Abstract Hilbert-Schmidt Operators
Hilbert-Schmidt operators are a special class of compact operators that play an important role in functional analysis and quantum mechanics. They can be thought of as infinite-dimensional analogues of matrices with finite Frobenius norm.
Hilbert-Schmidt Operator
An operator
on a Hilbert space is called Hilbert-Schmidt if its Hilbert-Schmidt norm
e.g.
- In
, the Hilbert–Schmidt norm is identical to the Frobenius norm.
Proposition
The product of two Hilbert-Schmidt operators is a trace class operator. Thus we can define the Hilbert–Schmidt inner product of two Hilbert-Schmidt operators as
Proposition
Suppose
is a complex Hilbert space. Then the space of all Hilbert-Schmidt operators, denoted or , is an two-sided -ideal in the Banach algebra . Moreover, is isometrically isomorphic to the Hilbert space tensor product , where is the conjugate Hilbert space of . This means for each Hilbert-Schmidt operator , we can define a bilinear form
Proof
Integral Operators
Integral Operator
On
, we can define an operator by the formula we call it an integral operator with associated kernel , where is a measurable function such that for almost every .
Proposition
An integral operator
is Hilbert-Schmidt if the associated kernel is an element of .
Proposition
Let
be a Hilbert-Schmidt operator on with kernel . Then
- for every
, and almost every , the function is integrable. is bounded, and . - the adjoint of
is also a Hilbert-Schmidt operator with kernel .