Hilbert-Schmidt operator

An operator $A$ acting on a Hilbert space $H$ such that for any orthonormal basis $\{ x _ {i} \}$ in $H$ the following condition is met:

$$\| A \| ^ {2} = \ \sum _ { i } \| Ax _ {i} \| ^ {2} < \infty$$

(however, this need be true for some basis only). A Hilbert–Schmidt operator is a compact operator for which the condition

$$\sum _ { i } | \lambda _ {i} ( A) | ^ {2} \leq \sum _ { i } s _ {i} ^ {2} ( A) = \ \| A \| ^ {2} = \mathop{\rm Tr} ( A ^ {*} A);$$

applies to its $s$- numbers $s _ {i} ( A)$ and its eigen values $\lambda _ {i} ( A)$; here $A ^ {*} A$ is a trace-class operator ( $A ^ {*}$ is the adjoint of $A$ and $\mathop{\rm Tr} C$ is the trace of an operator $C$). The set of all Hilbert–Schmidt operators on a fixed space $A$ forms a Hilbert space with scalar product

$$\langle A, B \rangle = \ \mathop{\rm Tr} ( AB ^ {*} ).$$

If $R _ \lambda ( A) = ( A - \lambda E ) ^ {-} 1$ is the resolvent of $A$ and

$$\mathop{\rm det} _ {2} ( E - zA) = \ \prod _ { i } ( 1 - z \lambda _ {i} ( A)) e ^ {z \lambda _ {i} ( A) }$$

is its regularized characteristic determinant, then the Carleman inequality

$$\left \| \mathop{\rm det} _ {2} \left ( E - { \frac{1} \lambda } A \right ) R _ \lambda ( A) \right \| \leq | \lambda | \mathop{\rm exp} \left [ { \frac{1}{2} } \left ( 1 + \frac{\| A \| ^ {2} }{| \lambda | ^ {2} } \right ) \right ]$$

holds.

A typical representative of a Hilbert–Schmidt operator is a Hilbert–Schmidt integral operator (which explains the origin of the name).

Comments

The $s$- numbers or singular values of $A$ are the (positive) eigen values of the self-adjoint operator $A ^ {*} A$. Instead of Hilbert–Schmidt operator one also says "operator of Hilbert–Schmidt class" . A bounded operator $T$ on a Hilbert space is said to be of trace class if $\sum \langle T \phi _ {j,\ } \psi _ {j} \rangle < \infty$ for arbitrary complete orthonormal systems $\{ \phi _ {j} \}$, $\{ \psi _ {j} \}$. Equivalently, $T$ is of trace class if $\sum s _ {i} ( T) < \infty$. The trace of such an operator is defined as $\sum \langle T \phi _ {j} , \phi _ {j} \rangle$, where $\phi _ {j}$ is any orthonormal basis. The product of two Hilbert–Schmidt operators is of trace class and the converse is also true.

The norm $\| A \|$ in the above article is not the usual operator norm of $A$ but its Hilbert–Schmidt norm.

References

[a1]	M. Reed, B. Simon, "Methods of modern mathematical physics" , 1. Functional analysis , Acad. Press (1972)
[a2]	I.C. Gohberg, S. Goldberg, "Basic operator theory" , Birkhäuser (1977)
[a3]	N.I. Akhiezer, I.M. Glazman, "Theory of linear operators in Hilbert space" , 1–2 , Pitman (1981) (Translated from Russian)