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On
-Quasiclass A Operators
Journal of Inequalities and Applications volume 2009, Article number: 921634 (2009)
Abstract
An operator is called
-quasiclass
if
for a positive integer
, which is a common generalization of quasiclass
. In this paper, firstly we prove some inequalities of this class of operators; secondly we prove that if
is a
-quasiclass
operator, then
is isoloid and
has finite ascent for all complex number
at last we consider the tensor product for
-quasiclass
operators.
1. Introduction
Throughout this paper let be a separable complex Hilbert space with inner product
. Let
denote the
-algebra of all bounded linear operators on
.
Let and let
be an isolated point of
. Here
denotes the spectrum of
. Then there exists a small enough positive number
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F921634/MediaObjects/13660_2009_Article_2031_Equ1_HTML.gif)
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F921634/MediaObjects/13660_2009_Article_2031_Equ2_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F921634/MediaObjects/13660_2009_Article_2031_IEq27_HTML.gif)
is called the Riesz idempotent with respect to , and it is well known that
satisfies
,
,
, and
for all positive integers
. Stampfli [1] proved that if
is hyponormal (i.e., operators such that
), then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F921634/MediaObjects/13660_2009_Article_2031_Equ3_HTML.gif)
After that many authors extended this result to many other classes of operators. Chō and Tanahashi [2] proved that (1.3) holds if is either
-hyponormal or log-hyponormal. In the case
, the result was further shown by Tanahashi and Uchiyama [3] to hold for
-quasihyponormal operators, by Tanahashi et al. [4] to hold for
-quasihyponormal operators and by Uchiyama and Tanahashi [5] and Uchiyama [6] for class A and paranormal operators. Here an operator
is called
-hyponormal for
if
, and log-hyponormal if
is invertible and
. An operator
is called
-quasihyponormal if
, where
and
is a positive integer; especially, when
,
, and
,
is called
-quasihyponormal,
-quasihyponormal, and quasihyponormal, respectively. And an operator
is called paranormal if
for all
; normaloid if
for all positive integers
.
-hyponormal, log-hyponormal,
-quasihyponormal,
-quasihyponormal, and paranormal operators were introduced by Aluthge [7], Tanahashi [8], S. C. Arora and P. Arora [9], Kim [10], and Furuta [11, 12], respectively.
In order to discuss the relations between paranormal and -hyponormal and log-hyponormal operators, Furuta et al. [13] introduced a very interesting class of bounded linear Hilbert space operators: class A defined by
, where
which is called the absolute value of
and they showed that class A is a subclass of paranormal and contains
-hyponormal and log-hyponormal operators. Class A operators have been studied by many researchers, for example, [5, 14–19].
Recently Jeon and Kim [20] introduced quasiclass A (i.e., ) operators as an extension of the notion of class A operators, and they also proved that (1.3) holds for this class of operators when
. It is interesting to study whether Stampli's result holds for other larger classes of operators.
In [21], Tanahashi et al. considered an extension of quasi-class A operators, similar in spirit to the extension of the notion of -quasihyponormality to
-quasihyponormality, and prove that (1.3) holds for this class of operators in the case
.
Definition 1.1.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F921634/MediaObjects/13660_2009_Article_2031_IEq77_HTML.gif)
is called a -quasiclass A operator for a positive integer
if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F921634/MediaObjects/13660_2009_Article_2031_Equ4_HTML.gif)
Remark 1.2.
In [21], this class of operators is called quasi-class (A, ).
It is clear that the class of quasi-classA operators
the class of k-quasiclass A operators and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F921634/MediaObjects/13660_2009_Article_2031_Equ5_HTML.gif)
We show that the inclusion relation (1.5) is strict, by an example which appeared in [20].
Example 1.3.
Given a bounded sequence of positive numbers , let
be the unilateral weighted shift operator on
with the canonical orthonormal basis
by
for all
, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F921634/MediaObjects/13660_2009_Article_2031_Equ6_HTML.gif)
Straightforward calculations show that is a
-quasiclass A operator if and only if
. So if
and
, then
is a
-quasiclass A operator, but not a
-quasiclass A operator.
