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 

hold for p₀ ≥ 0 and q₀ ≥ 1 with.

An operator is 1) paranormal if for all;

2) ^*paranormal if for all.

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

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

For p, q > 0, an operator is of class if it satisfies an operator inequality

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

For p > 0 a p-hyponormal operator if it satisfies an operator inequality

More generally, we define the class for p, q > 0 by an operator inequality

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

for all real k.

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

.