t
4k
,
and
(Ak B 2k Ak )
2k+2s
4k
≥ A2k+2s ,
B 2k−2s ≥ (B k A2k B k )
2k−2s
4k
.
So
Ak B 2k Ak = A4k ,
B k A2k B k = B 4k ,
by Lemma 2.6
A = B.
Lemma 2.8 ([11]). Let T be a class wF (p, r, q) operator, if Tep,r = |T |p U |T |r is normal, then
T is normal.
The following theorem have been shown by T. Huruya in [3], here we give a simple proof.
Theorem 2.9 (Furuta inequality [4]). If A ≥ B ≥ 0, then for each r > 0,
r
r 1
r
r 1
(i) B 2 Ap B 2 q ≥ B 2 B p B 2 q and
r
r 1
r
r 1
(ii) A 2 Ap A 2 q ≥ A 2 B p A 2 q
hold for p ≥ 0 and q ≥ 1 with (1 + r)q ≥ p + r.
Theorem 2.10. Let T be a p−hyponormal operator on H and let T = U |T | be the polar
decomposition of T , if Tes,t = |T |s U |T |t (s + t = 1) is normal, then T is normal.
Proof. First, consider the case max{s, t} ≥ p. Let A = |T |2p and B = |T ∗ |2p , p-hyponormality
of T ensures A ≥ B ≥ 0. Applying Theorem 2.9 to A ≥ B ≥ 0, since
t
s+t
s t
s+t
1+
≥ +
and
≥ 1,
p p + min(s, t)
p p
p + min(s, t)
we have
∗ e
(Tes,t
Ts,t )
p+min(s,t)
s+t
= (|T |t U ∗ |T |2s U |T |t )
p+min(s,t)
s+t
= (U ∗ U |T |t U ∗ |T |2s U |T |t U ∗ U )
= (U ∗ |T ∗ |t |T |2s |T ∗ |t U )
= U ∗ (|T ∗ |t |T |2s |T ∗ |t )
t
s
t
= U ∗ (B 2p A p B 2p )
≥ U ∗B
p+min(s,t)
p
p+min(s,t)
s+t
p+min(s,t)
s+t
p+min(s,t)
s+t
p+min(s,t)
s+t
U
U
U
= U ∗ |T ∗ |2(p+min(s,t)) U
= |T |2(p+min(s,t)) .
Similarly, we also have
∗
(Tes,t Tes,t
)
p+min(s,t)
s+t
J. Inequal. Pure and Appl. Math., 8(3) (2007), Art. 90, 7 pp.
≤ |T |2(p+min(s,t)) .
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4
C HANGSEN YANG AND Y ULIANG Z HAO
Therefore, we have
∗ e
(Tes,t
Ts,t )
p+min(s,t)
s+t
∗
≥ |T |2(p+min(s,t)) ≥ (Tes,t Tes,t
)
p+min(s,t)
s+t
.
If
Tes,t = |T |s U |T |t
(s + t = 1)
is normal, then
∗ e
(Tes,t
Ts,t )
p+min(s,t)
s+t
∗
= |T |2(p+min(s,t)) = (Tes,t Tes,t
)
p+min(s,t)
s+t
,
which implies
|T ∗ |t |T |2s |T ∗ |t = |T ∗ |2(s+t)
and |T |s |T ∗ |2t |T |s = |T |2(s+t) ,
then |T ∗ | = |T | by Lemma 2.7. Next, consider the case max{s, t} ≤ p. Firstly, p−hyponormality
of T ensures |T |2s ≥ |T ∗ |2s and |T |2t ≥ |T ∗ |2t for max{s, t} ≤ p by the Löwner-Heinz theorem. Then we have
Te∗ Tes,t = |T |t U ∗ |T |2s U |T |t ≥ |T |t U ∗ |T ∗ |2s U |T |t
s,t
= |T |2(s+t)
∗
Tes,t Tes,t
= |T |s U |T |2t U ∗ |T |s
≤ |T |2(s+t) .
If Tes,t = |T |s U |T |t (s + t = 1) is normal, then
∗ e
∗
,
Tes,t
Ts,t = |T |2((s+t) = Tes,t Tes,t
which implies
|T ∗ |t |T |2s |T ∗ |t = |T ∗ |2(s+t)
and |T |s |T ∗ |2t |T |s = |T |2(s+t) ,
then |T ∗ | = |T | by Lemma 2.7.
3. M AIN T HEOREM
Theorem 3.1. Assume that T is a class wF (p, r, q) operator with Ker(T ) ⊂ Ker(T ∗ ), and N
is a normal operator on H and K respectively. If X ∈ B(K, H) is injective with dense range
which satisfies T X = XN , then T ∗ X = XN ∗ .
Proof. Ker(T ) ⊂ Ker(T ∗ ) implies Ker(T ) reduces T . Also Ker(N ) reduces N since N is norL
L
mal. Using the orthogonal decompositions H = Ran(|T |) Ker(T ) and H = Ran(N ) Ker(N ),
we can represent T and N as follows.
