Squares of symmetric operators

2024/03/03 by Arlinskii, Yury
#47B44 #FOS: Mathematics #Functional Analysis (math.FA) #Primary 47B25 #Secondary 47A20

paper · doi:10.48550/arxiv.2403.01473

Abstract

Using the approach proposed in [5] , in an infinite-dimensional separable complex Hilbert space we give abstract constructions of families \\mathcal T_z\_\rm Im z>0 of closed densely defined symmetric operators with the properties: (I) the domain of \mathcal T_z² is a core of \mathcal T_z, (II) the domain of \mathcal T_z² is dense but note a core of \mathcal T_z, (III) the domain of \mathcal T_z² is nontrivial but non-dense. For this purpose a class of maximal dissipative operators is defined and studied. The case \rm dom \mathcal T_z²=\0\ has been considered in [5]. Given a densely defined closed symmetric operator S, in terms of the intersection of the domain of S with \rm ran (S-λI) and the projection of the domain of the adjoint S^* on \rm ran (S-λI), λ∈\mathbb C∖\mathbb R, necessary and sufficient conditions for the cases (I)--(III) related to the domain of S², are obtained.

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