• Journal of the Chungcheong Mathematical Society
• /
• v.24 no.2
• /
• pp.221-238
• /
• 2011

We prove the Hyers-Ulam stability of generalized Jensen's equations in Banach modules over a unital $C^{\ast}$-algebra. It is applied to show the stability of generalized Jensen's equations in a Hilbert module over a unital $C^{\ast}$-algebra. Moreover, we prove the stability of linear operators in a Hilbert module over a unital $C^{\ast}$-algebra.

• Korean Journal of Mathematics
• /
• v.15 no.2
• /
• pp.135-148
• /
• 2007

We prove the stability of generalized Jensen's equations in a Hilbert module over a unital $C^*$-algebra. This is applied to show the stability of a projection, a unitary operator, a self-adjoint operator, a normal operator, and an invertible operator in a Hilbert module over a unital $C^*$-algebra.

• Bulletin of the Korean Mathematical Society
• /
• v.40 no.1
• /
• pp.53-61
• /
• 2003

We prove the generalized Hyers-Ulam-Rassias stability of the invertible mapping in a Banach module over a unital Banach algebra, and prove the generalized Hyers-Ulam-Rassias stability of the Jensen's functional equations in a Hilbert module over a unital $C^{*}$-algebra.

• Journal of the Korean Mathematical Society
• /
• v.28 no.2
• /
• pp.293-308
• /
• 1991

• Journal of the Korean Mathematical Society
• /
• v.56 no.6
• /
• pp.1599-1611
• /
• 2019

Let $m,n{\in}{\mathbb{N}}$. We represent the additive subgroups of the ring ${\mathbb{Z}}_m{\times}{\mathbb{Z}}_n$, which are also (unital) subrings, and deduce explicit formulas for $N^{(s)}(m,n)$ and $N^{(us)}(m,n)$, denoting the number of subrings of the ring ${\mathbb{Z}}_m{\times}{\mathbb{Z}}_n$ and its unital subrings, respectively. We show that the functions $(m,n){\mapsto}N^{u,s}(m,n)$ and $(m,n){\mapsto}N^{(us)}(m,n)$ are multiplicative, viewed as functions of two variables, and their Dirichlet series can be expressed in terms of the Riemann zeta function. We also establish an asymptotic formula for the sum $\sum_{m,n{\leq}x}N^{(s)}(m,n)$, the error term of which is closely related to the Dirichlet divisor problem.

• Journal of the Chungcheong Mathematical Society
• /
• v.14 no.1
• /
• pp.15-19
• /
• 2001

We prove the Hyers-Ulam-Rassias stability of the Jensen's equation in left Banach B-modules over a unital Banach algebra B.

• Bulletin of the Korean Mathematical Society
• /
• v.48 no.4
• /
• pp.673-688
• /
• 2011

Let $n{\geq}2$ be an even integer. We investigate that if an odd mapping f : X ${\rightarrow}$ Y satisfies the following equation $2_{n-2}C_{\frac{n}{2}-1}rf\(\sum\limits^n_{j=1}{\frac{x_j}{r}}\)\;+\;{\sum\limits_{i_k{\in}\{0,1\} \atop {{\sum}^n_{k=1}\;i_k={\frac{n}{2}}}}\;rf\(\sum\limits^n_{i=1}(-1)^{i_k}{\frac{x_i}{r}}\)=2_{n-2}C_{{\frac{n}{2}}-1}\sum\limits^n_{i=1}f(x_i),$ then f : X ${\rightarrow}$ Y is additive, where $r{\in}R$. We also prove the stability in normed group by using shadowing property and the Hyers-Ulam stability of the functional equation in Banach spaces and in Banach modules over unital C-algebras. As an application, we show that every almost linear bijection h : A ${\rightarrow}$ B of unital $C^*$-algebras A and B is a $C^*$-algebra isomorphism when $h(\frac{2^s}{r^s}uy)=h(\frac{2^s}{r^s}u)h(y)$ for all unitaries u ${\in}$ A, all y ${\in}$ A, and s = 0, 1, 2,....

• Journal of the Chungcheong Mathematical Society
• /
• v.15 no.1
• /
• pp.25-33
• /
• 2002

We prove the generalized Hyers-Ulam-Rassias stability of linear operators in a Hilbert module over a unital $C^*$-algebra.

• Bulletin of the Korean Mathematical Society
• /
• v.39 no.3
• /
• pp.401-410
• /
• 2002

We prove the generalized Hyers-Ulam-Rassias stability of generalized Jensen's functional equations in Banach modules over a unital $C^{*}$-algebra. It is applied to show the stability of algebra homomorphisms between Banach algebras associated with generalized Jensen's functional equations in Banach algebras.

• Bulletin of the Korean Mathematical Society
• /
• v.44 no.4
• /
• pp.753-761
• /
• 2007

In this note we extend the results of [3] concerning subadditive separating maps from A=C(X) to B=C(Y), for compact Hausdorff spaces X and Y, to the case where A and B are regular Banach function algebras(not necessarily unital) with A satisfying Ditkin#s condition. In particular we describe the general form of these maps and get a result on continuity of separating linear functionals.