New York Journal of Mathematics
New York J. Math. 3 (1997) 1–31.
Norms of Inverses, Spectra, and Pseudospectra of Large Truncated Wiener-Hopf Operators and Toeplitz Matrices
A. B¨ottcher, S. M. Grudsky, and B. Silbermann
Abstract. This paper is concerned with Wiener-Hopf integral operators on Lp and with Toeplitz operators (or matrices) on lp. The symbols of the op- erators are assumed to be continuous matrix functions. It is well known that the invertibility of the operator itself and of its associated operator imply the invertibility of all sufficiently large truncations and the uniform boundedness of the norms of their inverses. Quantitative statements, such as results on the limit of the norms of the inverses, can be proved in the case p = 2 by means of C∗-algebra techniques. In this paper we replace C∗-algebra methods by more direct arguments to determine the limit of the norms of the inverses and thus also of the pseudospectra of large truncations in the case of general p.
Contents 1. Introduction 2 2. Structure of Inverses 4 3. Norms of Inverses 7 4. Spectra 15 5. Pseudospectra 17 6. Matrix Case 22 7. Block Toeplitz Matrices 23 8. Pseudospectra of Infinite Toeplitz Matrices 27 References 30
Received December 5, 1996. Mathematics Subject Classification. 47B35. Key words and phrases. Norms of inverses, Spectral approximation, Pseudospectra, Wiener- Hopf operators, Toeplitz matrices. The research of the first author was supported by the Alfried Krupp F¨orderpreis f¨ur junge Hochschullehrer of the Krupp Foundation. The research of the second author was supported by the Russian fund of fundamental investigations (Grants N 95-01-01285a and N 96-01-01195a).
c 1997 State University of New York ISSN 1076-9803/97 1 2 A. B¨ottcher, S. M. Grudsky, and B. Silbermann
1. Introduction Given a complex number c ∈ C and a function k ∈ L1(R), we consider the Wiener- Hopf integral operator W defined by Z∞ (1) (Wϕ)(x)=cϕ(x)+ k(x−t)ϕ(t)dt (0 0 on Lp(0, ∞)(1≤p≤∞). Let Fk be the Fourier transform of k, Z∞ (Fk)(ξ)= k(x)eiξxdx (ξ ∈ R). −∞ Many properties of the operator (1) can be read off from the function a = c + Fk. This function is referred to as the symbol of the operator (1), and we will henceforth denote this operator by W (a). The set {c + Fk : c ∈ C,k∈L1(R)}is denoted by C + FL1(R) and called the Wiener algebra. For τ ∈ (0, ∞), the truncated Wiener-Hopf integral operator Wτ (a) is the com- p pression of W (a)toL(0,τ). Thus, Wτ (a) acts by the rule Zτ (2) Wτ (a)ϕ (x)=cϕ(x)+ k(x−t)ϕ(t)dt (0 ε all belong to σp(A). In case W (a)orWτ(a) is invertible, we denote the inverse by −1 −1 W (a) and Wτ (a), respectively. −1 ε This paper concerns the limits of kWτ (a)kp,σp(Wτ(a)),σp(Wτ(a)) as τ goes −1 ε to infinity. We also study kAτ k and σp(Aτ ) in case {Aτ } is a more complicated family of operators, say X Y XY (4) Aτ = Wτ (ajk)orAτ=Pτ W(ajk)Pτ, j k j k where Pτ is given by ϕ(x)if0 p p here and in what follows we freely identify L (0,τ) and the image of Pτ on L (0, ∞). Let a ∈ C + FL1(R). The set a(R˙ ):={c}∪{a(ξ):ξ∈R}is a closed continuous curve in the complex plane. We give this curve the orientation induced Truncated Wiener-Hopf Operators and Toeplitz Matrices 3 by the change of ξ from −∞ to +∞.Forλ6∈ a(R˙ ), we denote by wind (a, λ) the winding number of a(R˙ ) about λ. A classical result says that n o ˙ ˙ (6) σp W (a) = a(R) ∪ λ ∈ C \ a(R) : wind (a, λ) 6=0 for 1 ≤ p ≤∞(see [19] and [14]). Now assume a ∈ C+FL1(R) and 1 ≤ p ≤∞. It is well known (see [14, Theorem III.3.1] for p ∈ [1, ∞) and [24, Theorem 4.58] for p = ∞) that −1 (7) lim sup kWτ (a)kp < ∞ τ→∞ if and only if W (a) is invertible on Lp(0, ∞). Note that (7) includes the requirement p that Wτ (a) is invertible on L (0,τ) for all sufficiently large τ>0. In order to say more about the upper limit in (7), we need the notion of the associated symbol. Given a = c + Fk ∈ C + FL1(R), the associated symbol 1 a˜ ∈ C + FL (R) is defined bya ˜(ξ):=a(−ξ). Thus, W (˜a) and Wτ (˜a) may be given by (1) and (2) with k(x − t) replaced by k(t − x). Sincea ˜(R˙ ) differs from a(R˙ ) only in the orientation, we infer from (6) that σp(W (a)) = σp(W (˜a)). In particular, W (˜a) is invertible if and only if W (a) is invertible. In this paper we prove the following results. Theorem 1.1. Let a ∈ C + FL1(R) and 1 ≤ p<∞.If(7)holds then the limit −1 limτ→∞ kWτ (a)kp exists and n o −1 −1 −1 (8) lim kWτ (a)kp = max kW (a)kp, kW (˜a)kp . τ→∞ Theorem 1.2. There exist a ∈ C + FL1(R) and p ∈ (1, ∞) such that −1 −1 (9) kW (˜a)kp > kW (a)kp. Given a family {Mτ }τ∈(0,∞) of subsets Mτ of C, we define the limiting set limτ→∞ Mτ as the set of all λ ∈ C for which there are τ1,τ2,... and λ1,λ2,... such that 0 <τ1 <τ2 <... ,τn →∞,λn∈Mτn,λn→λ. ε Here is what we will prove about the limiting sets of σp(Wτ (a)) and σp(Wτ (a)). Theorem 1.3. Let a ∈ C + FL1(R) and 1 ≤ p ≤∞. Then for every τ>0the spectrum σp(Wτ (a)) is independent of p. In particular, (10) lim σp Wτ (a) τ→∞ does not depend on p. There exist a ∈ C + FL1(R) such that (10) is not equal to the spectrum of W (a). Theorem 1.4. If a ∈ C + FL1(R) and 1 We remark that we will actually prove extensions of Theorems 1.1 and 1.4 which are also applicable to operators as in (4). Furthermore, we will extend Theorems 1.1 and 1.4 to the case of matrix-valued symbols. Finally, we will establish analogues of the above theorems for Toeplitz operators and matrices. 2. Structure of Inverses In what follows we have to consider the products W (a)W (b) and Wτ (a)Wτ (b) and are thus led to algebras generated by Wiener-Hopf operators. We henceforth exclude the case p = ∞ (but see the remark at the end of this section). The precipice between p ∈ [1, ∞) and p = ∞ comes from the fact that the projections Pτ given by (5) converge strongly to the identity operator I if and only if 1 ≤ p<∞. Given a Banach space X, we denote by L(X) and K(X) the bounded and com- pact (linear) operators on X, respectively. We need a few facts from [13, Section 2] and [8, Chapter 9]. If a ∈ L∞(R), then the operator ϕ 7→ F −1aF ϕ is bounded on L2(R). The set of all a ∈ L∞(R) for which there exists a constant C = C(a, p) < ∞ such that −1 2 p kF aF ϕkp ≤ Ckϕkp for all ϕ ∈ L (R) ∩ L (R) is denoted by Mp(R). The set Mp(R) is a Banach algebra with pointwise algebraic operations and the norm kF −1aF ϕk kak = sup p : ϕ ∈ L2(R) ∩ Lp(R),ϕ6=0 . Mp(R) kϕkp 1 ∞ We have M1(R)=C+FL (R) and M2(R)=L (R). If p ∈ (1, 2) ∪ (2, ∞) and 1/p +1/q = 1, then 1 ∞ C + FL (R) ⊂ Mp(R)=Mq(R)⊂L (R), both inclusions being proper. For a ∈ Mp(R), the Wiener-Hopf integral operator W (a) is defined by −1 p W (a)ϕ = χ+(F aF )ϕ ϕ ∈ L (0, ∞) , where (F −1aF ) is the continuous extension of F −1aF from L2 ∩ Lp to all of Lp 1 and χ+ denotes the characteristic function of (0, ∞). Of course, if a ∈ C + FL (R) then W (a)ϕ is given by (1). One can also associate a Hankel operator H(a) ∈L(Lp(0, ∞)) with every a ∈ Mp(R). We confine ourselves to stating that H(a) can be given by the formula Z∞ H(a)f (x)= k(x+t)ϕ(t)dt (0 0 if a = c + Fk ∈ C+FL1(R). ˙ 1 Let Cp(R) denote the closure of C + FL (R)inMp(R). We have ˙ 1 ˙ ˙ C1(R)=C+FL (R),C2(R)=C(R), where C(R˙ ) stands for the continuous functions on R which have finite and equal limits at +∞ and −∞.Ifp∈(1, 2) ∪ (2, ∞) and 1/p +1/q = 1, then 1 ˙ ˙ ˙ C + FL (R) ⊂ Cp(R)=Cq(R)⊂C(R)∩Mp(R); Truncated Wiener-Hopf Operators and Toeplitz Matrices 5 ˙ again both inclusions being proper. One can show that if a ∈ Cp(R) and