Abstract
Let be any bounded linear operator defined on a complex Hilbert space In this paper, we present some numerical radius inequalities involving the generalized Aluthge transform to attain upper bounds for numerical radius. Numerical computations are carried out for some particular cases of generalized Aluthge transform.
1. Introduction
Mathematical inequalities play an essential role in developing various areas of pure and applied mathematics. The usefulness of mathematical inequalities is to estimate the solutions of real-life problems in engineering and other fields of science. In mathematics, particularly in functional analysis, the study of numerical radius inequalities has become the attention of many researchers due to the applications of numerical radius in operator theory and numerical analysis, etc. (see [1–4]). Various mathematicians have developed number of numerical radius inequalities to estimate the upper and lower bounds for numerical radius. It is interesting for researchers to get the refinements and generalization of these inequalities. The aim of this paper is to study the generalization and refinements of existing inequalities for numerical radius. Now, we recall some notions to proceed our work.
Let be the -algebra of all bounded linear operators defined on a complex Hilbert space For the usual operator norm is defined as and the numerical radius is defined as
It is well known that the numerical radius defines an equivalent operator norm on and for we have
Many authors worked on the refinement of inequality (3) (see [5–7]). Kittaneh developed the following upper bound of numerical radius: which is a refinement of inequality (3) (see [5]). For having polar decomposition where is a partial isometry and , the Aluthge transform is defined as see [8]. Yamazaki developed an upper bound of numerical radius involving Aluthge transform given by which is an improvement of bounds (3) and (4) (see [9]). Bhunia et al. developed a bound of the numerical radius given by and proved that it is a refinement of bound (6) (see [10]). Okubo introduced a generalization of Aluthge transform which is defined as for known as -Aluthge transform (see [11]). After that, a number of numerical radius inequalities were established involving -Aluthge transform (see [12–14]). Abu Omar and Kittaneh using -Aluthge transform generalized bound (6) given by see [12]. Shebrawi and Bakherad introduced another generalization of Aluthge transform which is defined as where and are nonnegative and continuous functions such that , known as generalized Aluthge transform. The authors generalized inequality (9) given by see [15].
In this paper, we establish some new inequalities of the numerical radius using generalized Aluthge transform. Specifically, we generalize inequality (7) and improve the inequalities (3), (4), and (12). Some examples of operators are presented for which the bounds of numerical radius are computed from these inequalities for some choices of in (10).
2. Main Results
Now, we recall a lemma that will be used to achieve our goals.
Lemma 1 (see [9]). Let ; then, for we have where
Polarization identity: [15] For each , we have
Now, we establish an inequality of numerical radius which is a generalization of inequality (7) and a refinement of inequality (12).
Theorem 2. Let . Then, where and is nonnegative continuous functions defined on such that
Proof. Let be the polar decomposition of Then, by polarization identity, we have Therefore, Now taking supremum over in the last inequality and then applying Lemma 1, we obtain as desired.
Theorem 2 includes some particular cases of generalized Aluthge transform for different choices of continuous functions and in (10) as follows.
Corollary 3. Let . Then, for we have where and
Corollary 4. Let . Then, for we have where and
Corollary 5. Let . Then, for we have where In particular,
Remark 6. By using the inequality for all (see [16]) in inequality (15) obtained in Theorem 2, we have
Hence, where Thus, inequality (15) obtained in Theorem 2 is better than inequality (12).
Remark 7. For continuous functions and in (10), if , then inequality (15) becomes
In particular, if we take and , for this choice of and if , then inequality (15) becomes , and combined with inequality (3), we get
Theorem 8. Let . Then, we have where and is nonnegative continuous functions defined on such that
Proof. Since for all , then we have which yields
Since is self-adjoint, so Hence, using the properties of operator norm on we have where the equality holds because Now taking supremum over in last equality, then applying Lemma 1, yields
For different choices of and in (10), we obtain the following inequalities of numerical radius from Theorem 8.
