The other identities are proven the exact same way Comment If we work with one

# The other identities are proven the exact same way

• 24

This preview shows page 4 - 7 out of 24 pages.

The other identities are proven the exact same way.Comment.If we work with one Hilbert spaceH, then the above result givesthe fact thatB(H) is aunital involutive Banach algebra.(See Section 5 for theterminology.) This will be crucial for the development of the theory.Another important set of results deals with the kernel and the range. 304CHAPTER II: ELEMENTS OF FUNCTIONAL ANALYSISProposition 7.2(Kernel-Range Identities).LetH1andH2be Hilbert spaces.For any operatorTB(H1,H2), one has the equalities(i) KerT= (RanT);(ii)RanT= (KerT).Proof.(i). If we start with some vectorηKerT, then for everyξH1,we have(η|)H2= (T η|ξ)H1= 0,thus proving thatη,ξH1, i.e.η(RanT). This proves the inclusionKerT= (RanT).To prove the inclusion in the other direction, we start with some vectorηKerT= (RanT), and we prove thatT η= 0.This is however pretty clearsince we haveη(TT η), i.e.0 = (η|TT η)H2= (T η|T η)H1=T η2,which forcesT η= 0.(ii). This follows immediately from part (i) applied toT:RanT=([RanT])= (KerT).Example 7.1.Letn1 be some integer, and consider the Hilbert spaceCn,whose inner product is the standard one:(ξ|η) =nk=1¯ξkηk,ξ= (ξ1, . . . , ξn),η= (η1, . . . , ηn)Cn.The Banach algebraB(Cn) is obviously identified with the algebra Matn×n(C) ofn×nmatrices with complex coefficients. The adjoint operation then correspondsto a (familiar) operation in linear algebra. ForAMatn×n(C), sayA= [ajk]nj,k=1,one takesA= [bjk]nj,k=1to be theconjugate transpose ofA, i.e.theb’s aredefined asbjk= ¯akj.A similar identification works withB(Cm,Cn), identifiedwith Matn×m(C).The adjoint operation is used for defining certain types of operators.Definitions.LetHbe a Hilbert space.A. We say that an operatorTB(H) isnormal, ifT T=TT.B. We say that an operatorTB(H) isself-adjoint, ifT=T.C. We say that an operatorTB(H) ispositive, if(|ξ)H0,ξH.Remarks 7.2.A. Every self-adjoint operatorTB(H) is normal.B. The set{TB(H) :Tnormal}is closed inB(H), in the norm topology.Indeed, if we start with a sequence (Tn)n=1of normal operators, which converges(in norm) to someTB(H), then (Tn)n=1converges toT, and since the multi-plication mapB(H)×B(H)(X, Y)-→XYB(H)is continuous, haveT T= limn→∞TnTnandTT= limn→∞TnTn, so we imme-diately getT T=TT.C. ForTB(H), the following are equivalent (see Remark 3.1):Tis self-adjoint; §7.Operator Theory on Hilbert spaces305the sesquilinear mapφT:H×H(ξ, η)-→(|η)HCis sesqui-symmetric, i.e. (|η) =(|ξ),ξ, ηH;(|ξ)R,ξH.In particular, we see that every positive operatorTis self-adjoint.  • • • 