ON CLASS wF (p, r, q) OPERATORS AND
QUASISIMILARITY
CHANGSEN YANG
YULIANG ZHAO
College of Mathematics and Information Science
Henan Normal University,
Xinxiang 453007,
People’s Republic of China
EMail: yangchangsen117@yahoo.com.cn
Department of Mathematics
Anyang Institute of Technology
Anyang City, Henan Province 455000
People’s Republic of China
EMail: zhaoyuliang512@163.com
Class wF (p, r, q) Operators
and Quasisimilarity
Changsen Yang and Yuliang Zhao
vol. 8, iss. 3, art. 90, 2007
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Contents
Received:
17 October, 2006
Accepted:
15 June, 2007
Communicated by:
JJ
II
C.K. Li
J
I
2000 AMS Sub. Class.:
47B20, 47A30.
Page 1 of 15
Key words:
Class wF (p, r, q) operators, Fuglede-Putnam’s theorem, Property (β)ε , Subscalar, Subdecomposable.
Abstract:
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.
Acknowledgements:
The authors are grateful to the referee for comments which improved the paper.
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Contents
1
Introduction
3
2
Preliminaries
5
3
Main Theorem
10
Class wF (p, r, q) Operators
and Quasisimilarity
Changsen Yang and Yuliang Zhao
vol. 8, iss. 3, art. 90, 2007
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1.
Introduction
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.
Definition 1.1 ([10]). For p > 0, r ≥ 0, and q ≥ 1, an operator T belongs to class
wF (p, r, q) if
1
(|T ∗ |r |T |2p |T ∗ |r ) q ≥ |T ∗ |
and
2(p+r)
q
Class wF (p, r, q) Operators
and Quasisimilarity
Changsen Yang and Yuliang Zhao
vol. 8, iss. 3, art. 90, 2007
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2(p+r)(1− 1q )
|T |
p
∗ 2r
p 1− 1q
≥ (|T | |T | |T | )
.
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.
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The following Fuglede-Putnam’s theorem is famous. We extend this theorem for
class wF (p, r, q) operators.
Theorem 1.2 (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 ∗ .
Class wF (p, r, q) Operators
and Quasisimilarity
Changsen Yang and Yuliang Zhao
vol. 8, iss. 3, art. 90, 2007
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2.
Preliminaries
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
max{s, t} ≥ p.
p+min(s,t)
-hyponormal
s+t
for any s > 0 and t > 0 such that
(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)∗
1
for all λ in C, 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 ,
n
n
oo
p
1
1
where m = min q , max p+r , 1 − q , 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
+δ
β
αβ0+β
β0
0
0 ≤ δ ≤ α0 and B 2 Aα0 B 2 0 0 ≥ B β0 +δ , then
β
2
α
B A B
β
2
β+δ
α+β
≥ B β+δ ,
Class wF (p, r, q) Operators
and Quasisimilarity
Changsen Yang and Yuliang Zhao
vol. 8, iss. 3, art. 90, 2007
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and
α
α
Aα−δ̄ ≥ A 2 B β A 2
α−δ̄
α+β
hold for each α ≥ α0 , β ≥ β0 and 0 ≤ δ̄ ≤ α.
1
1
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.
s
2t
s
Lemma 2.7. Let A, B ≥ 0, s, t ≥ 0, if B A B = B
A = B.
2s+2t
t
2s
t
,AB A =A
2s+2t
then
Class wF (p, r, q) Operators
and Quasisimilarity
Changsen Yang and Yuliang Zhao
vol. 8, iss. 3, art. 90, 2007
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:
(B k A2k B k )
2k+2t
4k
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≥ B 2k+2t ,
A2k−2t ≥ (Ak B 2k Ak )
and
(Ak B 2k Ak )
2k+2s
4k
2k−2t
4k
,
≥ A2k+2s ,
B 2k−2s ≥ (B k A2k B k )
2k−2s
4k
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So
Ak B 2k Ak = A4k ,
B k A2k B k = B 4k ,
by Lemma 2.6
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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
Class wF (p, r, q) Operators
and Quasisimilarity
hold for p ≥ 0 and q ≥ 1 with (1 + r)q ≥ p + r.
Changsen Yang and Yuliang Zhao
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.
2p
= (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
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∗ 2p
Proof. First, consider the case max{s, t} ≥ p. Let A = |T | and B = |T | ,
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
p+min(s,t)
p+min(s,t)
∗ e
(Tes,t
Ts,t ) s+t = (|T |t U ∗ |T |2s U |T |t ) s+t
= (U ∗ U |T |t U ∗ |T |2s U |T |t U ∗ U )
vol. 8, iss. 3, art. 90, 2007
U
U
U = U ∗ |T ∗ |2(p+min(s,t)) U = |T |2(p+min(s,t)) .
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Similarly, we also have
∗
(Tes,t Tes,t
)
p+min(s,t)
s+t
≤ |T |2(p+min(s,t)) .
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
.
Class wF (p, r, q) Operators
and Quasisimilarity
If
Tes,t = |T |s U |T |t
(s + t = 1)
Changsen Yang and Yuliang Zhao
vol. 8, iss. 3, art. 90, 2007
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
,
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which implies
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|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
∗ e
Ts,t = |T |t U ∗ |T |2s U |T |t ≥ |T |t U ∗ |T ∗ |2s U |T |t
Tes,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
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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.
