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}$

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

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 \ |}$${\ displaystyle L (H)}$
• Be with conjugated numbers . If then and , then the product is a trace class operator and it holds . Each therefore defined by a continuous linear functional on . One can show that the mapping is an isometric isomorphism of onto the dual space of , or in short . So here, too, the conditions are very similar to those in the sequence spaces. In particular, the shadow classes are for reflexive , they are even uniformly convex . As with the sequence rooms, this is not the case. The relationships for are described in more detail in the article Lane Class Operator .${\ displaystyle 1 ${\ displaystyle {\ tfrac {1} {p}} + {\ tfrac {1} {q}} = 1}$ ${\ displaystyle T \ in {\ mathcal {S}} _ {p} (H)}$${\ 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 ${\ 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 .