Hilbert-Schmidt operator

In mathematics , a Hilbert-Schmidt operator (after David Hilbert and Erhard Schmidt ) is a continuous linear operator on a Hilbert space for which a certain number, the Hilbert-Schmidt norm , is finite. The Hilbert-Schmidt class , that is, the set of all these operators, forms a Banach algebra with the Hilbert-Schmidt norm , which is at the same time a Hilbert space. Hilbert-Schmidt operators can be characterized by infinite-dimensional matrices.

Motivation and Definition

Let and two orthonormal bases be in the Hilbert space . be a continuous linear operator on . Then applies ${\displaystyle (e_{i})_{i}}$ ${\displaystyle (f_{i})_{i}}$ ${\displaystyle H}$ ${\displaystyle A}$ ${\displaystyle H}$

${\displaystyle \sum _{i}\|Ae_{i}\|^{2}\,=\,\sum _{i,k}|\langle Ae_{i},f_{k}\rangle |^{2}\,=\,\sum _{i,k}|\langle e_{i},A^{*}f_{k}\rangle |^{2}\,=\,\sum _{k}\|A^{*}f_{k}\|^{2}}$.

By using two equal orthonormal bases,, this calculation shows that the left hand side remains unchanged when replaced by . This also applies to the right side. If you replace there with at different orthonormal bases and note , you can see that the size is independent of the selected orthonormal base. If this quantity is finite, a Hilbert-Schmidt operator is called and ${\displaystyle (e_{i})_{i}\,=\,(f_{i})_{i}}$ ${\displaystyle A}$ ${\displaystyle A^{*}}$ ${\displaystyle A}$ ${\displaystyle A^{*}}$ ${\displaystyle A^{**}=A}$ ${\displaystyle \textstyle \sum _{i}\|Ae_{i}\|^{2}}$ ${\displaystyle A}$

${\displaystyle \|A\|_{2}:=\left(\sum _{i}\|Ae_{i}\|^{2}\right)^{\frac {1}{2}}}$

is his Hilbert-Schmidt norm . Instead you can also find the spelling . ${\displaystyle \|A\|_{2}}$ ${\displaystyle \|A\|_{HS}}$

The Hilbert-Schmidt class , that is, the set of all Hilbert-Schmidt operators , is closed with regard to the algebraic operations of addition, multiplication and adjoint . So it is an algebra and is denoted by. ${\displaystyle H}$ ${\displaystyle HS(H)}$

An operator between two Hilbert spaces is called the Hilbert-Schmidt operator if for an orthonormal basis of is finite. Similar to the above, it is considered that this number is independent of the special choice of the orthonormal basis, and the square root of this number is also denoted by . ${\displaystyle A\colon H_{1}\rightarrow H_{2}}$ ${\displaystyle \textstyle \sum _{i}\|Ae_{i}\|^{2}}$ ${\displaystyle (e_{i})_{i}}$ ${\displaystyle H_{1}}$ ${\displaystyle \|A\|_{HS}}$

Infinite matrices

Sets in determining an orthonormal basis, it can be any continuous linear operator on an infinite matrix with interpret. is uniquely determined by this matrix and the selected orthonormal basis, because it is mapped to. It applies . Therefore, the Hilbert-Schmidt operators are precisely those continuous, linear operators whose matrix coefficients can be summed to the square. With the help of the Hölder inequality , the submultiplicativity of the Hilbert-Schmidt norm results , that is . The Hilbert-Schmidt norm therefore generalizes the Frobenius norm to the case of infinite-dimensional Hilbert spaces. ${\displaystyle H}$ ${\displaystyle (a_{i,j})_{i,j}}$ ${\displaystyle a_{i,j}=\langle Ae_{j},e_{i}\rangle }$ ${\displaystyle A}$ ${\displaystyle Ae_{i}}$ ${\displaystyle \textstyle \sum _{j}\langle Ae_{i},e_{j}\rangle e_{j}}$ ${\displaystyle \textstyle \sum _{i,j}|a_{i,j}|^{2}\,=\,\|A\|_{2}^{2}}$ ${\displaystyle \textstyle \|AB\|_{2}\leq \|A\|_{2}\|B\|_{2}}$

