Let $H = L\_2(S)$ be the complex Hilbert space over $S$ with the counting measure. (There might be another term for this concept, but) I define a continuous linear operator $L$ on $H$ with matrix representation $A$ to be Frobenius-finite if and only if $\displaystyle\sum\_{(i,j) \: \in \: S \times S} |a\_{i,j}|^2 < \infty$
  
  

1.   
 Is Frobenius-finiteness invariant under unitary similarity?
  
  

2.   
 If yes, is there a categorical (i.e., not using the concept of matrix representation) characterization of when a continuous linear operator is Frobenius-finite?
  
  

3.   
 Is there another term for what I am calling Frobenius-finite?



"Frobenius-finite" linear operators on a Hilbert Space
Since $\sum\_{(i,j)∈S×S}|a\_{i ,j}|^2 =$ Trace$(A^\*A)$ the answers to 1. and 2. are both affirmative, and as has already been said, the answer to 3. is "Hilbert-Schmidt."


