On the Orthogonal Stability of the Pexiderized Quadratic Equation

The Hyers--Ulam stability of the conditional quadratic functional equation of Pexider type f(x+y)+f(x-y)=2g(x)+2h(y), x\perp y is established where \perp is a symmetric orthogonality in the sense of Ratz and f is odd.Comment: 10 pages, Latex; Changed conten

Approximately Vanishing of Topological Cohomology Groups

In this paper, we establish the Pexiderized stability of coboundaries and cocycles and use them to investigate the Hyers--Ulam stability of some functional equations. We prove that for each Banach algebra $A$, Banach $A$-bimodule $X$ and positive integer $n, H^n(A,X)=0$ if and only if the $n$-th cohomology group approximately vanishes.Comment: 18 pages, minor correction

Approximate Homomorphisms of Ternary Semigroups

A mapping $f:(G_1,[ ]_1)\to (G_2,[ ]_2)$ between ternary semigroups will be called a ternary homomorphism if $f([xyz]_1)=[f(x)f(y)f(z)]_2$. In this paper, we prove the generalized Hyers--Ulam--Rassias stability of mappings of commutative semigroups into Banach spaces. In addition, we establish the superstability of ternary homomorphisms into Banach algebras endowed with multiplicative norms.Comment: 10 page

Orthogonalities and functional equations

In this survey we show how various notions of orthogonality appear in the theory of functional equations. After introducing some orthogonality relations, we give examples of functional equations postulated for orthogonal vectors only. We show their solutions as well as some applications. Then we discuss the problem of stability of some of them considering various aspects of the problem. In the sequel, we mention the orthogonality equation and the problem of preserving orthogonality. Last, but not least, in addition to presenting results, we state some open problems concerning these topics. Taking into account the big amount of results concerning functional equations postulated for orthogonal vectors which have appeared in the literature during the last decades, we restrict ourselves to the most classical equations

Matrix KSGNS construction and a Radon–Nikodym type theorem

In this paper, we introduce the concept of completely positive matrix of linear maps on Hilbert A-modules over locally C∗-algebras and prove an analogue of Stinespring theorem for it. We show that any two minimal Stinespring representations for such matrices are unitarily equivalent. Finally, we prove an analogue of the Radon–Nikodym theorem for this type of completely positive n×n matrices. © 2017 Royal Dutch Mathematical Society (KWG

Matrix KSGNS construction and a Radon–Nikodym type theorem

In this paper, we introduce the concept of completely positive matrix of linear maps on Hilbert A-modules over locally C∗-algebras and prove an analogue of Stinespring theorem for it. We show that any two minimal Stinespring representations for such matrices are unitarily equivalent. Finally, we prove an analogue of the Radon–Nikodym theorem for this type of completely positive n×n matrices. © 2017 Royal Dutch Mathematical Society (KWG