In this paper, firstly we consider some inequalities of -quasiclass A operators; secondly we prove that if
is a
-quasiclass A operator, then
is isoloid and
has finite ascent for all complex number
; at last we give a necessary and sufficient condition for
to be a
-quasiclass A operator when
and
are both non-zero operators.
2. Results
In the following lemma, Tanahashi, Jeon, Kim, and Uchiyama studied the matrix representation of a -quasiclass A operator with respect to the direct sum of
and its orthogonal complement.
Lemma 2.1 (see [21]).
Let be a
-quasiclass A operator for a positive integer
and let
on
be
matrix expression. Assume that ran
is not dense, then
is a class A operator on
and
. Furthermore,
.
Proof.
Consider the matrix representation of with respect to the decomposition
:
Let
be the orthogonal projection of
onto
. Then
. Since
is a
-quasiclass A operator, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F921634/MediaObjects/13660_2009_Article_2031_Equ7_HTML.gif)
Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F921634/MediaObjects/13660_2009_Article_2031_Equ8_HTML.gif)
by Hansen's inequality [22]. On the other hand
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F921634/MediaObjects/13660_2009_Article_2031_Equ9_HTML.gif)
Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F921634/MediaObjects/13660_2009_Article_2031_Equ10_HTML.gif)
That is, is a class A operator on
.
For any ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F921634/MediaObjects/13660_2009_Article_2031_Equ11_HTML.gif)
which implies .
Since , where
is the union of the holes in
which happen to be subset of
by [23, Corollary 7], and
and
has no interior points, we have
.
Theorem 2.2.
Let be a
-quasiclass A operator for a positive integer
. Then the following assertions hold.
for all
and all positive integers
.
If for some positive integer
, then
.
for all positive integers
, where
denotes the spectral radius of
.
To give a proof of Theorem 2.2, the following famous inequality is needful.
Lemma 2.3 (Hölder-McCarthy's inequality [24]).
Let . Then the following assertions hold.
for
and all
.
for
and all
.
Proof of Theorem 2.2.
-
(1)
Since it is clear that
-quasiclass A operators are
-quasiclass A operators, we only need to prove the case
. Since
(2.6)
by Hölder-McCarthy's inequality, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F921634/MediaObjects/13660_2009_Article_2031_Equ13_HTML.gif)
for is a
-quasiclass A operator.
-
(2)
If
, it is obvious that
. If
, then
by (
). The rest of the proof is similar.
-
(3)
We only need to prove the case
, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F921634/MediaObjects/13660_2009_Article_2031_Equ14_HTML.gif)
If for some
, then
by (2) and in this case
. Hence (3) is clear. Therefore we may assume
for all
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F921634/MediaObjects/13660_2009_Article_2031_Equ15_HTML.gif)
by (), and we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F921634/MediaObjects/13660_2009_Article_2031_Equ16_HTML.gif)
Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F921634/MediaObjects/13660_2009_Article_2031_Equ17_HTML.gif)
By letting , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F921634/MediaObjects/13660_2009_Article_2031_Equ18_HTML.gif)
that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F921634/MediaObjects/13660_2009_Article_2031_Equ19_HTML.gif)
Lemma 2.4 (see [21]).
Let be a
-quasiclass A operator for a positive integer
. If
and
for some
, then
.
Proof.
We may assume that . Let
be a span of
. Then
is an invariant subspace of
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F921634/MediaObjects/13660_2009_Article_2031_Equ20_HTML.gif)
Let be the orthogonal projection of
onto
. It suffices to show that
in (2.14). Since
is a
-quasiclass A operator, and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F921634/MediaObjects/13660_2009_Article_2031_Equ21_HTML.gif)
We remark
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F921634/MediaObjects/13660_2009_Article_2031_Equ22_HTML.gif)
Then by Hansen's inequality and (2.15), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F921634/MediaObjects/13660_2009_Article_2031_Equ23_HTML.gif)
Hence we may write
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F921634/MediaObjects/13660_2009_Article_2031_Equ24_HTML.gif)
We have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F921634/MediaObjects/13660_2009_Article_2031_Equ25_HTML.gif)
This implies and
. On the other hand,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F921634/MediaObjects/13660_2009_Article_2031_Equ26_HTML.gif)
Hence and
. Since
is a
-quasiclass A operator, by a simple calculation we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F921634/MediaObjects/13660_2009_Article_2031_Equ27_HTML.gif)
Recall that if and only if
and
for some contraction
. Thus we have
. This completes the proof.
