EXTREMAL DOMAINS FOR SELF-COMMUTATORS IN THE BERGMAN SPACE

EXTREMAL DOMAINS FOR SELF-COMMUTATORS IN THE BERGMAN SPACE MATTHEW FLEEMAN AND DMITRY KHAVINSON Abstract. In [10], the authors have shown that Putnam's inequality for the norm of self-commutators can be improved by a factor of 1 for Toeplitz operators with analytic symbol ϕ acting on the Bergman space A Ω). This improved upper bound is sharp when ϕω) is a disk. In this paper we show that disks are the only domains for which the upper bound is attained. 1. Introduction Let H be a complex Hilbert Space with inner product, : H H C and T be a bounded linear operator with adjoint T. Assume [T, T ] := T T T T 0, i.e. T is hyponormal, then Putnam's inequality states [T, T ] AreaspT )), where spt ) denotes the spectrum of T cf. []). In [8], it was proved that when H is the Hardy space, H Ω), or the Smirnov space, E Ω) cf. [4, p. and p.173]), where Ω is a domain bounded by nitely many rectiable curves, this inequality is sharp. Indeed, if we take T = T ϕ to be the Toeplitz operator with symbol ϕ analytic in a neighborhood of Ω, then by the Spectral Mapping Theorem cf. [13, p.63]) spt ϕ ) = ϕω), and the following lower bound holds: 1.1) [T ϕ, T ϕ ] 4Area ϕω)) ϕ E Ω) P Ω), where P Ω) denotes the perimeter of Ω. Since [Tϕ, T ϕ ] is a positive operator, an interesting consequence follows from 1.1) by setting ϕz) = z, so that ϕ E Ω) = 1 E Ω) = P Ω), and combining 1.1) with Putnam's inequality, we obtain 1.) P Ω) 4AreaΩ), which is the classical isoperimetric inequality. The equality in 1.) holds if and only if Ω is a disk. We are interested in exploring similar questions in a Bergman space setting. Recall that the Bergman space A D) is dened by: A D) := {f HolD) : fz) daz) < }, The authors acknowledge support from the NSF grant DMS - 0855597. 1 D

EXTREMAL DOMAINS FOR SELF-COMMUTATORS IN THE BERGMAN SPACE where da denotes the area measure on D. A Ω) is dened accordingly cf. [5]). The orthogonal projection from L D, da) = L D) onto A D) is called the Bergman projection and has integral representation P fz) = D fω) 1 ωz) daω). Recently, Bell, Ferguson, and Lundberg in [3] obtained a dierent lower bound for [T ϕ, T ϕ ] when T ϕ acts on the Bergman space A Ω). This lower bound turned out to be connected with the torsional rigidity of Ω cf. [11, p.]). Intuitively, if we imagine a cylindrical object with cross-section Ω, then the torsional rigidity quanties the resistance to twisting. There are several equivalent denitions cf. [11, pp.87-89]). The one used in [3] and [10] is the following: Denition. If Ω is a simply connected domain, the torsional rigidity ρ = ρω) is ρ = ν, where ν is the unique solution to the Dirichlet problem { ν = ν Ω = 0. In [3] it is shown that for T z acting on A Ω), [T z, T z ] Ω ρω) AreaΩ). The authors also conjectured that in the Bergman space setting Putnam's inequality could be improved by a factor of 1. This conjecture was recently proven by Olsen and Reguera in [10]. Combined with the lower bound given by Bell, Ferguson, and Lundberg, this yields a new proof of the St. Venant inequality ρω) AreaΩ). In this note we show that in the Bergman space setting disks are the only extremal domains for which Tz, T z achieves it's maximal bound of AreaΩ). More precisely, in section, we present a simple argument illustrating that the upper bound in Putnam's inequality can only be attained when Ω is a disk in any Hilbert space, while in the Bergman space setting Tϕ, T ϕ = AreaϕΩ)) = 1 when ϕω) = D, the unit disk. In section 3, we give a sketch of Olsen and Reguera's proof of the improved upper bound in the Bergman setting. This is needed for our argument in section 4 where we show that the upper bound for Tϕ, T ϕ is achieved if and only if ϕω) is a disk. This gives another proof of the well known fact that St. Venant's inequality becomes equality only for disks.. Non-sharpness of Putnam's Inequality In this section, we illustrate why Putnam's inequality is not sharp in a Bergman space setting. We start with the following elementary Lemma found in [5, p.13]. Lemma 1. Suppose ω = ϕz) maps a domain D conformally onto a domain Ω. Then the linear map T f) = g dened by gz) = fϕz))ϕ z)

