Abstract
In this paper, by the use of the famous Kato’s inequality for bounded linear operators, we establish some inequalities for ntuples 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 inequality1 Introduction
The ‘square root’ of a positive bounded selfadjoint operator on H can be defined as follows (see, for instance, [[1], p.240]).
If the operator is selfadjoint and positive, then there exists a unique positive selfadjoint operator such that . If A is invertible, then so is B.
If , then the operator is selfadjoint and positive. Define the ‘absolute value’ operator by .
In 1952, Kato [2] 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:
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 1.1Letbe anntuple of bounded linear operators on the Hilbert spaceandbe anntuple of nonnegative weights not all of them equal to zero. Then we have
He also obtained the following result.
Theorem 1.2With the assumptions in Theorem 1.1, we have
For various related results, see the papers [2131].
Motivated by the above results, we establish in this paper other similar inequalities for ntuples 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 ntuples of bonded linear operators on Hilbert spaces.
2 Some inequalities for an ntuple of linear operators
Employing Kato’s inequality (1.2), we can state the following new result.
Theorem 2.1Letbe anntuple of bounded linear operators on the Hilbert spaceandbe anntuple of nonnegative weights, not all of them equal to zero. Then we have
for any, and, in particular, for
Proof Utilizing Kato’s inequality, we have
which by summation gives
for any and . By the elementary inequality
we have
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 CauchyBuniakowskiSchwarz inequality
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,
From a different perspective that involves quadratics, we can state the following result as well.
Theorem 2.2Letbe anntuple of bounded linear operators on the Hilbert spaceandbe anntuple of nonnegative weights, not all of them equal to zero. Then we have
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
By the HölderMcCarthy inequality that holds for the positive operator P, for and with , we also have
and
If we add (2.9) with (2.10) and make use of (2.11) and (2.12), we deduce
Now, if we multiply (2.13) with , sum over j from 1 to n, we get
Since and , , then we get from (2.14) the first inequality in (2.8).
Now, on making use of the weighted Hölder discrete inequality
and
Summing these two inequalities, we deduce the second inequality in (2.8).
Finally, on utilizing the Hölder inequality
and the proof is concluded. □
Remark 2.2 For , we get from (2.8) that
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 3.1Letbe a function defined by power series with complex coefficients and convergent on the open disk, . IfNis a normal operator on the Hilbert spaceH, for, we have that, then we have the inequality
for any. In particular, if, then
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
and
for any and . We also have the inequalities
and
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
and
for any . From a different perspective, we also have
Theorem 3.2With the assumption of Theorem 3.1 and ifNis a normal operator on the Hilbert spaceHandsuch that, then we have the inequalities
for anywithand. In particular, for, we have
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
and
for any and for any with . Then from (3.19), we have
for any with and . By the weighted CauchyBuniakowskiSchwarz inequality, we also have
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:
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 ntuple of operators .
It has been shown in [29] that the following basic inequality for theEuclidean norm holds true:
for any ntuple and the constants and 1 are best possible.
In the same paper [29], the author has introduced the Euclidean operator radius of an ntuple of operators by
and proved that is a norm on and satisfies the double inequality
As pointed out in [29], the Euclidean numerical radius also satisfies the double inequality
for any and the constants and 1 are best possible.
In [30], by utilizing the concept of hypoEuclidean norm on , we obtained the following representation for the Euclidean norm.
Proposition 4.1For any, we have
We can state now the following result.
and
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 s1norm and s1numerical radius
Following [20], we consider the spnorm of the ntuple of operators by
We are interested in this section in the case , namely, on the s1norm 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 spnumerical radius of the ntuple of operators by [20]
which for reduces to the Euclidean operator radius introduced previously.
We observe that the spnumerical radius is also a norm on for , and for it satisfies the basic inequality
We can state the following result.
and
Proof From (2.1) we have
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 meangeometric 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 ,
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors read and approved the final manuscript.
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 (20120008474).
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