a(ξ) 6=0 ˙ −1 ˙ for all ξ ∈ R = R ∪ {∞}, then a ∈ Cp(R). If a ∈ M (R) then kW (a)k = |W (a)| = kak , where |W (a)| is the p p p Mp(R) p essential norm of W (a)onLp(0, ∞), n o p |W (a)|p := inf kW (a)+Kkp :K∈K L (0, ∞) . ˙ p Further, if a ∈ Cp(R) and K ∈K(L(0, ∞)), then σp(W (a)) is the set (6) and σp(W (a)) ⊂ σp(W (a)+K). In particular, W (a) and a are always invertible in case W (a)+K is invertible. p We denote by Ap the smallest closed subalgebra of L(L (0, ∞)) containing the 1 ˙ set {W (a):a∈C+FL (R)}.Ifa, b ∈ Cp(R), then (12) W (a)W (b)=W(ab) − H(a)H(˜b) and H(a) and H(˜b) (where ˜b(ξ)=b(−ξ)) are compact. This and the equality |W (a)| = kak imply that every operator in A is of the form W (a)+K with p Mp(R) p ˙ p a ∈ Cp(R) and K ∈K(L (0, ∞)). It is well known that in fact n o ˙ p (13) Ap = W (a)+K:a∈Cp(R),K∈K L (0, ∞) . ˙ p Suppose a ∈ Cp(R),K∈K(L(0, ∞)), and W (a)+K is invertible. Then −1 ˙ −1 a ∈ Cp(R), and (12) with b = a gives W (a)+K W(a−1)=I−H(a)H(˜a−1)+KW(a−1), −1 which shows that W (a)+K equals −1 −1 W (a−1)+ W(a)+K H(a)H(˜a−1) − W(a)+K KW(a−1). Thus, (W (a)+K)−1 is W (a−1) plus a compact operator. Note that this observation p in particular implies that Ap is inverse closed in L(L (0, ∞)). The finite section analogue of (12) is Widom’s formula ˜ (14) Wτ (a)Wτ (b)=Wτ(ab) − Pτ H(a)H(b)Pτ − Rτ H(˜a)H(b)Rτ p p where Pτ is as in (5) and Rτ : L (0, ∞) → L (0, ∞) is defined by ϕ(τ−x)if0 −1 −1 −1 Wτ (a)+PτNPτ = Wτ(a )+ Wτ(a)+PτNPτ PτYPτ −1 −1 −1 + Wτ(a)+PτNPτ RτZRτ + Wτ(a)+PτNPτ PτNPτWτ(a ). The space Lp(0, ∞) is the dual space of Lq(0, ∞)(1/p+1/q = 1) in case 1 (1) where kCτ kp → 0asτ→∞. Further, as Rτ Wτ (a)Rτ = Wτ (˜a) and Rτ NRτ → 0 strongly, we get analogously that −1 −1 Wτ (a)+PτNPτ RτZRτ = Rτ Wτ(˜a)+RτNRτ PτZRτ −1 (2) = RτW (˜a)ZRτ + Cτ (2) with kCτ kp → 0asτ→∞. Finally, in the same way we obtain −1 −1 −1 −1 (3) Wτ (a)+PτNPτ PτNPτWτ(a )=Pτ Wτ(a)+N NW(a )Pτ +Cτ (3) where kCτ kp → 0asτ→∞. In summary, we have shown the following result. ˙ p Proposition 2.1. Let 1 ≤ p<∞,a∈Cp(R),N∈K(L (0, ∞)).IfW(a)+N is invertible, then for all sufficiently large τ −1 −1 (17) Wτ (a)+PτNPτ = Wτ(a )+PτKPτ + RτLRτ + Cτ p where K, L ∈K(L (0, ∞)) and kCτ kp → 0 as τ →∞. Formula (17) is well known to workers in the field and extraordinarily useful in −1 connection with all questions concerning the behavior of Wτ (a) for large τ. Such formulas were first established and employed in [34] and [30] (also see [18]). Let Sp denote the set of all families {Aτ } = {Aτ }τ∈(0,∞) of operators Aτ ∈ p L(L (0,τ)) with supτ>0 kAτkp < ∞. With obvious algebraic operations and the norm k{Aτ }kp = sup kAτkp, τ>0 the set Sp is a Banach algebra. We let Cp stand for the collection of all elements {Cτ }∈Sp such that kCτ kp → 0asτ→∞. Obviously, Cp is a closed two-sided ideal of Sp and if {Aτ }∈Sp, then the norm of the coset {Aτ } + Cp in the quotient algebra Sp/Cp is k{Aτ } + Cpkp = lim sup kAτ kp. τ→∞ Truncated Wiener-Hopf Operators and Toeplitz Matrices 7 It is clear that {Wτ (a)}∈Sp for every a ∈ Mp(R). We denote by Fp the smallest 1 closed subalgebra of Sp containing all families {Wτ (a)} with a ∈ C + FL (R). If ˙ a, b ∈ Cp(R) then all Hankel operators in (14) are compact and hence n o Wτ (a) Wτ (b) = Wτ (ab)+PτKPτ + RτLRτ with certain compact operators K and L. This simple observation anticipates the following result. Proposition 2.2. If 1 ≤ p<∞then n o (18) Fp = Wτ (a)+PτKPτ + RτLRτ + Cτ : ˙ p a ∈ Cp(R),K,L∈K L (0, ∞) , {Cτ }∈Cp and Cp is a closed two-sided ideal of Fp. The discrete analogue of Proposition 2.2 is proved in [6] and [8, Proposition 7.27]. On first approximating arbitrary families {Cτ }∈Cp by piecewise constant families, Proposition 2.2 can be proved in the same way as its