Universal Toeplitz operators on the Hardy space over the polydisk
aa r X i v : . [ m a t h . F A ] S e p UNIVERSAL TOEPLITZ OPERATORS ON THE HARDY SPACEOVER THE POLYDISK
MARCOS FERREIRA, S. WALEED NOOR
Abstract.
The
Invariant Subspace Problem (ISP) for Hilbert spaces asksif every bounded linear operator has a non-trivial closed invariant subspace.Due to the existence of universal operators (in the sense of Rota), the ISPmay be solved by describing the invariant subspaces of these operators alone.We characterize all anaytic Toeplitz operators T φ on the Hardy space H ( D n )over the polydisk D n for n > Caradus criterion foruniversality, that is, when T ∗ φ is surjective and has infinite dimensional kernel.In particular if φ in a non-constant inner function on D n , or a polynomial inthe ring C [ z , . . . , z n ] that has zeros in D n but is zero-free on T n , then T ∗ φ isuniversal for H ( D n ). The analogs of these results for n = 1 are not true. Introduction
One of the most important open problems in operator theory is the
InvariantSubspace Problem (ISP), which asks: Given a complex separable Hilbert space H and a bounded linear operator T on H , does T have a nontrivial invariant sub-space? An invariant subspace of T is a closed subspace E ⊂ H such that T E ⊂ E .The recent monograph by Chalendar and Partington [3] is a reference for somemodern approaches to the ISP. In 1960, Rota [11] demonstrated the existence ofoperators that have an invariant subspace structure so rich that they could model every Hilbert space operator.
Definition . Let B be a Banach space and U a bounded linear operator on B . Then U is said to be universal for B , if for any bounded linear operator T on B thereexists a constant α = 0 and an invariant subspace M for U such that the restriction U | M is similar to αT . If U is universal for a separable, infinite dimensional Hilbert space H , then theISP is equivalent to the assertion that every minimal invariant subspace for U isone dimensional. The main tool thus far for identifying universal operators hasbeen the following criterion of Caradus [2]. The Caradus Criterion . Let H be a separable infinite dimensional Hilbert spaceand U a bounded linear operator on H . If ker( U ) is infinite dimensional and U issurjective, then U is universal for H . Mathematics Subject Classification.
Primary 30H10, 47B35; Secondary 47A15.
Key words and phrases.
Hardy space over the polydisk, Toeplitz operator, universal operator,invariant subspace.
Let T n be the Cartesian product of n copies of T = ∂ D equiped with the nor-malized Haar measure σ . Let L ( T n ) denote the usual Lebesgue space and L ∞ ( T n )the essentially bounded functions with respect to σ . The Hardy space H ( D n ) isthe Hilbert space of holomorphic functions f on D n satisfying || f || := sup
1. We now state our main result.
Theorem.
Let φ ∈ H ∞ ( D n ) for n > . Then T ∗ φ satisfies the Caradus criterion foruniversality if and only if φ is invertible in L ∞ ( T n ) but non-invertible in H ∞ ( D n ).This shows that analytic Toeplitz operators with universal adjoints are far moreubiquitous in the higher dimensional case n > Corollary . If φ is a non-constant inner function or a polynomial in the ring C [ z , . . . , z n ] that has zeros in D n but no zeros in T n , then T ∗ φ is universal for H ( D n ) with n > . In particular the backward shift operators T ∗ z , . . . , T ∗ z n are universal when n > not true in general for H ( D ). A closed subspace E ⊂ H ( D n ) issaid to be z i -invariant if z i E ⊂ E for some i = 1 , . . . , n . A T φ -invariant subspace E is called maximal if E ( H ( D n ) and there is no T φ -invariant subspace L suchthat E ( L ( H ( D n ). Since every maximal T φ -invariant subspace E correspondsto a minimal T ∗ φ -invariant subspace E ⊥ , we obtain the following version of the ISP. An equivalent version of the ISP . Suppose n > and i=1,. . . ,n. Then theISP has a positive solution if and only if every maximal z i -invariant subspace hascodimension in H ( D n ) . The maximal z -invariant subspaces in H ( D ) indeed have codimension one byBeurling’s Theorem. Hedenmalm [8] proved that this is also true in the classicalBergman space L a ( D ). A closed subspace E ⊂ H ( D n ) that is z i -invariant for all i = 1 , . . . , n simultaneously is called a shift-invariant subspace. The description ofall shift-invariant subspaces for n > Proof of theorem
Cowen and Gallardo-Gutirrez ([4],[5],[6],[7]) have frequently observed that if φ ∈ H ∞ ( D ) and there is an ℓ > | φ ( e iθ )) | > ℓ almost everywhere on T , then T /φ is a left inverse for T φ on H ( D ) and therefore that T ∗ φ is surjective. Ourfirst goal is to characterize the left-invertibility of analytic Toeplitz operators on H ( D n ) for all n ≥
1. The following general result can be found in functionalanalysis textbooks, but we state it here for completeness.
Lemma 1.
