Universal shifts and composition operators

Transcription

1 Universal shifts and composition operators Jonathan R. Partington and Elodie Pozzi Abstract It is shown that a large class of weighted shift operators T have the property that for every λ in the interior of the spectrum of T the operator U = T λid is universal in the sense of Caradus; i.e., every Hilbert space operator has a nonzero multiple similar to the restriction of U to an invariant subspace. As an application, composition operators induced by power mappings on the L 2 and Sobolev spaces of the unit interval are shown to have the same property: thus a complete knowledge of their minimal invariant subspaces would imply a solution to the invariant subspace problem for Hilbert space. A new Müntz-like theorem is proved: this is used to show that generalized polynomials are cyclic vectors for these operators in the L 2 case but not in the Sobolev case. Keywords: invariant subspace, universal operator, weighted shift, composition operator, cyclic vector, Müntz theorem. Mathematics Subject Classification (2): 47A5, 47B33, 47A6 Introduction In the field of operator theory, one of the most prominent open problems is the invariant subspace problem, sometimes optimistically known as the invariant subspace conjecture. It is the question whether the following statement is true: Given a complex Hilbert space H of dimension > and a bounded linear operator T : H H, then H has a non-trivial closed T -invariant subspace, i.e., there exists a closed linear subspace M of H which is different from {} and H such that T M M. While the general case of the invariant subspace problem is still open, many special cases have been settled (see, for example, [CE, RR3]). If the solution of the invariant subspace problem is positive, then at first sight it is necessary to prove a theorem that applies to all Hilbert space operators simultaneously. In fact, the situation is somewhat simplified by the existence of universal operators these have the property that if we could describe their lattice of subspaces precisely enough, then we could solve the invariant subspace problem. Accordingly, we recall the following definition. School of Mathematics, University of Leeds, Leeds LS2 9JT, U.K. J.R.Partington@leeds.ac.uk. Université Lyon, CNRS, Université Lyon, Institut Camille Jordan, 43 boulevard du novembre Villeurbanne Cedex, France. or-pozzi@math.univ-lyon.fr.

2 Definition.. Let X be a Banach space. Then an operator U L(X ) is said to be universal for X, if for each T L(X ) there is a constant λ and an invariant subspace M for U such that U M is similar to λt, i.e., λjt = UJ, where J : X M is a linear isomorphism. This is most useful in the case of a separable Hilbert space, since all its closed infinitedimensional subspaces are automatically isometric to the space itself (and we can then use the terminology universal for Hilbert space ). Clearly, we have the fact that if U L(H) is a universal operator, then the invariant subspace problem for Hilbert spaces is equivalent to the assertion that every infinite-dimensional invariant subspace for U contains a nontrivial proper closed invariant subspace. This may be rewritten in the form the minimal nontrivial invariant subspaces for U are one-dimensional. We now recall a theorem due to Caradus [Car69] that will enable us, in the case of Hilbert space, to show the existence of many natural operators that are universal. Theorem.. [Car69] Let H be a separable infinite-dimensional Hilbert space and U L(H). Suppose that. ker(u) is infinite-dimensional; and 2. im(u) is H. Then U is universal for H. An obvious example of such a universal operator is a backward shift of infinite multiplicity, but there are further examples where function theory can be use to obtain information on the invariant subspace lattice. For example, the Caradus result has been used by Nordgren, Rosenthal and Wintrobe [NRW87] to show that for the composition operator C φ on the Hardy space H 2, where φ is a hyperbolic automorphism of the disc, the operator C φ λid is universal for any λ in the interior of the spectrum σ(c φ ). The paper is organized as follows: in Section 2 we discuss bilateral weighted shifts T on l 2 (Z, L 2 ((a, a ))), providing sufficient conditions on the weight and the complex number λ so that T λid is universal. Then, in Section 3, as an application of the previous section, we find some very simple new universal composition operators on L 2 ((, )) and the universality of the adjoints of composition operators on the Sobolev space W ((, )). Finally, in Section 4, a new Müntz-type theorem is derived, and used to show the cyclicity of linear combinations of functions of the form x x α for the composition operator C φ on L 2 ((, )), where φ is defined on [, ] by φ(x) = x s with s <, and the non-cyclicity of all linear combinations of these functions for the composition operator C φ on W ((, )), where φ is defined on [, ] by φ(x) = x s with s >. 2 Shift on l 2 (Z, L 2 ((a, a ))) Let a < a be real, and let T : l 2 (Z, L 2 ((a, a ))) l 2 (Z, L 2 ((a, a ))) be the weighted right bilateral shift defined by (T x) n = k n x n for n Z where {k n } is a sequence 2