Lemma 2.5 (see [25]).
If satisfies
for some complex number
, then
for any positive integer
.
Proof.
It suffices to show by induction. We only need to show
since
is clear. In fact, if
, then we have
by hypothesis. So we have
, that is,
. Hence
.
An operator is said to have finite ascent if for some positive integer
.
Theorem 2.6.
Let be a
-quasiclass A operator for a positive integer
. Then
has finite ascent for all complex number
.
Proof.
We only need to show the case because the case
holds by Lemmas 2.4 and 2.5.
In the case , we shall show that
. It suffices to show that
since
is clear. Now assume that
. We may assume
since if
, it is obvious that
. By Hölder-McCarthy's inequality, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F921634/MediaObjects/13660_2009_Article_2031_Equ28_HTML.gif)
So we have , which implies
. Therefore
.
In the following lemma, Tanahashi, Jeon, Kim, and Uchiyama extended the result (1.3) to -quasiclass A operators in the case
.
Lemma 2.7 (see [21]).
Let be a
-quasiclass A operator for a positive integer
. Let
be an isolated point of
and
the Riesz idempotent for
. Then the following assertions hold.
If , then
is self-adjoint and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F921634/MediaObjects/13660_2009_Article_2031_Equ29_HTML.gif)
If , then
.
An operator is said to be isoloid if every isolated point of
is an eigenvalue of
.
Theorem 2.8.
Let be a
-quasiclass A operator for a positive integer
. Then
is isoloid.
Proof.
Let be an isolated point. If
, by (
) of Lemma 2.7,
for
. Therefore
is an eigenvalue of
. If
, by (
) of Lemma 2.7,
for
. So we have
. Therefore
is an eigenvalue of
. This completes the proof.
Let denote the tensor product on the product space
for nonzero
,
. The following theorem gives a necessary and sufficient condition for
to be a
-quasiclass A operator, which is an extension of [20, Theorem 4.2].
Theorem 2.9.
Let ,
be nonzero operators. Then
is a
-quasiclass A operator if and only if one of the following assertions holds
(1) or
.
(2) and
are
-quasiclass A operators.
Proof.
It is clear that is a
-quasiclass A operator if and only if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F921634/MediaObjects/13660_2009_Article_2031_Equ30_HTML.gif)
Therefore the sufficiency is clear.
To prove the necessary, suppose that is a
-quasiclass A operator. Let
,
be arbitrary. Then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F921634/MediaObjects/13660_2009_Article_2031_Equ31_HTML.gif)
It suffices to prove that if () does not hold, then (
) holds. Suppose that
and
. To the contrary, assume that
is not a
-quasiclass A operator, then there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F921634/MediaObjects/13660_2009_Article_2031_Equ32_HTML.gif)
From (2.25) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F921634/MediaObjects/13660_2009_Article_2031_Equ33_HTML.gif)
that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F921634/MediaObjects/13660_2009_Article_2031_Equ34_HTML.gif)
for all . Therefore
is a
-quasiclass A operator. As the proof in Theorem 2.2 (
), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F921634/MediaObjects/13660_2009_Article_2031_Equ35_HTML.gif)
So we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F921634/MediaObjects/13660_2009_Article_2031_Equ36_HTML.gif)
for all by (2.28). Because
is a
-quasiclass A operator, from Lemma 2.1 we can write
on
, where
is a class A operator (hence it is normaloid). By (2.30) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F921634/MediaObjects/13660_2009_Article_2031_Equ37_HTML.gif)
So we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F921634/MediaObjects/13660_2009_Article_2031_Equ38_HTML.gif)
where equality holds since is normaloid.
This implies that . Since
for all
, we have
. This contradicts the assumption
. Hence
must be a
-quasiclass A operator. A similar argument shows that
is also a
-quasiclass A operator. The proof is complete.
References
Stampfli JG: Hyponormal operators and spectral density. Transactions of the American Mathematical Society 1965, 117: 469–476.
