Theoretical Foundations of Functional Data Analysis, with an Introduction to Linear Operators

4.5 Trace class operators

In the previous section, we introduced HS operators that provide an important subset of the compact operators. Another such subset is the trace class or nuclear operators that we examine in this section.

Let be any bounded linear operator on some Hilbert space . By Theorem 3.4.3, we can define the square root of as a bounded, nonnegative, self-adjoint operator such that .

Definition 4.5.1

Let for separable Hilbert spaces and . Then, is trace class if for some CONS of , the quantity

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is finite. In this case, is said to be the trace norm of .

An argument ...