Related papers: The Resolvent Mean and The Parametrized $\mathcal{…
We define a new average - termed the resolvent average - for positive semidefinite matrices. For positive definite matrices, the resolvent average enjoys self-duality and it interpolates between the harmonic and the arithmetic averages,…
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…
In this paper the geometric mean of partial positive definite matrices with missing entries is considered. The weighted geometric mean of two sets of positive matrices is defined, and we show whether such a geometric mean holds certain…
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,…
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…
To provide generalized solutions if a given problem admits no actual solution is an important task in mathematics and the natural sciences. It has a rich history dating back to the early 19th century when Carl Friedrich Gauss developed the…
Let $A, B$ be positive definite matrices, $p=1, 2$ and $r\ge 0$. It is shown that \begin{equation*} ||A+ B + r(A\sharp_t B+A\sharp_{1-t} B)||_p \le ||A+ B + r(A^{t}B^{1-t} + A^{1-t}B^t)||_p. \end{equation*} We also prove that for positive…
In the 1980s, Kubo and Ando introduced operator means on $\mathbb{P}$, the open convex cone of positive definite operators. One significant example is the weighted geometric mean $$ A \sharp_{t} B = A^{1/2} (A^{-1/2} B A^{-1/2})^{t}…
We show that the set of fixed points of the average of two resolvents can be found from the set of fixed points for compositions of two resolvents associated with scaled monotone operators. Recently, the proximal average has attracted…
In this paper, we obtain some new matrix inequalities involving Hadamard product. Also some Hadamard product inequalities for accretive matrices involving the matrix means, positive unital linear maps and matrix concave functions are…
A generalized word in two positive definite matrices A and B is a finite product of nonzero real powers of A and B. Symmetric words in positive definite A and B are positive definite, and so for fxed B, we can view a symmetric word, S(A,B),…
The generalization of the geometric mean of positive scalars to positive definite matrices has attracted considerable attention since the seminal work of Ando. The paper generalizes this framework of matrix means by proposing the definition…
In this paper, we generalize the geometric mean of two positive definite matrices to that of third-order tensors using the notion of T-product. Specifically, we define the geometric mean of two T-positive definite tensors and verify several…
This paper is concerned with the problem of approximating the determinant of A for a large sparse symmetric positive definite matrix A. It is shown that an efficient solution of this problem is obtained by using a sparse approximate inverse…
To better understand the algebra $\mathcal{M}_n$ of all $n\times n$ complex matrices, we explore the class of accretive matrices. This class has received renowned attention in recent years due to its role in complementing those results…
Let $A_i$ and $B_i$ be positive definite matrices for all $i=1,\cdots,m.$ It is shown that $$\left|\left|\sum_{i=1}^m(A_i^2\sharp…
For a tuple $A=(A_1,\ A_2,\ ...,\ A_n)$ of elements in a unital Banach algebra ${\mathcal B}$, its projective joint spectrum $P(A)$ is the collection of $z\in {\mathbb C}^n$ such that $A(z)=z_1A_1+z_2A_2+\cdots +z_nA_n$ is not invertible.…
In this work, we consider symmetric positive definite pencils depending on two parameters. That is, we are concerned with the generalized eigenvalue problem $A(x)-\lambda B(x)$, where $A$ and $B$ are symmetric matrix valued functions in…
Resolvent compositions were recently introduced as monotonicity-preserving operations that combine a set-valued monotone operator and a bounded linear operator. They generalize in particular the notion of a resolvent average. We analyze the…
In this paper, we present new results on holomorphically accretive mappings and their resolvents defined on the open unit ball of a complex Banach space. We employ a unified approach to examine various properties of non-linear resolvents by…