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Norm inequalities for submultiplicative functions involving contraction sector \(2 \times 2\) block matrices
Journal of Inequalities and Applications volume 2020, Article number: 247 (2020)
Abstract
In this article, we show unitarily invariant norm inequalities for sector \(2\times 2\) block matrices which extend and refine some recent results of Bourahli, Hirzallah, and Kittaneh (Positivity, 2020, https://doi.org/10.1007/s11117-020-00770-w).
1 Introduction
Let \(\mathbb{M}_{n}\) be a set of all \(n\times n\) complex matrices. A matrix \(A\in \mathbb{M}_{n}\) is said to be positive semidefinite if \(x^{*}Ax\geq 0\) for all \(x\in \mathbb{C}^{n}\). If the eigenvalues \(\lambda _{1}(A), \dots , \lambda _{n}(A)\) of A are all real, we arrange them in nonincreasing order \(\lambda _{1}(A)\ge \cdots \ge \lambda _{n}(A)\). Singular values of A are the eigenvalues of \(|A|\) and are arranged in nonincreasing order \(s_{1}(A)\ge \cdots \ge s_{n}(A)\). For \(A \in \mathbb{M}_{n}\), we denote by \(|A|=(A^{*}A)^{\frac{1}{2}}\), \(A^{*}\), \(\|A\|\), and \(\|A\|_{\infty }=s_{1}(A)\) the absolute value, the conjugate transpose, the unitarily invariant norm, and the operator norm, respectively. We say A is a contraction if \(\|A\|_{\infty }\leq 1\). By convention, the \(n \times n\) identity matrix is denoted by \(I_{n}\). \(\|A\|\) and \(\|A\|_{\infty }=s_{1}(A)\) are unitarily invariant, i.e., \(\|UAV\|=\|A\|\) for all unitary matrices U, V. For \(A, B\in \mathbb{M}_{n}\), the weak majorization relation \(s(A)\prec _{w} s(B)\) means
For \(A\in \mathbb{M}_{n}\), recall the Cartesian (or Toeplitz) decomposition (see, e.g., [2, p. 6] and [3, p. 7])
where
The Cartesian decomposition of a matrix is unique. There are many interesting properties for such a decomposition. A celebrated result due to Fan and Hoffman (see, e.g., [2, p.73]) states that
The numerical range of \(A\in \mathbb{M}_{n}\) is defined by
which is a compact convex set (see, e.g., [4, Chap. 1]). For \(\alpha \in [0, \pi /2)\), a sector on the complex plane is
A sector matrix \(A\in \mathbb{M}_{n}\) is a matrix whose numerical range is contained in \(S_{\alpha }\) for some \(\alpha \in [0, \pi /2)\). It is clear that if \(A\in \mathbb{M}_{n}\) is a sector matrix, then ReA is positive semidefinite. The interested readers can refer to [5–10], and [4] for recent results on sector matrices. If \(W(A)\) is contained in the first quadrant of the complex plane, then ReA and ImA are positive semidefinite. We call such a matrix A accretive-dissipative. Note that if A is accretive-dissipative, then \(W(e^{-\frac{i\pi }{4}}A)\subseteq S_{\frac{\pi }{4}}\). Recently this class of matrices has been studied by researchers partly due to the fact that it contains the class of positive semidefinite matrices (see, e.g., [1, 11–17]).
Next we introduce a special class of functions. Let \(\mathcal{C}\) be the class of all nonnegative increasing functions f on \([0, \infty )\) preserving the weak-log majorization, i.e., for two nonincreasing sequences of nonnegative real numbers \((x_{1}, x_{2},\ldots , x_{n})\) and \((y_{1}, y_{2}, \ldots , y_{n})\), \(\prod_{j=1}^{k} x_{j} \leq \prod_{j=1}^{k} y_{j}\) for \(k=1,\ldots , n\) implies \(\prod_{j=1}^{k} f(x_{j}) \leq \prod_{j=1}^{k} f(y_{j})\) for \(k=1,\ldots , n\). There are many other properties on this class of functions; see [12, 18]. A nonnegative function \(f\in \mathcal{C}\) on the interval \([0, \infty )\) is said to be submultiplicative if \(f(ab)\leq f(a)f(b)\) whenever \(a, b\in [0, \infty )\). Recently, some unitarily invariant norm inequalities for submultiplicative functions of accretive-dissipative matrices have been shown in [12] and [13].
Bourahli et al. [1, Lemma 3.4] showed that if is a positive semidefinite contraction and s, t are positive real numbers such that \(\frac{1}{s}+\frac{1}{t}=1\), then
where f is an increasing submultiplicative function on \([0, \infty )\) with \(f(0)=0\). Moreover, if \(A\in \mathbb{M}_{2n}\) is just positive semidefinite (not necessarily contraction matrices), then they presented a result related to (2) in [1, Remark 3.5] as follows:
2 Unitarily invariant norms for submultiplicative functions
In [12] and [13], some unitarily invariant norms for accretive-dissipative matrices involving a special class of functions have been shown. In this section, we present inequalities for sector block matrices involving the class of function.
