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单词 ExistenceOfAdjointsOfBoundedOperators
释义

existence of adjoints of bounded operators


Let be a Hilbert spaceMathworldPlanetmath and let T:𝒟(T) be a densely defined linear operatorMathworldPlanetmath.

Theorem - If T is bounded (http://planetmath.org/ContinuousLinearMapping) then its adjointPlanetmathPlanetmathPlanetmath T* is everywhere defined and is also bounded.

Proof : Since T is densely defined and bounded, it extends uniquely to a bounded (everywhere defined) linear operator on , which we denote by T~.

For each z, the function f: defined byf(x)=T~x,z defines a bounded linear functionalMathworldPlanetmath on . By the Riesz representation theoremMathworldPlanetmath there exists u such that

f(x)=x,u

i.e.

T~x,z=x,u.

Since T~ extends T, we also have that for every z there exists u such that

Tx,z=x,ufor everyx𝒟(T).

We conclude that T* is everywhere defined. To see that it is bounded one just needs to check that

supz0T*zz=supz0T*z0|T*z,T*z|T*zzsupz0x0|x,T*z|xz=supz0x0|Tx,z|xzT

where the last inequality comes from the Cauchy-Schwarz inequality and the fact that T is bounded.

Remark - This theorem shows in particular that bounded linear operators T: have bounded adjoints T*:.

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更新时间:2025/5/4 22:47:47