The shadow classes, named after Robert Schatten , are special algebras of operators that are examined in the mathematical sub-area of functional analysis. They share many properties with the sequence spaces .
If there is a compact linear operator between infinite-dimensional (in finite-dimensional the sequence breaks off) Hilbert spaces , there is a monotonically falling sequence of non-negative real numbers with and orthonormal sequences in and in , so that
- applies to all and
- the operators for converge against in the operator norm .
This is the so-called Schmidt representation. In contrast to the orthonormal sequences , the sequence of numbers is clearly determined by . We therefore write for the -th term in the sequence and call this number the -th singular value of . One can show that the squares of these numbers form the monotonically decreasing eigenvalue sequence of the compact and positive operator .
For is the pth shadow class of compact operators from to through
The -norm of the operator is exactly the -norm of the associated sequence of singular values of the operator.
For the case one writes abbreviated . Often only these rooms are called shadow classes.
For the space corresponds to the set of trace class operators .
For corresponds to the Hilbert space of the Hilbert-Schmidt operators .
- The shadow classes have many properties in common with the rooms. is a Banach space with the norm. For true and therefore . Furthermore , where the operator norm is always applies .
- is with the operator multiplication even a Banach algebra with isometric involution , where the involution is the adjunction . If and are continuous linear operators , then it is and it holds . The shadow classes are therefore two-sided ideals in .
- Be with conjugated numbers . If then and , then the product is a trace class operator and it holds . Each therefore defined by a continuous linear functional on . One can show that the mapping is an isometric isomorphism of onto the dual space of , or in short . So here, too, the conditions are very similar to those in the sequence spaces. In particular, the shadow classes are for reflexive , they are even uniformly convex . As with the sequence rooms, this is not the case. The relationships for are described in more detail in the article Lane Class Operator .
- R. Shade: Norm Ideals of Completely Continuous Operators. Results of mathematics and its border areas, 2nd part, ISBN 3-540-04806-5 .
- Dunford, Schwartz: Linear Operators, Part II, Spectral Theory. ISBN 0-471-60847-5 .
- R. Meise, D. Vogt: Introduction to functional analysis. Vieweg, 1992 ISBN 3-528-07262-8 .