discrete version. Remark. For 1 ≤ p ≤∞, let Q(Lp(0, ∞)) be the so-called quasi-commutator ideal of Lp(0, ∞), i.e., let Q(Lp(0, ∞)) stand for the smallest closed two-sided ideal of L(Lp(0, ∞)) containing the set n o W (ab) − W (a)W (b):a, b ∈ C + FL1(R) . Proposition 2.1 easily implies that Q(Lp(0, ∞)) = K(Lp(0, ∞)) if 1 ≤ p<∞. One can show that Q(L∞(0, ∞)) is a proper subset of K(L∞(0, ∞)) and that, ∞ although Pτ does not converge strongly on L (0, ∞), one has kK −Pτ Kk→0 and ∞ kK −KPτk→0asτ→∞for every K ∈Q(L (0, ∞)) (see [24, Proposition 4.55]). ∞ Furthermore, if K ∈Q(L (0, ∞)), then Rτ KRτ goes to zero in the sense of P- convergence (see [24, Section 4.36]). These properties can be used to show that Propositions 2.1 and 2.2 remain true for all p ∈ [0, ∞] if only K(Lp(0, ∞)) is replaced by Q(Lp(0, ∞)). This in turn yields the validity of some results proved below in ∞ the case p = ∞ (e.g., of Corollary 3.3 and Theorem 6.2 with N ∈Q(Lm(0, ∞)) ∞ and Theorem 7.2 with N ∈Q(lm)). 3. Norms of Inverses Let Ap (1 ≤ p<∞) be the algebra (13). We equip the direct sum Ap ⊕Ap with the norm k(A, B)kp = max kAkp, kBkp . By virtue of Proposition 2.2, we may consider the quotient algebra Fp/Cp. Here is the key result of this paper. 8 A. B¨ottcher, S. M. Grudsky, and B. Silbermann Theorem 3.1. If 1 ≤ p<∞, then the map Symp : Fp/Cp →Ap⊕Ap given by n o Wτ (a)+PτKPτ + RτLRτ + Cτ + Cp 7→ W (a)+K, W(˜a)+L is an isometric Banach algebra homomorphism. Moreover, if {Aτ }∈Fp then lim sup kAτ kp = lim kAτ kp. τ→∞ τ→∞ Before giving a proof, a few comments are in order. We know from Proposition 2.2 that every family {Aτ }∈Fp is of the form Aτ = Wτ (a)+PτKPτ + RτLRτ + Cτ p with K, L ∈K(L(0, ∞)) and {Cτ }∈Cp. Clearly, W (a)+K and W (˜a)+L are nothing but the strong limits of Aτ and Rτ Aτ Rτ , respectively. Thus, we could equivalently also define Symp : {Aτ } + Cp 7→ s−lim Aτ , s−lim Rτ Aτ Rτ . τ→∞ τ→∞ This identification of Symp shows in particular that Symp is a well-defined contin- uous Banach algebra homomorphism and that (19) kSymp {Aτ } + Cp kp ≤ lim inf kAτ kp. τ→∞ Since the only compact Wiener-Hopf operator is the zero operator, it follows that Symp is injective. Thus, Symp is a Banach algebra homomorphism and the only thing we must show is that Symp is an isometry. ∗ If p = 2, then F2/C2 and A2 ⊕A2 are C -algebras and Sym2 is easily seen to be a ∗-homomorphism. Since injective ∗-homomorphisms of C∗-algebras are always isometric, the proof is complete. In the p 6= 2 case, the latter conclusion requires hand-work. ˙ Proof. Fix ε>0. Since a ∈ Cp(R), there is a number c ∈ C and a function k ∈ L1(R) with finite support, say supp k ⊂ (−s, s), such that k{Wτ (a − c − Fk)}kp ≤kW(a−c−Fk)kp <ε. Furthermore, if τ is large enough, then kP KP −Kk <ε,kP LP − Lk <ε. 0 τ0 τ0 p τ0 τ0 p As ε>0 may be chosen as small as desired, it follows that we are left with proving that n o ˜ (20) lim kBτ kp = max kW (b)+Kkp, kW(b)+Lkp τ→∞ where Bτ := Wτ (b)+PτKPτ + RτLRτ , b := c + Fk ∈ C+FL1(R), supp k ⊂ (−s, s), and the compact operators K, L are subject to the condition (21) P KP = K, P LP = L. τ0 τ0 τ0 τ0 Here s>0 and τ0 > 0 are certain fixed numbers. p Pick l>max{τ0,s} and put τ =4l. Let ϕ ∈ L (0,τ) be any function such that kϕkp = 1. We claim that there exists a number β ∈ (l + s, 3l − s) Truncated Wiener-Hopf Operators and Toeplitz Matrices 9 such that βZ+s l −1 (22) |ϕ(t)|pdt ≤ s β−s where [l/s] stands for the integral part of l/s. Indeed, since Z3l l+2(Zd+1)s dX0−1 p p p 1=kϕkp ≥ |ϕ(t)| dt ≥ |ϕ(t)| dt l d=0 l+2ds with d0 := [l/s], there exists a d1 such that 0 ≤ d1 ≤ d0 − 1 and l+2(Zd1+1)s l −1 |ϕ(t)|pdt ≤ , s l+2d1s which proves our claim with β := l +(2d1 +1)s. Given 0 <τ <τ ,weputP := P − P . With β as