Trace class

In mathematics, specifically functional analysis, a trace-class operator is a linear operator for which a trace may be defined, such that the trace is a finite number independent of the choice of basis used to compute the trace. This trace of trace-class operators generalizes the trace of matrices studied in linear algebra. All trace-class operators are compact operators. In quantum mechanics, mixed states are described by density matrices, which are certain trace class operators. Trace-class operators are essentially the same as nuclear operators, though many authors reserve the term "trace-class operator" for the special case of nuclear operators on Hilbert spaces and use the term "nuclear operator" in more general topological vector spaces (such as Banach spaces). Note that the trace operator studied in partial differential equations is an unrelated concept. Suppose is a Hilbert space and a bounded linear operator on which is non-negative (I.e., semi—positive-definite) and self-adjoint. The trace of , denoted by is the sum of the serieswhere is an orthonormal basis of . The trace is a sum on non-negative reals and is therefore a non-negative real or infinity. It can be shown that the trace does not depend on the choice of orthonormal basis. For an arbitrary bounded linear operator on we define its absolute value, denoted by to be the positive square root of that is, is the unique bounded positive operator on such that The operator is said to be in the trace class if We denote the space of all trace class linear operators on H by (One can show that this is indeed a vector space.) If is in the trace class, we define the trace of bywhere is an arbitrary orthonormal basis of . It can be shown that this is an absolutely convergent series of complex numbers whose sum does not depend on the choice of orthonormal basis. When H is finite-dimensional, every operator is trace class and this definition of trace of T coincides with the definition of the trace of a matrix.