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ON THE TOPOLOGY OF INVARIANT SUBSPACES OF A SHIFT OF HIGHER MULTIPLICITY 1 1 GIORGISHONIA 0 2 Abstract. Following Beurling’s theorem and a study of the topology of in- n a variant subspaces by R. Douglas and C. Pearcy [3] description of path con- J nected components of invariant subspace lattice for shift of multiplicity one 6 1 hasbeengivenbyR.Yang[6]. Thispapergeneralizesresulttoarbitraryfinite multiplicity. Weshowthatthereexistsonetoonecorrespondencebetweenthe ] A invariantsubspacelatticeofshiftofarbitraryfinitemultiplicityandthespace F ofinnerfunctions. . h t a m [ 1 v 4 3 0 3 . 1 0 1 1 : v i X r a 1 Contents 1. Introduction 3 1.1. Shift operator and Beurling’s theorem 3 1.2. Shifts of higher multiplicity 4 1.3. The dimension of the wandering space. 7 2. continuity in L2 norm 8 3. L∞ continuity 12 3.1. Main result 15 References 18 2 1. Introduction Beurling’s theorem gave an analytic description of invariant subspace for a shift operator of multiplicity one. Later this result has been generalized to shifts of arbitrary multiplicity. Through the study of the topology of invariant subspace lattice Douglas and Pearcy have proposed a problem of path connected compo- nent of the invariant subspace lattice. Yang studied the relationship between one parameter families of invariant subspaces of shift of multiplicity one and their re- spectiveanalyticdescription. Presentpapergeneralizestheresulttoarbitraryfinite multiplicity. 1.1. Shift operator and Beurling’s theorem. Completeanalyticdescriptionof invariant subspaces of shift operator of multiplicity one was given by Beurling [2]. We bring main highlights of the theory (cf. Rosenthal and Radjavi [4]). LetH beaseparable,complex,infinite-dimensionalHilbertspacewithorthonor- mal basis {en}n∈N. Let B(H) denote collection of all bounded linear operators on H. For anoperatorT ∈B(H) a(closed)subspaceM ⊂H is invariantifTM ⊂M and reducing if in addition TM⊥ ⊂ M⊥. For given operator T collection of all invariant subspaces ordered by inclusion form a lattice denoted Lat(T). A unilateralforwardshift of multiplicity one is an isometry operatorS ∈B(H), whoseactiononHilbertspacebasisvectors{e } isashiftforwardS(e )=e . n n∈N n n+1 Ifinstead we enumeratethe basisof H as {en}n∈Z then isometryU ∈B(H) acting on H by U(e ) = e is called a bilateral forward shift of multiplicity one. U n n+1 is a surjective isometry, S is an isometry, but not surjective. Both operators have adjoints, backwards shifts of respective basis. Description of invariant subspaces for shift operators is based on their analytic interpretation,whereHilbertspaceisidentifiedwithL2(T),squareintegrablefunc- tionsonthecircle(withnormalizedarc-lengthmeasure). Collection{zn}n∈Z where zn is a polynomial on T is a Hilbert basis and bilateral shift is acting as operator M , multiplication by z. For unilateral shift we have to consider Hardy space z 3 H2 = zn, a subspace of L2 restricted to the span of positive powers. Unilat- n∈N W eral shift is identified with M . With this interpretation we can now formulate z H2 Beurling’s theorem: (cid:12) (cid:12) Theorem 1. (Beurling): A subspace M ⊆ L2(T) is invariant and non-reducing for the bilateral shift U if and only if there exists a measurable function G(z) on T with |G(z)|=1 ae, such that M =GH2. Furthermore two spaces of the form G H2 and G H2with |G | = |G | = 1 z-ae 1 2 1 2 are equal if and only if the ratio G1 is constant z-ae. G2 ae The functions G∈H2 with |G(z)|=1 are called an inner functions. 