above, we have 1 2 (τ1,τ2) τ2 τ1 Bτ = Pβ Bτ Pβ + P(β,τ)Bτ P(β,τ) + PβBτ P(β,τ) + P(β,τ)Bτ Pβ. Since β + τ0 < 3l − s + l<4l=τ and Pτ0 LPτ0 = L, the equality PβRτ LRτ Pβ =0 holds. Hence, PβBτ Pβ = Pβ W (b)+K Pβ, which imples that (23) kPβBτ Pβkp ≤kW(b)+Kkp =: M1. and P KP = K, we see that P P KP P = 0. Therefore, As β>τ0 τ0 τ0 (β,τ) τ τ (β,τ) P(β,τ)Bτ P(β,τ) = P(β,τ) Wτ (b)+RτLRτ P(β,τ) ˜ = P(β,τ)Rτ Wτ (b)+L RτP(β,τ) ˜ = Rτ Pτ−β Wτ (b)+L Pτ−βRτ ˜ =RτPτ−β W(b)+L Pτ−βRτ, whence ˜ (24) kP(β,τ)Bτ P(β,τ)kp ≤kW(b)+Lkp =: M2 Again using (21)weget P(β,τ)Bτ Pβ + PβBτ P(β,τ) = P(β,τ)Wτ (b)Pβ + PβWτ (b)P(β,τ). Since supp k ⊂ (−s, s), we have Zβ Zβ Wτ (b)Pβ ϕ (x)= k(x−t)ϕ(t)dt = k(x − t)ϕ(t) dt 0 β−s 10 A. B¨ottcher, S. M. Grudsky, and B. Silbermann for x ∈ (β,τ) and Zτ βZ+s Wτ (b)P(β,τ)ϕ (x)= k(x−t)ϕ(t)dt = k(x − t)ϕ(t) dt β β for x ∈ (0,β). Consequently, p (25) P(β,τ)Bτ Pβϕ + PβBτ P(β,τ)ϕ p Zτ Zβ Zβ βZ+s p p = k(x − t)ϕ(t) dt dx + k(x − t)ϕ(t) dt dx β β−s 0 β Zβ βZ+s p p p ≤kkk1 |ϕ(t)| dt + |ϕ(t)| dt β−s β βZ+s l −1 = kkkp |ϕ(t)|pdt ≤kkkp , 1 1 s β−s the last two estimates resulting from the inequality kk ∗ ϕkp ≤kkk1kϕkp and from (22), respectively. Put f1 = PβBτ Pβϕ, f2 = P(β,τ)Bτ P(β,τ)ϕ, f3 =(P(β,τ)Bτ Pβ + PβBτ P(β,τ))ϕ. Then Bτ ϕ = f1 + f2 + f3 and thus, τ Z 1/p p kBτ ϕkp ≤kf1+f2kp+kf3kp = |f1(t)+f2(t)| dt + kf3kp 0 Zβ Zτ 1/p 1/p p p p p = |f1(t)| dt + |f2(t)| dt + kf3kp = kf1kp + kf2kp + kf3kp. 0 β From (23), (24), (25) we therefore get 1/p p p p p kBτ ϕkp ≤ M1 kPβϕkp + M2 kP(β,τ)ϕkp + kf3kp 1/p p p ≤ max{M1,M2} kPβϕkp +kP(β,τ)ϕkp + kf3kp = max{M1,M2}kϕkp + kf3kp −1 ≤ max{M1,M2}+kkk1[l/s] . If τ →∞then [l/s]−1 =[τ/(4s)]−1 → 0, which gives the estimate (26) lim sup kBτ kp ≤ max{M1,M2}. τ→∞ Since max{M1,M2} = kSymp({Bτ } + Cp)kp, we finally obtain (20) from (26) and (19). Truncated Wiener-Hopf Operators and Toeplitz Matrices 11 Corollary 3.2. Let 1 ≤ p<∞and {Aτ }∈Fp. Put A =s−lim Aτ ,B=s−lim Rτ Aτ Rτ . τ→∞ τ→∞ p and suppose A − λI and B − λI are invertible on L (0, ∞). Then Aτ − λI (= p Aτ − λPτ ) is invertible on L (0,τ) for all sufficiently large τ and n o −1 −1 −1 lim k(Aτ − λI) kp = max k(A − λI) kp, k(B − λI) kp . τ→∞ Proof. Since (A−λI, B−λI) = Symp {Aτ −λI}+Cp , we deduce from Theorem 3.1 p and the inverse closedness of Ap in L(L (0, ∞)) that {Aτ − λI} + Cp is invertible in Fp/Cp. Let {Dτ } + Cp ∈Fp/Cp be the inverse. Then −1 −1 Symp {Dτ } + Cp = (A − λI) , (B − λI) , −1 k{Dτ } + Cpkp = lim sup k(Aτ − λI) kp, τ→∞ and therefore the assertion follows from Theorem 3.1. The following corollary of Theorem 3.1 clearly contains Theorem 1.1. ˙ p Corollary 3.3. Let 1 ≤ p<∞,a∈Cp(R),N∈K(L(0, ∞)).IfW(a)+N is invertible on Lp(0, ∞) then the limit −1 (27) lim k Wτ (a)+PτNPτ kp τ→∞ exists and is equal to n o −1 −1 (28) max k W (a)+N kp, kW (˜a)kp . Proof. Theorem 3.1 and Proposition 2.1 show that the limit (27) exists and equals n o −1 −1 max kW (a )+Kkp, kW(˜a )+Lkp . −1 Passing in (17) to the strong limit τ →∞we get W (a)+N =W(a−1)+K, and multiplying (17) from both sides by Rτ and then passing to the strong limit τ →∞we see that W −1(˜a)=W(˜a−1)+L. Corollary 3.4. Let 1 ≤ p<∞and let ajk be a finite collection of functions in ˙ Cp(R). Put X Y XY XY A = W (ajk),B1= W(˜ajk),B2=W a˜jk . j k j k j k and suppose A is invertible on Lp(0, ∞) p (a) If B1 is invertible on L (0, ∞) then X Y −1 n o −1 −1 (29) lim Wτ (ajk) = max kA kp, kB1 kp , τ→∞ p j k 12 A. B¨ottcher, S. M. Grudsky, and B. Silbermann p while if B1 is not invertible on L (0, ∞) then X Y −1 (30) lim Wτ (ajk) = ∞. τ→∞ p j k (b) The operator B2 is invertible and X Y −1 n o −1 −1 (31) lim Pτ W (ajk)Pτ = max kA kp, kB kp . τ→∞ 2 j k P Q Proof. Let h = j k ajk. ˜ (a) We know from Section 2 that A = W (h)+M and B1 = W (h)+N with