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In this paper we present how spectral properties of certain linear operatorsvary when operators are considered in different Hilbert spaces having commondense domain as the space of polynomials in one real variable with complexcoefficients. This is done taking differential operator and matrix operatorrepresentation of dilation operator $E_{p,d}$ dilating $p$, a polynomialsequence, by $d$, a non-constant sequence of non-zero complex scalars. For thepurpose of identification of $E_{p,d}$ with formal differential operator, whichincludes finite order differential operators as special cases, we deriveconditions under which a formal differential operator has a polynomial sequenceas a sequence of eigen functions corresponding to non-zero eigen values. As aconsequence we get, for instance,  $$ ((1/72)x^4-3x) y^{(4)}-x y^{(2)}-(1/3)x y^{(1)} + (4/3)y.$$ does not haveany polynomial sequence as such a sequence of eigen functions, since no fourthorder polynomial is such an eigen svector.

Effect of change in inner product on spectral analysis of certain linear operators in Hilbert Spaces arising from orthogonal polynomials