Class wF (p, r, q) Operators
and Quasisimilarity
Changsen Yang and Yuliang Zhao
vol. 8, iss. 3, art. 90, 2007
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3.
Main Theorem
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
L
since N is normal. Using the orthogonal decompositions H = Ran(|T |) Ker(T )
L
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
Class wF (p, r, q) Operators
and Quasisimilarity
Changsen Yang and Yuliang Zhao
vol. 8, iss. 3, art. 90, 2007
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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 range. Put W1 = |T1 |p X1 , then
W1 ispalso injective with dense
(n) g
range and satisfies (T1 ) W1 = W1 N . Put Wn = (Te1 )p,r W(n−1) , then Wn is also
p,r
(n)
injective with dense range and satisfies (Te1 )p,r Wn = Wn N . From Lemma 2.2 and
(α )
Lemma 2.4, if there is an integer α0 such that (Te1 )p,r0 is a hyponormal operator, then
Close
(n)
(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)
e
T1
− λ f (λ) ≡ x, for every
p,r
∗ (n) (n) ∗ 12
(n)
(n)
e
e
x∈
T1
T1
− Te1
Te1
H.
p,r
p,r
p,r
p,r
Class wF (p, r, q) Operators
and Quasisimilarity
Changsen Yang and Yuliang Zhao
Hence
∗
(n)
∗
∗
e
Wn x = Wn
T1
− λ f (λ)
=
(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
(n)
implies that (Te1 )p,rLis 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 FugledePutnam’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 .
vol. 8, iss. 3, art. 90, 2007
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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 → ∞),
Class wF (p, r, q) Operators
and Quasisimilarity
in ε(U, X), then
Changsen Yang and Yuliang Zhao
r
r
(T − z)U |T | fn = U |T | (Tep,r − z)fn → 0
(n → ∞).
vol. 8, iss. 3, art. 90, 2007
Since T satisfies property (β)ε , we have U |T |r fn → 0 (n → ∞). and therefore
Tep,r fn → 0
(3.2)
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(n → ∞).
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(3.1) and (3.2) imply that
zfn = Tep,r fn − (Tep,r − z)fn → 0
(3.3)
(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
fn → 0
(3.4)
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(n → ∞).
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(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
(T − z)fn → 0
(3.5)
(n → ∞),
in ε(U, X). Then
(Tep,r − z)|T |p fn = |T |p (T − z)fn → 0
(n → ∞).
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Since Tep,r satisfies property (β)ε , we have |T |p fn → 0 (n → ∞), and therefore
T fn → 0
(3.6)
(n → ∞).
(3.5) and (3.6) imply
zfn = T fn − (T − z)fn → 0
(n → ∞).
So fn → 0 (n → ∞). Hence T satisfies property (β)ε .
Lemma 3.4 ([1]). Suppose that T is a p−hyponormal operator, then T is subscalar.
Class wF (p, r, q) Operators
and Quasisimilarity
Changsen Yang and Yuliang Zhao
vol. 8, iss. 3, art. 90, 2007
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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References
[1] L. CHEN, R. YINGBIN AND Y. ZIKUN, w-Hyponormal operators are subscalar, Integr. Equat. Oper. Th., 50 (2004), 165–168.
[2] L. CHEN AND Y. ZIKUN, Bishop’s property (β) and essential spectra of quasisimilar operators, Proc. Amer. Math. Soc., 128 (2000), 485–493.
[3] I. COLOJOARĀ AND C. FOIAS, Theory of Generalized Spectral Operators,
New York, Gordon and Breach, 1968.
Class wF (p, r, q) Operators
and Quasisimilarity
Changsen Yang and Yuliang Zhao
vol. 8, iss. 3, art. 90, 2007
[4] T. FURUTA, Invitation to Linear Operators – From Matrices to Bounded Linear Operators on a Hilbert Space, London: Taylor & Francis, 2001.
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[5] T. HURUYA, A note on p−hyponormal operators, Proc. Amer. Math. Soc., 125
(1997), 3617–3624.
[6] M. ITO AND T. YAMAZAKI, Relations between two inequalities
p
p
p
r
r r
B 2 Ap B 2 p+r ≥ B r and A 2 B r A 2 p+r ≤ Ap and its applications, Integr.
Equat. Oper. Th., 44 (2002), 442–450.
[7] C.R. PUTNAM, On normal operators in Hilbert space, Amer. J. Math., 73
(1951), 357–362.
[8] C.R. PUTNAM, Hyponormal contractions and strong power convergence, Pacific J. Math., 57 (1975), 531–538.
[9] C.R. PUTNAM, Ranges of normal and subnormal operators, Michigan Math.
J., 18 (1971) 33–36.
[10] C. YANG AND J. YUAN, Spectrum of class wF (p, r, q) operators for p+r ≤ 1
and q ≥ 1, Acta. Sci. Math. (Szeged), 71 (2005), 767–779.
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[11] C. YANG AND J. YUAN, On class wF (p, r, q) operators, preprint.
Class wF (p, r, q) Operators
and Quasisimilarity
Changsen Yang and Yuliang Zhao
vol. 8, iss. 3, art. 90, 2007
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