3 of positive and continuous functions on [a, a ] such that { uniformly a as n k n b as n + where a < b. We denote by. 2 the norm. L 2 ((a,a )). Theorem 2.. Let T be the bilateral weighted shift defined above. Then, σ(t ) = σ(t ) = {z C : a z b}. If a < λ < b, then λ is an eigenvalue of T of infinite multiplicity, but T λid is bounded below. Proof: Suppose that a >. Note that if f = m Z g me m where {e m } is the standard orthonormal basis of l 2 (Z), then T f 2 sup k m f 2. m Z Let ε >, m Z, A ε [a, a ], µ(a ε ) > such that and for x A ε, k m sup k m ε m Z 2, Also, if we take f = χ Aε e m, then we have k m (x) > k m ε 2. T f > ( k m ε) µ(a ε ). So, we have that T = sup m Z k m. By an inductive argument, we obtain that for n N, T n = sup m Z k m k m+... k m+n. Since k n converges uniformly to b as n tends to +, it follows that sup k m k m+... k m+n n b. m Z n Since T is unitarily equivalent to a bilateral right shift with weights k n = k n, in the same way, one can show that T n n /a. As does not lie in the spectrum n of T, σ(t ) = σ(t ). Hence, we have that σ(t ) {z C : a z b}; since σ(t ) = σ(t ), we obtain that σ(t ) {z C : a z b}. Note that if a =, then, we have σ(t ) D(, b). Let λ be such that a < λ < b. If f = n Z g ne n, where for all n Z, g n is in L 2 ((a, a )), the equation T f = λf gives that 3

4 λ + n= g n e n = + n= g n k n e n = + n= g n+ k n e n which implies that for all n Z, λg n = k n g n+. Setting g to be any function of norm in L 2 (a, a ), and defining on (a, a ), { λ g n = n g /(k k... k n ) for n >, λ n g k n k n+... k for n <, n= we see easily that n Z g n 2 2 converges. Hence, f = g n e n is an eigenvector of T and λ σ p (T ) with infinite multiplicity. We conclude that σ(t ) = σ(t ) = {z C : a z b}. It remains to check that for λ (a, b), T λid is bounded below. Here our method generalizes some ideas due to Ridge [Rid7]. Suppose towards a contradiction that λ is an approximate eigenvalue of T. So, for each i N, there is a unit vector f(i) = {f j (i)} j Z such that T f(i) λf(i) < i. Suppose first that lim inf i f (i) 2 =. So, for all ε >, there is an index i such that f (i) 2 < ε and T f(i) λf(i) < ε. Denoting by h(i) = f(i) f (i)e (that is, setting to zero the component corresponding to j = ), we have T h(i) λh(i) T f(i) λf(i) + T f (i)e λf (i)e < ε + ( T + λ )ε and thus there is an approximate eigenvector h(i) of norm such that h (i) = and T h(i) λh(i) < ε. We can write h(i) = l(i)+r(i) where l(i) is supported on the negative integers and r(i) is supported on the positive integers. Since their supports are disjoint, T h(i) λh(i) is the orthogonal sum of T l(i) λl(i) and T r(i) λr(i). Since h(i) is of norm, one of l(i) and r(i) has norm greater than and thus, we may find a sequence of approximate 2 eigenvectors supported entirely on either the positive or negative integers. We will denote it by p(i), and may suppose without loss of generality that p(i) = and T p(i) λp(i) < /i. Now T n p(i) λ n p(i) = (T n + λt n λ n Id)(T p(i) λp(i)) and so T n p(i) λ n p(i) < C n /i, where C n depends on λ and the weights but not on i. If p(i) = {p j (i)} is supported on the positive integers, then 4

