Іздер класы операторлар және олардың қасиеттері

Compact operator for which a finite trace can be defined
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.

Let be a bounded self adjoint operator on a Hilbert space. Then is trace class if and only if has a pure point spectrum with eigenvalues such that

Mercer's theorem provides another example of a trace class operator. That is, suppose is a continuous symmetric positive definite kernel on , defined as

then the associated Hilbert–Schmidt integral operator is trace class, i. e.,

Recall that the dual of is In the present context, the dual of trace class operators is the bounded operators More precisely, the set is a two sided ideal in So given any operator we may define a continuous linear functional on by This correspondence between bounded linear operators and elements of the dual space of is an isometric isomorphism. It follows that is the dual space of This can be used to define the weak * topology on