1.2. Shifts of higher multiplicity. LetK beaseparableHilbertspaceofdimen- sion n = dim(K). For the purpose of construction of generalization of Beurl- ing’s theorem dimension can be countable, though the main result of present paper is only covering arbitrary finite dimension. Let H = K be an or- i∈Z i L thogonal sum of infinite copies of K, with the derived summary inner product h v ; w i = hv ;w i . Bilateral shift of multiplicity n is an unitary oper- i i H i i Ki P P P ator U which is shifting each copy in the sum one position forward, UK = K . i i+1 Therestrictionofthisonspace K isaunilateral shiftofmultiplicityn. Sim- i∈N i L ilar to the shift of multiplicity one, shift of higher multiplicity also has analytic interpretationandthe descriptionofits invariantsubspacecanalsobe generalized. For analytic interpretation consider K-valued functions from unit circle f : T → K. Define f to be measurable, if z → hf(z),xi is measurable for all K x ∈ K. It is not hard to show that function z → kf(z)k is also measurable and alsofor twoK valuedmeasurablefunctions f, g the inner productz →hf(z),g(z)i is measurable as well. This defines a Hilbert space L2(K) with inner product hf,giL2(K) = Thf(z),g(z)iKdµ. Completeness of this space easily follows from R the representation of its elements with Fourier series. Every f ∈ L2(K) can be uniquely identified with series, f = x e with x ∈K and this representation i∈Z i i i has the property that hf(z),xi z−=aeP hx ,xizi is true for almost everywhere for i P 4 z ∈T andfor allx∈K. Itis alsoeasyto show that forf = x e we havethe i∈Z i i P familiar formula kfk2 = kx k2. i∈Z i P As in one dimensional case, analytic interpretation of a bilateral shift of multiplicity n is a multiplication by z on space L2(K), operator M . Unilateral shift of z multiplicity n is an operator S :=U|H2(K) where the space H2(K):={f ∈L2(K)|f = x e } i i nX∈N is subspace of L2(K) with coefficients of negative powers being zero. To generalize the notion of the inner functions we will consider operator valued functions defined on the unit circle T. A function F : T → B(K) is said to be measurable if for all x ∈ K the K-valued function z → F(z)x is measurable (as previouslydefined). LetkFk :=esssup kF(z)k . WhenkFk <∞wecan ∞ z∈T B(K) ∞ define an operator M on L2(K) by (M f)(z)=F(z)f(z). Let F F L∞(T,B(K)):={F :T→B(K)|kFk <∞} ∞ be a collection of all bounded measurable operator valued functions from the unit circle and let L∞(B(K)):={M |F ∈L∞(T,B(K))} F be a collection of respective operators on L2(K). It is easy to show that map F →M is an algebra isomorphism which respects adjoint operation. As a conse- F quence M is unitary iff F is z ∈ T ae-unitary. We define a collection of analytic F elements of L∞(T,B(K)), described by H∞(T,B(K)):={F ∈L∞(T,B(K))|H2(K)∈Lat(M )}, F which respectively defines class of operators H∞(B(K)):={M |F ∈H∞(T,B(K))}. F 5 OperatorsinH∞(B(K))havepowerseriesrepresentationwhich,thoughnotcrucial forourexposition,ishelpfulforanadditionalintuitiveperspectiveontheargument (cf Sz-Nagy and Foais [5]). Consider an operator G ∈ B(N;K) for i ∈ N and let i G(z)= ziG whereseriesconverge(weakly,strongly,innorm,allarethesame i∈N i P for separable Hilbert space) for all z ∈ D in the open unit disk. Such series with a condition that kG(z)k ≤ C for all z ∈ D is called a bounded analytic function. Eachseriescanbeassociatedwiththeoperator(N,K,G)onH2(K)throughstrong limitG(etθ)=lim G(z),whereconvergenceisalongnontangentialpath. This z→eiθ operator(N,K,G) is the same as multiplication operator previously defined based onoperatorvaluedfunctions,withinitialspaceH2(N)andfinalspaceH2(K),both definitionsformingcollectionH∞(B(K)). Suchoperator(N,K,G)(oralternatively M ) is an isometry on H2(K) if and only if G(etθ)∈B(N,K) is isometry ae-θ, in G which case it is called inner function and has initial space H2(N) and final space H2(K). In our main result, we will identify N with Cm, K with Cn and consider G to be an isometry from H2(Cm) to H2(Cn). Generalization of Beurling’s