EXTREMAL DOMAINS FOR SELF-COMMUTATORS IN THE BERGMAN SPACE 3 denes an isometry of A Ω) onto A D). Proof. That T is an isometry is clear from the fact that fw) daω) = fϕz)) ϕ z) daz), Ω D where ϕ z) is the Jacobian of the conformal map ϕ. To see that T is onto, let g A D) and let z = ψw) be the inverse mapping. Then fω) = gψω))ψ ω) is in A Ω) and T f) = g since T f) = fϕz))ϕ z) = gψϕz))ϕ ψω))ψ ω), and we can write ψ 1 ω) = ϕ ψω)), which is well dened on D because ϕ D 0. So T f) = g and T is onto as claimed. The following statement is now straightforward. Theorem. Suppose Ω is a bounded Jordan domain and ϕ : D Ω is a conformal mapping. Then [T ϕ, T ϕ ] A D) A D) = [T z, T z ] A Ω) A Ω). Proof. We start with the following straightforward calculation cf. []). If we take A 1D) to be the unit ball of A D), we have that Thus we have that [Tϕ, T ϕ ] = sup [Tϕ, T ϕ ]f, f f A 1 D) ) = sup T ϕ f A D) T ϕf A D) f A 1 D) = sup f A 1 D) = sup f A 1 D) [T ϕ, T ϕ ] = = sup ) ϕf A D) P ϕf) A D) ϕf L D) P ϕf) A D) ). ) sup ϕf g A 1 D) L D) P ϕg) A D) g A 1 D) { inf { ϕg f A D) f L D) }}. Fixing f, g A D), with g A 1D), and letting ψ = ϕ 1, we see that ϕg f L D) = ϕg f da = zgψω)) fψω)) ψ ω) daω) Ω = zgψω))ψ ω) fψω))ψ ω) daω). Ω By Lemma 1, T f) = fψω))ψ ω) is a surjective isometry from A D) onto A Ω). So, we have that sup { inf { ϕg g A 1 D) f A D) f L D) }} = and the proof is complete. D sup { inf { zg g A 1 Ω) f A Ω) f L Ω) }},

EXTREMAL DOMAINS FOR SELF-COMMUTATORS IN THE BERGMAN SPACE 4 This leads to the following interesting observation. Theorem 3. Let ϕ and Ω be as in Theorem. Then [T ϕ, T ϕ ] can only achieve the upper bound stated in Putnam's inequality cf. [1]) if ϕd) is a disk. Proof. We argue as follows. Let A 1Ω) be the unit ball in A Ω). By Theorem, we have that [Tϕ, T ϕ ] A D) A D) = [Tz, T z ] A Ω) A Ω) = sup { inf { zg g A 1 Ω) f A Ω) f A Ω) }}. Fix g A 1Ω), we have inf zg f A Ω) f L Ω) = inf z f g da h: gh A Ω) Ω inf zg f da f A Ω) Ω inf h: gh A Ω) z h since g A 1Ω). Further, since the polynomials P are dense in H Ω) for any bounded Jordan domain Ω, and since for all g A 1Ω), and all p P, we have that gp A Ω), we obtain from the last inequality that inf zg f A Ω) f L Ω) inf z h L Ω) h RΩ) where RΩ) is the uniform closure of the algebra of rational functions in Ω with poles outside Ω. In [1], Alexander proved that AreaΩ) inf z f L Ω), f RΩ) and further that equality is achieved if, and only if, Ω is a disk cf. [, 6]). The theorem now immediately follows. Remark. If we take H to be any Hilbert space and T to be any subnormal operator with a rationally cyclic vector, then there is a positive nite Borel measure µ on spt ) such that T is unitarily equivalent to multiplication by z on R spt ), µ) which is the closure of RspT )) in L spt ), µ) cf. []). From this, repeating the above argument word for word, we obtain that if [T AreaspT )), T ] =, then spt ) must be a disk. The case when T does not have a rationally cyclic vector follows from the above case as in [], so that the above theorem extends to all Hilbert spaces and any subnormal operator T. The following example shows that the converse fails, and in particular fails for Bergman spaces. Example 4. Let ϕz) = z k for some k N, and let T ϕ : A D) A D), and recall that P : L D) A D) is the orthogonal projection of L D) onto A D). As we showed in Theorem, [T ϕ, T ϕ ] = sup g A 1 D) ) ϕg L D) P ϕg) A D).

