Hilbert-Schmidt norm

The norm of a linear operator $T$ acting from a Hilbert space $H$ into a Hilbert space $H_1$, given by $$ |T| = \left({\sum_{\alpha\in A} \Vert Te_\alpha \Vert^2}\right)^{1/2} \,, $$ where $\{e_\alpha : \alpha \in A \}$ is an orthonormal basis in $H$. The Hilbert–Schmidt norm satisfies all the axioms of a norm and is independent of the choice of the basis. Its properties are: $\Vert T \Vert \le |T|$, $|T| = |T^*|$, $|T_1T_2| \le \Vert T_1\Vert \cdot |T_2|$, where $\Vert T\Vert$ is the operator norm of $T$ in the Hilbert space. If $H_1 = H$, then $$ |T|^2 = \sum_{\alpha,\beta\in A} (Te_\alpha,e_\beta)^2 \ . $$

References

[1]	N. Dunford, J.T. Schwartz, "Linear operators. Spectral theory" , 2 , Interscience (1963)
[2]	I.M. Gel'fand, N.Ya. Vilenkin, "Generalized functions. Applications of harmonic analysis" , 4 , Acad. Press (1968) (Translated from Russian)

Comments

A Hilbert–Schmidt operator, or operator of Hilbert–Schmidt class, is one for which the Hilbert–Schmidt norm is well-defined: it is necessarily a compact operator.

References

[a1]	N.I. Akhiezer, I.M. Glazman, "Theory of linear operators in Hilbert space" , 1–2 , Pitman (1981) (Translated from Russian)