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Let $H$ be a separable Hilbert space. Let $(e_i)_i$ be an orthonormal basis. For any bounded linear map $T$ we write, whenever possible

$$\operatorname{tr} T := \sum_{i}^{\infty} \langle T e_i, e_i \rangle$$

Now let $L(H)$ be the set of bounded operators, $N(H)$ the set of nuclear operators. We want to show

$$i: L(H) \rightarrow N(H)' , \quad T \mapsto ( S \mapsto \operatorname{tr}(TS))$$

is an isometric isomorphism.

By the ideal property of the nuclear operators, it is easy to see that $i$ is well-defined, linear and bounded with $|i| \leq 1$.

Recall that the operators $\langle \cdot, e_i \rangle e_j$ form a basis of $N(H)$. You easily see the operator is injective (if the induced functional is zero, it's preimage must have been zero,too) and surjective (simply build up an operator $T \in L(H)$ which induces a given functional in $N(H)$).

Question: How can I finish the proof with showing that $i$ is an isometry?

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2 Answers 2

up vote 2 down vote accepted

I would do it as follows:

  1. For $(x,y) \in H \times H$ define an element of $N(H)$ by $x \,\tilde{\otimes}\, y = \langle \cdot , y \rangle x$ and note that $\|x \, \tilde{\otimes}\,y\|_{1} = \|x\|\,\|y\|$.
  2. For a functional $\varphi \in N(H)'$ define a sesquilinear form on $H$ by $B_{\varphi}(x,y) = \varphi(x \,\tilde{\otimes}\, y)$. The map $\varphi \mapsto B_{\varphi}$ is clearly linear and of norm $\leq 1$ since $|B_{\varphi}(x,y)| = |\varphi(x \, \tilde{\otimes}\,y)| \leq \|\varphi\|\,\|x\|\,\|y\|$ by 1.
  3. Use the version of the Riesz representation theorem stating that every bounded sesquilinear form $B$ is of the form $B(x,y) = \langle x,T y \rangle$ for a unique $T$. Moreover, $\|B\| = \|T\|$. In other words, the map $B \mapsto T$ from bounded sesquilinear forms to $L(H)$ is a linear isometry.
  4. Define $j(\varphi)$ by $\varphi \mapsto B_{\varphi} \mapsto T_{\varphi}$ by combining the maps from 2. and 3. Verify that $ji = 1_{L(H)}$ and $ij = 1_{N(H)'}$. Since both $i$ and $j$ have norm $\leq 1$, they must be isometries.
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+1: I prefer this approach. I somehow didn't notice your answer until I was nearly done writing my answer. We didn't really overlap, but my answer does much less, assuming the rest has been done. –  Jonas Meyer Feb 22 '11 at 20:58
    
@Jonas: Thanks! I think it is good for the OP to have these complementary answers. For once our answers didn't overlap that much... –  t.b. Feb 22 '11 at 21:25

I think it helps to think of the commutative analogue of this, which is the fact that $\ell^\infty$ is the dual of $\ell^1$. If $(a_n)_n\in\ell^\infty$, then the corresponding functional is $(x_n)_n\mapsto\sum_n a_nx_n$. To show that the map from $\ell^\infty$ to the dual of $\ell^1$ does not decrease norm, let $\varepsilon\gt0$, take $m$ such that $|a_m|\gt \|(a_n)_n\|_\infty-\varepsilon$, and let $(x_n)_n\in\ell^1$ be such that $x_m=1$ and $x_n=0$ for $n\neq m$. Then $|\sum_n a_nx_n|=|a_m|=|a_m|\|(x_n)_n\|_1 \geq(\|(a_n)_n\|_\infty-\varepsilon)\|(x_n)_n\|_1$.

In the noncommutative case, you have a bounded operator $T$, and given small $\varepsilon$ you can find a unit vector $e\in H$ such that $\|Te\|\geq\|T\|-\varepsilon$, and a unit vector $f\in H$ such that $\langle Te,f\rangle = \|Te\|$. Consider the rank one operator $S=\langle\cdot,f\rangle e$, which has norm $1$ in $N(H)$. If my calculations are correct, the trace of TS is $\langle Te,f\rangle=\|Te\|$, showing that the norm of $i(T)$ is at least $\|T\|-\varepsilon$.

Note that on Hilbert space, nuclear operators are often called "trace class", and that a proof that $L(H)$ is the dual of the trace class operators can be found in standard texts on operator theory.

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