Operator Theory: Advances and Applications, Vol. 199, 191–202 c 2009 Birkh¨auser Verlag Basel/Switzerland Variable-coefficient Toeplitz Matrices with Symbols beyond the Wiener Algebra Albrecht B¨ottcher and Sergei Grudsky In Memory of Georg Heinig Abstract. Sequences of so-called variable-coefficient Toeplitz matrices arise in many problems, including the discretization of ordinary differential equations with variable coefficients. Such sequences are known to be bounded if the generating function satisfies a condition of the Wiener type, which is far away from the minimal requirement in the case of constant coefficients. The purpose of this paper is to uncover some phenomena beyond the Wiener condition. We provide counterexamples on the one hand and prove easy-to-check sufficient conditions for boundedness on the other. Mathematics Subject Classification (2000). Primary 47B35; Secondary 15A60, 65F35. Keywords. Toeplitz matrix, variable coefficients, matrix norm. 1. Introduction Let a be a complex-valued continuous function on [0, 1] × [0, 1] × T,whereT is the complex unit circle, a :[0, 1] × [0, 1] × T → C, (x, y, t) → a(x, y, t). For n ∈ Z,we −n puta ˆn(x, y)= T a(x, y, t) t |dt|/(2π) and so have the Fourier series ∞ n a(x, y, t)= aˆn(x, y)t , (1) n=−∞ 2 where equality holds at least in the L sense. Let AN (a)bethematrix N j k AN (a)= aˆj−k , . (2) N N j,k=0 Occasionally we allow us to write a(x, y, t)andAN (a(x, y, t)) for a and AN (a). This work was partially supported by CONACYT project U46936-F, Mexico. 192 A. B¨ottcher and S. Grudsky We refer to AN (a) as a Toeplitz matrix with variable coefficients. Clearly, if a does not depend on x and y,thenAN (a) is a pure Toeplitz matrix. It is easily seen that every (N +1)× (N + 1) matrix may be written as AN (a)withsomea.Thus, the notion of a Toeplitz matrix with variable coefficients is rather an asymptotic ∞ notion which makes sense for the entire sequences {AN (a)}N=0 but not for a single (N +1)× (N +1)matrix. Variable-coefficient Toeplitz matrices and their modifications and generaliza- tions are currently emerging in many applications (see, for example, [1], [4], [5], [6], [7], [8], [11], [12]) and go under various names, such as generalized Toeplitz [9], locally Toeplitz [11] or generalized locally Toeplitz matrices [8], Berezin-Toeplitz matrices [1], twisted Toeplitz matrices [12], or generalized discrete convolutions [10], [13]. In our opinion, variable-coefficient Toeplitz matrices [3] is the perhaps best name, at least when considering matrices of the form (2). The problem of pri- mary interest is the understanding of the spectral and pseudospectral properties of AN (a)asN →∞. Accordingly, the works cited above and also [3], [9], [14] deal with extensions of the Szeg¨o and Avram-Parter theorems, that is, with asymp- totic formulas for tr f(AN (a)), and with asymptotic inverses and pseudospectra of AN (a). This paper addresses the problem of the uniform boundedness of AN (a), that is, the question whether AN (a)∞ remains bounded as N →∞.Hereandinwhat follows, A∞ is the spectral norm of A. ∞ n If a is independent of x and y, a(t)= n=−∞ aˆnt ,wedenoteAN (a)by N TN (a). Thus, TN (a)=(ˆaj−k)j,k=0.ItiswellknownthatTN (a)∞ ≤TN+1(a)∞ for all N and that lim TN (a)∞ = M∞(a):=sup|a(t)|. (3) N→∞ t∈T If a is of the form a(x, y, t)=b(x, y)tn,then N n j + n j AN (a)=TN (t )diag b , N N j=0 (where j + n is taken modulo N + 1) and hence n AN (b(x, y)t )∞ ≤ M∞,∞(b):= sup sup |b(x, y)|. x∈[0,1] y∈[0,1] Consequently, for a given by (1) we have ∞ AN (a)∞ ≤ M∞,∞(ˆan). (4) n=−∞ Thus, if ∞ M∞,∞(ˆan) < ∞, (5) n=−∞ Toeplitz Matrices with Variable Coefficients 193 ∞ then {AN (a)∞}N=0 is a bounded sequence. Condition (5) is a condition of the Wiener type. In the case where a does not depend on x and y, it amounts to saying that TN (a)∞ remains bounded as N →∞if a belongs to the Wiener algebra, ∞ that is, if n=−∞ |aˆn| < ∞. This is clearly