We shall show relation between two operator inequalities and for positive, invertible operators A and B, where f and g are non-negative continuous invertible functions on satisfying f(t)g(t)=t-1 .
We denote by capital letter A, B et al. the bounded linear operators on a complex Hilbert space H. An operator T on H is said to be positive, denoted by
M. Ito and T. Yamazaki [
and Yamazaki and Yanagida [
for (not necessarily invertible) positive operators A and B and for fixed
for (not necessarily invertible) positive operators A and B, where f and g are non-negative continuous functions on
Remarks (1.1): The two inequalities in (1.1) are closely related to Furuta inequalities [
The inequalities in (1.1) and (1.2) are equivalent, respectively, if A and B are invertibles; but they are not always equivalent. Their equivalence for invertible case was shown in [
Motivated by the result (1.3) of M. Ito [
We denote by
Theorem 1: Let A and B be positive invertible operators, and let f and g be non-negative invertible continuous functions on
1)
2)
Here
The following Lemma is helpful in proving our results:
Lemma 2: If
Proof of Lemma: Since
sequence of polynomials on
Hence the result.
Proof of Theorem 1: For
1) We suppose that
Let
We have
Further since
we have
Then
i.e.
2) We suppose that
With
Now as
we have
Then
thus completing the proof of 2.
Corollary 3. Let A and B be positive invertible operators, and let f and g be non-negative continuous invertible functions on
1) If
2) If
Proof 1) This result follows from 1) of Theorem 1 because each of the conditions
2) This result follows from 2) of Theorem (1) because
Hence the proof is complete.
Remark (3.1) 1) If
2) The invertibility of positive operators A and B is necessary condition.
3) We have considered
We have the following results as a consequence of corollary 3.
Theorem 4: Let A and B be positive invertible operators. Then for each
1) If
2) If
In Theorem 4 we consider that
Theorem 5: Let A and B be positive invertible operators. Then for each
1) If
2) If
Proof of Theorem 4: 1) First we consider the case when
if
If
if
i.e., if
i.e., if
i.e., if
or in other words,
But, since
2) Again first we consider the case
Since
If p = 0 and r > 0, (5.2) means that
ensures
which implies that
Hence (5.3) means that
Hence the result.
Proof of Theorem 5: We can prove by the similar way to Theorem 4 for
Corollary 4: Let A and B be positive invertible operators, and let f and g be non-negative continuous invertible functions on
Proof: The proof
MohammadIlyas,ReyazAhmad,ShadabIlyas, (2015) Relation between Two Operator Inequalities . Advances in Pure Mathematics,05,93-99. doi: 10.4236/apm.2015.52012