MATH 47/Wter 00 Notes o the adjot ad o ormal operators I these otes, V s a fte dmesoal er product space over, wth gve er * product uv, T, S, T, are lear operators o V U, W are subspaces of V Whe we say subspace, we mea oe of the fxed space V ab,,, μ, λ, deote complex umbers uvw,, deote elemets of V U s T -varat, or U s varat uder T, f for all u U, we have Tu ( ) U If U s T -varat, we have the operator T U o U, called the rescto of T to U, whch s defed by ( T U)( u) = T( u) Ideed, T U :U U, sce Tu ( ) U wheever u U, ad the learty of T U follows from the learty of T Note that the total space V ad the val subspace { 0 } are always T -varat A partcular type of T -varat subspace s a egespace of E [ T] μ def = Ker( T μi ) = { u V : T( u) = μu} T For ay μ, U = s a T -varat subspace: f u U, the Tu ( ) U, sce TT ( ( u)) = T( μu) = μtu ( ) E [ T 0 precsely whe μ s a egevalue of T ; the o-zero elemets of E μ [ T], f ay (whe μ s a of T ), are the egevectors of T for the egevalue μ Eμ [ T ] s called a egespace of T f E [ T] 0 μ { } μ ] { } We say that T ad S commute f T S = S T A bass = (,, ) of V s a dagoalzg bass for T (t could also be called a egebass for T ) f the max [ T ] s dagoal; equvaletly, f each bass elemet s a egevector of T Theorem For every T, there s a uque detcally for all uv V, T * such that Tu ( ), v = ut, *( v) The proof of ths theorem s omtted here It ca be foud our textbook: Theorem 3; proved roblem 34 (The reaso why Theorem ue s smple To determe T *, we have to determe each value T * ( v) Fx v, ad seek w= T*( v) It has to satsfy Tu ( ), v = uw, for
all u The left-had-sde, Tu ( ), v, s a fucto of u ; fact, t s a lear fucto F : V, a lear fuctoal, meag just that the codoma the feld tself (the vector space of dmeso =) Now, t turs out, that just the learty of F esures the uque exstece of w such that Fu ( ) = uw, for all u : all lear fuctoals ca be represeted the form u u, w If you follow ths, ad so determe w= T*( v), the secod ad fal step s to show that T * : V V s lear, whch s easy) A very easy but very mportat cosequece of the defto of the adjot s that the adjot of the adjot s the orgal: ( T*) * = T** = T Note also the followg smple cosequece of Theorem If U s varat uder both T ad T *, the the adjot ( T U)* of the rescted operator T U equals T* U Ideed, frst of all, T* U s a well-defed operator o the space U by the assumpto that U s T * -varat The equalty ( T U )( u), v = u,(t* U)( v) for uv, U follows from the equalty Tu ( ), v = ut, *( v ), sce ( T U)(u) = Tu ( ) ad ( T* U)( v) =T *( v ) Thus, T* U satsfes all the characterstc propertes of the adjot of T U ; sce, by the theorem, there s oly oe adjot to T U, T* U must be the adjot of T U Lemma Suppose T ad S commute The ay egespace of T s S varat roof Let U = E μ [ T] Let u U ; we wat to show that Su ( ) U(?) u U meas that Tu ( ) = μ u Therefore, ( ST )( u) = STu ( ( )) = S ( μu ) = μs( u) But also ( TS )( u ) = ( ST )( u ) Thus, TS ( ( u )) = ( TS) ( u) =( ST )( u) = μ Su ( ), whch meas that Su ( ) U as desred Lemma 3 Suppose that U s T -varat The U s T * -varat roof Let w U, to show that T*( w) U, or other words, that ut, *( w ) = 0 for all u U Let u U We have ut, *( w) = T( u), w by the defto of the adjot T * Sce U s T -varat, we have Tu ( ) U, ad sce w U, we have Tu ( ), w = 0 Therefore, ut, *( w ) = 0 as desred roposto 4 Suppose S s a set of lear operators o V such that, for ay * ST, S, we have that S ad T commute, as well as S ad T commute The
