The numerical radius of a matrix is a scalar quantity that has many applications in the study of matrix analysis. Due to the difficulty in computing the numerical radius, inequalities bounding it have received a considerable attention in the literature. In this article, we present many new bounds for the numerical radius of accretive matrices. The importance of this study is the presence of a new approach that treats a specific class of matrices, namely the accretive ones. The new bounds provide a new set of inequalities, some of which can be considered as refinements of other existing ones, while others present new insight to some known results for positive matrices.
In this paper, we discuss new inequalities for accretive matrices through non standard domains. In particular, we present several relations for $A^r$ and $A\sharp_rB$, when $A,B$ are accretive and $r\in (-1,0)\cup (1,2).$ This complements the well established discussion of such quantities for accretive matrices when $r\in [0,1],$ and provides accretive versions of known results for positive matrices.
The concept of Loewner (partial) order for general complex matrices is introduced. After giving the definition of arithmetic, geometric, and harmonic mean for accretive-dissipative matrices, we study their basic properties. An AM-GM-HM inequality is established for two accretive-dissipative matrices in the sense of this extended Loewner order. We also compare the harmonic mean and parallel sum of two accretive-dissipative matrices, revealing an interesting relation between them. A number of examples are included.
The main goal of this paper is to discuss the recent advancements of operator means for accretive matrices in a more general setting. In particular, we present the general form governing the well established definition of geometric mean, then we define arbitrary operator means and functional calculus for accretive matrices. Applications of this new discussion involve generalizations of known inequalities from the setting of positive matrices to that of accretive matrices. This includes the arithmetic-harmonic mean comparisons, monotony of operator means, Ando's inequality, Choi's inequality, A…
Let $A={bmatrix} A_{11} &A_{12} A_{21} & A_{22} {bmatrix}$ be an $n\times n$ accretive-dissipative matrix, $k$ and l be the orders of $A_{11}$ and $A_{22}$, respectively, and let $m=\min\{k,l\}$. Then $$|\det A|\le a|\det A_{11}|\cdot|\det A_{22}|,$$ where $a=\{{array}{l l} 2^{3m/2}, & \text{if} m\le n/3; 2^{n/2}, & \text{if} n/3<m\le n/2. {array}.$ This improves a result of Ikramov.
To better understand the algebra $\mathcal{M}_n$ of all $n\times n$ complex matrices, we explore the class of accretive matrices. This class has received renowned attention in recent years due to its role in complementing those results known for positive definite matrices. More precisely, we have several results that allow a better understanding of accretive matrices. Among many results, we present order-preserving results, Choi-Davis-type inequalities, mean-convex inequalities, sub-multiplicative results for the real part, and new bounds of the absolute value of accretive matrices. These resu…
For $n$-by-$n$ and $m$-by-$m$ complex matrices $A$ and $B$, it is known that the inequality $w(A\otimes B)\le\|A\|w(B)$ holds, where $w(\cdot)$ and $\|\cdot\|$ denote, respectively, the numerical radius and the operator norm of a matrix. In this paper, we consider when this becomes an equality. We show that (1) if $\|A\|=1$ and $w(A\otimes B)=w(B)$, then either $A$ has a unitary part or $A$ is completely nonunitary and the numerical range $W(B)$ of $B$ is a circular disc centered at the origin, (2) if $\|A\|=\|A^k\|=1$ for some $k$, $1\le k<\infty$, then $w(A)\ge\cos(π/(k+2))$, and, moreover, …
Several spectral radii formulas for infinite bounded nonnegative matrices in max algebra are obtained. We also prove some Perron-Frobenius type results for such matrices. In particular, we obtain results on block triangular forms, which are similar to results on Frobenius normal form of $n \times n$ matrices. Some continuity results are also established.
Operator matrices have played a significant role in studying Hilbert space operators. In this paper, we discuss further properties of operator matrices and present new estimates for the operator norms and numerical radii of such operators. Moreover, operator matrices whose real and imaginary parts are positive will be discussed, and sharper bounds will be shown for such class.
In this paper, we present several sharp upper bounds for the numerical radii of the diagonal and off-diagonal parts of the $2\times2$ block operator matrix $\begin{bmatrix}A&B\\ C&D\end{bmatrix}$. Among extensions of some results of Kittaneh et al., it is shown that if $T=\begin{bmatrix}A&0\\ 0&D\end{bmatrix}$, and $f$ and $g$ are non-negative continuous functions on $[0,\infty)$ such that $f(t)g(t)=t\,\,(t\geq 0)$, then for all nonnegative nondecreasing convex functions $h$ on $[0,\infty)$ , we obtain that \begin{align*}h\left(w^r(T)\right)\leq \max\left(\left\|\frac{1}{p}h\left(f^{pr}(\left|…
