Hilbert–Schmidt operator - Imported Knowlpedia Content

Definition

Let $H$ and $K$ be Hilbert spaces. A bounded operator $T:H\to K$ is Hilbert–Schmidt if, for one—and hence every— orthonormal basis $(e_i)$ of $H$ ,

\sum_i\lVert Te_i\rVert_K^2<\infty.

The basis-independent quantity

\lVert T\rVert_{\mathrm{HS}} =\left(\sum_i\lVert Te_i\rVert_K^2\right)^{1/2}

is the Hilbert–Schmidt norm. The Hilbert–Schmidt operators form $\mathcal S^2(H,K)$ , the $p=2$ Schatten class. Every Hilbert–Schmidt operator is compact, but a compact operator need not be Hilbert–Schmidt. The condition measures square-summability across orthogonal input directions and is independent of all choices of basis.

Equivalent characterizations

For $T:H\to K$ , the following conditions are equivalent:

$T$ is Hilbert–Schmidt;
its singular values satisfy $\sum_n s_n(T)^2<\infty$ ;
the positive operator $T^*T$ has finite trace.

With the trace understood as the basis-independent sum of diagonal coefficients,

\lVert T\rVert_{\mathrm{HS}}^2 =\operatorname{Tr}(T^*T) =\operatorname{Tr}(TT^*).

These formulas also show that $T$ is Hilbert–Schmidt exactly when $T^*$ is.

Hilbert-space and ideal structure

For $S,T\in\mathcal S^2(H,K)$ , the formula

\langle S,T\rangle_{\mathrm{HS}}=\operatorname{Tr}(T^*S)

makes $\mathcal S^2(H,K)$ a Hilbert space. If $A$ and $B$ are bounded operators of compatible sizes, then

\lVert ATB\rVert_{\mathrm{HS}} \leq \lVert A\rVert\,\lVert T\rVert_{\mathrm{HS}}\,\lVert B\rVert.

Consequently, $\mathcal S^2(H)$ is a two-sided operator ideal. The product of two Hilbert–Schmidt operators is trace class, a fact behind many operator-valued integral formulas.

Examples and near misses

On $\ell^2$ , the diagonal operator $T(x_n)=(a_nx_n)$ is Hilbert–Schmidt exactly when $(a_n)\in\ell^2$ , and then $\lVert T\rVert_{\mathrm{HS}}=\lVert(a_n)\rVert_{\ell^2}$ . Every finite-rank operator is Hilbert–Schmidt.

The diagonal operator with $a_n=n^{-1/2}$ is compact but not Hilbert–Schmidt. Thus convergence of singular values to zero is insufficient; their squares must be summable.

Canonical scope and legacy entry

This general $H\to K$ definition is the canonical corpus entry for Hilbert–Schmidt operators. The older Shale-paper Hilbert–Schmidt entry records the same notion only in the specialized $H\to H$ setting and is retained for compatibility with links in that project. New general-purpose content should link to the present functional-analysis entry.

References

Barry Simon, Trace Ideals and Their Applications, 2nd ed., American Mathematical Society, 2005. DOI record. Relevant: Chapters 1–2 on Hilbert–Schmidt operators, trace ideals, and ideal inequalities.
Rajendra Bhatia, Matrix Analysis, Springer, 1997. DOI record. Relevant: Chapter IV on unitarily invariant norms and Schatten classes.