On a Hilbert space, $H$, the standard matrix trace can be generalized to a special class of bounded operators on $H$ called the trace class operators. In this expository article on trace class operators, we show how to arrive at theorems about these operators via a well known analogy with $l_p$ spaces.

 

$l_p$ Spaces

First we present well known results on $l_p$ spaces. We say $\lambda := (\lambda_1, \lambda_2,...)$  is a sequence where each $\lambda_n \in \mathbb{C}$. For $1 \leq p \lt \infty$, we define the norm $l_p$-norm of $\lambda$ by:

\begin{equation*} ||\lambda||_p := (\sum_{n=1}^{\infty} |\lambda_n|^p)^{\frac{1}{p}} \end{equation*}

and the corresponding sequence space, $l_p$, by:

\begin{equation} l_p = \{ \lambda = (\lambda_1, \lambda_2, ...) \; | \; ||\lambda||_p \lt \infty \} \end{equation}

One can show that the $l_p$-norm is in fact a norm on $l_p$, and makes it into a Banach space.

For $p = \infty$, we define the norm $l_\infty$-norm of $\lambda$ by:

\begin{equation*} ||\lambda||_\infty := \sup_{n=1,...,\infty} |\lambda_n| \end{equation*}

and the corresponding sequence space, $l_\infty$, by:

\begin{equation} l_\infty = \{ \lambda = (\lambda_1, \lambda_2, ...) \; | \; ||\lambda||_\infty \lt \infty   \} \end{equation}

Similarly, the $l_\infty$-norm makes $l_\infty$ into a Banach space. We additionally let $l^\infty$ be the set of sequences where all but finitely many components are non-zero. Then $l^\infty$ forms a subspace of $l_\infty$. We define $l_0$ to be the closure of $l^\infty$ in $l_\infty$. Note that $\lambda \in l_0$ iff $|\lambda_n| \rightarrow 0$ as $n \rightarrow \infty$.

For $p < \infty$ and $q$ satisfying $\frac{1}{p} + \frac{1}{q} = 1$, one can show that $l_p^* = l_q$, where the map $l_q \rightarrow l_p^*$, sending $\mu \in l_q$ into $\mu^* \in l_p^*$ is given by: \begin{equation} \mu^*(\lambda) = \sum_{n=1}^{\infty} \lambda_n \mu_n, \quad \lambda \in l_p \end{equation}

Additionally, by the same pairing, one can show that $l_0^* = l_1$ and $l_1^* = l_\infty$.

Spaces of operators on $H$

We now study spaces of operators on a Hilbert space $H$. Any given bounded operator, $A \in \mathcal{B}(H)$, is not necessarily diagonalizable, hence there\'s no obvious way to associate a sequence to $A$. However, if $A$ is a compact operator, we can always associate a sequence to $A$ by using its singular value decomposition.

Indeed, if $A$ is compact, it\'s readily seen that $A^* A$ is a non-negative self-adjoint operator, and it can also be shown that $A^* A$ is compact [Reed1980, theorem VI.12]. Thus by the spectral theorem for compact self-adjoint operators [Bollobas1999, theorem 14.3], there exists an (possibly finite) orthonormal basis $(e_n)_{n=1}^{\infty}$ for $(ker A)^\perp$ such that $A^* A$ takes the form:

\begin{equation}A^* A e_n = (\lambda_n)^2 e_n \end{equation}

where $\lambda_n > 0$ are known as the singular values of $A$. Another consequence of the spectral theorem is that $\lambda := (\lambda_1,\lambda_2,...) \in l_0$, and $||\lambda||_\infty = ||A||$.

Since the singular values of a compact operator are in $l_0$, we denote the space of compact operators on $H$ by $\mathcal{B}_0(H)$. For every $p \in [1, \infty)$, motivated by the analogy with $l_p$, we define a norm on $\mathcal{B}_0(H)$ by:

\begin{equation} ||A||_{p} := ||\lambda||_p = (\sum_{n=1}^{\infty} |\lambda_n|^p)^{\frac{1}{p}} \end{equation}

The set of operators where the above sum converges is denoted $\mathcal{B}_p(H)$. Even though many of the functional analytic properties of $l_p$ carry over to $\mathcal{B}_p(H)$, as we will soon see, because the map taking $\mathcal{B}_0(H)$ into $l_\infty$ is non-linear, one has to prove many of these properties manually. In fact, it can be shown that $\mathcal{B}_p(H)$ is a Banach space [Pedersen1989, exercise 3.4.2] with the following special cases: $A \in B_0(H)$ iff $A$ is a compact operator, $A \in B_1(H)$ iff $A$ is a trace-class operator, and $A \in B_2(H)$ iff $A$ is a Hilbert-Schmidt operator. The embeddings of the $l_p$ spaces $l_1 \subset l_2 \subset ... \subset l_0 \subset l_\infty$ have corresponding embeddings $B_1(H) \subset B_2(H) \subset ... \subset B_0(H) \subset B(H)$ of the operator spaces.

For trace-class operators $A \in B_1(H)$, we define the trace of $A$, denoted $tr(A)$, to be

\begin{equation} tr(A) := \sum_{n=1}^{\infty} \lambda_n \lt e_n, f_n \gt \; = \sum_{n=1}^{\infty} \lt e_n, A e_n \gt \end{equation}

where $f_n := \frac{Ae_n}{\lambda_n}$ for non-zero $\lambda_n$, and one readily sees that the above sum converges absolutely. An application of Parseval\'s theorem shows that the second equation for the trace is independent of the orthonormal basis $\{ e_n \}_{n=1}^{\infty}$ (see [Reed1980, theorem VI.18]). This definition for the trace shares several properties with the standard matrix trace (e.g. see [Pedersen1989, section 3.4]).

We also have the generalized Hölder inequality, called the Hölder-von Neumann inequality, which says that for $A \in B_p(H)$, $B \in B_q(H)$ where $q$ is conjugate to $p$, and $p$ satisfies $1 \lt p \lt \infty$, one has

\begin{equation} ||A B||_1 \leq ||A||_p ||B||_q \end{equation}

which generalizes Hölder\'s inequality for $l_p$ spaces [Pedersen1989, exercise 3.4.3]. From this inequality one can see that many of the functional analytic properties of $l_p$ carry over to these operator spaces:  $(B_p(H))^* = B_q(H)$, $(B_0(H))^* = B_1(H)$ and $(B_1(H))^* = B(H)$ [Pedersen1989, section 3.4].

References

[Bollobas1999] Linear Analysis: An Introductory Course (), Cambridge University Press, .
[Pedersen1989] Analysis Now (), Springer-Verlag, .
[Reed1980] Methods of Modern Mathematical Physics: Functional analysis ( and ), Academic Press, .