there s a orthoormal bass of V whch s a dagoalzg bass for every S S : a sgle commo orthoormal dagoalzg bass for all operators S at oce roof Note that the assumpto mples that each S S s ormal: take S = T the assumpto The proof s by ducto o the dmeso of V Whe =, the the asserto s val: ay bass ( ) s a dagoalzg bass for ay operator o V Suppose >, ad suppose that the asserto s ue for all er product spaces V of dmeso less tha Let V be a er product space of dmeso equal to There are two cases I Case, we assume that every T S s a scalar multple of the detty operator I o V : T = μ I for some μ ; equvaletly, Eμ [ T ] =V The, aga, the asserto s val, for the same reaso as before: every bass of V s a dagoalzg bass for all T S Case s whe Case does ot hold The: there s T S whch s ot of the form T = μ I Let μ be a egevalue of T There s such μ by the Fudametal Theorem of Algebra: the polyomal char T ( λ ) has at least oe root (Ths s the pot ths proof where we use complex scalars a essetal way) We have assumed that Eμ[ T] V Let U = Eμ[ T], ad W = U The U W = V, ad U { 0}, ad W { 0} ; the frst equalty because μ s a egevalue, the secod because U V Therefore, 0 < dm( U) <, ad 0 < dm( W) < I clam that both U ad W are varat uder both S ad S *, for ay S S Ideed, sce U s a egespaces of (the chose) T, ad both S ad S * commute wth T, by Lemma, we have that U s S -varat ad S * -varat Next, by Lemma 3, W = U s S * -varat, sce U s S -varat; ad W = U s S = ( S *)*-varat, sce U s S * -varat We therefore have the stuato o both of the subspaces V = U ad V = W that we have a set of operators S = { S V : S S } wth the property that for ay S = S V (wth S S ) ad T = T V (wth T S ), both S, S = S V ad T = T V commute, as well as ( S )* = S* V ad T = T V commute, drectly followg from the facts that S ad T commute as well as S * ad T commute Sce, for both V = U ad V = W, we have that dm(v ) <, we ca apply the ducto hypothess, to coclude that there s a sgle commo orthogoal dagoalzg bass for all operators the set S = S U S, ad aother oe,, for S { : } { S W S S} = : S 3
Let k = dm( U), ad l = dm( W) ; let =(,, k ) ad =(,, k+ k+= l ) Sce U W = V, we have that = =(,, ) s a bass of V It s a orthogoal bass: ay two of the frst k elemets of are orthogoal sce s a orthogoal system; ay two of the last l elemets of are orthogoal sce s a orthogoal system; ad ay elemet amog the frst k ad ay oe amog the last l are orthogoal sce the frst s U, the secod s W, ad U W Let S S Every bass vector s a egevector of S : f =,, k, the s a egevector of S U, ( S U)( ) = λ for some λ, ad thus S ( ) = λ, ad smlarly for = k+,,k+ l = Icdetally, the egevalues of S are thus see to be λ,, λ, k λk +,, λ k + l =, where λ,, λ k are the egevalues of S U, ad λ, k+, λk+ l= are the egevalues of S W Ths completes the proof of the roposto roposto 4 already cotas the Theorem 5 (Specal Theorem for fte dmesoal Hlbert spaces) Every ormal operator (o a fte dmesoal Hlbert space) has a orthoormal dagoalzg bass roof Apply roposto 4 to the set S = { T} sce T commutes wth T (obvously), ad of T ) The hypothess of roposto 4 holds T * commutes wth T (by the ormalty However, we ca also use roposto 4 to prove a soger result Frst, aother proposto, oe that s terestg tself -- whose proof uses Theorem 5 (Ths s terestg because the statemet has othg to do wth dagoalzato) roposto 6 Suppose S s a ormal operator, T s ay operator (o V, of course) If S ad T commute, the S * ad T commute as well roof Let, by Theorem 5, be a orthoormal dagoalzg bass for S Therefore, for D= [ S], A= [ T], we have that ( ) D= d s dagoal, d = 0 wheever j, ad for ( ) A= a, we have D A= A D Ths meas that, for ay, j from to, we have dk akj = ak d kj, whch, by d = 0 wheever j, reduces to k= k= 4