compact operators M and N. Obviously, X Y (32) Aτ := Wτ (ajk) → A strongly j k and X Y (33) Rτ Aτ Rτ = Wτ (˜ajk) → B1 strongly. j k Suppose the limit in (30) does not exist or is finite. Then there are τk →∞ such that k(R A R )−1k = kA−1k ≤ m<∞. This implies that kP ϕk ≤ τk τk τk p τk p τk p mkR A R ϕk and thus kϕk ≤ mkB ϕk for all ϕ ∈ Lp(0, ∞). Consequently, τk τk τk p p 1 p B1 is injective. Since A = W (h)+M is assumed to be invertible, the operator ˜ B1 = W (h)+N must be Fredholm of index zero (see [14]). This in conjunction with the injectivity of B1 shows that B1 is actually invertible. Thus, if B1 is not invertible, then (30) holds. Now suppose B1 is invertible. From (32) and (33) we infer that Symp {Aτ }+Cp equals (A, B1), so that (29) is immediate from Theorem 3.1. ˜ (b) If A = W (h)+M is invertible, then so also is B2 = W (h) and (31) is nothing but the conclusion of Corollary 3.2. For p =2,Theorem 3.1 was established in [31], and Corollaries 3.2 to 3.4 ap- peared explicitly in [4] for the first time. It should be noted that [4] and [31] deal with Wiener-Hopf operators generated by piecewise continuous symbols, in which case the analogues of the algebras Ap and Fp/Cp do not admit such simple de- scriptions as in (13) and (18). The proofs given in [4] and [31] make heavy use of the C∗-algebra techniques which have their origin in [6] (also see [5], [8], [17], [28], [32]). We remark that the aforementioned works raised some psychological barrier, in view of which the validity of such p 6= 2 results as Theorem 3.1 is fairly surprising. The psychological barrier was broken only in [16], where (the discrete analogue of) the N = 0 version of Corollary 3.3 was established for p = 1 by different methods. In particular, in view of the proofs of [16] and the proof of Theorem 3.1 given here, one of the authors withdraws his too enthusiastic statement that “formulas like −1 −1 lim sup kWτ (a)k2 = kW (a)k2 τ→∞ cannot be proved by bare hands”, which is on p. 274 of [4]. Finally, it should be mentioned that the lp version of Theorem 3.1 was de facto predicted already in [27, Truncated Wiener-Hopf Operators and Toeplitz Matrices 13 Remark 3 on p. 303] and is in very disguised form also contained in the results of [17, pp. 186–205]. We remark that Corollary 3.3 can be stated in slightly different terms. Define the sesquilinear operator R on Lp(0, ∞)by(Rϕ)(x)=ϕ(x). Obviously, R2 = I and W (˜a)=RW (a)R 1 where a(ξ)=a(ξ). Notice that if a = c + Fk ∈ C+FL (R), then W (a) and Wτ (a) are given by (1) and (2) with k(t − x) in place of k(x − t). Since R is an isometry, we have −1 −1 (34) kW (˜a)kp = kW (a)kp for 1 ≤ p ≤∞. On identifying the dual space of Lp (1 kfkp,µ := kΛµfkp < ∞, 1,µ and let L (R) be the space of all measurable functions on R for which Λµf is in 1 1,|µ| p,µ L (R). If a ∈ C + FL (R), then W (a) and Wτ (a) are bounded on L (0, ∞) p,µ and L (0,τ) for all p ∈ [1, ∞) and all µ ∈ R. The algebras Ap,µ and Fp,µ are defined in the natural manner. Taking into account [24, Remark 4.51, 2◦] one can construct an isometric Banach algebra isomorphism of Fp,µ/Cp,µ onto the direct 1,|µ| sum Ap,µ ⊕Ap. It follows in particular that if a ∈ C + FL (R), a(ξ) 6= 0 for all ξ ∈ R˙ , and wind (a, 0) = 0, then n o −1 −1 −1 −1 lim sup kWτ (a)kp,µ = lim kWτ (a)kp,µ = max kW (a)kp,µ, kW (˜a)kp . τ→∞ τ→∞ Proof of Theorem 1.2. For ξ ∈ R, put i 2i 6 a (ξ)=1+ ,a(ξ)=1+ − − 2(ξ − i) + ξ+i (ξ+i)2 −1 −1 1 and consider a = a− a+ . Since a ∈ C+FL (R) and these functions have no zeros on R, the symbol a belongs to C + FL1(R) by Wiener’s theorem. The functions a− and a+ have no zeros in the lower and upper complex half-planes, respectively, which implies (see, e.g., [14]) that W (a) is invertible