theorem to higher dimensions also involves Wold decomposition: Proposition2. (Wolddecomposition): EveryinvariantsubspaceM ofthebilateral shift U has a unique decomposition of the form M = M ⊕ M where M is a 1 2 1 reducing subspace for U, M is an invariant subspace of U and UiM ={0} 2 i∈N 2 T With this now we are able to formulate generalizationof Beurling’s theorem: Theorem 3. (Generalization of Beurling): A subspace M of a Hilbert space H = K (where K is a Hilbert space) is an invariant subspace of the bi- i∈Z i i L lateral shift U if and only if M = M ⊕M H2(N) where M reduces U, N ⊆ K 1 V 1 is a subspace, V ∈L∞(T,B(K)) is a z-ae partial isometry on K with initial space N. Further: Subspace M is uniquely determined by subspace M; and 1 6 If V ∈ L∞(T,B(K)) is a partial isometry with an initial space N and 1 1 M H2(N)=M H2(N ), then there is a partial isometry W with an initial space V V1 1 N and final space N such that M =M W. 1 V1 V Since unilateralshift hasno reducingsubspace,we alsohavefollowingcorollary: Corollary 4. A subspace M of a Hilbert space H = K (where K is a i∈Z i i L Hilbert space) is an invariant subspace of the unilateral shift S if and only if M = M H2(N) where N is a subspace of K and G ∈ H∞(T,B(K)) is an op- G erator valued function such that G(z) is z−ae partial isometry with initial space N. There is also a uniqueness result similar to the bilateral shift. At this point we are ready to formulate the main result of the present paper. Theorem 5. Let H2(Cn) be Hardy space for some finite n and let {p } be a t t∈[0,1] family of orthogonal projections such that: 1. Family is continuous in norm, lim kp −p k=0; t→τ t τ 2. Each projection p H2(Cn) produces an invariant subspace of a unilateral shift t operator (of multiplicity n). Then there exists integer m ≤ n and a choice of a family of inner functions G ∈H∞(T,B(Cm, Cn)) for t∈[0,1] such that: t 1. p H2 =G H2 ; and t Cn t Cm 2. {G } are sup-norm continuous, t t∈[0,1] esssupθkGt(eiθ)−Gτ(eiθ)kB(Cm;Cn) t→=τ 0 Present paper covers above result for an arbitrary finite n. In section two we demonstrates continuity in L2norm, third section demonstrates continuity in sup- norm. 1.3. The dimension of the wandering space. Given a family of norm continuous projections the first question is whether the dimensions of respective inner 7 functions, dimension of the wandering space, of initial space N in Beurling’s theorem is consistent across the family. This amounts to the existence of an integer m in the theorem 5. The following proposition addresses this question affirmatively. Proposition 6. Consider Hardy space H2(Cn) for some finite n and a family of projections {p } such that: t t∈[0,1] 1. Family is continuous in norm, lim kp −p k=0; t→τ t τ 2. Each projection p H2(Cn) produces an invariant subspace of a unilateral shift t operator (of multiplicity n). Let Wt = I−S(S|ptH2)∗ ptHC2m be respective wandering space. (cid:0) (cid:1) Then dim(W )=const for all t∈[0,1]. t Proof. We will use the fact, that if distance between two projections is less then one, then they have the same rank. tl→imτk I−S(S|ptH2)∗ pt− I −S(S|pτH2)∗ pτk≤ (cid:0) (cid:1) (cid:0) (cid:1) ≤kSktl→imτ (S|ptH2)∗pt−(S|pτH2)∗pτ +tl→imτkpt−pτk=0<1 (cid:13) (cid:13) (cid:13) (cid:13) thatis(eventually)W andW havethesamedimensionm,thusrankisconstant t τ on the whole t∈[0,1] interval. (cid:3) 2. continuity in L2 norm Beurling’s theorem only defines inner function G up to a unitary rotation (for t each t ∈ [0,1]). In order to construct continuous path G for t ∈ [0,1] we need a t consistent method of selecting unique G for each t. t For the clarity of exposition, we will first consider a basic case m =n and then the general case m ≤ n, whose treatment is only different from that of the basic case by an