Concept

Hilbert–Schmidt operator

Summary
In mathematics, a Hilbert–Schmidt operator, named after David Hilbert and Erhard Schmidt, is a bounded operator A \colon H \to H that acts on a Hilbert space H and has finite Hilbert–Schmidt norm |A|^2_{\operatorname{HS}} \ \stackrel{\text{def}}{=}\ \sum_{i \in I} |Ae_i|^2_H, where {e_i: i \in I} is an orthonormal basis. The index set I 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 |\cdot|_\text{HS} is identical to the Frobenius norm. ||·||HS is well defined The Hilbert–Schmidt norm does not depend on the choice of orthonormal basis. Indeed, if {e_i}{i\in I} and {f_j}{j\in I} are such bases, then
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