The definition of trace-class operators

Question

A trace-class operators are defined on a separable Hilbert space. However, is it possible to define trace or trace-class operators on an arbitrary Hilbert spaces? I am just curious...

Answer 1

In Chapter 16 of "Introduction to Functional Analysis" by Meise & Vogt (1997), the trace class gets introduced on arbitrary infinite-dimensional Hilbert spaces ($\mathcal G,\mathcal H,\ldots)$. Basically, every compact operator $A:\mathcal G\to\mathcal H$ has a Schmidt representation

$$ Ax=\sum_{n=0}^\infty s_n(A)\langle e_n,x\rangle f_n $$

with a non-negative, decreasing null sequence $(s_n(A))_{n\in\mathbb N_0}$ (singular values) and orthonormal systems $(e_n)_{n\in\mathbb N_0}$ in $\mathcal G$, $(f_n)_{n\in\mathbb N_0}$ in $\mathcal H$. Then the trace class is defined as

$$ \mathcal B_1(\mathcal G,\mathcal H):=\Big\lbrace A:\mathcal G\to\mathcal H\text{ compact}\,|\,\Vert A\Vert_1:=\sum_{n=0}^\infty s_n(A)<\infty\Big\rbrace.\tag1 $$

One then can define a trace on $\mathcal B_1(\mathcal H)$ as usual. Now

• the trace norm $\Vert\cdot\Vert_1$ actually is a norm. (Corollary 16.15 in Meise/Vogt)
• $\mathcal B_1(\mathcal G,\mathcal H)$ is a Banach space with respect to the trace norm. (Corollary 16.24)
• The dual space $\mathcal B_1(\mathcal G,\mathcal H)'$ is isometrically isomorphic to $\mathcal B(\mathcal H,\mathcal G)$ by means of the map $\Psi:B\mapsto\operatorname{tr}(B(\cdot))$. (Proposition 16.26)

One can also show that in the case of separable Hilbert spaces, (1) is equivalent to the usual definition

$$ \mathcal B_1(\mathcal H)=\big\lbrace A\in\mathcal B(\mathcal H)\,|\,\operatorname{tr}\sqrt{A^\dagger A}<\infty \big\rbrace $$

see e.g. Theorem VI.21 in "Methods of Modern Mathematical Physics. I: Functional Analysis" by Reed & Simon (1980).

Answer 2

Now this looks like a separable space: you pick a basis that is infinite but countable. A non separable space has uncountable bases. Now, provided you have a proper measure theory, your sum might generalise to an integral. But as Keith is, I'm eager to see examples.

The reason why is that operators on non-separable spaces are likely to be in type III von Neumann (W*-)algebras. In particular, such non-separable spaces decompose as the direct sum of uncountably many different separable subspaces. The result is that the rank of projectors on these subspaces is infinite, just as the rank of the identity operator is. As a result, it seems difficult to even define a trace on these operators.

But this is mainly gut feeling. So if examples were identified, that would be great.