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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.

A Characterization of the Spaces $\mathfrak{S}_{{1/{k + 1}}}^{{k /{k + 1}}} $ by Means of Holomorphic Semigroups