Theorem of Kadets-Snobar

The Kadets-Snobar theorem , named after MI Kadets and MG Snobar , is a theorem from the mathematical branch of functional analysis . It represents an estimate for minimal norms of projection operators with finite-dimensional image space . This estimate proves to be optimal in a certain way.

Formulation of the sentence

Let it be a real, standardized space and a finite-dimensional subspace . Then there is a continuous , linear operator with ${\displaystyle X}$ ${\displaystyle Y\subset X}$ ${\displaystyle P:X\rightarrow X}$

• ${\displaystyle P\circ P=P}$, ( is a projection operator), ${\displaystyle P}$
• ${\displaystyle \mathrm {Bild} (P)=Y}$, (is the image space of the projection operator ), ${\displaystyle Y}$
• ${\displaystyle \|P\|\leq {\sqrt {\mathrm {dim} (Y)}}}$, (the operator norm is bounded by the square root of the dimension).

Remarks

From Auerbach's lemma one can easily derive an analogous proposition with the estimation . The above sentence by Kadets-Snobar therefore represents an improved assessment of possible projector standards and the question arises as to whether further improvements are possible. In the following it will be explained briefly that the Kadets-Snobar theorem is already optimal in an asymptotic sense. ${\displaystyle \|P\|\leq \mathrm {dim} (Y)}$

One defines for a finite-dimensional subspace of a Banach space ${\displaystyle Y}$ ${\displaystyle X}$

${\displaystyle \lambda (Y,X):=\inf\{\|P\|\,\mid P:X\rightarrow X{\text{ stetiger linearer Projektor mit Bild }}Y\}}$

and calls this quantity the projection constant of relative to . Without restriction, one can understand a finite-dimensional normed space as a subspace of , the sequence space of restricted sequences. Then you can show that ${\displaystyle Y}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle \ell _{\infty }}$

${\displaystyle \lambda (Y):=\sup\{\lambda (Y,X)\mid X{\text{ Oberraum von }}Y\}=\lambda (Y,\ell _{\infty })}$.

This quantity is called the absolute projection constant of . These terms essentially go back to B. Grünbaum . The Kadets-Snobar theorem says precisely that ${\displaystyle Y}$

${\displaystyle \lambda (Y)\leq {\sqrt {n}}}$, for every -dimensional normalized space . ${\displaystyle n}$ ${\displaystyle Y}$

If one understands the Kadets-Snobar theorem as an asymptotic estimate, one obtains

${\displaystyle \limsup _{n\to \infty }{\frac {\lambda (Y_{n})}{\sqrt {n}}}\leq 1}$for every sequence of -dimensional spaces. ${\displaystyle (Y_{n})_{n\in \mathbb {N} }}$ ${\displaystyle n}$

The order of this asymptotic estimation cannot be improved any further, because H. König constructed a sequence of finite-dimensional spaces in 1985, for which the equals sign stands in the above formula. In 1987, König and DR Lewis were able to show that they always did ${\displaystyle {\sqrt {n}}}$

${\displaystyle \lambda (Y)<{\sqrt {\mathrm {dim} (Y)}}}$

applies. If you set

${\displaystyle \lambda _{n}:=\sup\{\lambda (Y)\mid Y{\text{ normierter Raum mit }}\mathrm {dim} (Y)=n\}}$,

so the Kadets-Snobar theorem gives the inequality . The exact determination of the has proven difficult. In 2010 B. Chalmers and G. Lewicki were able to prove the already formulated and later so-called Grünbaum Conjecture . For dimension 2 we have a significantly better estimate than from the Kadets-Snobar theorem, which only provides. ${\displaystyle \lambda _{n}\leq {\sqrt {n}}}$ ${\displaystyle \lambda _{n}}$ ${\displaystyle \lambda _{2}={\tfrac {4}{3}}}$ ${\displaystyle \lambda _{2}={\tfrac {4}{3}}=1,3333\ldots }$ ${\displaystyle \lambda _{2}\leq {\sqrt {2}}=1,4142\ldots }$

Individual evidence

1. ↑ MI Kadets, MG Snobar: Certain functionals on the Minkowski compactum , Mathematical Notes 10 (1971), pages 694-696
2. ^ Hermann König: Eigenvalue Distribution of Compact Operators , Springer-Verlag 1986, ISBN 978-3-0348-6280-6 , Theorem 4. b. 6th
3. ^ R. Meise, D. Vogt: Introduction to functional analysis . Vieweg, 1992, ISBN 3-528-07262-8 , sentence 12.14
4. ^ Albrecht Pietsch : History of Banach Spaces and Linear Operators , Birkhäuser Boston (2007), ISBN 978-0-8176-4367-6 , Section 6.1.1.6
5. ↑ ^{a b} B. Grünbaum: Projection constants , Trans. Amer. Math. Soc. 95 (1960), pp. 451-465
6. ^ Albrecht Pietsch: History of Banach Spaces and Linear Operators , Birkhäuser Boston (2007), ISBN 978-0-8176-4367-6 , Section 6.1.1.7
7. ↑ H. König, DR Lewis: A strict inequality for projection constants Journal of Functional Analysis 74 (2), 1987, pages 328-332
8. ^ B. Chalmers, G. Lewicki: A proof of the Grünbaum conjecture , Studia Mathematica (2010), Volume 200 (2), pp. 103-129