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The Ulam stability (see, e.g., [4]) for the composition of operators was investigated in the framework of Frechet spaces in [5], where the proofs had a "topological" character. Proofs with a "metric" character were given in [1] in the framework of Banach spaces. A $C_0$-semigroup $(T(t))_{t\geq 0}$ with each $T(t)$ Ulam stable was presented in [1]. The paper [2] is devoted to a $C_0$-semigroup $(R_t)$ such that each $R_t$ with $t>0$ is Ulam unstable. A crucial property was the injectivity of $R_t$, derived from the injectivity of the Weierstrass transform, see [3] In this talk we give details concerning the above facts and add some new results. In particular, two open problems are presented.

[1] Ana-Maria Acu, Ioan Ra\c sa, Ulam stability for the composition of operators, Symmetry 2020, 12, 1159

[2] Ana Maria Acu, Ioan Raşa, A $C_0$-semigroup of Ulam unstable operators, Symmetry 2020, 12, 1844

[3] John A. Baker, Functional Equations and Weierstrass Transform, Results in Mathematics, Vol. 26 (1994), 199-204.

[4]  J. Brzdek,  D. Popa,  I. Raşa, B. Xu,  Ulam stability of operators, In Mathematical Analysis and Its Applications, 1st ed.; Academic Press: Dordrecht, The Netherlands, 2018.

[5] P.S. Johnson, S. Balaji, Hyers-Ulam stability of linear operators in Frechet spaces, Appl. Math. Inf. Sci. 2012, 6, 525–528.