Hilbert–Schmidt operator

In mathematics, a Hilbert-Schmidt operator ( by David Hilbert and Erhard Schmidt) is a continuous linear operator on a Hilbert space, for a certain number, the Hilbert-Schmidt norm is finite. The Hilbert-Schmidt class, i.e. the set of all of these operators, together with the Hilbert-Schmidt norm a Banach, which is a Hilbert space simultaneously. Hilbert - Schmidt operators can be characterized by infinite-dimensional matrices.

Motivation

Let and be two orthonormal bases in Hilbert space. is a continuous linear operator on H. Then

By using two same orthonormal bases used, this bill shows that the left side remains unchanged when is replaced by. This then also applies to the right side. Substituting there through at different orthonormal bases and observed, one can see that the size is independent of the chosen orthonormal basis. If this size is finite, then is called a Hilbert-Schmidt operator and

Is its Hilbert-Schmidt norm. Instead we find the spelling.

The Hilbert-Schmidt class, i.e. the set of all the Hilbert - Schmidt operators to H is completed in terms of algebraic operations of addition, multiplication, or adjunction. So it is an algebra and is denoted by.

An operator between two Hilbert spaces is called Hilbert-Schmidt operator if is an orthonormal basis of finite. Similar to the above considered to be that number of the particular choice of the orthonormal basis is independent, and refers to the square root of this number is also counted.

Infinite matrices

If we specify an orthonormal basis, so you can definitely continuous linear operator as an infinite matrix conceive with. is uniquely determined by this matrix and the selected orthonormal basis, because is mapped to. It is true. Therefore, the Hilbert - Schmidt operators are precisely those continuous linear operators whose matrix coefficients are square summable. With the help of Hölder 's inequality gives the Submultiplikativität the Hilbert-Schmidt norm, ie. The Hilbert-Schmidt norm generalizes therefore the Frobenius norm to the case of infinite-dimensional Hilbert spaces.

Integral operators

Many of Fredholm integral operators are Hilbert - Schmidt operators. Indeed, let a bounded operator from to, then it can be shown that a Hilbert-Schmidt operator if and only if there is an integral kernel with

Almost everywhere. In this case, If the Hilbert-Schmidt norm and the norm of match of, it is therefore

A similar statement is also true for arbitrary measure spaces instead of the unit interval.

HS ( H) as a Hilbert space

The product of two Hilbert - Schmidt operators is always a trace class operator. And two of the Hilbert - Schmidt operators, it is therefore defined by an inner product on the area of the Hilbert - Schmidt operators. is with this scalar product is a Hilbert space and it is, ie the Hilbert-Schmidt norm is a Hilbert space norm. In the finite dimensional case, this Hilbert-Schmidt scalar product corresponds to the Frobenius inner product for matrices.

HS ( H) as a Banach algebra

The operators -algebra with the Hilbert-Schmidt norm not only be a Hilbert space, but because of the inequality at the same time is a Banach algebra. is a two-sided ideal in the algebra of all continuous linear operators on H, and it applies to everyone. Each Hilbert-Schmidt operator is a compact operator. Therefore, a two-sided ideal in the C * - algebra of compact operators on, here is dense in in the operator norm. The trace class is included as a two -sided ideal in tight. Therefore, it has the inclusions

.

Save and itself contains no other - closed two -sided ideals. The algebra of Hilbert - Schmidt operators is simple in this sense, it forms the basic building block of the structure theory of H * - algebras.