TraceClassOperator

A trace class operator is a mathematical concept found within the field of functional analysis, particularly in the context of operator theory. It's a generalization of finite-dimensional linear operators, which are matrices.

More specifically, an operator $A$ on a Hilbert space $H$ is said to be in the trace class (or is a trace class operator) if it is a compact operator for which the trace can be well-defined.

Here are the precise conditions:

1. Compactness: $A$ is a compact operator if, for every bounded sequence $(x_n)$ in $H$, there exists a subsequence $(x_{n_k})$ such that $(Ax_{n_k})$ is a convergent sequence in $H$.

2. Trace Class: Suppose $A$ is a compact operator on $H$. We say $A$ is of trace class if there exists an orthonormal basis $(e_n)$ for $H$ such that the series

$$\sum_{n=1}^{\infty} |Ae_n|^2$$

converges. In this case, the trace of $A$, denoted $Tr(A)$, is defined by the absolutely convergent series

$$Tr(A) = \sum_{n=1}^{\infty} \langle Ae_n, e_n \rangle$$ where $\langle , \rangle$ denotes the inner product in $H$.

Note: An important feature of the trace class operators is that their trace is independent of the choice of orthonormal basis. That is, if $A$ is of trace class and $(e_n)$ and $(f_n)$ are orthonormal bases for $H$, then

$$\sum_{n=1}^{\infty} \langle Ae_n, e_n \rangle = \sum_{n=1}^{\infty} \langle Af_n, f_n \rangle$$

Properties of Trace Class Operators:

1. Trace class operators form a two-sided ideal in the algebra of all bounded linear operators. That is, if $A$ is a trace class operator and $B$ is any bounded linear operator, then both $AB$ and $BA$ are trace class.

2. If $A$ and $B$ are trace class operators, then their sum $A + B$ and their product $AB$ are also trace class, and

$$Tr(A + B) = Tr(A) + Tr(B)$$

and

$$Tr(AB) = Tr(BA)$$ hold.