A characterization of the spaces $S^{k/k+1}_{1/k+1}$ by means of holomorphic semigroups

The Gel’fand–Shilov spaces $\mathfrak{S}_\alpha ^\beta ,{\alpha = 1}/ {(k + 1)},{\beta = k} /{(k + 1)}$, are special cases of a general type of test function spaces introduced by de Graaf. We give a self-adjoins operator so that the test functions in those $\mathfrak{S}_\alpha ^\beta $ spaces can be expanded in terms of the eigenfunctions of that self-adjoins operator.