generalized frames in hilbert spaces

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1、BulletinoftheIranianMathematicalSocietyVol.35No.1(2009),pp97-109.GENERALIZEDFRAMESINHILBERTSPACES∗A.NAJATIANDA.RAHIMICommunicatedbyHeydarRadjaviAbstract.Here,wedevelopthegeneralizedframetheory.Wein-troducetwomethodsforgeneratingg-framesofaHilbertspaceH.Thefirstmethodusesboundedlinearoperatorsbetween

2、Hilbertspaces.Thesecondmethodusesboundedlinearoperatorson`2togenerateg-framesofH.Wecharacterizealltheboundedlinearmappingsthattransformg-framesintootherg-frames.Wealsocharacterizesimilarandunitaryequivalentg-framesintermoftherangeoftheirlinearanalysisoperators.Finally,wegeneralizethefundamentalfram

3、eidentitytog-framesandderivesomenewre-sults.1.IntroductionThroughoutthispaper,HandKareseparableHilbertspacesand{Hi:i∈I}isasequenceofseparableHilbertspaces,whereIisasubsetofZ.L(H,Hi)isthecollectionofallboundedlinearoperatorsfromHtoHi,andΛ={Λi∈L(H,Hi):i∈I},Θ={Θi∈L(H,Hi):i∈I}.Λiscalledageneralizedfram

4、eorsimplyg-frameoftheHilbertspaceHwithrespectto{Hi:i∈I}ifforanyvectorf∈H,X(1.1)Akfk2≤kΛfk2≤Bkfk2,ii∈IMSC(2000):Primary:41A58;Secondary:42C15Keywords:Frames,g-frames,g-Rieszbases,g-orthonormalbases.Received:11February2007,Accepted:12June2008∗Correspondingauthorc2009IranianMathematicalSociety.9798Naj

5、atiandRahimiwheretheg-frameboundsAandBarepositiveconstants.ΛiscalledaParsevalg-frameofHwithrespectto{Hi:i∈I}ifA=B=1in(1.1).Wesayasequence{Λi∈L(H,K):i∈I}isag-frameofHwithrespecttoKwheneverHi=K,foreachi∈I.Wealsosimplysayag-frameforHwheneverthespacesequence{Hi:i∈I}isclear.Thisnotationhasbeenintroduced

6、byW.Sunin[6].Itisanextensionofframesthatconcludeallpreviousextensionsofframes.Specially,ifΛisag-frameofH,thenanyvectorf∈Hcanberepresentedas[6]:X(1.2)f=Λ∗ΛS−1f,iii∈IwhereS−1istheinverseofthepositivelinearoperatorSonH,definedby:X(1.3)Sf:=Λ∗Λf.iii∈ISiscalledtheg-frameoperatorforΛ.Definition1.1.LetΛbeag-

7、frameofH.Ag-frameΘofHiscalledadualg-frameofΛifitsatisfies:Xf=Λ∗Θf,∀f∈H.iii∈IItiseasytoshowthatifΘisadualg-frameofΛ,thenΛwillbeadualg-frameofΘ.LetΛbeag-frameofHwithg-frameoperatorS.Then,(1.2)showsthat{ΛiS−1∈L