## Journal of Inequalities and Applications

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# Applications of Kato’s inequality for n-tuples of operators in Hilbert spaces, (I)

Journal of Inequalities and Applications20132013:21

DOI: 10.1186/1029-242X-2013-21

Accepted: 28 December 2012

Published: 16 January 2013

## Abstract

In this paper, by the use of the famous Kato’s inequality for bounded linear operators, we establish some inequalities for n-tuples of operators and apply them for functions of normal operators defined by power series as well as for some norms and numerical radii that arise in multivariate operator theory.

MSC:47A63, 47A99.

### Keywords

bounded linear operators functions of normal operators inequalities for operators norm and numerical radius inequalities Kato’s inequality

## 1 Introduction

The ‘square root’ of a positive bounded self-adjoint operator on H can be defined as follows (see, for instance, [[1], p.240]).

If the operator is self-adjoint and positive, then there exists a unique positive self-adjoint operator such that . If A is invertible, then so is B.

If , then the operator is self-adjoint and positive. Define the ‘absolute value’ operator by .

In 1952, Kato [2] proved the following generalization of Schwarz inequality:
(1.1)

for any , and T is a bounded linear operator on H.

Utilizing the modulus notation introduced before, we can write (1.1) as follows:
(1.2)

For results related to the Kato’s inequality, see [218] and [19].

In the recent paper [20], by employing Kato’s inequality (1.2), Dragomir established the following results for sequences of bonded linear operators on complex Hilbert spaces.

Theorem 1.1 Let be an n-tuple of bounded linear operators on the Hilbert space and be an n-tuple of nonnegative weights not all of them equal to zero. Then we have
(1.3)

for any with and .

He also obtained the following result.

Theorem 1.2 With the assumptions in Theorem  1.1, we have
(1.4)

for any .

For various related results, see the papers [2131].

Motivated by the above results, we establish in this paper other similar inequalities for n-tuples of bounded linear operators that can be obtained from Kato’s result (1.2) and apply them to functions of normal operators defined by power series as well as to some norms and numerical radii that can be associated with these n-tuples of bonded linear operators on Hilbert spaces.

## 2 Some inequalities for an n-tuple of linear operators

Employing Kato’s inequality (1.2), we can state the following new result.

Theorem 2.1 Let be an n-tuple of bounded linear operators on the Hilbert space and be an n-tuple of nonnegative weights, not all of them equal to zero. Then we have
(2.1)
for any , and, in particular, for
(2.2)

for any .

Proof Utilizing Kato’s inequality, we have
and by replacing α with ,
which by summation gives
(2.3)
for any and . By the elementary inequality
(2.4)
we have
which by (2.3) produces
(2.5)
for any and . Multiplying the inequalities (2.5) with the positive weights , summing over j from 1 to n and utilizing the weighted Cauchy-Buniakowski-Schwarz inequality
where , we have
(2.6)

for any , and the inequality in (2.1) is proved. □

Remark 2.1 In order to provide some applications for functions of normal operators defined by power series, we need to state the inequality (2.1) for normal operators , , namely,
(2.7)

for any and for any .

From a different perspective that involves quadratics, we can state the following result as well.

Theorem 2.2 Let be an n-tuple of bounded linear operators on the Hilbert space and be an n-tuple of nonnegative weights, not all of them equal to zero. Then we have
(2.8)

for any with and .

Proof We must prove the inequalities only in the case , since the case or follows directly from the corresponding case of Kato’s inequality.

Utilizing Kato’s inequality for the operator , , we have
(2.9)
and, by replacing α with ,
(2.10)

for any .

By the Hölder-McCarthy inequality that holds for the positive operator P, for and with , we also have
(2.11)
and
(2.12)

for any with , and .

If we add (2.9) with (2.10) and make use of (2.11) and (2.12), we deduce
(2.13)

for any with , and .

Now, if we multiply (2.13) with , sum over j from 1 to n, we get
(2.14)

for any with and .

Since and , , then we get from (2.14) the first inequality in (2.8).

Now, on making use of the weighted Hölder discrete inequality
where , we also have
and

Summing these two inequalities, we deduce the second inequality in (2.8).

Finally, on utilizing the Hölder inequality
where and , we have

and the proof is concluded. □

Remark 2.2 For , we get from (2.8) that
(2.15)

for any with .

## 3 Inequalities for functions of normal operators

Now, by the help of power series , we can naturally construct another power series which will have as coefficients the absolute values of the coefficient of the original series, namely, . It is obvious that this new power series will have the same radius of convergence as the original series. We also notice that if all coefficients , then .

As some natural examples that are useful for applications, we can point out that if
(3.1)
then the corresponding functions constructed by the use of the absolute values of the coefficients are as follows:
(3.2)

The following result is a functional inequality for normal operators that can be obtained from (2.1).

Theorem 3.1 Let be a function defined by power series with complex coefficients and convergent on the open disk , . If N is a normal operator on the Hilbert space H, for , we have that , then we have the inequality
(3.3)
for any . In particular, if , then
(3.4)

for any .

