Acta Scientiarum Mathematicarum

Trinh Tuan

Le Van Hien

Nguyen Thi Hong Phuong

$$L_p$$-boundedness for weighted Fourier convolution by Hermite polynomial and their applications

加权傅里叶卷积的$L_p$有界性及Hermite多项式的应用

Jānis Cīrulis

Order structure of U-semiabundant semigroups and rings. Part II: two-sided Lawson’s order

U-半丰富半群与环的序结构。第二部分：双边 Lawson 序

Abstract Using works of T. Ando and L. Gurvits, the well-known theorem of P.R. Halmos concerning the existence of unitary dilations for contractive linear operators acting on Hilbert spaces is recast as a result for d -tuples of contractive Hilbert space operators satisfying a certain matrix-positivity condition. Such operator d -tuples satisfying this matrix-positivity condition are called, herein, Toeplitz-contractive, and a characterisation of the Toeplitz-contractivity condition is presented. The matrix-positivity condition leads to definitions of new distance-measures in several variable operator theory, generalising the notions of norm, numerical radius, and spectral radius to d -tuples of operators (commuting, for the spectral radius) in what appears to be a novel, asymmetric way. Toeplitz contractive operators form a noncommutative convex set, and a scaling constant $$c_d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>c</mml:mi> <mml:mi>d</mml:mi> </mml:msub> </mml:math> for inclusions of the minimal and maximal matrix convex sets determined by a stretching of the unit circle $$S^1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:math> across d complex dimensions is shown to exist.

Douglas Farenick

Matrix convexity and unitary power dilations of Toeplitz-contractive operator tuples

矩阵凸性与Toeplitz压缩算子元组的酉幂扩张

Recently the Karcher mean has been extended to the case of probability measures of positive operators on infinite-dimensional Hilbert spaces as the unique solution of a nonlinear operator equation on the convex Banach-Finsler manifold of positive operators. Let $$(\Omega ,\mu )$$ be a probability space, and let $$\tau :\Omega \rightarrow \Omega $$ be a totally ergodic map. The main result of this paper is a new ergodic theorem for functions $$ F\in L^1(\Omega ,\mathbb {P})$$ , where $$\mathbb {P}$$ is the open cone of the strictly positive operators acting on a (separable) Hilbert space. In our result, we use inductive means to average the elements of the orbit, and we prove that almost surely these averages converge to the Karcher mean of the push-forward measure $$F_*(\mu )$$ . From our result, we recover the strong law of large numbers and the "no dice" results proved by the third and fourth authors in the article Strong law of large numbers for the $$L^1$$ -Karcher mean, Journal of Func. Anal. 279 (2020). From our main result, we also deduce an ergodic theorem for Markov chains with state space included in $$\mathbb {P}$$ .

Jorge Antezana

Eduardo Ghiglioni

Yongdo Lim

Miklós Pálfia

Ergodic theorems for the $$L^1$$-Karcher mean

$$L^1$$-Karcher 均值的遍历定理

Lajos Molnár

Foreword

前言

If $\sigma$ is a symmetric mean and $f$ is an operator monotone function on $[0, \infty)$, then $$f(2(A^{-1}+B^{-1})^{-1})\le f(A\sigma B)\le f((A+B)/2).$$ Conversely, Ando and Hiai showed that if $f$ is a function that satisfies either one of these inequalities for all positive operators $A$ and $B$ and a symmetric mean different than the arithmetic and the harmonic mean, then the function is operator monotone. 
In this paper, we show that the arithmetic and the harmonic means can be replaced by the geometric mean to obtain similar characterizations. Moreover, we give characterizations of operator monotone functions using self-adjoint means and general means subject to a constraint due to Kubo and Ando.

Abbreviation	Acta Sci. Math.
Journal Impact	0.44
Quartiles(Global)	MATHEMATICS(Q3)

ISSN	0001-6969, 2064-8316
h-index	16

Publication Information	Publisher: Springer International Publishing AG，Publishing cycle: ，Journal Type: journal，Open Access Journals: No
Basic data	Year of publication: 2009，Proportion of original research papers: 97.22%，Self Citation Rate:20.00%， Gold OA Rate: 10.43%

Acta Scientiarum Mathematicarum

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$$L_p$$-boundedness for weighted Fourier convolution by Hermite polynomial and their applications

Order structure of U-semiabundant semigroups and rings. Part II: two-sided Lawson’s order

Matrix convexity and unitary power dilations of Toeplitz-contractive operator tuples

Ergodic theorems for the $$L^1$$-Karcher mean

Foreword

Order structure of U-semiabundant semigroups and rings. Part II: two-sided Lawson’s order

$$L_p$$-boundedness for weighted Fourier convolution by Hermite polynomial and their applications

Matrix convexity and unitary power dilations of Toeplitz-contractive operator tuples

New characterizations of operator monotone functions

Ergodic theorems for the $$L^1$$-Karcher mean