EXTREMAL DOMAINS FOR SELF-COMMUTATORS IN THE BERGMAN SPACE 5 Let ψ n z) = n+1 ) 1 z n, where n = 0, 1,,.... The collection {ψ n z)} forms an orthonormal basis for A D) cf. [5, p. 11]). For g A 1D), we may write gz) = ĝn)ψ n z), where ĝn) := g, ψ n and ĝn) = 1. Since we have an orthonormal basis at hand, we may calculate P ϕg) explicitly. P z k g) = z k g, ψ n ψ n. Calculating z k g, ψ n, we nd that z k g, ψ n = z k m=0 ĝm)ψ m, ψ n, where m + 1)) z k ĝm)ψ m, ψ n = ĝm)z m z n+k da D m + 1)) = ĝm) m + n + k + δ m,n+k. Where δ i,j is the Kronecker symbol. Thus, ) 1 z k g, ψ n = ĝn + k), n + k + 1 and so we obtain that.1) P z k g) A D) = n + k + 1 ĝn + k). Similarly, when we calculate z k g A D), we nd that m + 1)) z k ĝm)ψ m, ψ n = ĝm)z m+k z n da D m + 1)) = ĝm) m + n + k + δ m+k,n. Thus { z k n k+1 g, ψ n = n+1 ĝn k) n k 0 n < k. Hence,.) z k g L D) = n k + 1 ĝn k). n=k Combining.1) and.), we obtain that [Tϕ, n k + 1 T ϕ ] = sup { ĝn k) g A 1 D) n=k k 1 sup { g A 1 D) n + k + 1 ĝn) + n + k + 1 ĝn + k) } n + k + 1 n k + 1 ) ĝn) } n=k

since EXTREMAL DOMAINS FOR SELF-COMMUTATORS IN THE BERGMAN SPACE 6 k 1 sup { g A 1 D) n + k + 1 ĝn) + n=k k n + k + 1 ĝn) } n + k + 1 n k + 1 n + k + 1 n k + 1 n + k + 1 = k n + k + 1, k 0. Further, since n+1 n+k+1 k k for 0 n k 1, we obtain that [Tϕ, k T ϕ ] ĝn) = 1 k. sup g A 1 D) This upper bound is achieved if we take g = ψ k 1, so that [Tϕ, T ϕ ] = 1, whenever ϕz) = z k for any k N. Thus, we see that the converse to Theorem 3 fails. This calculation, independently done by T. Ferguson, leads to the conjecture, following Bell et. al. that in the Bergman space setting, Putnam's inequality can be improved by a factor of 1. This conjecture was recently proven by Olsen and Reguera in [10]. In the following section we give a sketch of their proof, which will be needed in Ÿ4. 3. Sketch of the Olsen and Reguera proof In their paper, Olsen and Reguera worked with the Hankel operator on A D) with symbol ϕ L D) dened by H ϕ f) := I P )ϕf), They then proved the following theorem. f A D). Theorem 5. Let ϕ A D) be in the Dirichlet space D, i.e ϕ A D). Then H ϕ 1 ϕ A D). Proof. For the reader's convenience, we give here a sketch of their proof. For full details, cf. [10, Ÿ]. For f A D),we write fz) = n 0 a nz n, and without loss of generality we assume that f A D) = 1, and set ϕz) = k 1 c kz k we can also assume without loss of