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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.

190 Bounded Linear Operators on a Hilbert Space is an orthogonal projection of L2(R) onto the subspace of functions with support contained in A. A frequently encountered case is that of projections onto a one-dimensional

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