Question

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 $\mathcal{H}$.

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 $\mathcal{B}[\mathcal{K}]$.

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.

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 $B\in\mathcal{L}$ ?

Answer 1

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 $B \in F$.

I would probably just define $C \otimes L$ to be the set of contractions in $B(H) \otimes B(K) \cong B(H\otimes K)$.