generality that ϕ0) = 0). The basic strategy is to calculate H ϕ f in terms of these Taylor coecients and obtain the desired norm estimate by working directly with the coecients. Crucial to our purposes is the fact that the only inequality used in [10] is the arithmetic-geometric inequality ab a +b. First, by computing P ϕz n ) for each n, we nd that H ϕ f = ϕz)fz) P ϕf)z) 3.1) = c l a n z l z n l 0 n 0 n 1 n 1 k=0 k + 1 a nc n k z k. Then, after rewriting the above expression to take advantage of the orthogonality, we let z = re iθ and integrate the modulus squared with respect to dθ. This yields that H ϕ f A D) is equal to 3.) r k a n c n+k r n + r k a n c n k r n k) k + 1 ). k 1 n 0 k 0 n k+1

EXTREMAL DOMAINS FOR SELF-COMMUTATORS IN THE BERGMAN SPACE 7 This expression is once again rewritten and then integrated with respect to rdr. If we set a n a m c k+m c k+n I) := n + m + k + 1, n,m 1,k 0 II) := a n a m c m k c n k m k)n k) )m + 1)n + m k + 1), k 0 n,m k+1 then we obtain that H ϕ f A D) = I) + II). Relabeling the indices slightly, and setting a n = b n+1 ), we nd that nm I) = b n b m c k+m c k+n n + m + k, II) = n,m 1,k 0 n,m,k 1 mn b n+k b m+k c m c n n + m + k. Using the symmetry in m and n we may interpret each term as being half that of its real part so that the inequality Reab) a + b may be applied to each term of the above expressions, and this is the only place where inequalities occur, which yields I) II) n,m 1,k 0 n,m,k 1 bn c k+m + b m c k+n ) nm n + m + k) = bn+k c m + b m+k c n ) nm n + m + k) = n,m 1,k 0 n,m,k 1 b n c k+m nm n + m + k =: I ), b n+k c m mn m + n + k =: II ). By changing the order of summation properly with the goal of isolating unique pairs of indices, we arrive at the expression I ) + II ) = b n c m nm. n,m 1 Finally, replacing a n = b n+1 ), we now see that the right hand side exactly equals 1 b n c m mn = 1 a n c m m = 1 f A D) ϕ A D). n,m 0 n 0 m 1 which was to be shown. Remark 6. From here, the inequality 3.3) [T ϕ, T ϕ ] ϕ A Ω) is seen as a corollary by showing that if ψ is the conformal map from Ω to D, then [T ϕ, T ϕ ] A Ω) A Ω) = H ϕ A Ω) L Ω) = H ϕ ψ A D) L D), and thus we can apply Theorem 5 and the result follows. Again, refer to [10] for more details. Taking ϕz) = z, and combining 3.3) with the result of Bell, Ferguson, and Lundberg, one arrives at a proof of the sharp St. Venant inequality 3.4) ρω) Area Ω).

