ON UNBOUNDED SUBNOMAL OPERATORS

• Published : 1993.02.01

Abstract

In this paper we will extend some notions of bounded linear operators to some unbounded linear operators. Let H be a complex separable Hilbert space and let B(H) denote the algebra of bounded linear operators. A closed densely defind linear operator S in H, with domain domS, is called subnormal if there is a Hilbert space K containing H and a normal operator N in K(i.e., $N^{*}$N=N $N^*/)such that domS .subeq. domN and Sf=Nf for f .mem. domS. we will show that the Radjavi and Rosenthal theorem holds for some unbounded subnormal operators; if$S_{1}$and$S_{2}$are unbounded subnormal operators on H with dom$S_{1}$= dom$S^{*}$$_{1} and dom S_{2}=dom S^{*}$$_{2}$and A .mem. B(H) is injective, has dense range and$S_{1}$A .coneq. A$S^{*}$$_{2}, then S_{1} and S_{2} are normal and S_{1}.iden. S^{*}$$_{2}$.2}$.X>.