Properties
| 1. | If A is a non-negative self-adjoint, A is trace class if and only if Tr(A) < ∞. Therefore a self adjoint operator A is trace class if and only if its positive part A+ and negative part A− are both trace class. (The positive and negative parts of a self adjoint operator are obtained via the continuous functional calculus.) |
| 2. | The trace is a linear functional over the space of trace class operators, i.e.
The bilinear map is an inner product on the trace class; the corresponding norm is called the Hilbert-Schmidt norm. The completion of the trace class operators in the Hilbert-Schmidt norm are called the Hilbert-Schmidt operators. |
| 3. | If is bounded and is trace class, and are also trace class and
besides, under the same hypothesis, |
| 4. | If is trace class, then one can define the Fredholm determinant of
for the elements of the spectrum of ; the trace class condition on guarantees that the infinite product is finite: indeed it also guarantees that if and only if is invertible. |
Read more about this topic: Trace Class
Famous quotes containing the word properties:
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)