Inverse Young inequality in quaternion matrices

Document Type : Research Paper

Authors

Department of Mathematical Sciences, Isfahan University of Technology, Isfahan, Islamic Republic of Iran

Abstract

Inverse Young inequality asserts that if $nu >1$, then $|zw|ge nu|z|^{frac{1}{nu}}+(1-nu)|w|^{frac{1}{1-nu}}$, for all complex numbers $z$ and $w$, and equality holds if and only if $|z|^{frac{1}{nu}}=|w|^{frac{1}{1-nu}}$. In this paper the complex representation of quaternion matrices is applied to establish the inverse Young inequality for matrices of quaternions. Moreover, a necessary and sufficient condition for equality is given.

Keywords

References

[1] T. Ando, Matrix Young inequalities, Oper. Theory Adv. Appl. 75 (1995), 33–38.
[2] M. Argerami and D. R. Farenick, Young’s inequality in trace-class operators, Math. Ann., 325 (2003), 727–744.
[3] R. Bhatia and F. Kittaneh, On the singular values of a product of operators, SIAM J. Matrix Anal. Appl. 11
(1990), 727–277.
[4] J. Erlijman, D. R. Farenick and R. Zeng, Young’s inequality in compact operators, Oper. Theory Adv. Appl., 130(2001), 171–184.
[5] D. R. Farenick and S. M. Manjegani, Young’s inequality in operator algebras, J. Ramanujan Math. Soc., 20(2)(2005), 107–124.
[6] D. R. Farenick and B. A .F. Pidkowich, The spectral theorem in quaternions, Linear Algebra Appl., 371 (2003), 75–102.
[7] H. Glockner, Functions operating on positive semidefinite quaternionic matrices, Monatsh. Math., 132 (2001), 303–324.
[8] H. C. Lee, Eigenvalues and canonical forms of matrices with quaternion coecients, Proc. Roy. Irish. Acad.
Sec. A., 52 (1949), 253–260.
[9] S. M. Manjegani and A. Norouzi, Martix form for the inverse Young inequality, Linear Algebra Appl., 486
(2015), 484 - 493..
[10] R. C. Thompson, Convex and concave functions of singular values of matrix sums, Pacific J. Math., 66 (1976), 285–290.
[11] R. C. Thompson, The case of equality in the matrix-valued triangle inequality, Pacific J. Math., 82 (1979),
279–280.
[12] R. C. Thompson, Matrix-valued triangle inequality: quaternion version, Linear and multilinear Algebra,
25(1989), 85–91.
[13] R. Zeng, The quaternion matrix-valued Young’s inequality, J. Inequal. Pure Appl. Math. (6) , Art. 89, (2005).
Wavelet and Linear Algebra
Volume 3, Issue 1
spring and summer
2016
Pages 45-52

Files

Share

How to cite

Statistics