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Applications of Kato’s inequality for n-tuples of operators in Hilbert spaces, (I)
Journal of Inequalities and Applications volume 2013, Article number: 21 (2013)
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.
The ‘square root’ of a positive bounded self-adjoint operator on H can be defined as follows (see, for instance, [, 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  proved the following generalization of Schwarz inequality:
for any , and T is a bounded linear operator on H.
Utilizing the modulus notation introduced before, we can write (1.1) as follows:
In the recent paper , 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
for any with and .
He also obtained the following result.
Theorem 1.2 With the assumptions in Theorem 1.1, we have
for any .
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
for any , and, in particular, for
for any .
Proof Utilizing Kato’s inequality, we have
and by replacing α with ,
which by summation gives
for any and . By the elementary inequality
which by (2.3) produces
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
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,
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
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
and, by replacing α with ,
for any .
By the Hölder-McCarthy inequality that holds for the positive operator P, for and with , we also have
for any with , and .
If we add (2.9) with (2.10) and make use of (2.11) and (2.12), we deduce
for any with , and .
Now, if we multiply (2.13) with , sum over j from 1 to n, we get
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
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
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
then the corresponding functions constructed by the use of the absolute values of the coefficients are as follows:
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
for any . In particular, if , then
for any .
Proof If N is a normal operator, then for any , we have that
Now, utilizing the inequality (2.9), we can write
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
for any , which in its turn produces the norm inequality
for any . Making use of the examples in (3.1) and (3.2), we can state the vector inequalities
for any and . We also have the inequalities
for any and N a normal operator.
If we utilize the following function as power series representations with nonnegative coefficients:
where Γ is the gamma function, then we can state the following vector inequalities:
for any and N a normal operator. If , then we also have the inequalities
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
for any with and . In particular, for , we have
for any with .
Proof If we use the second and third inequality from (2.8) for powers of operators, we have
for any with and . Since N is a normal operator on the Hilbert space H, then
for any and for any with . Then from (3.19), we have
for any with and . By the weighted Cauchy-Buniakowski-Schwarz inequality, we also have
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 , 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:
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  that the following basic inequality for theEuclidean norm holds true:
for any n-tuple and the constants and 1 are best possible.
In the same paper , the author has introduced the Euclidean operator radius of an n-tuple of operators by
and proved that is a norm on and satisfies the double inequality
for each n-tuple .
As pointed out in , the Euclidean numerical radius also satisfies the double inequality
for any and the constants and 1 are best possible.
In , 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
We can state now the following result.
Theorem 4.1 For any , we have
for any .
Proof We have from the second inequality in (2.8)
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 , we consider the s-p-norm of the n-tuple of operators by
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
Similarly, we can also consider the s-p-numerical radius of the n-tuple of operators by 
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
We can state the following result.
Theorem 5.1 For any , we have
Proof From (2.1) we have
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
and then we get the following refinement of the generalized triangle inequality:
From (5.6) we also have, for ,
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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).
The authors declare that they have no competing interests.
All authors read and approved the final manuscript.
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Dragomir, S.S., Cho, Y.J. & Kim, Y. Applications of Kato’s inequality for n-tuples of operators in Hilbert spaces, (I). J Inequal Appl 2013, 21 (2013). https://doi.org/10.1186/1029-242X-2013-21
- bounded linear operators
- functions of normal operators
- inequalities for operators
- norm and numerical radius inequalities
- Kato’s inequality