Chapter 8 Bounded Linear Operators on a Hilbert Space

Chapter8 BoundedLinearOperatorsona HilbertSpaceIn thischapterwe describe someimportant classesof boundedlinearoperatorsonHilbertspaces,in cludingprojections,unitaryoperators,ands elf-adjoint alsoprove theRieszrepresentationtheorem,which characterizestheboundedlinearfunctionals on a Hilbertspace,anddiscussweakconvergencein beginby describingsomealgebraicpropertiesof a linearspaceXsuch thateveryx2 Xcanbe writtenuniquelyasx=y+zwithy2 Mandz2N, thenwe say thatX=M Nis thedirect sumofMandN, andwe callNacomplementarysubspaceofMinX. Thedecompositionx=y+zwithy2 Mandz2 Nis uniqueif andonlyifM\N=f0g. A givensubspaceMhasmany ,ifX=R3andMis a planethroughtheorigin,thenany linethroughtheoriginthatdoes notlieinMis a ,andthedimensionof a complementarysubspaceis N, thenwe de netheprojectionP:X!XofXontoMalongNbyP x=y, wherex=y+zwithy2 Mandz2N. Thisprojectionis Linear ,withranP=MandkerP=N, andsatis esP2=P.

188 Bounded Linear Operators on a Hilbert Space (a) If P : X ! X is a projection, then X = ranP kerP. (b) If X = M N, where M and N are linear subpaces of X, then there is a projection P : X ! X with ranP = M and kerP = N. Proof. To prove (a), we rst show that x 2 ranP if and only if x = Px. If x = Px, then clearly x 2 ranP.

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  Linear, Chapter, Operator, Bounded, Hilbert, Chapter 8 bounded linear operators on a hilbert, Bounded linear operators on a hilbert

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