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
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
and
Now,
for all 
for all 
for all 
for all 
We know that if a > 0, b and c are real numbers then 
for every real t if and only if 
. Hence
for all 
Since T be p-hyponormal, by Remark (2.1) (§2) 
i.e.
Hence
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
, 
and
. Now
for all 
for all 
for all 
i.e., 
(3.2)
Since T is p-hyponormal so
, i.e.
i.e. 
(3.3)
From (3.2) and (3.3), we have
for all 
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.
We apply it to (1.2) (§1), in the case when
, 
, 
We have
and
Hence
, so that
i.e. 
i.e. 
i.e.
.
Hence
.