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
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
Are defined. The sequence space is the sequence that can be summed to the -th power. For one defines the -norm of the operator by this norm on the sequence:
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.
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 .