1. Introduction
Let
denote the algebra of bounded liner operators on a Hilbert space H. An operator
is positive if
for all
. An operator
is hyponormal if
and p-hyponormal if
for p > 0. By the well known Lowner-Heinz theorem “
ensures
for
”, every p-hyponormal operator is q-hyponormal for
. The Furuta’s inequalities [2] are as follows:
If
then for each
(1.1)
(1.2)
hold for p0 ≥ 0 and q0 ≥ 1 with
.
An operator
is 1) paranormal if
for all
;
2) *paranormal if
for all
.
2. Preliminaries and Background
M. Fujii, D. Jung, S. H. Lee, M. Y. Lee and R. Nakamoto [1] introduced the following classes of operators:
An operator
is p-paranormal for p > 0, if
(2.1)
holds for all
, where U is the partial isometry appearing in the polar decomposition
of T with
.
For p > 0, an operator
is of class
if it satisfies an operator inequality
(2.2)
For p, q > 0, an operator
is of class
if it satisfies an operator inequality
(2.3)
In this sequel we introduce *p-paranormal operator, classes of operators
for p > 0 and
for p, q > 0 as follows:
A p-hyponormal operator is *p-paranormal if
(2.4)
For p > 0 a p-hyponormal operator
if it satisfies an operator inequality
(2.5)
More generally, we define the class
for p, q > 0 by an operator inequality
(2.6)
Remark (2.1). If T is p-hyponormal then using Furuta inequality (1.1) (§1) it can be proved easily that
.
Remark (2.2). By inequality (2.6) we have
![](https://www.scirp.org/html/11-5300265\024c6d46-ab50-4418-b01b-353db1286446.jpg)
The well known theorem of T. Ando [3] for paranormal operator is required in the proof of our main result.
Theorem (2.3). (Ando’s Theorem): An operator T is paranormal if and only if
(2.7)
for all real k.
3. Main Results
M. Fujii, et al. [1] proved the following theorem [1; Theorem 3.4].
Theorem (3.1). If
for p > 0 then T is p-paranormal.
In the following first we present an alternative way in which Theo (3.1) is proved in [1]. For this we have considered a quadratic form analogous to inequation (2.7) (§2). We also present a necessary and sufficient condition for a p-hyponormal operator T to be a *p-paranormal operator and the monotonicity of class
.
Theorem (3.2). A p-hyponormal operator
is p-paranormal if and only if
for all
and p > 0.
Proof. Let T =
be p-hyponormal where U is partial isometry, hence
.
We have
![](https://www.scirp.org/html/11-5300265\1a863d4c-3996-4b41-a799-e05665002d0c.jpg)
and
![](https://www.scirp.org/html/11-5300265\f721a1a0-2d7e-4510-821f-55afef6915fa.jpg)
Now,
![](https://www.scirp.org/html/11-5300265\529244a2-349d-4b9c-9756-c5f163ff2cbc.jpg)
for all ![](https://www.scirp.org/html/11-5300265\20ea32c0-83a8-4434-9664-0eea9732a41b.jpg)
![](https://www.scirp.org/html/11-5300265\8db22e65-b28a-420a-9a60-d517dfaf38dd.jpg)
for all ![](https://www.scirp.org/html/11-5300265\658de8f9-2ed5-4692-b5f1-08f4b673bec1.jpg)
![](https://www.scirp.org/html/11-5300265\47c49b1e-c89a-4385-9db1-7d18f49754dd.jpg)
for all ![](https://www.scirp.org/html/11-5300265\97bcd4f3-d671-45fa-8fd4-cfd3cc5053b2.jpg)
![](https://www.scirp.org/html/11-5300265\36114525-b8b8-49ae-b4b6-bcc5175b1baa.jpg)
for all ![](https://www.scirp.org/html/11-5300265\ebd3f207-8132-4cf0-b0cb-c2fa3dd4c694.jpg)
We know that if a > 0, b and c are real numbers then
for every real t if and only if
. Hence
![](https://www.scirp.org/html/11-5300265\029fae1d-f049-43a6-8be9-7ceaec164c95.jpg)
for all ![](https://www.scirp.org/html/11-5300265\33523a55-d7e3-418d-8db2-b62d2c9b1fc1.jpg)
![](https://www.scirp.org/html/11-5300265\cd54e2eb-8d24-44ce-89b1-c6746c032ea6.jpg)
![](https://www.scirp.org/html/11-5300265\c79ddbfd-a63a-42c2-9617-8b3ab8cde1e8.jpg)
Since T be p-hyponormal, by Remark (2.1) (§2)
i.e.