5 T n p(i) 2 = = ( k j k j+... k j+n p j (i) 2 2 ( j= inf j> [ inf j> ) /2 ( ) 2 min k j(u)k j+ (u)... k j+n (u) p j (i) 2 2 u [a,a ] j= ( )] min k j(u)k j+ (u)... k j+n (u). u [a,a ] Without loss of generality, one can suppose that b >. So, for n sufficiently large, we have ( ) inf min k j(u)k j+ (u)... k j+n (u) > λ n + 2. j> u [a,a ] Choosing i larger than C n, we obtain a contradiction. Applying similar arguments to T, we obtain a contradiction when T has an approximate eigenvector supported on the negative integers. Suppose that lim inf i f (i) 2 = d >. Since we have an approximate eigenvector f(i) = {f n (i)} of norm such that T f(i) λf(i) <, then a simple inductive i argument shows that there exist constants {D n } n independent of i such that f n+(i) k k... k n f λ n+ (i) D n, for n N. 2 i If f (i) 2 d/2, then, Since f n+ (i) 2 k k... k n f λ n+ (i) D n 2 i min u [a,a ] k j (u)k j+ (u)... k j+n (u) f λ n+ (i) 2 D n i. min u [a,a ] k j (u)k j+ (u)... k j+n (u) λ n+, n + we may find an index n such that f n+ (i) 2 2 D n /i. But, as f(i) is a vector of norm, if we choose i larger than D n, we obtain a contradiction. Corollary 2.. Let T : l 2 (Z, L 2 ((a, a ))) l 2 (Z, L 2 ((a, a ))) be the weighted right bilateral shift defined by (T x) n = k n x n for n Z where {k n } is a sequence of positive and continuous functions on [a, a ] such that k n converges uniformly to b as n and a as n + where a < b. Then, for any complex number a < λ < b, T λid is a universal operator for Hilbert space. In particular, every operator has an invariant subspace if and only if the minimal nontrivial invariant subspaces of T are all one-dimensional. 5 ) /2

6 Proof: T is unitarily equivalent to a weighted right shift where the weights is a sequence of positive and continuous functions such that k n tends uniformly to a at and uniformly to b at +. By Theorem 2., it follows that T λid has an infinite dimensional kernel and, T λid is bounded below which implies that T λid is surjective. So, we conclude that T λid is universal for Hilbert space. The last statement follows immediately from the remarks after Definition., noting that T and T λid have the same lattice of invariant subspaces. Remark 2.. Note that, in this case, the point spectrum of T is empty. 3 Applications to composition operators 3. Composition operators on L 2 ((, )) Definition 3.. A mapping φ : [, ] [, ] will be called L 2 -admissible, if it is strictly increasing and differentiable, with φ() = and φ() =, and has the properties that φ has no other fixed points, φ(x) x on [, ], φ is continuous on (, ) and (φ ) is bounded. We then define the composition operator C φ by C φ (f) = f φ. Such operators have been studied by Spalsbury [Spa8] in the context of C[, ] and C [, ], for functions of the form φ(x) = x s, and generalizations of the Müntz theorem were obtained in order to obtain information on cyclic vectors. Here we work in a Hilbertian context, and with a more general range of symbols φ. Proposition 3.. Let φ be L 2 -admissible. Then, C φ is bounded on L 2 ((, )) and its adjoint is given by C φ (f) = (φ ) (f φ ). Proof: Let f be in L 2 ((, )). Then C φ (f)(x) 2 dx = sup u [,] f(φ(x)) 2 dx = (φ ) (u) f 2 L 2 ((,)). f(u) 2 (φ ) (u) du Thus, C φ (φ ) /2. The computation of the adjoint is elementary, and we omit it. Now, fix a (, ). Let f be a function in L 2 ((, )). We denote by {a n } n Z the sequence {φ n (a )} n Z, where for n <, φ n = (φ ) n. If we define g n (u) such that g n (u) = f(φ n (u)) for n Z and u (a, a ), 6

