(1). There is a natural isomorphism between the spaces $scrHotimes scrH^*$ and $scrL(scrH,scrK)$ in which the elementary tensor $kotimes h^*$corresponds to the linear map that takes a vector $u$ of $scrH$ to $sef{h,u}k$. This linear transformation has rank one and all rank one transformations can be obtained in this way.
(2). An explicit transformation of this isomorphism $varphi$ is outlined below. Let $e_1,cdots,e_n$ be an orthonormal basis for $scrH$ and for $scrH^*$. Let $f_1,cdots,f_m$ be an orthonormal basis of $scrK$. Identify each element of $scrL(scrH,scrK)$ with it matrix with respect to these bases. Let $E_{ij}$ be the matrix all whose entries are zero except the $(i,j)$-entry, which is $1$. Show that $varphi(f_iotimes e_j)=E_{ij}$ for all $1leq ileq m$, $1leq jleq n$. Thus, if $A$ is any $m imes n$ matrix with entries $a_{ij}$, then $$ex varphi^{-1}(A)=sum_{i,j}a_{ij}(f_iotimes e_j) =sum_{i,j}(Ae_j)otimes e_j. eex$$
(3). the space $scrL(scrH,scrK)$ is a Hilbert space with inner product $$ex sef{A,B}= r A^*B. eex$$ The set $E_{ij}$, $1leq ileq m$, $1leq jleq n$ is an orthonormal basis for this space. Show that the map $varphi$ is a Hilbert space isomorphism; i.e., $$ex sef{varphi^{-1}(A),varphi^{-1}(B)} =sef{A,B},quadforall A,B. eex$$
Solution.
(1). $$eex a{rcl} scrKotimes scrH^*& o&scrL(scrH,scrK)\ kotimes h^*&mapsto&sex{umapsto sef{h,u}k}. ea eeex$$ On the other hand, if $fin scrL(scrH,scrK)$ is of rank one, then there exists some $0 eq vin scrK$ such that $$ex f(u)=a_uv. eex$$ Since $$eex ea a_{bu}v=f(bu)=ba_uv a a_{bu}=ba_u,\ a_{u_1+u_2}v=f(u_1+u_2)=a_{u_1}v+a_{u_2}v& a a_{u_1+u_2}=a_{u_1}+a_{u_2}, eea eeex$$ we have $$ex scrH i umapsto a_uin bC eex$$ is linear, and thus there exists some $hin scrH$ such that $$ex a_u=sef{h,u} a f(u)=sef{h,u}k. eex$$
(2). As noticed in (1), $$ex varphi(f_iotimes e_j)(e_k)=sef{e_j,e_k}f_i=delta_{jk}f_i, eex$$ and thus $$ex varphi(f_iotimes e_j)(e_1,cdots,e_n) =(f_1,cdots,f_m)E_{ij}. eex$$
(3). $$eex ea sef{A,B}&=sum_{i,j} ar a_{ji}b_{ji},\ sef{E_{ij},E_{kl}} &=sum_{p,q}delta_{pi}delta_{qj}cdot delta_{pk}delta_{ql}\ &=delta_{ik}delta_{jl}sum_{p,q}delta_{pi}delta_{qj},\ sef{varphi^{-1}(A),varphi^{-1}(B)} &=sum_{j,k} sef{(Ae_j)otimes e_j,(Be_k)otimes e_k}\ &=sum_{j,k} sef{Ae_j,Be_k}sef{e_j,e_k}\ &=sum_{j,k} sef{Ae_j,Be_j}\ &=sum_{i,j}ar a_{ij}b_{ij}\ &=sef{A,B}. eea eeex$$