Lemma 2.1
([19, p. 280])
Let A, X, B be \(m\times p\), \(p\times q\), \(q\times n\) matrices, respectively. Then
Lemma 2.2
([8, Theorem 2.1])
Let \(A\in \mathbb{M}_{n}\) be \(n\times n\) such that \(W(A)\subseteq S_{\alpha }\) for some \(\alpha \in [0, \pi /2)\). Then there exist an invertible matrix X and a unitary and diagonal matrix \(Z=\operatorname{diag}(e^{i\theta _{1}},\ldots , e^{i\theta _{n}})\) with all \(|\theta _{j}|\leq \alpha \) such that \(A=XZX^{*}\). Moreover, such a matrix Z is unique up to permutation.
Lemma 2.3
([8, Corollary 2.3 (ii)])
Let \(A\in \mathbb{M}_{n}\) be such that \(W(A)\subseteq S_{\alpha }\) for some \(\alpha \in [0, \pi /2)\), and let \(A=XZX^{*}\) be a sectoral decomposition of A, where X is invertible and Z is unitary and diagonal. Then
for every matrix \(R\in \mathbb{M}_{n}\).
We are ready to present our main result of this section.
Theorem 2.4
Let \(f\in \mathcal{C}\) be an increasing submultiplicative function on \([0, \infty )\) and \(A\in \mathbb{M}_{2n}\) be a contraction matrix partitioned as
with \(W(A)\subseteq S_{\alpha }\) for some \(\alpha \in [0, \pi /2)\). Then, for all \(r, s, t>0\) with \(\frac{1}{s}+\frac{1}{t}=1\) and all unitarily invariant norms,
and
Proof
Note that A is a sector matrix with \(W(A)\subseteq S_{\alpha }\). By Lemma 2.2, we have \(A=XZX^{*}\), where X is invertible and Z is unitary and diagonal. We partition X as , \(X_{1},X_{2}\in \mathbb{M}_{n \times 2n}\). Thus, \(\operatorname{Re}A_{11}=X_{1}(\operatorname{Re}Z)X_{1}^{*}\), \(\operatorname{Re}A_{22}=X_{2}(\operatorname{Re}Z)X_{2}^{*}\), and \(A_{12}=X_{1}ZX_{2}^{*}\). Consider the Cartesian decomposition
where R is positive semidefinite and S is Hermitian. Since A is a contraction matrix,
and
Thus, R and S are also contraction matrices, which implies that both \(\operatorname{Re}A_{11}=R_{11}\) and \(\operatorname{Re}A_{22}=R_{22}\) are positive semidefinite contractions.
Now
for \(l=1,2, \ldots , n\).
Since \(\operatorname{Re}A_{22}\) is also a contraction, it follows from (1) and (9) that
for \(l=1,2, \ldots , n\).
Multiplying inequalities (10) and (11) by each other implies that
for \(l=1,2, \ldots , n\).
So,
for \(l=1,2, \ldots , n\). Let \(\alpha =(\alpha _{1},\alpha _{2}, \ldots , \alpha _{n})\) be a decreasing sequence of nonnegative real numbers. The α-norm of a matrix \(B\in \mathbb{M}_{n}\) is defined by
The α-norms are unitarily invariant [4, p. 204].
Actually, inequality (13) means that
for \(k=1,2, \ldots , n\), which implies that
for \(k=1,2, \ldots , n\). Thus,
for all decreasing sequences \(\alpha =(\alpha _{1},\ldots , \alpha _{n})\) of nonnegative real numbers. It follows from the above inequality that
The inequality for \(A_{21}\) is similarly proved. □
Remark 1
In particular, when A is a positive semidefinite contraction (\(\alpha =0\)) and \(r=1\), Theorem 2.4 gives
which is due to Bourahli et al. [1, Lemma 3.4]. Thus, our result (5) is a generalization of (14).
Remark 2
If A is just a general sector matrix with \(W(A)\subseteq S_{\alpha }\) for \(\alpha \in [0, \frac{\pi }{2})\) (not a contraction matrix), then we have the following result: Let \(f\in \mathcal{C}\) be an increasing submultiplicative function on \([0, \infty )\) and be a sector matrix with \(W(A)\subseteq S_{\alpha }\) for some \(\alpha \in [0, \pi /2)\). Then, for all \(r, s, t>0\) with \(\frac{1}{s}+\frac{1}{t}=1\) and all unitarily invariant norms,
By (9), (10), and Lemma 2.1, we have
and
for \(l=1,2, \ldots , n\).
