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 Frobeniusfinite if and only if $\displaystyle\sum_{(i,j) \: \in \: S \times S} a_{i,j}^2 < \infty$
1.
Is Frobeniusfiniteness 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 Frobeniusfinite?
3.
Is there another term for what I am calling Frobeniusfinite?

1$\begingroup$ I wish people wouldn't use "categorical" as a synonym for "nice & algebraiclooking & coordinate invariant"... $\endgroup$– Yemon ChoiSep 17 '10 at 5:05

$\begingroup$ (In case you were still wondering what happens when $S$ is uncountable  one of your replies to Bill Johnson seemed to imply you thought this could be an issue  I left a comment to his answer. Perhaps you might have received a less terse response if you'd given a bit more detail in your question about what courses you have & haven't taken in analysis, since it's rather hard to infer that from your profile page.) $\endgroup$– Yemon ChoiSep 18 '10 at 9:29
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 "HilbertSchmidt."

$\begingroup$ Although in infinite dimensions, making sure that intuitive calculations with trace actually work can be a little tricky for the firsttime user/learner  at least that's my dim recollection $\endgroup$ Sep 17 '10 at 5:27

2$\begingroup$ Well, perhaps in general that's so, but here the terms are all positive, and that simplifies things a lot. $\endgroup$ Sep 17 '10 at 5:38

$\begingroup$ @Dick Palais: Down voting because, IMO, such elementary questions are not appropriate for MO and should not be answered. I intentionally gave no argument but only enough info so that the OP could easily find the answers. $\endgroup$ Sep 17 '10 at 11:45

6$\begingroup$ With all due respect, I would like to differ with you. It was a perfectly natural question for a nonexpert and I think it is only polite to give an appropriate hint that will allow an OP to figure out an answer for him or herself rather than telling him to go look it up in some book or Wikipedia. My answer was under two lines shorter than your own (and a lot less insulting). $\endgroup$ Sep 17 '10 at 12:49
Google hilbert schmidt operator.

$\begingroup$ Well, that certainly answers #3, but the pages that brings up either assume the space is separable or do not conclude that the result is independent of basis, so they don't quite answer #1. (And they certainly don't answer #2.) $\endgroup$– user5810Sep 17 '10 at 2:28

2$\begingroup$ Look at most any book on operator theory or Banach space theory to find that the answers to (1) and (2) are yes and yes. Your question is pretty elementary for MO and I will now vote to close. $\endgroup$ Sep 17 '10 at 2:35

$\begingroup$ Are they elementary enough that there is an online reference for (1) and (2)? $\endgroup$– user5810Sep 17 '10 at 2:39

$\begingroup$ I don't think MO is meant to be the Open University... $\endgroup$ Sep 17 '10 at 5:04

$\begingroup$ Any operator satisfying your "Frobeniusfinite" condition is supported on $T \times T$ where $T$ is a countable subset of $S$. Hence once you have looked up the separable case, the supposedly inseparable case should follow without undue worry or need for consultation $\endgroup$ Sep 17 '10 at 5:07