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Let Tn be the Cartesian product of n copies of T = ∂D equiped with the nor- malized Haar measure σ. Let L2(Tn) denote the usual Lebesgue space and L∞(Tn) the essentially bounded functions with respect to σ. The H2(Dn) is the of holomorphic functions f on Dn satisfying

||f||2 := sup |f(rζ)|2dσ(ζ) < ∞. 0

Tφf = P (φf) 2 n 2 n for f ∈ H (D ). Then Tφ is a bounded linear operator on H (D ). The adjoint of ∗ ∞ n 2 n Tφ is simply given by Tφ = Tφ¯. If φ ∈ H (D ), then Tφf = φf for all f ∈ H (D ) and Tφ is called an analytic Toeplitz operator. The best known examples of universal operators are all adjoints of analytic 2 ∗ Toeplitz operators on H (D), or are equivalent to one of them. For example Tφ when φ is a singular inner function or infinite Blaschke product. Another well known example was discovered by Nordgren, Rosenthal and Wintrobe [10] in the 1980’s. They proved that if φ is a hyperbolic automorphism of the unit disk D, then Cφ − λI is universal for all λ in the interior of the spectrum of Cφ. In the last few years, Cowen and Gallardo-Gutirrez ([4],[5],[6],[7]) have undertaken a thorough analysis of adjoints of analytic Toeplitz operators that are universal for H2(D). The objective of this article is to consider analytic Toeplitz operators Tφ whose adjoints are universal on H2(Dn) for n> 1. We now state our main result. ∞ n ∗ Theorem. Let φ ∈ H (D ) for n> 1. Then Tφ satisfies the Caradus criterion for universality if and only if φ is invertible in L∞(Tn) but non-invertible in H∞(Dn). This shows that analytic Toeplitz operators with universal adjoints are far more ubiquitous in the higher dimensional case n> 1.

Corollary. If φ is a non-constant inner function or a polynomial in the ring n n ∗ C[z1,...,zn] that has zeros in D but no zeros in T , then Tφ is universal for H2(Dn) with n> 1.

∗ ∗ In particular the backward shift operators Tz1 ,...,Tzn are universal when n> 1. These results are not true in general for H2(D). A closed subspace E ⊂ H2(Dn) is said to be zi-invariant if ziE ⊂ E for some i = 1,...,n. A Tφ-invariant subspace 2 n E is called maximal if E ( H (D ) and there is no Tφ-invariant subspace L such 2 n that E ( L ( H (D ). Since every maximal Tφ-invariant subspace E corresponds ∗ ⊥ to a minimal Tφ -invariant subspace E , we obtain the following version of the ISP.

An equivalent version of the ISP. Suppose n > 1 and i=1,. . . ,n. Then the ISP has a positive solution if and only if every maximal zi-invariant subspace has codimension 1 in H2(Dn). 3

The maximal z-invariant subspaces in H2(D) indeed have codimension one by Beurling’s Theorem. Hedenmalm [8] proved that this is also true in the classical 2 2 n Bergman space La(D). A closed subspace E ⊂ H (D ) that is zi-invariant for all i =1,...,n simultaneously is called a shift-invariant subspace. The description of all shift-invariant subspaces for n > 1 is a central open problem of multivariable operator theory (see the recent surveys of Sarkar [13] and Yang [14]) which could therefore lead to a solution of the ISP.

1. Proof of theorem Cowen and Gallardo-Gutirrez ([4],[5],[6],[7]) have frequently observed that if φ ∈ H∞(D) and there is an ℓ > 0 so that |φ(eiθ))| > ℓ almost everywhere on T, then 2 ∗ T1/φ is a left inverse for Tφ on H (D) and therefore that Tφ is surjective. Our first goal is to characterize the left-invertibility of analytic Toeplitz operators on H2(Dn) for all n ≥ 1. The following general result can be found in textbooks, but we state it here for completeness. Lemma 1. Let T be a 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 injective and has closed range, then it has a bounded inverse T −1 : T (H) → H by the open mapping theorem [12, Corollary 2.12]. Therefore T −1T = I and (3) =⇒ (1).  Recently Koca and Sadik [9] proved that if f ∈ H∞(Dn), then the subspace E = fH2(Dn) is shift-invariant if and only if f is invertible in L∞(Tn). Using this result we obtain the following characterization of left-invertibility for analytic Tφ. 2 n Proposition 2. Let Tφ be an analytic Toeplitz operator on H (D ) for n ≥ 1. ∞ n Then Tφ is left-invertible if and only if φ is invertible in L (T ). ∞ n Proof. If φ is invertible in L (T ), then T1/φ is a (not necessarily analytic) Toeplitz operator and

T1/φTφf = T1/φ(φf)= Pf = f. 2 n for all f ∈ H (D ). Therefore Tφ is left-invertible. Conversely if Tφ is left-invertible, then it has closed range by Lemma 1. Then E = φH2(Dn) is closed and satisfies ziE ⊂ E for all i =1,...,n. Hence E is a shift-invariant subspace and therefore φ is invertible in L∞(Tn) by the result of Koca and Sadik [9, Theorem 2].  Therefore by Lemma 1 and Proposition 2 we know precisely when the adjoint of an analytic Toeplitz operator is surjective. The key ingredient in the proof of the main theorem is a result of Ahern and Clark [1, Theorem 3].

2 n The Ahern and Clark Theorem. If f1,...,fk ∈ H (D ) with k

By the shift-invariant subspace E generated by f1,...,fk means the smallest shift-invariant subspace containing f1,...,fk. So the shift-invariant subspaces of the form φH2(Dn) for some invertible φ ∈ L∞(Tn) are singly generated. We arrive at our main result.

∞ n ∗ Theorem. Let φ ∈ H (D ) for n > 1. Then Tφ is surjective and has infinite dimensional kernel if and only if φ is invertible in L∞(Tn) but not in H∞(Dn). ∗ ∗ Proof. Suppose Tφ is surjective and Ker(Tφ ) is infinite dimesnional. Then φ is invertible in L∞(Tn) by Lemma 1 and Proposition 2. It follows that the closed 2 n ⊥ ∗ subspace E = φH (D ) has infinite codimension since E = Ker(Tφ ). In particular φH2(Dn) 6= H2(Dn) and hence 1/φ∈ / H∞(Dn). Conversely suppose φ is invertible ∞ n ∞ n ∗ in L (T ) but not in H (D ). Then Tφ is surjective by Lemma 1 and Proposition 2. The assumption that 1/φ∈ / H∞(Dn) means that E = φH2(Dn) is a proper 2 n ⊥ ∗ subspace of H (D ). Therefore E = Ker(Tφ ) is infinite dimensional by the Ahern and Clark Theorem.  We immediately obtain the following dichotomy.

2 n Corollary. Let Tφ be a left-invertible analytic Toeplitz operator on H (D ) for ∗ some n> 1. Then either Tφ is invertible or Tφ is universal.

Acknowledgement This work constitutes a part of the doctoral thesis of the first author, which is supervised by the second author.

References

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14. R. Yang, A brief survey of operator theory in H2(D2). 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: [email protected] (1st author).

IMECC, Universidade Estadual de Campinas, Campinas-SP, Brazil. E-mail address: [email protected] (2nd author).