Proof If N is a normal operator, then for any , we have that
Now, utilizing the inequality (2.9), we can write
(3.5)

for any and . Since , then it follows that the series and are absolute convergent in , and by taking the limit over in (3.5), we deduce the desired result (3.3). □

Remark 3.1 With the assumptions in Theorem 3.1, if we take the supremum over , , then we get the vector inequality
(3.6)
for any , which in its turn produces the norm inequality
(3.7)
for any . Making use of the examples in (3.1) and (3.2), we can state the vector inequalities
(3.8)
and
(3.9)
for any and . We also have the inequalities
(3.10)
and
(3.11)

for any and N a normal operator.

If we utilize the following function as power series representations with nonnegative coefficients:
(3.12)
where Γ is the gamma function, then we can state the following vector inequalities:
(3.13)
for any and N a normal operator. If , then we also have the inequalities
(3.14)
(3.15)
and
(3.16)

for any . From a different perspective, we also have

Theorem 3.2 With the assumption of Theorem  3.1 and if N is a normal operator on the Hilbert space H and such that , then we have the inequalities
(3.17)
for any with and . In particular, for , we have
(3.18)

for any with .

Proof If we use the second and third inequality from (2.8) for powers of operators, we have
(3.19)
for any with and . Since N is a normal operator on the Hilbert space H, then
and
for any and for any with . Then from (3.19), we have
(3.20)
for any with and . By the weighted Cauchy-Buniakowski-Schwarz inequality, we also have
(3.21)

for any with .

Now, since the series , , are convergent, then by (3.20) and (3.21), on letting , we deduce the desired result (3.17). □

Similar inequalities for some particular functions of interest can be stated. However, the details are left to the interested reader.

## 4 Applications for the Euclidean norm

In [29], the author has introduced the following norm on the Cartesian product , where denotes the Banach algebra of all bounded linear operators defined on the complex Hilbert space H:
(4.1)

where and is the Euclidean closed ball in .

It is clear that is a norm on and, for any , we have

where is the adjoint operator of , . We call this the Euclidean norm of an n-tuple of operators .

It has been shown in [29] that the following basic inequality for theEuclidean norm holds true:
(4.2)

for any n-tuple and the constants and 1 are best possible.

In the same paper [29], the author has introduced the Euclidean operator radius of an n-tuple of operators by
(4.3)
and proved that is a norm on and satisfies the double inequality
(4.4)

for each n-tuple .

As pointed out in [29], the Euclidean numerical radius also satisfies the double inequality
(4.5)

for any and the constants and 1 are best possible.

In [30], by utilizing the concept of hypo-Euclidean norm on , we obtained the following representation for the Euclidean norm.

Proposition 4.1 For any , we have
(4.6)

We can state now the following result.

Theorem 4.1 For any , we have
(4.7)
and
(4.8)

for any .

Proof We have from the second inequality in (2.8)
(4.9)
for any with and . Taking the supremum over , we have
which proves the first part of (4.7). The second part follows by the elementary inequality

for and . The inequality (4.8) follows from (4.9) by taking and then the supremum over . □

## 5 Applications for s-1-norm and s-1-numerical radius

Following [20], we consider the s-p-norm of the n-tuple of operators by
(5.1)
For , we get
We are interested in this section in the case , namely, on the s-1-norm defined by
Since for any we have , then by the properties of the supremum, we get the basic inequality
(5.2)
Similarly, we can also consider the s-p-numerical radius of the n-tuple of operators by [20]
(5.3)

which for reduces to the Euclidean operator radius introduced previously.

We observe that the s-p-numerical radius is also a norm on for , and for it satisfies the basic inequality
(5.4)

We can state the following result.

Theorem 5.1 For any , we have
(5.5)
and
(5.6)
Proof From (2.1) we have
(5.7)

for any .

Taking the supremum over , in (5.7), we have

and the first inequality in (5.5) is proved. The second part follows by the arithmetic mean-geometric mean inequality.

Now, if we take in (5.7), then we get

Taking the supremum over , we deduce the desired result (5.6). □

Remark 5.1 If we take in the first inequality in (5.5), then we deduce
(5.8)
and then we get the following refinement of the generalized triangle inequality:
From (5.6) we also have, for ,
(5.9)

## Declarations

### Acknowledgements

The authors wish to thank the anonymous referees for their valuable comments. Also, this research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2012-0008474).

## Authors’ Affiliations

(1)
School of Computer Science and Mathematics, MCMC, Victoria University of Technology
(2)
School of Computational & Applied Mathematics, University of the Witwatersrand
(3)
Department of Mathematics Education and the RINS, Gyeongsang National University
(4)
Department of Mathematics, Changwon National University