![](https://www.scirp.org/html/11-5300265\a6d6fd86-94f9-445d-aa84-b99ca2f40704.jpg)
Hence
![](https://www.scirp.org/html/11-5300265\9edaadac-3c4c-44ab-979e-ecc78829155b.jpg)
![](https://www.scirp.org/html/11-5300265\fc5522d5-672a-4594-8066-8e4193f39505.jpg)
i.e. if and only if T is p-paranormal.
Remark (3.3). Theorem (3.2) is independent of
being taken as unit vector where as M. Fujii, et al. [1] have considered
as unit vector in the result [1, Theo. 3.4].
The following result presents a necessary and sufficient condition for p-hyponormal operator T to be a *pparanormal operator.
Theorem (3.4). A p-hyponormal operator T is *pparanormal if and only if
for all
(3.1)
Proof. Let
be p-hyponormal operator where U is a partial isometry also let
so that
, ![](https://www.scirp.org/html/11-5300265\af73e18d-6b8b-43b3-90b5-f64cced7c6c3.jpg)
and
. Now
![](https://www.scirp.org/html/11-5300265\1219ed8b-3cd3-4b55-ad20-a26bc7b76e87.jpg)
for all ![](https://www.scirp.org/html/11-5300265\53dde316-3553-491e-b61f-aae4b4d30d1f.jpg)
![](https://www.scirp.org/html/11-5300265\64c6ee5d-6168-464f-90c8-49ebef1120f8.jpg)
for all ![](https://www.scirp.org/html/11-5300265\481848e3-9e08-4095-bb65-694c2c1dc8b3.jpg)
![](https://www.scirp.org/html/11-5300265\e8edcbbb-b979-4d2b-ae7b-5e8dc18160b8.jpg)
for all ![](https://www.scirp.org/html/11-5300265\9fc9f0f1-cf0c-4213-a774-328a922b1128.jpg)
![](https://www.scirp.org/html/11-5300265\3a46eb88-70a2-48b2-a6d5-29af627ce4b9.jpg)
i.e.,
(3.2)
Since T is p-hyponormal so
, i.e.
![](https://www.scirp.org/html/11-5300265\7980e33e-7493-4df9-b511-ff50c74ec7ec.jpg)
i.e.
(3.3)
From (3.2) and (3.3), we have
![](https://www.scirp.org/html/11-5300265\39c4b95f-474e-4234-9f89-e167dd351632.jpg)
for all ![](https://www.scirp.org/html/11-5300265\82d75557-3d48-407d-a155-d98d598042ab.jpg)
![](https://www.scirp.org/html/11-5300265\c299edb4-c2e7-4424-8751-516426dcd299.jpg)
i.e. if and only if T is *p-paranormal.
In the following we present monotonicity of
. We need Furuta inequality [2,4] to prove the following theorem, see also [5,6].
Theorem (3.5). If
and 0 < q then
.
Proof. Let
where
and 0 < t then by the definition of class
for p, q > 0.
![](https://www.scirp.org/html/11-5300265\1b6e55e5-caaa-4788-93d8-0bbed99a71f4.jpg)
We apply it to (1.2) (§1), in the case when
,
, ![](https://www.scirp.org/html/11-5300265\60861660-88c9-4157-89e4-320a13cc65f0.jpg)
We have
![](https://www.scirp.org/html/11-5300265\0cd47f71-ed42-4243-b589-2bbd04042af8.jpg)
and
![](https://www.scirp.org/html/11-5300265\c8f7ac8d-5e85-4db8-930b-9c902cf0a9db.jpg)
Hence
, so that
![](https://www.scirp.org/html/11-5300265\07728e8b-c396-44c0-8d0c-0c50825c045a.jpg)
i.e. ![](https://www.scirp.org/html/11-5300265\fd153e50-e8cd-4463-9688-9579a04fe6b8.jpg)
i.e. ![](https://www.scirp.org/html/11-5300265\6e71100b-1b06-4625-8178-6de6cc69b851.jpg)
i.e.
.
Hence
.