# A doubt about tensor product on Hilbert Spaces

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 $B\in\mathcal{L}$ ?

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If the answer to the last question is yes (I have to admit I don't know this) then it should be formulated more carefully, e.g. $T$ is in that set iff there exist $A$ and $B$ with that property. Note that $A\hat{\otimes}B=rA\hat{\otimes}1/rB$ and your classes are not closed under multiplication with scalars (if a contraction is what I guess that it is, namely an operator of norm less than $1$). – user20266 Aug 22 '12 at 16:12