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A new theory of generalized functions has been developed by one of the authors (de Graaf). In this theory the analyticity domain of each positive self-adjoint unbounded operator $\mathcal{A}$ in a Hilbert space X is regarded as a test space denoted by $\mathcal{S}_{x,\mathcal{A}} $. In the first part of this paper, we consider perturbations $\mathcal{P}$ on $\mathcal{A}$ for which there exists a Hilbert space Y such that $\mathcal{A} + \mathcal{P}$ is a positive self-adjoint operator in Y. In particular, we investigate for which perturbations $\mathcal{P}$ and for which $\nu > 0,S_{X,\mathcal{A}^\nu } \subset \mathcal{S}_{Y,(\mathcal{A} + \mathcal{P})^\nu } $. The second part is devoted to applications. We construct Hankel invariant distribution spaces. The corresponding test spaces are described in terms of the $S_\alpha ^\beta $-spaces introduced by Gel’fand and Shilov. It turns out that the modified Laguerre polynomials establish an uncountable number of bases for the space of even entire functions in ...

Analyticity Spaces of Self-Adjoint Operators Subjected to Perturbations with Applications to Hankel Invariant Distribution Spaces