Shadow class

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 . ${\displaystyle \ell ^{p}}$

definition

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 ${\displaystyle T\colon H\rightarrow G}$ ${\displaystyle (s_{n})_{n}}$ ${\displaystyle s_{n}\rightarrow 0}$ ${\displaystyle (e_{n})_{n}}$ ${\displaystyle H}$ ${\displaystyle (f_{n})_{n}}$ ${\displaystyle G}$

• ${\displaystyle \textstyle Tx=\sum _{n=1}^{\infty }s_{n}\langle x,e_{n}\rangle f_{n}}$applies to all and ${\displaystyle x\in H}$
• the operators for converge against in the operator norm . ${\displaystyle \textstyle \sum _{n=1}^{N}s_{n}\langle \cdot ,e_{n}\rangle f_{n}}$ ${\displaystyle N\to \infty }$ ${\displaystyle T}$

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 . ${\displaystyle (s_{n})_{n}}$ ${\displaystyle T}$ ${\displaystyle s_{n}(T)}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle T}$ ${\displaystyle T^{*}T\in L(H)}$

For is the pth shadow class of compact operators from to through ${\displaystyle 1\leq p<\infty }$ ${\displaystyle H}$ ${\displaystyle G}$

${\displaystyle {\mathcal {S}}_{p}(H,G)\,:=\,\{T:H\rightarrow G;\,T\,{\rm {kompakt}},\,(s_{n}(T))_{n}\in \ell ^{p}\}}$

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: ${\displaystyle \ell ^{p}}$ ${\displaystyle p}$ ${\displaystyle T\in {\mathcal {S}}_{p}(H,G)}$ ${\displaystyle p}$

${\displaystyle \|T\|_{p}:=\left(\sum _{n=1}^{\infty }s_{n}(T)^{p}\right)^{\frac {1}{p}}}$

The -norm of the operator is exactly the -norm of the associated sequence of singular values ​​of the operator. ${\displaystyle p}$ ${\displaystyle \ell ^{p}}$

For the case one writes abbreviated . Often only these rooms are called shadow classes. ${\displaystyle G=H}$ ${\displaystyle {\mathcal {S}}_{p}(H):={\mathcal {S}}_{p}(H,H)}$

Special cases

For the space corresponds to the set of trace class operators . ${\displaystyle p=1}$ ${\displaystyle {\mathcal {S}}_{1}(H,G)}$

For corresponds to the Hilbert space of the Hilbert-Schmidt operators . ${\displaystyle p=2}$ ${\displaystyle {\mathcal {S}}_{2}(H,G)}$

properties

• 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 . ${\displaystyle \ell ^{p}}$ ${\displaystyle {\mathcal {S}}_{p}(H)}$ ${\displaystyle p}$ ${\displaystyle p\leq q}$ ${\displaystyle \|\cdot \|_{p}\geq \|\cdot \|_{q}}$ ${\displaystyle {\mathcal {S}}_{p}(H)\subset {\mathcal {S}}_{q}(H)}$ ${\displaystyle \|T\|\leq \|T\|_{p}}$ ${\displaystyle \|T\|}$ ${\displaystyle T}$

• ${\displaystyle {\mathcal {S}}_{p}(H)}$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 . ${\displaystyle T\in {\mathcal {S}}_{p}(H)}$ ${\displaystyle A,B\in L(H)}$ ${\displaystyle H}$ ${\displaystyle ATB\in {\mathcal {S}}_{p}(H)}$ ${\displaystyle \|ATB\|_{p}\leq \|A\|\|T\|_{p}\|B\|}$ ${\displaystyle L(H)}$

• 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 . ${\displaystyle 1<p,q<\infty }$ ${\displaystyle {\tfrac {1}{p}}+{\tfrac {1}{q}}=1}$ ${\displaystyle T\in {\mathcal {S}}_{p}(H)}$ ${\displaystyle S\in {\mathcal {S}}_{q}(H)}$ ${\displaystyle TS}$ ${\displaystyle Sp(TS)\leq \|T\|_{p}\|S\|_{q}}$ ${\displaystyle S\in {\mathcal {S}}_{q}(H)}$ ${\displaystyle T\mapsto Sp(TS)}$ ${\displaystyle \psi _{S}}$ ${\displaystyle {\mathcal {S}}_{p}(H)}$ ${\displaystyle S\mapsto \psi _{S}}$ ${\displaystyle {\mathcal {S}}_{q}(H)}$ ${\displaystyle {\mathcal {S}}_{p}(H)}$ ${\displaystyle {\mathcal {S}}_{p}(H)\,'\cong {\mathcal {S}}_{q}(H)}$ ${\displaystyle 1<p<\infty }$ ${\displaystyle {\mathcal {S}}_{1}(H)}$ ${\displaystyle {\mathcal {S}}_{1}(H)}$

swell

• 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 .