Hilbert–Schmidt operator
In mathematics, a Hilbert–Schmidt operator, named after David Hilbert and Erhard Schmidt, is a bounded operator formula_1 that acts on a Hilbert space formula_2 and has finite Hilbert–Schmidt norm where formula_4 is an orthonormal basis. The index set formula_5 need not be countable. However, the sum on the right must contain at most countably many non-zero terms, to have meaning. This definition is independent of the choice of the orthonormal basis. In finite-dimensional Euclidean space, the Hilbert–Schmidt norm formula_6 is identical to the Frobenius norm. Section::::||·|| is well defined. The Hilbert–Schmidt  norm does not depend on the choice of orthonormal basis. Indeed, if formula_7 and formula_8 are such bases, then If formula_10 then formula_11 As for any bounded operator, formula_12 Replacing formula_13 with formula_14 in the first formula, obtain formula_15 The independence follows. Examples. An important class of examples is provided by Hilbert–Schmidt integral operators. Every bounded operator with a finite-dimensional range (these are called operators of finite rank) is a Hilbert–Schmidt operator. The identity operator on a Hilbert space is a Hilbert–Schmidt operator if and only if the Hilbert space is finite-dimensional. Given any formula_16  and "formula_17" in "formula_18", define formula_19 by formula_20, which is a continuous linear operator of rank 1 and thus a Hilbert–Schmidt operator; moreover, for any bounded linear operator "formula_21" on "formula_18" (and into "formula_18"), formula_24. If formula_25 is a bounded compact operator with eigenvalues formula_26 of formula_27, where each eigenvalue is repeated as often as its multiplicity, then "formula_28" is Hilbert–Schmidt if and only if formula_29, in which case the Hilbert–Schmidt norm of "formula_28" is formula_31. If formula_32, where formula_33 is a measure space, then the integral operator formula_34 with kernel "formula_35" is a Hilbert–Schmidt operator  and formula_36. Space of Hilbert–Schmidt operators. 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, denoted by "B"("H") or "B"("H"), 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 norm induced by this inner product is the  Hilbert–Schmidt norm under which the space of Hilbert–Schmidt operators is complete (thus making it into a Hilbert space). The space of all bounded linear operators of finite rank (i.e. that have a finite-dimensional range) is a dense subset of the space of Hilbert–Schmidt operators (with the Hilbert–Schmidt norm). The set of Hilbert–Schmidt operators is closed in the norm topology if, and only if, "H" is finite-dimensional.