# zbMATH — the first resource for mathematics

Norm inequalities involving accretive-dissipative $$2\times 2$$ block matrices. (English) Zbl 1398.15020
Summary: Let $$T_{11}, T_{12}, T_{21}$$, and $$T_{22}$$ be $$n \times n$$ complex matrices, and let $$H = \left (\begin{matrix} T_{11} & T_{12} \\ T_{21} & T_{22} \end{matrix} \right)$$ be accretive-dissipative. It is shown that if $$f$$ is an increasing convex function on $$[0, \infty)$$ such that $$f(0) = 0$$, then $$||| f(| T_{12} |^2) + f(| T_{21}^\ast |^2) ||| \leq ||| f(| T |^2) |||$$ for every unitarily invariant norm $$||| \cdot |||$$. Moreover, if $$f$$ is an increasing concave function on $$[0, \infty)$$ such that $$f(0) = 0$$, then $$||| f(| T_{12} |^2) + f(| T_{21}^\ast |^2) ||| \leq 4 ||| f(\frac{| T |^2}{4}) |||$$ for every unitarily invariant norm $$||| \cdot |||$$. Among other inequalities for the Schatten $$p$$-norms, it is shown that$$\| T_{12} \|_p^p + \| T_{21} \|_p^p \leq 2^{p - 1} \| T_{11} \|_p^{p / 2} \| T_{22} \|_p^{p / 2}$$ for $$p \geq 2$$.

##### MSC:
 15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory 15A18 Eigenvalues, singular values, and eigenvectors 47A30 Norms (inequalities, more than one norm, etc.) of linear operators 47B15 Hermitian and normal operators (spectral measures, functional calculus, etc.)
Full Text:
##### References:
 [1] Aujla, J. S.; Silva, F. C., Weak majorization inequalities and convex functions, Linear Algebra Appl., 369, 217-233, (2003) · Zbl 1031.47007 [2] Bhatia, R., Matrix analysis, (1997), Springer-Verlag New York [3] Bourin, J.-C.; Uchiyama, A matrix subadditivity inequality for $$f(A + B)$$ and $$f(A) + f(B)$$, Linear Algebra Appl., 423, 512-518, (2007) · Zbl 1123.15013 [4] Fan, Ky; Hoffman, A. J., Some metric inequalities in the space of matrices, Proc. Amer. Math. Soc., 6, 111-116, (1955) · Zbl 0064.01402 [5] Fu, X.; Liu, Y., Rotfel’d inequality for partitioned matrices with numerical ranges in a sector, Linear Multilinear Algebra, 64, 105-109, (2016) · Zbl 1336.47016 [6] George, A.; Ikramov, Kh. D., On the properties of accretive-dissipative matrices, Math. Notes, 77, 767-776, (2005) · Zbl 1079.15021 [7] George, A.; Ikramov, Kh. D.; Kucherov, A. B., On the growth factor in Gaussian elimination for generalized higham matrices, Numer. Linear Algebra Appl., 9, 107-114, (2002) · Zbl 1071.65033 [8] Gohberg, I. C.; Krein, M. G., Introduction to the theory of linear nonselfadjoint operators, Transl. Math. Monogr., vol. 18, (1969), Amer. Math. Soc. Providence, RI · Zbl 0181.13504 [9] Gunzburger, M. D.; Plemmons, R. J., Energy conserving norms for the solution of hyperbolic systems of partial differential equations, Math. Comp., 33, 1-10, (1979) · Zbl 0395.35060 [10] Hiai, F., Log-majorization and norm inequalities for exponential functions, (Linear Operators, Banach Center Publ., vol. 38, (1997)), 119-181 · Zbl 0885.47003 [11] Higham, N. J., Factorizing complex symmetric matrices with positive real and imaginary parts, Math. Comp., 67, 1591-1599, (1998) · Zbl 0909.65016 [12] Hirzallah, O.; Kittaneh, F., Non-commutative clarkson inequalities for unitarily invariant norms, Pacific J. Math., 202, 363-369, (2002) · Zbl 1054.47011 [13] Hirzallah, O.; Kittaneh, F., Non-commutative clarkson inequalities for n-tuples of operators, Integral Equations Operator Theory, 60, 369-379, (2008) · Zbl 1155.47013 [14] Horn, R. A.; Johnson, C. R., Topics in matrix analysis, (1991), Cambridge University Press Cambridge · Zbl 0729.15001 [15] Kosem, T., Inequalities between $$\| f(A + B) \|$$ and $$\| f(A) + f(B) \|$$, Linear Algebra Appl., 418, 153-160, (2006) · Zbl 1105.15016 [16] Lin, M., Reverse determinantal inequalities for accretive-dissipative matrices, Math. Inequal. Appl., 12, 955-958, (2012) · Zbl 1260.15033 [17] Lin, M., Fischer type determinantal inequalities for accretive dissipative matrices, Linear Algebra Appl., 438, 2808-2812, (2013) · Zbl 1261.15029 [18] Lin, M., A note on the growth factor in Gaussian elimination for accretive-dissipative matrices, Calcolo, 51, 363-366, (2014) · Zbl 1320.65045 [19] Lin, M.; Zhou, D., Norm inequalities for accretive-dissipative matrix matrices, J. Math. Anal. Appl., 407, 436-442, (2013) · Zbl 1306.47081 [20] Siegel, C. L., Topics in complex function theory, vol. III, (1973), Wiley New York · Zbl 0257.32002 [21] Tao, Y., More results on singular value inequalities of matrices, Linear Algebra Appl., 416, 724-729, (2006) · Zbl 1106.15013 [22] Uchiyama, M., Subadditivity of eigenvalue sums, Proc. Amer. Math. Soc., 134, 1405-1412, (2006) · Zbl 1089.47010 [23] Zhan, X., Matrix theory, Grad. Stud. Math., vol. 147, (2013), Amer. Math. Soc. Providence, RI [24] Zhang, F., A matrix decomposition and its applications, Linear Multilinear Algebra, 63, 2033-2042, (2015) · Zbl 1361.15030 [25] Zhang, P., A further extension of rotfel’d theorem, Linear Multilinear Algebra, 63, 2511-2517, (2015) · Zbl 1393.15028 [26] Zhang, Y., Unitarily invariant norm inequalities for accretive-dissipative operators, J. Math. Anal. Appl., 412, 564-569, (2014) · Zbl 1308.47043
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.