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The Ulam stability of the composition of two Ulam stable operators has been investigated by several authors. Composition of operators is a key concept when speaking about <inline-formula><math display="inline"><semantics><msub><mi>C</mi><mn>0</mn></msub></semantics></math></inline-formula>-semigroups. Examples of <inline-formula><math display="inline"><semantics><msub><mi>C</mi><mn>0</mn></msub></semantics></math></inline-formula>-semigroups formed with Ulam stable operators are known. In this paper, we construct a <inline-formula><math display="inline"><semantics><msub><mi>C</mi><mn>0</mn></msub></semantics></math></inline-formula>-semigroup <inline-formula><math display="inline"><semantics><msub><mrow><mo>(</mo><msub><mi>R</mi><mi>t</mi></msub><mo>)</mo></mrow><mrow><mi>t</mi><mo>≥</mo><mn>0</mn></mrow></msub></semantics></math></inline-formula> on <inline-formula><math display="inline"><semantics><mrow><mi>C</mi><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></semantics></math></inline-formula> such that for each <inline-formula><math display="inline"><semantics><mrow><mi>t</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>, <inline-formula><math display="inline"><semantics><msub><mi>R</mi><mi>t</mi></msub></semantics></math></inline-formula> is Ulam unstable. Moreover, we compute the central moments of <inline-formula><math display="inline"><semantics><msub><mi>R</mi><mi>t</mi></msub></semantics></math></inline-formula> and establish a Voronovskaja-type formula. This enables to prove that <inline-formula><math display="inline"><semantics><mrow><msup><mi>C</mi><mn>2</mn></msup><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></mrow></semantics></math></inline-formula> is contained in the domain <inline-formula><math display="inline"><semantics><mrow><mi>D</mi><mo>(</mo><mi>A</mi><mo>)</mo></mrow></semantics></math></inline-formula> of the infinitesimal generator of the semigroup. We raise the problem to fully characterize the domain <inline-formula><math display="inline"><semantics><mrow><mi>D</mi><mo>(</mo><mi>A</mi><mo>)</mo></mrow></semantics></math></inline-formula>.