Hilbert–Schmidt operator

In mathematics, a Hilbert–Schmidt operator, named after David Hilbert and Erhard Schmidt, is a bounded operator ${\displaystyle A\colon H\to H}$ that acts on a Hilbert space ${\displaystyle H}$ and has finite Hilbert–Schmidt norm

$${\displaystyle \|A\|_{\operatorname {HS} }^{2}\ {\stackrel {\text{def}}{=}}\ \sum _{i\in I}\|Ae_{i}\|_{H}^{2},}$$

where ${\displaystyle \{e_{i}:i\in I\}}$ is an orthonormal basis. The index set ${\displaystyle I}$ 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 ${\displaystyle \|\cdot \|_{\text{HS}}}$ is identical to the Frobenius norm.