\delimiter"026A30C\delimiter"426830ATv,v\delimiter"526930B\delimiter"026A30Cμ\delimiter"026B30Dv\delimiter"026B30D2 for all v implies \delimiter"026B30DT\delimiter"026B30Dμ


Let H be a unitary space, T be a self-adjointPlanetmathPlanetmathPlanetmath linear operatorMathworldPlanetmath and μ0. If |Tv,v|μv2 for all vH then T is a bounded operatorMathworldPlanetmathPlanetmath and Tμ.


We will show that Tvμv for all vH. This is trivial if Tv or v is zero, so assume they are not. Let λ be any positive number.

Tv2 =Tv,Tv

Now if we put λ2=Tvv we get Tv2μTvv hence Tvμv. ∎


F. Riesz and B. Sz-Nagy, Functional AnalysisMathworldPlanetmath, F. Ungar Publishing, 1955, chap VI.

Title \delimiter"026A30C\delimiter"426830ATv,v\delimiter"526930B\delimiter"026A30Cμ\delimiter"026B30Dv\delimiter"026B30D2 for all v implies \delimiter"026B30DT\delimiter"026B30Dμ
Canonical name vertlangleTvvranglevertleqmuVertVVert2ForAllVImpliesVertTVertleqmu
Date of creation 2013-03-22 15:25:33
Last modified on 2013-03-22 15:25:33
Owner Gorkem (3644)
Last modified by Gorkem (3644)
Numerical id 16
Author Gorkem (3644)
Entry type Theorem
Classification msc 46C05