Corollary 9. Let . Then, for , we have where and
Corollary 10. Let . Then, for we have where and
Corollary 11. Let . Then, for we have where In particular,
Remark 12. It is easy to check that (see [9] for details). Using the following inequality for all (see [14]), Corollary 11 yields
We know that (see [17]). Hence,
Thus, the bound given in Corollary 11 is better than bound (4).
Remark 13. If in inequality (27) obtained in Theorem 8 for different choices of and in (10), then inequality (27) becomes
If and are equivalent conditions, then inequality (27) becomes
Theorem 14. Let . Then, we have where is nonnegative continuous functions defined on such that
Proof. Since for all , then we have which implies
In the last equality, . Hence,
The first inequality holds because where the second inequality holds because and and the third equality holds because and Now taking supremum over in above equality the using Lemma 1, we obtain as desired.
Corollary 15. Let . Then, for we have where
Corollary 16. Let . Then, for we have where
Corollary 17. Let . Then, for we have In particular,
Remark 18. Yan et al. proved that see [18]. Inequality (41) obtained in Theorem 14 gives better bounds of numerical radius of for different choices of and in (10) when Then, is a polar decomposition of , where and is partial isometry.
Bounds (41) and (50) are computed for some choices of and in (10) for the given in Table 1, whereas the numerical radius of is
The spectral radius of an operator is defined as where denotes the spectral radius. For further inforamtion on spectral radius, see [19]. The following theorem will be used to develop the next inequality of numerical radius.
Theorem 19 (see [17]). Let Then,
Theorem 20. Let . Then, where is nonnegative continuous functions defined on such that
Proof. Since for all , then we have which implies
Now by using the properties of operator norm on we have where , and ; the first equality holds for Hermitian operator satisfying Now applying Theorem 19 on last equality with and we obtain
Taking supremum over in last inequality, then applying Lemma 1, we obtain
Corollary 21. Let . Then, for we have where
Corollary 22. Let . Then, for we have where
Corollary 23. Let . Then, for we have In particular,
Remark 24. It is easy to observe that inequality (57) obtained in Theorem 20 is better than inequality (41).
Now, we exhibit some examples where numerical radius bounds are computed from inequalities (15), (27), (41), and (57) for some choices of pair in (10) and for a given operator .
Example 1. Given Then, is a polar decomposition of , where and is partial isometry.
Bounds (15), (27), (41), and (57) are computed for some choices of and in (10) for the given in Table 2, whereas the numerical radius of is
Example 2. Let Then, is a polar decomposition of , where and
Bounds (15), (27), (41), and (57) are computed for some choices of and in (10) for the given in Table 3, whereas the numerical radius of is
3. Conclusion
From the results of this paper, we conclude that the inequalities of numerical radius involving generalized Aluthge transform have variety of upper bounds for numerical radius due to the choice of in generalized Aluthge transform (10). The inequalities (15), (27), (41) and (57) obtained in Theorem 2, Theorem 8, Theorem 20, and Theorem 14 are new and generalized upper bounds for numerical radius. These generalized upper bounds can be useful to find better bounds of numerical radius already existing in literature for some choices of in generalized Aluthge transform (10) and certain operators. It is proved that inequality (15) of Theorem 2 generalizes inequality (7) and improves inequality (12) for any choice of in (10). Inequality (27) of Theorem 8 is sharper than inequality (12) for the choice of in (10). Inequality (41) of Theorem 14 is better than inequality (57) of Theorem 20. But for inequality (57) of Theorem 20, we can find such matrix and pairs of for which the inequality of Theorem 20 can give better bound of numerical radius available in literature. To support theoretical investigations, some examples are presented where numerical radius and its upper bounds are computed for the pairs in generalized Aluthge transform. Examples 1 and 2 show that there is no comparison between the bounds obtained from the inequalities (15), (27), and (41) of Theorem 2, Theorem 8, and Theorem 20; however, generalized Aluthge transform has choices of the pair in (10) for which better upper bounds can be computed for certain operators.
Data Availability
There is no data required for this paper.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors would like to thank the Deanship of Scientific Research of King Abdulaziz University, Jeddah, Saudi Arabia, for technical and financial support.