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单词 OperatorNormOfMultiplicationOperatorOnL2
释义

operator norm of multiplication operator on L2


The operator norm of the multiplication operator Mϕ is theessential supremumMathworldPlanetmath of the absolute valueMathworldPlanetmathPlanetmathPlanetmathPlanetmath of ϕ. (This may beexpressed as Mϕop=ϕL.)In particular, if ϕ is essentially unboundedPlanetmathPlanetmath, the multiplicationoperator is unbounded.

For the time being, assume that ϕ is essentially bounded.

On the one hand, the operator norm is bounded by the essentialsupremum of the absolute value because, for any ψL2,

MϕψL2=ψ(x)2ϕ(x)2𝑑μ(x)
(esssupϕ2)ψ(x)2𝑑μ(x)
=(esssup|ϕ|)ψL2

and, hence

Mϕop=supMϕψL2ψL2(esssup|ϕ|).

On the other hand, the operator norm bounds by the essential supremumof the absolute value . For any ϵ>0, the measureMathworldPlanetmathPlanetmath of theset

A={x|ϕ(x)|esssup|ϕ|-ϵ}

is greater than zero. If μ(A)<, set B=A, otherwise letB be a subset of A whose measure is finite. Then, if χB isthe characteristic functionMathworldPlanetmathPlanetmathPlanetmath of B, we have

MϕχBL2=ϕ(x)2χB(x)2𝑑μ(x)
=Bϕ(x)2𝑑μ(x)
μ(B)(esssup|ϕ|-ϵ)

and, hence

Mϕop=supMϕψL2ψL2MϕχBL2χBL2=esssup|ϕ|-ϵ.

Since this is true for every ϵ>0, we must have

Mϕopesssup|ϕ|.

Combining with the inequality in the opposite direction,

Mϕop=esssup|ϕ|.

It remains to consider the case where |ϕ| is essentiallyunbounded. This can be dealt with by a variation on the preceedingargumentPlanetmathPlanetmath.

If ϕ is unbounded, then μ({x|ϕ(x)|R})>0for all R>0. Furthermore, for any R>0, we can find N>R suchthat μ(A)>0, where

A={xN+1|ϕ(x)|N}.

If μ(A)<, set B=A, otherwise letB be a subset of A whose measure is finite. Then, if χB isthe characteristic function of B, we have

MϕχBL2=ϕ(x)2χB(x)2𝑑μ(x)
=Bϕ(x)2𝑑μ(x)
μ(B)N

and, hence

Mϕop=supMϕψL2ψL2MϕχBL2χBL2=NR.

Since this is true for every R, we see that the operator norm isinfiniteMathworldPlanetmathPlanetmath, i.e. the operator is unbounded.

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更新时间:2025/5/4 13:25:02