Multiplying inequalities (16) and (17) by each other implies that
for \(l=1,2, \ldots , n\).
So,
for \(l=1,2, \ldots , n\). Based on (19), we can obtain the desired result by a proof similar to that given for inequality (13). Therefore, when \(\alpha =0\) and \(r=1\), our result (15) is (2).
Theorem 2.5
Let \(f\in \mathcal{C}\) be an increasing submultiplicative function on \([0, \infty )\) and \(A\in \mathbb{M}_{2n}\) be a contraction matrix partitioned as
with \(W(A)\subseteq S_{\alpha }\) for some \(\alpha \in [0, \pi /2)\). Then, for all \(r, s, t>0\) with \(\frac{1}{s}+\frac{1}{t}=1\) and all unitarily invariant norms,
Proof
By Theorem 2.4, we can have the desired result. □
Remark 3
When \(f(t)=t\), \(r=1\), and \(\alpha =\frac{\pi }{4}\), result (20) becomes
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References
Bourahli, A., Hirzallah, O., Kittaneh, F.: Unitarily invariant norm inequalities for functions of accretive-dissipative \(2 \times 2\) block matrices. Positivity (2020). https://doi.org/10.1007/s11117-020-00770-w
Bhatia, R.: Matrix Analysis. GTM, vol. 169. Springer, Berlin (1997)
Horn, R.A., Johnson, C.R.: Matrix Analysis, 2nd edn. Cambridge University Press, New York (2013)
Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis. Cambridge University Press, Cambridge (1991)
Drury, S.W., Lin, M.: Singular value inequalities for matrices with numerical ranges in a sector. Oper. Matrices 8, 1143–1148 (2014)
Fu, X., Liu, Y.: Rotfel’d inequality for partitioned matrices with numerical ranges in a sector. Linear Multilinear Algebra 64, 105–109 (2015)
Li, C.K., Sze, N.: Determinantal and eigenvalue inequalities for matrices with numerical ranges in a sector. J. Math. Anal. Appl. 410, 487–491 (2014)
Zhang, F.: A matrix decomposition and its applications. Linear Multilinear Algebra 63, 2033–2042 (2015)
Yang, J., Lu, L., Chen, Z.: Schatten q-norms and determinantal inequalities for matrices with numerical ranges in a sector. Linear Multilinear Algebra 67, 221–227 (2019)
Yang, J., Lu, L., Chen, Z.: A refinement of Rotfel’d type inequality for partitioned matrices with numerical ranges in a sector. Linear Multilinear Algebra 67, 1719–1726 (2019)
George, A., Ikramov, Kh.D.: On the properties of accretive–dissipative matrices. Math. Notes 77, 767–776 (2005)
Gumus, I.H., Hirzallah, O., Kittaneh, F.: Norm inequalities involving accretive-dissipative \(2 \times 2\) block matrices. Linear Algebra Appl. 528, 76–93 (2017)
Jabbarzadeh, M.R., Kaleibary, V.: Inequalities for accretive-dissipative block matrices involving convex and concave functions Linear Multilinear Algebra (2020). https://doi.org/10.1080/03081087.2020.1726277
Kittaneh, F., Sakkijha, M.: Inequalities for accretive-dissipative matrices. Linear Multilinear Algebra 67, 1037–1042 (2019)
Lin, M.: Reversed determinantal inequalities for accretive-dissipative matrices. Math. Inequal. Appl. 12, 955–958 (2012)
Lin, M., Zhou, D.: Norm inequalities for accretive-dissipative operator matrices. J. Math. Anal. Appl. 407, 436–442 (2013)
Lin, M.: Fischer type determinantal inequalities for accretive-dissipative matrices. Linear Algebra Appl. 438, 2808–2812 (2013)
Hiai, F.: Log-majorizations and norm inequalities for exponential operators. In: Linear Operators. Banach Center Publ., vol. 38, pp. 119–181 (1997)
Zhang, F.: Matrix Theory: Basic Results and Techniques, 2nd edn. Springer, New York (2011)
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The author wants to express her sincere thanks to the referee for his or her valuable remarks and suggestions, which made this paper more readable.
Funding
Zhou’s work was supported by the High-level Talents Program of Natural Science Foundation of Hainan Province, China (2019RC193) and the Scientific Research Project of Colleges and Universities of Hainan Province, China (Hnky2020-31).
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Zhou, X. Norm inequalities for submultiplicative functions involving contraction sector \(2 \times 2\) block matrices. J Inequal Appl 2020, 247 (2020). https://doi.org/10.1186/s13660-020-02514-6
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DOI: https://doi.org/10.1186/s13660-020-02514-6