T1 0
T =
,
0 0
N1 0
N=
,
0 0
where T1 is an injective class wF (p, r, q) operator on Ran(|T |) and N1 is injective normal on
Ran(N ). The assumption T X = XN asserts that X maps Ran(N ) to Ran(T ) ⊂ Ran(|T |)
and Ker(N ) to Ker(T ), hence X is of the form:
X1 0
X=
,
0 X2
where X1 ∈ B(Ran(N ), Ran(|T |)), X2 ∈ B(Ker(N ), Ker(T )). Since T X = XN , we have
that T1 X1 = X1 N1 . Since X is injective with dense range, X1 is also injective with dense
J. Inequal. Pure and Appl. Math., 8(3) (2007), Art. 90, 7 pp.
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O N C LASS wF (p, r, q) O PERATORS A ND Q UASISIMILARITY
5
g
range. Put W1 = |T1 |p X1 , then W1 is also injective with dense range and satisfies (T
1 )p,r W1 =
p
e (n) W1 N . Put Wn = (T1 )p,r W(n−1) , then Wn is also injective with dense range and satisfies
(n)
(Te1 )p,r Wn = Wn N . From Lemma 2.2 and Lemma 2.4, if there is an integer α0 such that
(α )
(n)
(Te1 )p,r0 is a hyponormal operator, then (Te1 )p,r is a hyponormal operator for n ≥ α0 . It follows
from Lemma 2.3 that there exists a bounded function f : C → H such that
∗
(n)
Te1
− λ f (λ) ≡ x, for every
p,r
∗ (n) (n) ∗ 12
(n)
(n)
x∈
Te1
Te1
− Te1
Te1
H.
p,r
p,r
p,r
p,r
Hence
Wn∗ x
∗
(n)
Te1
− λ f (λ)
=
Wn∗
=
(N1∗
−
p,r
∗
λ)Wn f (λ)
∈ Ran(N1∗ − λ) for all λ ∈ C
By Lemma 2.1, we have Wn∗ x = 0, and hence x = 0 because Wn∗ is injective. This implies that
L
(n)
(Te1 )p,r is normal. By Lemma 2.8 and Theorem 2.10, T1 is nomal and therefore T = T1 0 is
also normal. The assertion is immediate from Fuglede-Putnam’s theorem.
Let X be a Banach space, U be an open subset of C. ε(U, X) denotes the Fréchet space of
all X−valued C ∞ −functions, i.e., infinitely differentiable functions on U ([3]). T ∈ B(X) is
said to satisfy property (β)ε if for each open subset U of C, the map
Tz : ε(U, X) → ε(U, X),
f 7→ (T − z)f
is a topological monomorphism, i.e., Tz fn → 0 (n → ∞) in ε(U, X) implies fn → 0 (n → ∞)
in ε(U, X) ([3]).
Lemma 3.2 ([1]). Let T ∈ B(X). T is subscalar if and only if T satisfies property (β)ε .
Lemma 3.3. Let T ∈ B(X). T satisfies property (β)ε if and only if Tep,r satisfies property (β)ε .
Proof. First, we suppose that T satisfies property (β)ε , U is an open subset of C, fn ∈ ε(U, X)
and
(Tep,r − z)fn → 0
(3.1)
(n → ∞),
in ε(U, X), then
(T − z)U |T |r fn = U |T |r (Tep,r − z)fn → 0
(n → ∞).
Since T satisfies property (β)ε , we have U |T |r fn → 0 (n → ∞). and therefore
Tep,r fn → 0
(3.2)
(n → ∞).
(3.1) and (3.2) imply that
(3.3)
zfn = Tep,r fn − (Tep,r − z)fn → 0
(n → ∞)
in ε(U, X). Notice that T = 0 is a subscalar operator and hence satisfies property (β)ε by
Lemma 3.2. Now we have
(3.4)
fn → 0
J. Inequal. Pure and Appl. Math., 8(3) (2007), Art. 90, 7 pp.
(n → ∞).
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6
C HANGSEN YANG AND Y ULIANG Z HAO
(3.1) and (3.4) imply that Tep,r satisfies property (β)ε . Next we suppose that Tep,r satisfies property (β)ε , U is an open subset of C, fn ∈ ε(U, X) and
(3.5)
(T − z)fn → 0
(n → ∞),
in ε(U, X). Then
(Tep,r − z)|T |p fn = |T |p (T − z)fn → 0
(n → ∞).
Since Tep,r satisfies property (β)ε , we have |T |p fn → 0 (n → ∞), and therefore
(3.6)
T fn → 0
(n → ∞).
(3.5) and (3.6) imply
zfn = T fn − (T − z)fn → 0
So fn → 0 (n → ∞). Hence T satisfies property (β)ε .
(n → ∞).
Lemma 3.4 ([1]). Suppose that T is a p−hyponormal operator, then T is subscalar.
Theorem 3.5. Let T ∈ wF (p, r, q) and p + r = 1, then T is subdecomposable.
Proof. If T ∈ wF (p, r, q), then Tep,r is a m-hyponormal operator by Lemma 2.4, and it follows
from Lemma 3.4 that Tep,r is subscalar. So we have T is subscalar by Lemma 3.2 and Lemma
3.3. It is well known that subscalar operators are subdecomposable operators ([3]). Hence T is
subdecomposable.
Recall that an operator X ∈ B(H) is called a quasiaffinity if X is injective and has dense
range. For T1 , T2 ∈ B(H), if there exist quasiaffinities X ∈ B(H2 , H1 ) and Y ∈ B(H1 , H2 )
such that T1 X = XT2 and Y T1 = T2 Y then we say that T1 and T2 are quasisimilar.
Lemma 3.6 ([2]). Let S ∈ B(H) be subdecomposable, T ∈ B(H). If X ∈ B(K, H) is
injective with dense range which satisfies XT = SX, then σ(S) ⊂ σ(T ); if T and S are
quasisimilar, then σe (S) ⊆ σe (T ).
Theorem 3.7. Let T1 , T2 ∈ wF (p, r, q). If T1 and T2 are quasisimilar then σ(T1 ) = σ(T2 ) and
σe (T1 ) = σe (T2 ).
Proof. Obvious from Theorem 3.5 and Lemma 3.6.
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