Chō M, Tanahashi K: Isolated point of spectrum of -hyponormal, log-hyponormal operators. Integral Equations and Operator Theory 2002,43(4):379–384. 10.1007/BF01212700
Tanahashi K, Uchiyama A: Isolated point of spectrum of -quasihyponormal operators. Linear Algebra and Its Applications 2002, 341: 345–350. 10.1016/S0024-3795(01)00476-1
Tanahashi K, Uchiyama A, Chō M: Isolated points of spectrum of -quasihyponormal operators. Linear Algebra and Its Applications 2004, 382: 221–229.
Uchiyama A, Tanahashi K: On the Riesz idempotent of class operators. Mathematical Inequalities & Applications 2002,5(2):291–298.
Uchiyama A: On the isolated points of the spectrum of paranormal operators. Integral Equations and Operator Theory 2006,55(1):145–151. 10.1007/s00020-005-1386-0
Aluthge A: On -hyponormal operators for . Integral Equations and Operator Theory 1990,13(3):307–315. 10.1007/BF01199886
Tanahashi K: On log-hyponormal operators. Integral Equations and Operator Theory 1999,34(3):364–372. 10.1007/BF01300584
Arora SC, Arora P: On -quasihyponormal operators for . Yokohama Mathematical Journal 1993,41(1):25–29.
Kim IH: On -quasihyponormal operators. Mathematical Inequalities & Applications 2004,7(4):629–638.
Furuta T: On the class of paranormal operators. Proceedings of the Japan Academy 1967, 43: 594–598. 10.3792/pja/1195521514
Furuta T: Invitation to Linear Operators: From Matrices to Bounded Linear Operators on a Hilbert Space. Taylor & Francis, London, UK; 2001:x+255.
Furuta T, Ito M, Yamazaki T: A subclass of paranormal operators including class of log-hyponormal and several related classes. Scientiae Mathematicae 1998,1(3):389–403.
Chō M, Giga M, Huruya T, Yamazaki T: A remark on support of the principal function for class operators. Integral Equations and Operator Theory 2007,57(3):303–308. 10.1007/s00020-006-1463-z
Chō M, Yamazaki T: An operator transform from class to the class of hyponormal operators and its application. Integral Equations and Operator Theory 2005,53(4):497–508. 10.1007/s00020-004-1332-6
Ito M: Several properties on class including -hyponormal and log-hyponormal operators. Mathematical Inequalities & Applications 1999,2(4):569–578.
Ito M, Yamazaki T: Relations between two inequalities and and their applications. Integral Equations and Operator Theory 2002,44(4):442–450. 10.1007/BF01193670
Uchiyama A: Weyl's theorem for class operators. Mathematical Inequalities & Applications 2001,4(1):143–150.
Wang D, Lee JI: Spectral properties of class operators. Trends in Mathematics Information Center for Mathematical Sciences 2003,6(2):93–98.
Jeon IH, Kim IH: On operators satisfying . Linear Algebra and Its Applications 2006,418(2–3):854–862. 10.1016/j.laa.2006.02.040
Tanahashi K, Jeon IH, Kim IH, Uchiyama A: Quasinilpotent part of class or -quasihyponormal operators. Operator Theory: Advances and Applications 2008, 187: 199–210.
Hansen F: An operator inequality. Mathematische Annalen 1980,246(3):249–250. 10.1007/BF01371046
Han JK, Lee HY, Lee WY: Invertible completions of upper triangular operator matrices. Proceedings of the American Mathematical Society 2000,128(1):119–123. 10.1090/S0002-9939-99-04965-5
McCarthy CA: “
” . Israel Journal of Mathematics 1967, 5: 249–271. 10.1007/BF02771613
Han YM, Lee JI, Wang D: Riesz idempotent and Weyl's theorem for -hyponormal operators. Integral Equations and Operator Theory 2005,53(1):51–60. 10.1007/s00020-003-1313-1
Acknowledgments
The authors would like to express their cordial gratitude to the referee for his useful comments and Professor K. Tanahashi and Professor I. H. Jeon for sending them [21]. This research is supported by the National Natural Science Foundation of China (no. 10771161).
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Gao, F., Fang, X. On -Quasiclass A Operators.
J Inequal Appl 2009, 921634 (2009). https://doi.org/10.1155/2009/921634
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DOI: https://doi.org/10.1155/2009/921634