Norm Conditions for Separability in $${\mathbb M}_m\otimes {\mathbb M}_n$$ M m ⊗ M n

Tsuyoshi Ando ()
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Tsuyoshi Ando: Hokkaido University (Emeritus)

A chapter in Analysis and Operator Theory, 2019, pp 35-45 from Springer

Abstract: Abstract An element $$\mathbf{S}$$ S of the tensor product $${\mathbb M}_m\otimes {\mathbb M}_n$$ M m ⊗ M n is said to be separable if it admits a (separable) decomposition $$ \mathbf{S}\ =\ \sum _pX_p\otimes Y_p \quad \exists \ \ 0 \le X_p \in {\mathbb M}_m,\ \exists \ 0 \le Y_p \in {\mathbb M}_n. $$ S = ∑ p X p ⊗ Y p ∃ 0 ≤ X p ∈ M m , ∃ 0 ≤ Y p ∈ M n . This decomposition is not unique. We present some conditions on suitable norms of $$\mathbf{S}$$ S which guarantee its separability. Even when separability of $$\mathbf{S}$$ S is guaranteed by some method, its separable decomposition itself is difficult to construct. We present a general condition which makes it possible to find a way of an explicit separable decomposition.

Date: 2019
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Persistent link: https://EconPapers.repec.org/RePEc:spr:spochp:978-3-030-12661-2_2

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DOI: 10.1007/978-3-030-12661-2_2

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