Results Math 72 (2017), 875–891
2017 Springer International Publishing AG
published online July 13, 2017
Results in Mathematics
A Factorization Approach to the Extension
Theory of the Tensor Product of Nonnegative
Marcel Roman and Adrian Sandovici
Abstract. Certain characterizations of the Friedrichs and the Kre˘ın von-
Neumann extensions of the tensor product of two nonnegative linear re-
lations A and B in terms of the Friedrichs and the Kre˘ın-von Neumann
extensions of A and B are provided. A characterization of the extremal
extensions of the tensor product of A and B is also given.
Mathematics Subject Classiﬁcation. 47A06, 47A80, 46M05.
Keywords. Hilbert space, linear relation, tensor product.
This note is devoted to tensor products of multi-valued linear operators (lin-
ear relations) in separable complex Hilbert spaces. The main objective is to
investigate the links between the extremal extensions of the tensor product of
two nonnegative linear relations and the extremal extensions of the linear rela-
tions themselves. In particular, the cases of Friedrichs and Kre˘ın von-Neumann
extensions are considered.
More precisely, assume that A and B are two nonnegative linear relations
in the separable complex Hilbert spaces H and K, respectively. Then the tensor
product A ⊗ B of A and B is a nonnegative linear relation in the separable
complex Hilbert space H
⊗H. Its closure is denoted by A
⊗B. It will be proven
that the Friedrichs and the Kre˘ın von-Neumann extensions of A
⊗B are given