Hilbert–Schmidt operator

In mathematics, a Hilbert–Schmidt operator, named for David Hilbert and Erhard Schmidt, is a bounded operator A on a Hilbert space H with finite Hilbert–Schmidt norm

where ∥   ∥ is the norm of H, { e i : i ∈ I } an orthonormal basis of H, and Tr is the trace of a nonnegative self-adjoint operator. Note that the index set need not be countable. This definition is independent of the choice of the basis, and therefore

for A i , j = ⟨ e i , A e j ⟩ and ∥ A ∥ 2 the Schatten norm of A for p = 2. In Euclidean space ∥   ∥ H S is also called Frobenius norm, named for Ferdinand Georg Frobenius.

The product of two Hilbert–Schmidt operators has finite trace class norm; therefore, if A and B are two Hilbert–Schmidt operators, the Hilbert–Schmidt inner product can be defined as

The Hilbert–Schmidt operators form a two-sided *-ideal in the Banach algebra of bounded operators on H. They also form a Hilbert space, which can be shown to be naturally isometrically isomorphic to the tensor product of Hilbert spaces

where H^∗ is the dual space of H.

The set of Hilbert–Schmidt operators is closed in the norm topology if, and only if, H is finite-dimensional.

An important class of examples is provided by Hilbert–Schmidt integral operators.

Hilbert–Schmidt operators are nuclear operators of order 2, and are therefore compact.