Go Hirasawa

Hyers-Ulam-Rassias stability of Jordan homomorphisms on Banach algebras

We prove that a Jordan homomorphism from a Banach algebra into a semisimple commutative Banach algebra is a ring homomorphism. Using a signum effectively, we can give a simple proof of the Hyers-Ulam-Rassias stability of a Jordan homomorphism between Banach algebras. As a direct corollary, we show that to each approximate Jordan homomorphism from a Banach algebra into a semisimple commutative Banach algebra there corresponds a unique ring homomorphism near to.

HYERS-ULAM STABILITY OF A CLOSED OPERATOR IN A HILBERT SPACE

We give some necessary and su-cient conditions in order that a closed operator in a Hilbert space into another have the Hyers-Ulam stability. Moreover, we prove the existence of the stability constant for a closed operator. We also determine the stability constant in terms of the lower bound.

Additively spectral-radius preserving surjections between unital semisimple commutative Banach algebras

AbstractLet A and B be unital, semisimple commutative Banach algebras with the maximal ideal spaces MA and MB, respectively, and let r(a) be the spectral radius of a. We show that if T: A → B is a surjective mapping, not assumed to be linear, satisfying r(T(a) + T(b)) = r(a + b) for all a; b ∈ A, then there exist a homeomorphism φ: MB → MA and a closed and open subset K of MB such that

GER-TYPE AND HYERS-ULAM STABILITIES FOR THE FIRST-ORDER LINEAR DIFFERENTIAL OPERATORS OF ENTIRE FUNCTIONS

\widehat{T\left( a \right)}\left( y \right) = \left\{ \begin{gathered} \widehat{T\left( e \right)}\left( y \right)\hat a\left( {\phi \left( y \right)} \right) y \in K \hfill \\ \widehat{T\left( e \right)}\left( y \right)\overline {\hat a\left( {\phi \left( y \right)} \right)} y \in M_\mathcal{B} \backslash K \hfill \\ \end{gathered} \right.

The Hyers–Ulam and Ger Type Stabilities of the First Order Linear Differential Equations

for all a ∈ A, where e is unit element of A. If, in addition,

A perturbation of ring derivations on Banach algebras

\widehat{T\left( e \right)} = 1