EXTREMAL DOMAINS FOR SELF-COMMUTATORS IN THE BERGMAN SPACE 8 It should be noted that when ϕ = z k many of the terms in 3.) become zero resulting in the value we found in Example 4 of 1 rather than the Olsen-Reguera upper bound of k. 4. Unique extremality of the Disk We now show that from the proof of Theorem 5, we may deduce that equality is obtained in 3.3) and only if ϕω) is a disk. This will come as a corollary to the following Theorem. Theorem 7. Suppose ϕz) is analytic in D such that ϕz) D, the Dirichlet space. Further suppose that Then ϕd) is a disk. [Tϕ, T ϕ ] A D) A D) = ϕ A Ω). Proof. Since ϕ D, H ϕ is compact cf.[14, Ÿ7.4, p.145]), and so attains its norm on A 1D). Recall from the proof of Theorem that ) [T ϕ, T ϕ ] A D) A D) = sup f A 1 D) ϕf L D) P ϕf) A D) = H ϕ A D) L D). We now examine the proof of Theorem 5 to nd exactly when equality may happen. Recall that if f A 1D), then H ϕ f A D) = I) + II) I ) + II ) = 1 f A D) ϕ A D), where I), II), I ), and II ) are as in Theorem 5. The only inequality at work here is Reab) a + b, where equality is achieved if, and only if, a = b. Thus we nd that equality is achieved if I) = I ) and II) = II ), which will only happen if the following innite system of equations is satised: 4.1) b i c j+k = b j c i+k i, j 1, k 0, 4.) b i+k c j = b j+k c k i, j, k 1, where ϕz) = k 1 c kz k is given and fz) = n 1 nb nz n 1 is an extremal function in A 1D) such that the above equations are satised. It is clear that if c k = 0 for all but a single k, that is if ϕz) = cz k then the above equations can be satised by a non-zero f A 1D). In fact, we know from Example 4 that if we take f = ψ k 1, then 4.1) and 4.) will be trivially satised. As we remarked above, in this case the formula 3.) is oversimplied, so that the resulting norm is c instead of our expected upper bound of c k It is also clear that the above equations are satised when ϕz) = k 1 rk z k for some r < 1. Here, 1 the extremal f = ϕ A D) k 0 rk z k. In both cases ϕd) is a disk. We will now show that for all other ϕ, 4.1) and 4.) only hold for f 0. We will do this by looking at two cases. First suppose that ϕz) has at least two non-zero Taylor coecients, c m, c n, with m < n, and at least one zero coecient c k0 such that k 0 > n. This encompasses all Taylor series which do not have an innite non-zero tail. Without loss of generality

EXTREMAL DOMAINS FOR SELF-COMMUTATORS IN THE BERGMAN SPACE 9 we can assume that k 0 = by taking c k0 to be the rst zero coecient after at least two non-zero coecients. We now assume that we have found an f A 1D) whose Taylor coecients satisfy 4.1) and 4.). By 4.), we have that 4.3) b n+k c m = b m+k c n k 1, 4.4) b n+k+1 c m = b m+k c n+k+1 k 1. Hence, we can conclude that b j = 0 for all j n + by 4.4), which implies that b m+k = 0 for all k by 4.3). We now let i = m + 1, j = m and choose k such that m + k = n. Then by 4.1) we have that b m+1 c m+k = b m+1 c n = b m c m+1+k = b m c n+1 = 0, which, shows that b m+1 = 0. Now choosing i < m + 1, j = m + 1 and choosing k such that m + 1 + k = n, then by 4.1) we have that b i c m+1+k = b i c n = b m+1 c n+k = 0. Hence, we have that in fact b i = 0 for all i 1, which means that f 0. Suppose now instead, that ϕz) is such that its Taylor series does have an innite non-zero tail, but the coecients do not exhibit a geometric progression. This means that we can nd three non-zero coecients, c m, c m+1, and c m+ such that c m 4.5) c m+1. c m+1 c m+ By 4.), we have that 4.6) b m+k c m+1 = b m+1+k c m k 1, 4.7) b m+1+k c m+ = b m++k c m+1 k 1. In particular, choosing k = in 4.6) and k = 1 in 4.7) we have that b m+ c m+1 = b m+3 c m, and b m+ c m+ = b m+3 c m+1, which by 4.5) means that b m+ = b m+3 = 0. In fact, the same argument shows that b j = 0 for all j m +. But then of course, by 4.6) we immediately get that b j = 0 for all j m + 1. Now once again simply let i < m + 1, j = m + 1, and k = 1, and then by 4.1) we once again have that b i = 0 for all i 1, and so f 0. Our result now follows as a corollary. Corollary 8. [T z, T z ] A Ω) A Ω) = AreaΩ) Proof. By Theorem, if, and only if, Ω is a disk. [T z, T z ] A Ω) A Ω) = [T ϕ, T ϕ ] A D) A D) where ϕ is the conformal map from D onto the simply connected domain Ω. The corollary now immediately follows from Theorem 6. Remark. Just as the results in [10] and [3] can be combined to give a new proof of the St. Venant inequality, corollary 8 gives another proof that equality in the St. Venant inequality characterizes disks.