7 then each g n lies in L 2 ((a, a )) and we have, writing f n = f (an,a n+ ), that f 2 L 2 ((,)) = n Z f n 2 L 2 ((a n,a n+ )) = n Z a a a f n (φ n (u)) 2 (φ n ) (u) du = g n (u) 2 (φ n ) (u) du n Z a = g n (φn ) 2. L 2 ((a,a )) n Z ) If we consider V : L 2 ([, ]) l 2 (Z, L 2 ((a, a ))) defined by V (f) = (g n (φn ) then V is a unitary operator. Now, if we compose f by the operator C φ, we have n Z, C φ (f) 2 L 2 ((,)) = n Z = n Z an+ a n a a f(φ(x)) 2 dx g n+ (u) 2 (φ n ) (u) du So, if f = n Z g n (φn ) e n, where (e n ) n Z is the standard orthonormal basis of l 2 (Z), then, C φ (f) = n Z g n+ (φn ) e n. Thus, C φ maps on L 2 ((, )) as a weighted left shift on l 2 (Z, L 2 ((a, a ))) with weights k n = (φ n ) = (φ n+ ) φ, for n Z. φ n Theorem 3.. Let φ be an L 2 -admissible function such that for some a (, ), the sequence φ converges uniformly on [a φ n, φ(a )] to a as n and b as n + where a < b. Then, σ(c φ ) = {z C : a z b} and {z C : a < z < b} σ p (C φ ). Likewise, for any complex numbers a < λ < b, C φ λid is a universal operator for Hilbert space. Proof: Since a weighted left shift with weights {k n } n Z is unitarily equivalent to a weighted right shift with weights { k n } n Z where k n = k n for n Z, then, C φ is unitarily equivalent to a weighted right shift on l 2 (Z, L 2 ((a, a )), denoted by S φ with weights k n =, n Z such that (φ ) φ n k n uniformly { b as n a as n + 7

8 Theorem 2. implies that σ(s φ ) = {z C : a z b} and σ p (S φ ) = {z C : a < z < b}. It follows from Corollary 2. that for any a < λ < b, S φ λid is a universal operator for Hilbert space. So, we have the conclusion for C φ λid. Remark 3.. If φ satisfies the hypothesis of Theorem 3., the point spectrum of C φ is empty. We now have an analogous version of Corollary 2. for C φ. Corollary 3.. Let φ and C φ be as in Theorem 3.. Then, every operator has an invariant subspace if and only if the minimal nontrivial invariant subspaces of C φ are all one-dimensional. Example 3.. Fix < s <. Let φ be the function defined on [, ] by φ(x) = x s. Then C φ is bounded on L 2 ((, )). Note that, for n Z and x (, ), Then we have φ φ n (x) = s x sn ( s)/2. φ φ n It follows from Theorem 3. that σ(c φ ) = D (, / s ) uniformly and { as n s as n + { z C : < z < / s } σp (C φ ), and for any complex number λ such that < λ < / s, C φ λid is a universal operator for Hilbert space. Thus every Hilbert space operator has an invariant subspace if and only if the minimal nontrivial invariant subspaces of C φ are all one dimensional. 3.2 Composition operators on Sobolev spaces W ((, )) Now let W (, ) be the space of absolutely continuous functions f defined on [, ] such that f() = and f L 2 ((, )), with norm where. 2 denotes the norm in L 2 ((, )). f W (,) = f 2. Definition 3.2. A mapping φ : [, ] [, ] will be called W -admissible, if it is a strictly increasing and continuously differentiable function, with φ() = and φ() =, such that φ(x) x for x [, ], and φ has no other fixed points. We denote by C φ the composition operator induced by φ. Proposition 3.2. Let φ be W -admissible. Then, C φ is bounded on W ((, )). 8

