Transcription of Chapter 8 Bounded Linear Operators on a Hilbert Space
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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.
8.2 The dual of a Hilbert space A linear functional on a complex Hilbert space H is a linear map from H to C. A linear functional ’ is bounded, or continuous, if there exists a constant M such that j’(x)j Mkxk for all x 2 H: (8.3) The dual of a Hilbert space 191 The norm of a …
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