additionalcare which is needed for pinning down an unique G for each t t from a unitarily equivalent family. Proposition 7. Consider Hardy space H2(Cn) for some finite n and a family of projections {p } such that: t t∈[0,1] 8 1. Family is continuous in norm, lim kp −p k=0; t→τ t τ 2. Each projection p H2(Cn) produces an invariant subspace of a unilateral shift t operator (of multiplicity n). Consider analytical description of this subspace G H2(Cm) for some inner func- t tion G from H2(Cm) to H2(Cn) where m≤n and p H2(Cn)=G H2(Cm) t t t Then the family {G } can be chosen to be L2 continuous, specifically t t∈[0,1] kG (eiθ)−G (eiθ)k2 dµt→=τ 0 Z t τ B(Cm;Cn) θ Proof: First we consider the case, when the dimension n from the construction ofHardyspaceH2(Cn)isthe sameasthe dimensionofthe wanderingspaceW. Case 1. If m = n we can go to a point λ ∈ D in the unit disk [1]. We use a reproducing kernel in order to relate projection p = Proj to the t GtH2 innerfunctionG . We consideraninnerproduct p f ; zkG (z)u where t t t 1−λz D E f ∈Cn and u∈Cm are arbitrary vectors. We get f f p ; zkG (z)u = ; zkG (z)u = f; λkG (λ)u = t t t t (cid:28) 1−λz (cid:29) (cid:28)1−λz (cid:29) (cid:10) (cid:11) G (λ)∗f G (z) G(λ)∗f = G (λ)∗f; λku = t ; zku = t t ; zkG(z)u t (cid:28) 1−λz (cid:29) (cid:28) 1−λz (cid:29) (cid:10) (cid:11) At this point we can replace f with e ,...e ∈Cn to get 1 n e e e G (z)G (λ)∗ (2.1) p 1 , p 2 ...p n = t t t t t (cid:18) 1−λz 1−λz 1−λz(cid:19) 1−λz Denote e e e Λ (z):= p 1 , p 2 ...p n . t t t t (cid:18) 1−λz 1−λz 1−λz(cid:19) Taking z =λ we get G (λ)G (λ)∗ =(1−|λ|2)Λ (λ). t t t Since the right hand side is expressed in terms of p , it is a continuous t function in t ∈ [0, 1]. The left hand side is also continuous as a function of 9 λ ∈ D. Hence rank(G (λ)) = rank(G (λ)G (λ)∗) is lower-semi-continuous t t t in λ and t. If G (λ ) has a full rank n for some λ then by lower-semi- t0 t0 t0 continuity G (λ ) must also have same rank n for t close to t . We claim t t0 0 that for every t ∈ [0, 1] there exists such λ where G (λ ) has a full rank t t t n. For if no such λ exists for some t, we must have detG (λ) = 0 for t t every λ ∈ D and therefore detG (z) = 0 a.e. for z ∈ T. This however is t impossiblesince G (z)isanisometryand|detG (z)|=1fora.e. z ∈T. The t t preceding remark and compactness of [0, 1] produces a collection of points λ ,λ ...λ ∈D andafinite opencover{(a , b )}N ofinterval[0, 1]such 1 2 N k k k=1 that G (λ ) is invertible for t∈(a , b ) for k =1,2...N. t k k k We set G (λ )= (1−|λ |2)Λ (λ ) k,t k k t k p for k = 1,2...N and t ∈ (a , b ). Within each interval (a , b ) this def- k k k k inition identifies one choice from a family of inner functions which so far was only defined up to an unitary rotation by Beurling’s theorem. This choice is also continuous, G (λ ) is continuous in t. Having picked G at k,t k k,t a single point λ ∈ D we consistently pick it from the unitarily equivalent k familyonthewholeunitcircle: G (z)=(1−λ z)Λ (z)G (λ )−1. Thusde- k,t k t t k finedG isL2continuousintfort∈(a , b )andG H2(Cn)=p H2(Cn). k,t k k k,t t Finally we patch up G functions defined in intervals t ∈ (a , b ), k,t k k k = 1,2...N into a single G˜ , adjusting by rotation so that definition on t every subsequent interval to be continuation in t of previous. Starting from t = 0 we put G˜ = G , for t ∈ [a = 0;b ]. Next, since b ∈ (a ;b ) we t 0,t 0 0 0 1 1 have G U =G˜ , where U is an unitary constant. We put G˜ =G U 1,b0 1 b0 1 t 1,t 1 for t ∈(b ;b ]. This extends G˜ to the next interval continuously. Carrying 0 1 t on the process in the same manner for all covering intervals {(a , b )}N k k k=1 we get single inner function G˜ which is L2 continuous in t for t∈[0, 1] and t produces desired invariant subspaces: G˜ H2(Cn)=p H2(Cn) t t 10