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Register a new userMaximal monotonicity, conjugation and the duality product
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Recently, the authors studied the connection between each maximal monotone operator T and a family H(T) of convex functions. Each member of this family characterizes the operator and satisfies two particular inequalities. The aim of this paper is to establish the converse of the latter fact. Namely, that every convex function satisfying those two particular inequalities is associated to a unique maximal monotone operator.
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We prove that the principal pivot transform (also known as the partial inverse, sweep operator, or exchange operator in various contexts) maps matrices with positive imaginary part to matrices with positive imaginary part. We show that the principal pivot transform is matrix monotone by establishing Hermitian square representations for the imaginary part and the derivative.
We provide an approach to maximal monotone bifunctions based on the theory of representative functions. Thus we extend to nonreflexive Banach spaces recent results due to A.N. Iusem and, respectively, N. Hadjisavvas and H. Khatibzadeh, where sufficient conditions guaranteeing the maximal monotonicity of bifunctions were introduced. New results involving the sum of two monotone bifunctions are also presented.
The maximal correlation coefficient is a well-established generalization of the Pearson correlation coefficient for measuring non-linear dependence between random variables. It is appealing from a theoretical standpoint, satisfying Renyis axioms for a measure of dependence. It is also attractive from a computational point of view due to the celebrated alternating conditional expectation algorithm, allowing to compute its empirical version directly from observed data. Nevertheless, from the outset, it was recognized that the maximal correlation coefficient suffers from some fundamental deficiencies, limiting its usefulness as an indicator of estimation quality. Another well-known measure of dependence is the correlation ratio which also suffers from some drawbacks. Specifically, the maximal correlation coefficient equals one too easily whereas the correlation ratio equals zero too easily. The present work recounts some attempts that have been made in the past to alter the definition of the maximal correlation coefficient in order to overcome its weaknesses and then proceeds to suggest a natural variant of the maximal correlation coefficient as well as a modified conditional expectation algorithm to compute it. The proposed dependence measure at the same time resolves the major weakness of the correlation ratio measure and may be viewed as a bridge between these two classical measures.
Let $Dinmathbb{N}$, $qin[2,infty)$ and $(mathbb{R}^D,|cdot|,dx)$ be the Euclidean space equipped with the $D$-dimensional Lebesgue measure. In this article, the authors establish the Fefferman-Stein decomposition of Triebel-Lizorkin spaces $dot{F}^0_{infty,,q}(mathbb{R}^D)$ on basis of the dual on function set which has special topological structure. The function in Triebel-Lizorkin spaces $dot{F}^0_{infty,,q}(mathbb{R}^D)$ can be written as the certain combination of $D+1$ functions in $dot{F}^0_{infty,,q}(mathbb{R}^D) bigcap L^{infty}(mathbb{R}^D)$. To get such decomposition, {bf (i),} The authors introduce some auxiliary function space $mathrm{WE}^{1,,q}(mathbb R^D)$ and $mathrm{WE}^{infty,,q}(mathbb{R}^D)$ defined via wavelet expansions. The authors proved $dot{F}^{0}_{1,q} subsetneqq L^{1} bigcup dot{F}^{0}_{1,q}subset {rm WE}^{1,,q}subset L^{1} + dot{F}^{0}_{1,q}$ and $mathrm{WE}^{infty,,q}(mathbb{R}^D)$ is strictly contained in $dot{F}^0_{infty,,q}(mathbb{R}^D)$. {bf (ii),} The authors establish the Riesz transform characterization of Triebel-Lizorkin spaces $dot{F}^0_{1,,q}(mathbb{R}^D)$ by function set $mathrm{WE}^{1,,q}(mathbb R^D)$. {bf (iii),} We also consider the dual of $mathrm{WE}^{1,,q}(mathbb R^D)$. As a consequence of the above results, the authors get also Riesz transform characterization of Triebel-Lizorkin spaces $dot{F}^0_{1,,q}(mathbb{R}^D)$ by Banach space $L^{1} + dot{F}^{0}_{1,q}$. Although Fefferman-Stein type decomposition when $D=1$ was obtained by C.-C. Lin et al. [Michigan Math. J. 62 (2013), 691-703], as was pointed out by C.-C. Lin et al., the approach used in the case $D=1$ can not be applied to the cases $Dge2$, which needs some new methodology.
Let $sigma(A)$, $rho(A)$ and $r(A)$ denote the spectrum, spectral radius and numerical radius of a bounded linear operator $A$ on a Hilbert space $H$, respectively. We show that a linear operator $A$ satisfying $$rho(AB)le r(A)r(B) quadtext{ for all bounded linear operators } B$$ if and only if there is a unique $mu in sigma (A)$ satisfying $|mu| = rho(A)$ and $A = frac{mu(I + L)}{2}$ for a contraction $L$ with $1insigma(L)$. One can get the same conclusion on $A$ if $rho(AB) le r(A)r(B)$ for all rank one operators $B$. If $H$ is of finite dimension, we can further decompose $L$ as a direct sum of $C oplus 0$ under a suitable choice of orthonormal basis so that $Re(C^{-1}x,x) ge 1$ for all unit vector $x$.
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