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:
“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)
“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)