Relation between Two Operator Inequalities and

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Received 8 February 2015; accepted 22 February 2015; published 26 February 2015

1. Introduction

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 if for all.

M. Ito and T. Yamazaki [1] obtained relations between two inequalities

and, (1.1)

and Yamazaki and Yanagida [2] obtained relation between two inequalities

and, (1.2)

for (not necessarily invertible) positive operators A and B and for fixed and. These results led M. Ito [3] to obtain relation between two operator inequalities

and, (1.3)

for (not necessarily invertible) positive operators A and B, where f and g are non-negative continuous functions on satisfying.

Remarks (1.1): The two inequalities in (1.1) are closely related to Furuta inequalities [4] .

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 [5] .

Motivated by the result (1.3) of M. Ito [3] , we obtain the results taking representing functions f and g as non-negative continuous invertible functions on satisfying.

2. Main Results

We denote by the kernel of an operator T.

Theorem 1: Let A and B be positive invertible operators, and let f and g be non-negative invertible continuous functions on satisfying. Then the following hold:

1) ensures

2) ensures.

Here and denote orthoprojections to and respectively.

The following Lemma is helpful in proving our results:

Lemma 2: If is a continuous function on and T is an invertible operator with, then

.

Proof of Lemma: Since is a continuous function on, it can be uniformly approximated by a

sequence of polynomials on. We may assume that itself is a polynomial. Then

Hence the result.

Proof of Theorem 1: For, let and;

1) We suppose that. Then

Let and then

We have.

Further since increases as decreases and

we have

.

Then

i.e.

2) We suppose that; i.e., then

.

With and, we have by Lemma 2

Now as and since

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 satisfying.

1) If or, then ensures.

2) If, then ensures.

Proof 1) This result follows from 1) of Theorem 1 because each of the conditions and

implies, so that

2) This result follows from 2) of Theorem (1) because, so that

Hence the proof is complete.

Remark (3.1) 1) If, then automatically since, so 1) of corollary 3 holds without any conditions.

2) The invertibility of positive operators A and B is necessary condition.

3) We have considered instead of because the requirement of the limit.

when is not fulfilled, rather it is fulfilled when because.

We have the following results as a consequence of corollary 3.

Theorem 4: Let A and B be positive invertible operators. Then for each and, the following hold

1) If then.

2) If and then.

In Theorem 4 we consider that for or when and we define for a positive invertible operator T.

Theorem 5: Let A and B be positive invertible operators. Then for each and, the following hold:

1) If, then.

2) If and, then .

Proof of Theorem 4: 1) First we consider the case when and.Replacing A with A^{p} and B with

and putting and in 1) of Corollary 3 so that, we have

if then. (5.1)

If and (5.1) means that

if then

i.e., if then

i.e., if then

i.e., if then

or in other words, ensures.

But, since implies, it follows an equivalent assertion ensures, i.e., which is further equivalent to the trivial assertion ensures.

2) Again first we consider the case and. Replacing A with and B with A^{p} and putting

and in 2) of Corollary 3.

Since, we have

ensures. (5.2)

If p = 0 and r > 0, (5.2) means that ensures i.e.,

ensures, (5.3)

which implies that.

Hence (5.3) means that ensures, i.e. ensures.

Hence the result.

Proof of Theorem 5: We can prove by the similar way to Theorem 4 for and, replacing A with A^{p} and B with and putting and for 1) in 1) of Corollary 3 and replacing A with and B with A^{p} and putting and for 2) in 2) of Corollary 3.

Corollary 4: Let A and B be positive invertible operators, and let f and g be non-negative continuous invertible functions on satisfying. If, then

.

Proof: The proof follows directly by applying the condition, in 1) of Corollary 3 and for the proof we have only to interchange the roles of A and B and those of f and g in 2) of Corollary 3, Since if.

References

[1] Ito, M. and Yamazaki, T. (2002) Relations between Two Inequalities and and Their Applications. Integral Equations and Operator Theory, 44, 442-450.

http://dx.doi.org/10.1007/BF01193670

[2] Yamazaki, T. and Yanagida, M. (to appear) Relations between Two Operator Inequalities and Their Application to Paranormal Operators. Acta Scientiarum Mathematicarum (Szeged).

[3] Ito, M. (2005) Relations between Two Operator Inequalities Motivated by the Theory of Operator Means. Integral Equations and Operator Theory, 53, 527-534.

http://dx.doi.org/10.1007/s00020-004-1321-9

[4] Furuta, T. (1987) Assures for with. Proceedings of the American Mathematical Society, 101, 85-88.

[5] Furuta, T. (1992) Applications of Order Preserving Operator Inequalities. Operator Theory: Advances and Applications, 59, 180-190.