on Lp(0, ∞) for 1 ≤ p ≤∞ −1 and that W (a)=W(a+)W(a−). Consequently, Z∞ (37) W −1(a)ϕ (x)=ϕ(x)+ γ(x, t)ϕ(t) dt (0 0 14 A. B¨ottcher, S. M. Grudsky, and B. Silbermann where minZ{t,x} γ(x, t)=γ+(x−t)+γ−(x−t)+ γ+(x − r)γ−(r − t) dr 0 and γ are the kernels defined by a =1+Fγ. Clearly, Z∞ 1 2i 6 2e−x +6xe−x if x>0, γ (x)= − e−iξxdξ = + 2π ξ + i (ξ + i)2 0ifx<0, −∞ Z∞ 1 i − 1 ex if x<0, γ (x)= e−iξxdξ = 2 − 2π 2(ξ − i) 0ifx>0, −∞ whence 1 (3 + 18(x − t))e−(x−t) +(5+6x)e−(x+t) if x>t, γ(x, t)= 4 −7ex−t+(5+6x)e−(x+t) if x 0 −x −2x Since S∞(x)=(1+3x)e +(2+6x)e >0 for all x ∈ (0, ∞), it follows that −1 (39) kW (a)k∞ = 1 + sup S∞(x)=1+S∞(∞)=8. x>0 We now show that if p ∈ (1, ∞) is sufficiently large and 1/p +1/q = 1, then −1 −1 (40) kW (a)kp < kW (a)kq. This together with (34) and (35) gives the desired inequality (9). Truncated Wiener-Hopf Operators and Toeplitz Matrices 15 Abbreviate W −1(a)toA.By(38), there is a function f ∈ L∞(R) with finite support, say supp f ⊂ (0,T), such that kAfk1/kfk1 > 9 − ε. Since ZT 1+δ 1+δ kfk1+δ = |f(x)| dx 0 1+δ depends continuously on δ ∈ [0, ∞), we see that kfk1+δ →kfk1 as δ → 0, whence kfk1+δ →kfk1 as δ → 0. Further, we know that ZT (Af)(x)=f(x)+ γ(x, t)f(t) dt. 0 R If x>T, then T γ(x, t)f(t) dt equals 0 ZT ZT 3 + 18(x − t) e−xetf(t) dt + (5+6x)e−xetf(t)dt, 0 0 which is of the form αe−x + βxe−x with α, β ∈ C. Consequently, (Af)(x)=g(x)+(α+βx)e−x ∞ 1+δ where g ∈ L (0, ∞) and supp g ⊂ (0,T). This easily implies that kAfk1+δ → kAfk1 and thus kAfk1+δ →kAfk1 as δ → 0. In summary, we have shown that kAfk1+δ/kfk1+δ > 9 − 2ε and thus kAk1+δ > 9 − 2ε whenever δ>0 is sufficiently small. From the Riesz-Thorin interpolation theorem and from (38), (39) we obtain that 1/p 1−1/p 1/p 1−1/p kAkp ≤kAk1 kAk∞ =9 8 < 8+ε if only p is sufficiently large. But if p is large enough, then q =1+δwith δ as small as desired. As 8 + ε<9−2εfor ε<1/3, we arrive at (40). 4. Spectra The comparision of the spectra of W (a) and Wτ (a) as well as the discrete ana- logue of this problem, the comparision of the spectra of infinite Toeplitz matrices and their large finite sections, has been the subject of extensive investigations for many decades and is nevertheless a field with still a lot of mysteries. Clearly, the Szeg¨o-Widom theorem and its continuous analogue, the Achiezer-Kac-Hirschman formula, pertain to this topic (see the books [7], [8], [15], [35]). The asymptotic spectral behavior of truncated Toeplitz and Wiener-Hopf operators is explicitly dis- cussed in the papers [1], [2], [4], [5], [9], [12], [23], [27], [29], [36], [33]; this list is incomplete. We here confine ourselves to pointing out only a couple of phenomena. ˙ The equality (6) holds for every a ∈ Cp(R) and shows that σp(W (a)) is indepen- dent of p (note that this is no longer true for piecewise continuous symbols). The following result confirms the same effect for truncated Wiener-Hopf operators. 1 Theorem 4.1. If a ∈ C + FL (R) and τ>0, then σp(Wτ (a)) does not depend on p ∈ [1, ∞]. 16 A. B¨ottcher, S. M. Grudsky, and B. Silbermann (−1) p Proof. Define the shift operators Vτ and Vτ on L (0, ∞)by ϕ(x−τ)ifτ d(ξ)=m−(ξ)b(ξ)m+(ξ) eiτξ(Fk)(ξ) c +(Fk)(ξ) = −c +(Fk)(ξ) e−iτξ(Fk)(ξ) (Fk )(ξ) c +(Fk)(ξ) = τ −c +(Fk)(ξ)(Fk−τ)(ξ) with kτ (x)=k(x−τ),k−τ(x)=k(x+τ). The operators W (m) are invertible, −1 the inverses being W (m ). Thus, B = W (b) is invertible if and only if W (d)is invertible. The operator W (d) is a block Wiener-Hopf operator and the symbols of the entries of W (d) belong to C + FL1(R). We have det d(ξ)=c2, and hence W (d)is invertible if and only if c 6= 0 and the partial indices of d are both zero (again see [14]). As the partial indices of a nonsingular matrix function with entries in the