9 Proof: Let f be in W ((, )). C φ (f) 2 W (,) = Thus, C φ W (,) φ /2. f (φ(x)) 2 φ (x) 2 dx = φ f 2 W ((,)). f(u) 2 φ (φ (u)) du Note that the operator C φ is also bounded on W (, ), the space of absolutely continuous functions f on [, ] such that f L 2 ((, )). As the mapping f f() is bounded on W, then f = (f f()) + f() and C φ f = C φ (f f()) + f(). Lemma 3.. Let φ be W -admissible. Then C φ on W ((, )) is unitarily equivalent to a weighted composition operator D φ on L 2 ((, )). Proof: Let f be in W ((, )). We denote by F the derivative of f which is in L 2 ((, )). We have C φ (f) 2 W ((,)) = F (φ(x)) 2 φ (x) 2 dx = φ.(f φ) 2 2 = D φ (F ) 2 where D φ is the weighted composition operator on L 2 ((, )) defined by D φ (g) = φ (g φ) for all g L 2 ((, )). If D is the mapping from W ((, )) onto L 2 ((, )) defined by D(f) = f for all f W ((, )), then we have the commutative diagram D f C φ f φ D f D φ φ (f φ) Since D is a unitary operator from W ((, )) to L 2 ((, )), we have the conclusion. Proposition 3.3. Let φ be W -admissible and suppose that for some a (, ), the sequence (φ ) φ n (x) converges uniformly on [a, φ (a )] to a as n and b as n + where a < b. Then σ(c φ ) = {z C : a z b} and σ p (C φ ) =. Likewise, for any complex numbers a < λ < b, Cφ λid is a universal operator for Hilbert space. 9

10 Proof: Note that if F, G L 2 ((, )), then D φ (F ), G 2 = = φ (x)f (φ(x))g(x) dx F (u)g(φ (u)) du = F, S φ (G) 2, where S φ denotes the composition operator on L 2 ((, )). So, we have Dφ = S φ. Thus, we deduce by Theorem 3. that σ(dφ ) = {z C : a z b} and σ p(dφ ) = {z C : a < z < b}. Likewise, for any complex numbers a < λ < b, Dφ λid is a universal operator for Hilbert space. By unitary equivalence from Lemma 3. and by Remark 3., we have the conclusion. Example 3.2. Fix s >. Let ψ be the function defined on [, ] by ψ(x) = x s. Then, C ψ is bounded on W ((, )). Note that for all x [, ], we have ψ (x) = x /s ; using similar arguments as in Example 3., one can prove that ψ satisfies the hypothesis of Proposition 3.3. It follows that σ(c ψ ) = D (, s ), σ p (C ψ ) = and for any complex number λ such that < λ < s, Cψ λid is a universal operator for Hilbert space. Thus, every operator has an invariant subspace if and only if the minimal nontrivial invariant subspaces of Cψ are all one-dimensional. 4 Cyclic vectors for C φ In the sequel, we will denote by φ the function x x s where < s <. Here, we are interested in cyclic vectors of composition operators induced by φ: in view of the universality results proved in Section 3, these are of particular significance in their application to the study of invariant subspaces. Proposition 4.. Let p be the function defined on [, ] by p(x) = cx α, where c C \ {} and α > 2. Then p is a cyclic vector for C φ in L 2 ((, )). Proof: Note that span{p(x sk ) : k } = span{x λ k : k }, where (λk ) k is the infinite sequence of positive and distinct real numbers defined by λ k = αs k, for k =,, 2,.... By the full Müntz theorem in L 2 ((, )), see [BE96], we have that span{x λ k, k } = L 2 ((, )) k 2λ k + (2λ k + ) 2 + =. Now

11 2αs k (2αs k + ) 2 + (2αs k + ) 2 + k + and k + αsk 2. So, k= 2αs k + (2αs k + ) 2 + is divergent, and we conclude that span{p(x sk ) : k } = L 2 ((, )). Using a more complicated method, we now prove that every generalized polynomial which maps zero to zero is also cyclic for C φ. Theorem 4.. Let P be the generalized polynomial on [, ] defined by P (x) = n k= a kx r k with < r <... < r n and the a k non-zero. Then P is cyclic in L 2 (, ) for C φ. Proof: To see this, we consider the isometry between L 2 (, ) and L 2 (, ) defined by (Jf)(t) = f(e t )e t/2, which takes a function x x r to t e rt t/2. Then a function f is orthogonal to C j φp if and only if the corresponding Jf is orthogonal to n P j : t a k exp( s j r k t t/2). k= Taking the Fourier Laplace transform and using the Paley Wiener theorem, we arrive at a function F H 2 (C + ), where C + = {z C : Im(z) > }, such that n a k F k= ( is j r k + i ) = (j =,, 2,...). 2 We have the completeness of the iterates C j φp if we can show that F is identically zero. Note that, in the well-known case n =, this now implies that F is identically zero since (is j r + i 2 ) j is not a Blaschke sequence, given that r > by assumption. In general, since s j r k as j for all k, we have, writing G(z) = F (i(z+/2)) for z in a neighbourhood of, that n a k G(zr k ) =, by the principle of isolated zeroes, and G is holomorphic near. k=