far away from (3) and is therefore a source of motivation for looking whether the uniform boundedness of AN (a)∞ can be guaranteed under weaker assumptions. The first question one might ask is whether the sole continuity of the gener- ating function a on [0, 1]2 ×T := [0, 1]×[0, 1]×T ensures the uniform boundedness of AN (a)∞. We show that, surprisingly, the answer to this question is no. 2 Theorem 1.1. There exist a ∈ C([0, 1] × T) such that sup AN (a)∞ = ∞. On the other hand, we will prove that AN (a)∞ remains bounded if a(x, y, t) has certain smoothness in x and y. (Notice that (5) is a requirement on the smooth- ness in t.) Our results will imply the following. Theorem 1.2. If a ∈ C4,0([0, 1]2 × T), which means that the function a(x, y, t) has continuous partial derivatives with respect to x and y up to the order 4,then sup AN (a)∞ < ∞. For a(x, y, t) independent of x and y, this is equivalent to the statement that TN (a) is uniformly bounded if a ∈ C(T). In fact, we can sharpen Theorem 1.2 as follows. Theorem 1.3. If a ∈ L∞(T,C4([0, 1]2)), that is, if a is an L∞ function on T with values in the Banach space of all functions on [0, 1]2 that have continuous partial derivatives up to the order 4,thensup AN (a)∞ < ∞. In the case of constant coefficients, this theorem is best possible: it says that ∞ sup TN (a)∞ < ∞ if a ∈ L (T). The next question is whether the exponent 4 is close to a kind of a minimum. Let 0 <α≤ 1. For a continuous function a(x, y, t)on[0, 1]2 × T we define |a(x2,y,t) − a(x1,y,t)| Mα,∞,∞(a)=sup sup sup α , t∈T y∈[0,1] x1,x2 |x2 − x1| |a(x, y2,t) − a(x, y1,t)| M∞,α,∞(a)=sup sup sup α , t∈T x∈[0,1] y1,y2 |y2 − y1| and |∆2a(x1,x2,y1,y2,t)| Mα,α,∞(a)=sup sup sup α α t∈T x1,x2 y1,y2 |x2 − x1| |y2 − y1| where ∆2a(x1,x2,y1,y2,t) is the second difference ∆2a(x1,x2,y1,y2,t)=a(x2,y2,t) − a(x2,y1,t) − (a(x1,y2,t) − a(x1,y1,t)). ∈ Here supz1,z2 means the supremum over all z1,z2 [0, 1] such that z1 = z2.We say that the function a(x, y, t) belongs to Hα,α,∞ if the three numbers Mα,α,∞(a), 194 A. B¨ottcher and S. Grudsky Mα,∞,∞(a), M∞,α,∞(a) are finite and we denote by H1+α,1+α,∞ the set of all functions a(x, y, t) that have continuous partial derivatives up to the order 2 in x and y and for which the three numbers M1+α,∞,∞(a):=Mα,∞,∞(∂xa),M∞,1+α,∞(a):=M∞,α,∞(∂ya), M1+α,1+α,∞(a):=Mα,α,∞(∂x∂ya) 4,0 2 are finite. Notice that C ([0, 1] × T) ⊂ H2,2,∞.Wehereprovethefollowing. Theorem 1.4. If β<1/2, there exist functions a in the space Hβ,β,∞ such that sup AN (a)∞ = ∞.Ifβ>1,thensup AN (a)∞ < ∞ for every function a in Hβ,β,∞. The theorem leaves a gap. We can actually remove this gap and prove that sup AN (a)∞ < ∞ whenever a ∈ Hβ,β,∞ and β>1/2, which is guaranteed if a ∈ C2,0([0, 1]2 × T). However, the proof of this result is very sophisticated. We see the purpose of this paper in revealing the delicacy of the problem of the uniform boundedness of AN (a)∞ and in providing results that might be sufficient for applications. Drawing down things to β>1/2 is a matter of mathematical ambition and will be the topic of our subsequent paper [2]. 2. H¨older continuity We already defined the space Hβ,β,∞ and the quantities Mβ,β,∞(a), Mβ,∞,∞(a), 2 M∞,β,∞(a) for a continuous function a(x, y, t)on[0, 1] × T and for 0 <β≤ 2. In addition, we put M∞,∞,∞(a)=sup sup sup |a(x, y, t)|. t∈T x∈[0,1] y∈[0,1] Note that if a(x, y, t)=xγ + yγ with 0 <γ<1, then Mβ,∞,∞(a)=M∞,β,∞(a)=∞,Mβ,β,∞(a)=0 for γ<β<1, which shows that the assumption Mβ,β,∞(a) < ∞ does not imply that Mβ,∞,∞(a)andM∞,β,∞(a) are finite. In the introduction we also mentioned 2,0 2 that C ([0, 1] × T) is contained in H1,1,0 and thus in Hβ,β,0 for 0 <β<1. This follows from the representations x2 a(x2,y,t) − a(x1,y,t)= ∂xa(ξ,y,t)dξ, x1 y2 a(x, y2,t) − a(x, y1,t)= ∂ya(x, η, t)dη, y1 x2 y2 ∆2a(x1,x2,y1,y2,t)= ∂x∂ya(ξ,η,t)dηdξ, x1 y1 which show that in fact the continuity of the partial derivatives ∂xa, ∂ya, ∂x∂ya 2,0 2 4,0 2 would suffice.