EXTREMAL DOMAINS FOR SELF-COMMUTATORS IN THE BERGMAN SPACE 10 5. Concluding Remarks It must be noted that Olsen-Reguera proof only applies to Toeplitz operators whose symbol is in the Dirichlet space, and the upper bound is in terms of ϕ A rather than the area of ϕd). Example 4 however leads us to believe that the upper bound in terms of the area ϕd) without multiplicity, i.e. the spectrum of T ϕ is, perhaps, the correct one. If so, the result should be extendable to all Toeplitz operators with bounded analytic symbol. One of the consequences of the Olsen- Reguera result is a unique sharp upper bound on all extremal problems of the type sup ϕuda u A D)), u 1 D since by a standard duality argument, we have that inf ϕ f L D) = sup{ ϕuda : u A D)), u 1} f A D) D cf. [7, Ÿ4]). We believe examining extremal problems of this type could very well lead to a proof of the Olsen-Reguera result which doesn't depend on power series calculations. It would be interesting to be able to determine the sharp isoperimetric bounds for these types of extremal problems, similar to Putnam's inequality and [9]. It would also be worthwhile to investigate nitely connected domains. While the result of Olsen-Reguera is certainly applicable, it would be nice to establish the deciency based on the number of boundary components, as in [9]. Finally, it would be interesting to see which, if any, other so-called isoperimetric sandwiches could be expressed in terms of [T, T ] acting on other function spaces, e.g. the Dirichlet space. References [1] Alexander, H., Projection of polynomial hulls, J. Funct. Anal. 3 1973), pp 13-19. [] Axler, S., Shapiro, J.H., Putnam's theorem, Alexander's spectral area estimate, and VMO, Math. Ann. 71 pp 161-183, 1985. [3] Bell, S., Ferguson, T., Lundberg, E., Self-commutators of Toeplitz operators and isoperimetric inequalities, arxiv:111.6937, 01 [4] Duren, P. Theory of H p Spaces, Academic Press, New York, 1970. [5] Duren, P., Schuster, A., Bergman Spaces, Mathematical Surveys and Monographs v.100, 004. [6] Gamelin, T.W., Khavinson, D., The isoperimetric inequality and rational approximation, The American Mathematical Monthly, Vol. 96, No. 1 Jan., 1989), pp 18-30. [7] Guadarrama, Z., Khavinson, D., Approximating z in Hardy and Bergman norms, Banach Spaces of Analytic Function, pp. 43-61, Contemp. Math., 454, Amer. Math. Soc., Providence, RI, 008. [8] Khavinson, D., A note on Toeplitz operators, Banach spaces Columbia, Mo., 1984) Lecture Notes in Math., vol. 1166, Springer, Berlin, 1985, pp. 89-94. [9] Khavinson, D., Lueking, D., On an extremal problem in the theory of rational approximation, Journal of Approximation Theory 50, pp17-137, 1987. [10] Olsen, J-F., Reguera, M.C., On a sharp estimate for Hankel operators and Putnam's inequality, arxiv:1305.5193v1, 013 [11] Pólya, G., Szegö, G., Isoperimetric Inequalities in Mathematical Physics, Annals of Mathematics Studies, no. 7, Princeton University Press, Princeton, N. J., 1951. [1] Putnam, C.R., An inequality for the area of hyponormal spectra. Math.Z. 116, 33-330, 1970. [13] Rudin, W., Functional Analysis, McGraw-Hill, 1973. [14] Zhu, K., Operator Theory in Function Spaces, Mathematical Surveys and Monographis v.138, 1990.