Let T be a bounded operator on H . Then the following are equivalent. (1) T is left-invertible. (2) T ∗ is surjective. (3) T is injective and has closed range.Proof. If T is left-invertible then there exists an bounded operator S with ST = I and hence T ∗ S ∗ = I . So T ∗ is surjective and (1) = ⇒ (2). The equivalence(2) ⇐⇒ (3) is the closed range theorem [12, Theorem 4.1.5]. If T is injectiveand has closed range, then it has a bounded inverse T − : T ( H ) → H by the openmapping theorem [12, Corollary 2.12]. Therefore T − T = I and (3) = ⇒ (1). (cid:3) Recently Koca and Sadik [9] proved that if f ∈ H ∞ ( D n ), then the subspace E = f H ( D n ) is shift-invariant if and only if f is invertible in L ∞ ( T n ). Using thisresult we obtain the following characterization of left-invertibility for analytic T φ . Proposition 2.
Let T φ be an analytic Toeplitz operator on H ( D n ) for n ≥ .Then T φ is left-invertible if and only if φ is invertible in L ∞ ( T n ) .Proof. If φ is invertible in L ∞ ( T n ), then T /φ is a (not necessarily analytic) Toeplitzoperator and T /φ T φ f = T /φ ( φf ) = P f = f. for all f ∈ H ( D n ). Therefore T φ is left-invertible. Conversely if T φ is left-invertible,then it has closed range by Lemma 1. Then E = φH ( D n ) is closed and satisfies z i E ⊂ E for all i = 1 , . . . , n . Hence E is a shift-invariant subspace and therefore φ is invertible in L ∞ ( T n ) by the result of Koca and Sadik [9, Theorem 2]. (cid:3) Therefore by Lemma 1 and Proposition 2 we know precisely when the adjoint ofan analytic Toeplitz operator is surjective. The key ingredient in the proof of themain theorem is a result of Ahern and Clark [1, Theorem 3].
The Ahern and Clark Theorem . If f , . . . , f k ∈ H ( D n ) with k < n , then theshift-invariant subspace E generated by f , . . . , f k is either all of H ( D n ) or E ⊥ isinfinite dimensional. MARCOS FERREIRA, S. WALEED NOOR
By the shift-invariant subspace E generated by f , . . . , f k means the smallestshift-invariant subspace containing f , . . . , f k . So the shift-invariant subspaces ofthe form φH ( D n ) for some invertible φ ∈ L ∞ ( T n ) are singly generated. We arriveat our main result. Theorem . Let φ ∈ H ∞ ( D n ) for n > . Then T ∗ φ is surjective and has infinitedimensional kernel if and only if φ is invertible in L ∞ ( T n ) but not in H ∞ ( D n ) .Proof. Suppose T ∗ φ is surjective and Ker( T ∗ φ ) is infinite dimesnional. Then φ isinvertible in L ∞ ( T n ) by Lemma 1 and Proposition 2. It follows that the closedsubspace E = φH ( D n ) has infinite codimension since E ⊥ = Ker( T ∗ φ ). In particular φH ( D n ) = H ( D n ) and hence 1 /φ / ∈ H ∞ ( D n ). Conversely suppose φ is invertiblein L ∞ ( T n ) but not in H ∞ ( D n ). Then T ∗ φ is surjective by Lemma 1 and Proposition2. The assumption that 1 /φ / ∈ H ∞ ( D n ) means that E = φH ( D n ) is a propersubspace of H ( D n ). Therefore E ⊥ = Ker( T ∗ φ ) is infinite dimensional by the Ahernand Clark Theorem. (cid:3) We immediately obtain the following dichotomy.
Corollary . Let T φ be a left-invertible analytic Toeplitz operator on H ( D n ) forsome n > . Then either T φ is invertible or T ∗ φ is universal. Acknowledgement
This work constitutes a part of the doctoral thesis of the first author, which issupervised by the second author.
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14. R. Yang, A brief survey of operator theory in H ( D ). Handbook of analytic operator theory,223258, CRC Press/Chapman Hall Handb. Math. Ser., CRC Press, Boca Raton, FL, 2019. Departamento de Cincias Exatas e Tecnolgicas, UESC, Ilhus, Brasil.
E-mail address : msferreira@uesc . br (1st author). IMECC, Universidade Estadual de Campinas, Campinas-SP, Brazil.
E-mail address : waleed@unicamp . brbr