Commuting is an important property in any two-step information merging procedure where the results should not depend on the order in which the single steps are performed. In the case of bisymmetric aggregation operators with the neutral elements, Saminger, Mesiar and Dubois, already reduced characterization of commuting $n$-ary operators to resolving the unary distributive functional equations. And then the full characterizations of these equations are obtained under the assumption that...

Since the common approach to defining membership functions of fuzzy numbers is rather subjective, another, more objective method is proposed. It is applicable in situations where two models, say $M_1$ and $M_2$, share the same uncertain input parameter $p$. Model $M_1$ is used to assess the fuzziness of $p$, whereas the goal is to assess the fuzziness of the $p$-dependent output of model $M_2$. Simple examples are presented to illustrate the proposed approach.

It is well known that, in forward inference in fuzzy logic, the generalized modus ponens is guaranteed by a functional inequality called the law of $T$-conditionality. In this paper, the $T$-conditionality for $T$-power based implications is deeply studied and the concise necessary and sufficient conditions for a power based implication $I^T$ being $T$-conditional are obtained. Moreover, the sufficient conditions under which a power based implication $I^T$ is $T^{*}$-conditional are discussed, this discussions...

We analyze the existence of fuzzy sets of a universe that are convex with respect to certain particular classes of fusion operators that merge two fuzzy sets. In addition, we study aggregation operators that preserve various classes of generalized convexity on fuzzy sets. We focus our study on fuzzy subsets of the real line, so that given a mapping $F:[0,1]\times [0,1]\to [0,1]$, a fuzzy subset, say $X$, of the real line is said to be $F$-convex if for any $x,y,z\in ℝ$ such that $x\le y\le z$, it holds that ${\mu }_X\left(y\right)\ge F({\mu }_X\left(x\right),{\mu }_X\left(z\right))$, where ${\mu }_X:ℝ\to [0,1]$ stands here for the...

A classification of weakly compact multiplication operators on $L\left(L_p\right),$1<p<$,isgiven.ThisanswersaquestionraisedbySaksmanandTylliin1992.Theclassificationinvolvestheconceptof$p$-strictlysingularoperators,andwealsoinvestigatethestructureofgeneral$p$-strictlysingularoperatorson$Lp$.Themainresultisthatifanoperator$T$on$Lp$,$1<p<2$,is$p$-strictlysingularand$T|X$isanisomorphismforsomesubspace$X$of$Lp$,then$X$embedsinto$Lr$forall$r<2$,but$X$neednotbeisomorphictoaHilbertspace.$It is also shown that if $T$ is convolution by a biased coin on $L_p$ of the Cantor group, $1\le p<2$, and $T_{|X}$ is an isomorphism for some reflexive subspace $X$ of $L_p$, then $X$ is isomorphic to a Hilbert space. The case $p=1$ answers a question asked by Rosenthal in 1976.

We introduce the definition of $p$-limited completely continuous operators, $1\le p<\infty$. The question of whether a space of operators has the property that every $p$-limited subset is relative compact when the dual of the domain and the codomain have this property is studied using $p$-limited completely continuous evaluation operators.

In this paper, we obtain new sufficient conditions for the operators $F_{{\alpha }_1,{\alpha }_2,...,{\alpha }_n,\beta }\left(z\right)$ and $G_{{\alpha }_1,{\alpha }_2,...,{\alpha }_n,\beta }\left(z\right)$ to be univalent in the open unit disc $𝒰$, where the functions $f_1,f_2,...,f_n$ belong to the classes $S^{*}(a,b)$ and $𝒦(a,b)$. The order of convexity for the operators $F_{{\alpha }_1,{\alpha }_2,...,{\alpha }_n,\beta }\left(z\right)$ and $G_{{\alpha }_1,{\alpha }_2,...,{\alpha }_n,\beta }\left(z\right)$ is also determined. Furthermore, and for $\beta =1$, we obtain sufficient conditions for the operators $F_n\left(z\right)$ and $G_n\left(z\right)$ to be in the class $𝒦(a,b)$. Several corollaries and consequences of the main results are also considered.

We show that for a linear space of operators $ℳ\subseteq ℬ({\mathscr{H}}_1,{\mathscr{H}}_2)$ the following assertions are equivalent. (i) $ℳ$ is reflexive in the sense of Loginov-Shulman. (ii) There exists an order-preserving map $\Psi =({\psi }_1,{\psi }_2)$ on a bilattice $\mathrm{Bil}\left(ℳ\right)$ of subspaces determined by $ℳ$ with $P\le {\psi }_1(P,Q)$ and $Q\le {\psi }_2(P,Q)$ for any pair $(P,Q)\in \mathrm{Bil}\left(ℳ\right)$, and such that an operator $T\in ℬ({\mathscr{H}}_1,{\mathscr{H}}_2)$ lies in $ℳ$ if and only if ${\psi }_2(P,Q)T{\psi }_1(P,Q)=0$ for all $(P,Q)\in \mathrm{Bil}\left(ℳ\right)$. This extends the Erdos-Power type characterization of weakly closed bimodules over a nest algebra to reflexive spaces.

The aim of this paper is to characterize the $L^p-L^q$ boundedness of two classes of integral operators from $L^p(𝒰,\mathrm{d}{V}_{\alpha })$ to $L^q(𝒰,\mathrm{d}{V}_{\beta })$ in terms of the parameters $a$, $b$, $c$, $p$, $q$ and $\alpha$, $\beta$, where $𝒰$ is the Siegel upper half-space. The results in the presented paper generalize a corresponding result given in C. Liu, Y. Liu, P. Hu, L. Zhou (2019).

Fuzzy algebra is a special type of algebraic structure in which classical addition and multiplication are replaced by maximum and minimum (denoted $\oplus$ and $\otimes$, respectively). The eigenproblem is the search for a vector $x$ (an eigenvector) and a constant $\lambda$ (an eigenvalue) such that $A\otimes x=\lambda \otimes x$, where $A$ is a given matrix. This paper investigates a generalization of the eigenproblem in fuzzy algebra. We solve the equation $A\otimes x=\lambda \otimes B\otimes x$ with given matrices $A,B$ and unknown constant $\lambda$ and vector $x$. Generalized eigenvectors...