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)