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

Abstract

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 $S_\mu ^\mu (\frac{1}{2} \leqq \mu \leqq 1)$. For an even entire function $\varphi $ we give necessary and sufficient conditions on the coefficients in the Fourier expansion with respect to each basis such that $\varphi \in S_\mu ^\mu $.