This is a very basic question and might be way too easy for MO. I am learning analysis in a very backwards way. This is a question about complex Hilbert spaces but here's how I came to it: I have in the past written a paper about (amongst other things) compact endomorphisms of $p$-adic Banach spaces (and indeed of Banach modules over a $p$-adic Banach algebra), and in this paper I continually used the notion of a "matrix" of an endomorphism as an essential crutch when doing calculations and proofs. I wondered at the time where more "conceptual" proofs existed, and probably they do, but I was too lazy to find them.
Now I find myself learning the basic theory of certain endomorphisms of complex separable Hilbert spaces (continuous, compact, Hilbert-Schmidt and trace class operators) and my instinct, probably wrong, is to learn the theory in precisely the same way. So this is the sort of question I find myself asking.
Say $H$ is a separable Hilbert space with orthonomal basis $(e_i)_{iinmathbf{Z}_{geq1}}$. Say $T$ is a continuous linear map $Hto H$. Then $T$ is completely determined by its "matrix" $(a_{ij})$ with $Te_i=sum_ja_{ji}e_j$. But are there completely "elementary" conditions which completely classify which collections of complex numbers $(a_{ij})$ arise as "matrices" of continuous operators?
I will ask a more precise question at the end, but let me, for the sake of exposition, tell you what the the answer is in the $p$-adic world.
In the $p$-adic world, $sum_na_n$ converges iff $a_nto 0$, and life is easy: the answer to the question in the $p$-adic world is that $(a_{ij})$ represents a continuous operator iff
(1) For all $i$, $sum_j|a_{ji}|^2<infty$ (equivalently, $a_{ji}to 0$ as $jtoinfty$), and
(2) there's a universal bound $B$ such that $|a_{ij}|leq B$ for all $i,j$.
[there is no inner product in the $p$-adic case, so no adjoint, and the conditions come out being asymmetric in $i$ and $j$].
See for example pages 8--9 of this paper of mine, although of course this isn't due to me---it's in Serre's paper on compact operators on $p$-adic Banach spaces from the 60s---see Proposition 3 of Serre's paper. In particular, in the $p$-adic world, one can identify the continuous maps $Hto H$ (here $H$ is a $p$-adic Banach space with countable ON basis $(e_i)$) with the collection of bounded sequences in $H$, the identification sending $T$ to $(Te_i)$.
In the real/complex world though, the analogue of this result fails: the sequence $(e_1,e_1,e_1,ldots)$ is a perfectly good bounded sequence, but there is no continuous linear map $Hto H$ sending $e_i$ to $e_1$ for all $i$ (where would $sum_n(1/n)e_n$ go?).
Let's consider the finite rank case, so $T$ is a continuous linear map $Hto H$ with image landing in $mathbf{C}e_1$. Then by Riesz's theorem, $T$ is just "inner product with an element of $H$ and then multiply by $e_1$". Hence we have an additional condition on the $a_{ij}$, namely that $sum_j|a_{ij}|^2<infty$. Furthermore a continuous linear map is bounded, as is its adjoint.
This makes me wonder whether the following is true, or whether this is still too naive:
Q) Say $(a_{ij})$ $(i,jinmathbf{Z}_{geq1})$ is a collection of complex numbers satisfying the following:
There is a real number $B$ such that
1) For all $i$, $sum_j|a_{ij}|^2leq B$
2) For all $j$, $sum_j|a_{ij}|^2leq B$
Then is there a unique continuous linear map $T:Hto H$ with $Te_i=sum_ja_{ji}e_i$?
My guess is that this is still too naive. Can someone give me an explicit counterexample? Or, even better, a correct "elementary" list of conditions characterising the continuous endomorphisms of a Hilbert space?
On the other hand, it clearly isn't a complete waste of time to think about matrix coefficients. For example there's a bijection between Hilbert-Schmidt operators $T:Hto H$ and collections $(a_{ij})$ of complexes with $sum_{i,j}|a_{ij}|^2<infty$, something which perhaps the experts don't use but which I find incredibly psychologically useful.
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