12 Now G is not a polynomial, unless it is identically zero (since F H 2 (C + )) and so if (g m ) m are its Taylor coefficients, then infinitely often we have g m and n a k rk m =, k= which cannot be true since this is asymptotic to a n rn m as m. Hence G must be identically zero and so the system of iterates is complete. For a contrasting result, we now denote by φ the function x x s where s > and by D φ the weighted composition operator in L 2 ((, )) introduced in Lemma 3.. Proposition 4.2. Let P be the function defined on [, ] by P (x) = r c i x α i, where each c i C and < α 2 < α <... < α r. Then, P is not a cyclic vector for D φ in r L 2 ((, )). Thus, if Q(x) = b i x β i, with b i C and < β 2 < β <... < β r, then the function Q is not a cyclic vector for C φ in W ((, )). i= i= Proof: Note that D k φ(p ) = r s k c j x sk α j +s k. j= Now, we have span{dφ k, k } span{xλ k, k } where {λk } k is a sequence of positive and distinct numbers such that there is (i, j) N {,..., r}, λ k = s i α j +s i. So, it follows from the full Müntz theorem in L 2 ((, )) that P is not a cyclic vector for D φ if and only if r 2(s k α j + s k ) converges. Now, for each j, we see 2(s k α k j= j + s k ) 2 + that s k α j + s k 2(s k α j + s k ) 2 + 2(s k α j + s k ) 2 + k + and k + α j + 4(α j + ) 2 s k 4(α j + ) 2 s 2k. We conclude that k vector for D φ. r j= 2(s k α j + s k ) 2(s k α j + s k ) 2 + converges and thus P is not a cyclic 2

13 To see that Q is not cyclic for C φ in W ((, )), we now set P = Q and use Lemma 3.. Let C ([, ]) := {f C ([, ]), f() = }. We then obtain a non-cyclicity result in C ([, ]) for a special class of functions which can be written as f(x) = bx + r(x), where r(x) as x tends to (see Theorem 3 in [Spa8]). Recall that φ(x) = x s for x s >. r Corollary 4.. Let P be the function defined on [, ] by P (x) = c i x α i, where each c i C and α < α <... < α r. Then, P is not a cyclic vector for C φ in C ((, )). Proof: The non-cyclicity of P follows easily from Proposition 4.2 and the following inequalities : i= for all f C ([, ]). f W ((,)) := f L 2 ((,)) f + f =: f C ([,]), Acknowledgements The authors are indebted to Isabelle Chalendar for many helpful discussions during the preparation of this paper. References [BE96] P. Borwein and T. Erdélyi. The full Müntz theorem in C[, ] and L [, ]. J. London Math. Soc. (2), 54():2, 996. [Car69] [CE] S. R. Caradus. Universal operators and invariant subspaces. Proc. Amer. Math. Soc., 23(): , 969. I. Chalendar and J. Esterle. Le problème du sous-espace invariant. In Development of mathematics 95 2, pages Birkhäuser, Basel, 2. [NRW87] E. Nordgren, P. Rosenthal, and F. S. Wintrobe. Invertible composition operators on H p. J. Funct. Anal., 73(2): , 987. [Rid7] [RR3] W. C. Ridge. Approximate point spectrum of a weighted shift. Trans. Amer. Math. Soc., 47: , 97. H. Radjavi and P. Rosenthal. Invariant subspaces. Dover Publications Inc., Mineola, NY, second edition, 23. 3

14 [Spa8] A. Spalsbury. Perturbations in Müntz s theorem. J. Approx. Theory, 5():48 68, 28. 4