A doubt about tensor product on Hilbert Spaces - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T11:55:28Z http://mathoverflow.net/feeds/question/105244 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/105244/a-doubt-about-tensor-product-on-hilbert-spaces A doubt about tensor product on Hilbert Spaces ashade 2012-08-22T14:59:29Z 2012-08-22T17:21:53Z <p>An operator is a bounded (i.e., continuous) linear transformation between Hilbert spaces. Let $\mathcal{B}[\mathcal{H}]$ be the set of all operators in the Hilbert space $\mathcal{H}$.</p> <p>Let $\mathcal{H}$ and $\mathcal{K}$ be any two Hilbert spaces. Consider $\mathcal{C}$ be the class of all strict contractions on $\mathcal{B}[\mathcal{H}]$ and let $\mathcal{L}$ be the class of all contractions on $\mathcal{B}[\mathcal{K}]$.</p> <p>Let $\mathcal{H}\hat{\otimes}\mathcal{K}$ be the tensor product space between the Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$, where $\hat{\otimes}$ denote the tensor product.</p> <p>Question: What is the definition of $\mathcal{C}\hat{\otimes}\mathcal{L}$ on $\mathcal{B}[\mathcal{H}\hat{\otimes}\mathcal{K}]$. On other words, what is the definition for the tensor product of operators classes? Moreover, $T\in\mathcal{C}\hat{\otimes}\mathcal{L}$ if and only if $T=(A\hat{\otimes}B)$, such that $A\in\mathcal{C}$ and $B\in\mathcal{L}$ ?</p> http://mathoverflow.net/questions/105244/a-doubt-about-tensor-product-on-hilbert-spaces/105251#105251 Answer by Nik Weaver for A doubt about tensor product on Hilbert Spaces Nik Weaver 2012-08-22T17:21:53Z 2012-08-22T17:21:53Z <p>There is no standard definition of the tensor product of subsets. If you want to take a tensor product of closed subspaces $E$ of $B(H)$ and $F$ of $B(K)$ there are a variety of choices (there's a nice survey at hrcak.srce.hr/file/1655). The simplest is probably the so-called "spatial" tensor product defined as the closed linear span of all operators of the form $A \otimes B \in B(H \otimes K)$ with $A \in E$ and $B \in F$.</p> <p>I would probably just define $C \otimes L$ to be the set of contractions in $B(H) \otimes B(K) \cong B(H\otimes K)$.</p>