Integral operators

Many Fredholm integral operators are Hilbert-Schmidt operators. Namely, let be a bounded operator from to , then it can be shown that a Hilbert-Schmidt operator is exactly if there is an integral kernel with ${\displaystyle T\in L(L^{2}([0,1]),L^{2}([0,1]))}$ ${\displaystyle L^{2}([0,1])}$ ${\displaystyle L^{2}([0,1])}$ ${\displaystyle T}$ ${\displaystyle k\in L^{2}([0,1]\times [0,1])}$

${\displaystyle T(x)(s)=\int _{0}^{1}k(s,t)x(t)\mathrm {d} t}$

almost everywhere . In this case the Hilbert-Schmidt norm of and the norm of coincide, so it applies ${\displaystyle T}$ ${\displaystyle L^{2}}$ ${\displaystyle k}$

${\displaystyle \|T\|_{HS}=\left(\int _{0}^{1}\int _{0}^{1}|k(s,t)|^{2}\mathrm {d} s\mathrm {d} t\right)^{\frac {1}{2}}=\|k\|_{L^{2}}.}$

An analogous statement also applies to any dimension spaces instead of the unit interval.

HS (H) as Hilbert space

The product of two Hilbert-Schmidt operators is always a track class operator . If and are two Hilbert-Schmidt operators, then the Hilbert-Schmidt operators are defined by a scalar product on the space. becomes a Hilbert space with this scalar product and it is , d. H. the Hilbert-Schmidt norm is a Hilbert space norm. In the finite-dimensional case, this Hilbert-Schmidt scalar product corresponds to the Frobenius scalar product for matrices. ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle \langle A,B\rangle :=Sp(B^{*}A)}$ ${\displaystyle HS(H)}$ ${\displaystyle \|A\|_{2}={\sqrt {\langle A,A\rangle }}}$

HS (H) as Banach algebra

With the Hilbert-Schmidt norm, the operator algebra is not only a Hilbert space, but also a Banach algebra because of the inequality . is a two-sided ideal in the algebra of all continuous linear operators on H, and it is for all , . Every Hilbert-Schmidt operator is a compact operator . Therefore, a two-sided ideal in the C ^* -algebra of compact operators on , lies close in respect. The operator norm . The trace class is a two-sided, thick Ideal to contain. So you have the inclusions ${\displaystyle HS(H)}$ ${\displaystyle \|AB\|_{2}\leq \|A\|_{2}\|B\|_{2}}$ ${\displaystyle HS(H)}$ ${\displaystyle B(H)}$ ${\displaystyle \|BAC\|_{2}\leq \|B\|\cdot \|A\|_{2}\cdot \|C\|}$ ${\displaystyle A\in HS(H)}$ ${\displaystyle B,C\in B(H)}$ ${\displaystyle HS(H)}$ ${\displaystyle K(H)}$ ${\displaystyle H}$ ${\displaystyle HS(H)}$ ${\displaystyle K(H)}$ ${\displaystyle N(H)}$ ${\displaystyle HS(H)}$

${\displaystyle N(H)\subset HS(H)\subset K(H)\subset B(H)}$.

Except and itself contains no further closed two-sided ideals. The algebra of the Hilbert-Schmidt operators is simple in this sense, it forms the basic building block of the structural theory of the H ^* -algebras . ${\displaystyle \{0\}}$ ${\displaystyle HS(H)}$ ${\displaystyle \|\cdot \|_{2}}$

See also

• The Hilbert-Schmidt operators are a special case of a shadow class .

literature

• RV Kadison , JR Ringrose : Fundamentals of the Theory of Operator Algebras , 1983