## References

1. Helmberg G: Introduction to Spectral Theory in Hilbert Space. Wiley, New York; 1969.MATH
2. Kato T: Notes on some inequalities for linear operators. Math. Ann. 1952, 125: 208–212. 10.1007/BF01343117
3. Fujii M, Lin C-S, Nakamoto R: Alternative extensions of Heinz-Kato-Furuta inequality. Sci. Math. 1999, 2(2):215–221.
4. Fujii M, Furuta T, Löwner-Heinz C: Heinz-Kato inequalities. Math. Jpn. 1993, 38(1):73–78.MATH
5. Fujii M, Kamei E, Kotari C, Yamada H: Furuta’s determinant type generalizations of Heinz-Kato inequality. Math. Jpn. 1994, 40(2):259–267.
6. Fujii M, Kim YO, Seo Y: Further extensions of Wielandt type Heinz-Kato-Furuta inequalities via Furuta inequality. Arch. Inequal. Appl. 2003, 1(2):275–283.
7. Fujii M, Kim YO, Tominaga M: Extensions of the Heinz-Kato-Furuta inequality by using operator monotone functions. Far East J. Math. Sci.: FJMS 2002, 6(3):225–238.
8. Fujii M, Nakamoto R: Extensions of Heinz-Kato-Furuta inequality. Proc. Am. Math. Soc. 2000, 128(1):223–228. 10.1090/S0002-9939-99-05242-9
9. Fujii M, Nakamoto R: Extensions of Heinz-Kato-Furuta inequality. II. J. Inequal. Appl. 1999, 3(3):293–302.
10. Furuta T: Equivalence relations among Reid, Löwner-Heinz and Heinz-Kato inequalities, and extensions of these inequalities. Integral Equ. Oper. Theory 1997, 29(1):1–9. 10.1007/BF01191475
11. Furuta T: Determinant type generalizations of Heinz-Kato theorem via Furuta inequality. Proc. Am. Math. Soc. 1994, 120(1):223–231.
12. Furuta T: An extension of the Heinz-Kato theorem. Proc. Am. Math. Soc. 1994, 120(3):785–787. 10.1090/S0002-9939-1994-1169027-6
13. Kittaneh F: Notes on some inequalities for Hilbert space operators. Publ. Res. Inst. Math. Sci. 1988, 24(2):283–293. 10.2977/prims/1195175202
14. Kittaneh F: Norm inequalities for fractional powers of positive operators. Lett. Math. Phys. 1993, 27(4):279–285. 10.1007/BF00777375
15. Lin C-S: On Heinz-Kato-Furuta inequality with best bounds. J. Korea Soc. Math. Educ. Ser. B Pure Appl. Math. 2008, 15(1):93–101.
16. Lin C-S: On chaotic order and generalized Heinz-Kato-Furuta-type inequality. Int. Math. Forum 2007, 2(37–40):1849–1858.
17. Lin C-S: On inequalities of Heinz and Kato, and Furuta for linear operators. Math. Jpn. 1999, 50(3):463–468.MATH
18. Lin C-S: On Heinz-Kato type characterizations of the Furuta inequality. II. Math. Inequal. Appl. 1999, 2(2):283–287.
19. Uchiyama M: Further extension of Heinz-Kato-Furuta inequality. Proc. Am. Math. Soc. 1999, 127(10):2899–2904. 10.1090/S0002-9939-99-05266-1
20. Dragomir SS: Some inequalities of Kato’s type for sequences of operators in Hilbert spaces. Publ. Res. Inst. Math. Sci. 2012, 48: 937–955. 10.2977/PRIMS/92
21. Cho YJ, Dragomir SS, Pearce CEM, Kim SS: Cauchy-Schwarz functionals. Bull. Aust. Math. Soc. 2000, 62: 479–491. 10.1017/S0004972700019006
22. Dragomir SS, Cho YJ, Kim SS: Some inequalities in inner product spaces related to the generalized triangle inequality. Appl. Math. Comput. 2011, 217: 7462–7468. 10.1016/j.amc.2011.02.046
23. Dragomir SS, Cho YJ, Kim JK: Subadditivity of some functionals associated to Jensen’s inequality with applications. Taiwan. J. Math. 2011, 15: 1815–1828.
24. Lin C-S, Cho YJ: On Hölder-McCarthy-type inequalities with power. J. Korean Math. Soc. 2002, 39: 351–361.
25. Lin C-S, Cho YJ: On norm inequalities of operators on Hilbert spaces. 2. In Inequality Theory and Applications. Edited by: Cho YJ, Kim JK, Dragomir SS. Nova Science Publishers, New York; 2003:165–173.
26. Lin C-S, Cho YJ: On Kantorovich inequality and Hölder-McCarthy inequalities. Dyn. Contin. Discrete Impuls. Syst. 2004, 11: 481–490.
27. Lin C-S, Cho YJ: Characteristic property for inequalities of bounded linear operators. 4. In Inequality Theory and Applications Edited by: Cho YJ, Kim JK, Dragomir SS. 2007, 85–92.
28. Lin C-S, Cho YJ:Characterizations of operator inequality . Math. Inequal. Appl. 2011, 14: 575–580.
29. Popescu G: Unitary invariants in multivariable operator theory. Mem. Am. Math. Soc. 2009., 200: Article ID 941
30. Dragomir SS: The hypo-Euclidean norm of an n -tuple of vectors in inner product spaces and applications. J. Inequal. Pure Appl. Math. 2007., 8(2): Article ID 52MathSciNet
31. McCarthy CA:. Isr. J. Math. 1967, 5: 249–271. 10.1007/BF02771613