d a = a d I other words, a ( d d ) = 0 Therefore, ether a = 0 (Case ), or d d = 0 (Case ) Sce s orthoormal, [ S* ] = D = ( d ) j I clam that D A= A D Ideed, sce D s dagoal, ths reduces, just as before, to the questo whether we ca see that a ( d d ) = 0 If Case holds, ths s ue But f Case holds, the the complex d beg d d a d d = cojugate of d, d d equals zero mples that d d equals zero, hece aga ( ) 0 D A= A D, A= [ T] ad [ S*] = D together mply that S* T = T S* as desred Theorem 7 (Geeralzed Specal Theorem for fte dmesoal Hlbert spaces) ) Suppose that S s a set of commutg ormal lear operators o V : every S S s ormal, ad, for ay ST S,, S ad T commute The there s a orthoormal bass of V whch s a dagoalzg bass for every S S : there s a sgle commo orthoormal dagoalzg bass for all operators S at oce ) I partcular, f S ad T are commutg ormal operators, the there s a orthoormal bass of V whch s a dagoalzg bass for both S ad T roof Ths follows from roposto 4 ad roposto 6 Ideed, the codtos of roposto 4 are fulflled Let S ad T be both from the set S The S ad T commute, drectly by the assumpto of the theorem But also, S * ad T commute, sce, by assumpto, S s ormal, ad thus, by roposto 6, the fact that S ad T commute mples that S * ad T commute roposto 8 ) ( S T) * = T* S* ) ( S+ T) * = S* + T* 3) ( at )* = at * 4) If S ad T are ormal ad commute wth each other, the 4) S T s ormal, ad 4) S + T s ormal 5) If T s ormal, so s at 6) Let f( x, x,) be a polyomal wth complex coeffcets ay umber of varables x,, x Assume that TT,, are commutg ormal operators 5
( T Tj = Tj T for all, j =,, ) The T = f( T, T,) s a ormal operator Moreover, f s a commo orthoormal dagoalzg bass for TT,,, the also dagoalzes T ; f = (,, ) ad T ( ) () () λ = f ( λ, λ,) ( k ) k = λ, the T( ) λ = where roof ): We have the detty ( ST)( u), v = S ( T( u )), v = T( u), S* ( v ) = u, T*( S*( v)) = u,( T* S*)( v) Ths mples the asserto ): ( S + T) ( u ), v = S( u) + T( u), v = S( u), v + T( u), v = = us, *() v + ut, *() v = us, *() v+ T*() v = u,( S* + T*)() v Aga, ths mples the asserto 3): ( a T)( u ), v = a ( T( u)), v = a T( u), v = a u, T*( v) = u, a T*( v), whch s suffcet 4): We eed to show that ( ST) (( ST)*) = (( ST )*) ( ST) We have, usg ): ad ( ST ) (( ST )*) = ( ST ) ( T * S*) = STT * S * (( ST )*) ( ST ) = ( T * S*) ( ST ) = T * S * ST But sce every oe of SS, *, TT, * commutes wth every other, as a cosequece of roposto 6, the two products are equal 43): Usg ) we have ad smlarly, ( S+ T) ( S+ T)* = ( S+ T) ( S* + T*) = S S* + S T* + T S* + T T*, ( S+ T) * ( S+ T) = ( S* + T*) ( S+ T) = S* S+ S* T + T* S+ T* T For the same reaso as 4), these values are the same 5): ( at ) (( at )*) = ( at ) ( a( T*)) = a a T T * 6
ad (( at )*) ( at ) = ( a( T*)) ( at ) = a a T * T Sce T s ormal, these are equal 6): Ths follows from Theorem 7, by applyg 4) ad 5) repeatedly, to buld up the polyomal f( x, x,) 7