Schatten class operator

The shadow classes, named after Robert shadows are special algebras of operators, which are studied in the mathematical branch of functional analysis. They share many properties with the following rooms.

Definition

Is a compact linear operator between infinite-dimensional ( in the finite breaks the sequence from ) Hilbert spaces, so there is a decreasing sequence of non-negative real numbers and orthonormal sequences in and so that

  • Holds for all and
  • The operators for the operator norm converge to.

This is the so-called Schmidt- representation. The number sequence is in contrast to the orthonormal sequences uniquely determined by. One therefore writes for the -th follower and called this number and the th singular value of. It can be shown that the squares of these numbers are the monotonically decreasing eigenvalue positive result of the compact and the operator.

For the p-th shadow class of compact operators from to is by

Defined. Here is the sequence space of the -th power summable sequences. For one defines the norm of the operator just by this standard on the following:

The norm of the operator is exactly the norm of the associated sequence of singular values ​​of the operator.

In the event you write for short. Often called just these spaces shadow classes.

Special cases

For corresponds to the space of the set of trace class operators.

For corresponds to the Hilbert space of the Hilbert-Schmidt operators.

Properties

  • The shadow classes share many properties with the rooms. is a Banach space with the norm. For valid and therefore. In addition, we always have, with the operator norm of is.
  • Is the operator multiplication even a Banach algebra with isometric involution, where the involution is the adjunction. Are and continuous linear operators on, it is and it is. The shadow classes are therefore two -sided ideals in.
  • Be with conjugated numbers. If then and so the product is a trace class operator, and it is. Therefore, each defined by a continuous linear functional on. It can be shown that the figure is an isometric isomorphism of the dual space of, or shortly. So you have here a very similar situation as in the follow areas. In particular, the shadow classes are reflexive, they are even uniformly convex. As in the following areas, this is not the case for. The conditions for this are described in the article trace class operator in more detail.

Swell

  • R. shadow: Norm Ideals of Completely Continuous Operators. Results of mathematics and its applications, 2nd episode, 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
  • Normed space
  • Functional Analysis
Robert Schatten Operatornorm Isometry Involution (mathematics) Hermitian adjoint Conjugate index
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