Wiener algebra do not depend on p, we arrive at the conclusion that the invertibility p of our original operator Wτ (a)onL(0, ∞) is independent of p. Since Wτ (a) − λI = Wτ (a − λ), we finally see that the spectrum σp(Wτ (a)) is the same for all p ∈ [1, ∞]. Thus, when considering the limiting set Λ(a) := lim σp Wτ (a) τ→∞ for a ∈ C + FL1(R) we can restrict ourselves to the case p =2. For Toeplitz matrices with rational symbols the discrete analogue of Λ(a)was completely identified by Schmidt and Spitzer [29] and Day [11], [12]. The Wiener- Hopf analogue of this result was established in [9] with the help of formulas for Truncated Wiener-Hopf Operators and Toeplitz Matrices 17 Wiener-Hopf determinants contained in [3]. We only remark that if a ∈ C+FL1(R) is a rational function, then Λ(a) is a nonempty bounded set which is comprised of a finite union of closed analytic arcs and which is in no obvious way related to the 2 spectrum of W (a). For example, if a(ξ)=(3+iξ)/(1 + ξ ), then σp(W (a)) is a set bounded by an ellipse while√ Λ(a) is the√ union of the circle {λ ∈ C : |λ − 1/12| = 1/12} and the interval [3/2− 2, 3/2+ 2] (see [1] and [9]). Note that Theorem 4.1 and this example completes the proof of Theorem 1.3. The situation is more delicate for non-rational symbols. Curiously, in this case the behavior of σp(Wτ (a)) is often more “canonical” than one might expect. Let, for example, a(ξ)=−sign ξ. This is a piecewise continuous function with two jumps, one at the origin and one at infinity. The Wiener-Hopf operator associated with this function is the Cauchy singular integral operator Z∞ 1 ϕ(t) W (a)ϕ (x)= dt (0 Of course, Wτ (a) is given by the same formula with ∞ replaced by τ. Itiswell known (see, e.g., [13]) that if 1 5. Pseudospectra Things are dramatically simpler when passing from spectra to pseudospectra. Henry Landau [20], [21], [22] was probably the first to study pseudospectra of truncated Toeplitz and Wiener-Hopf operators. For p = 2, the discrete analogue of Theorem 1.4 was established by Reichel and Trefethen [26], although their proof contained a gap. A completely different proof (based on C∗-algebra techniques) was given in [4]. Pseudospectra of Wiener-Hopf integral operators (with Volterra kernels) were also considered in [25]. We remark that in [4] the formula (11) and its discrete analogue are proved for p = 2 under the sole assumption that a is a locally normal function, which is, for example, the case if a is piecewise continuous. We will derive formula (11) from Corollary 3.2. In order to do this, we need the following result, which says that the norm of the resolvent of an operator on Lp cannot be locally constant. For p = 2, we learned this result from Andrzej Daniluk (private communication, see [10]); his proof is published in [4]. We here give a proof for general p. Theorem 5.1. Let (X, dµ) be a space with a measure and let 1 −1 −1 open subset U of C.Ifk(A−λI) kp ≤ M for all λ ∈ U, then k(A − λI) kp Remark. It should be noted that such a result is not true for general analytic 2 z operator-valued functions. To see this, equip C with the norm ω = p p p |z|p + |ω|p and consider the function λ 0 A : C →L(C2),λ7→ . 01 Clearly, kA(λ)kp = max{|λ|, 1} and thus kA(λ)kp = 1 for all λ in the unit disk. Proof. We may without loss of generality assume that p ≥ 2; otherwise we can pass to the adjoint operator. A little thought reveals that what we must show is the following: if U is an open −1 subset of C containing the origin and k(A − λI) kp ≤ M for all λ ∈ U, then −1 kA kp = C(r, x)+ Bl(r, x, ϕ) dµ(x) X l=1 with ∞ X 2 C(r, x)= r2j (A−(j+1)f)(x) , j=0 X∞ l+2k ilϕ −(l+k+1) −(k+1) Bl(r, x, ϕ)=2 r Re e (A f)(x)(A f)(x) . k=0 For n =0,1,2,... , put Z X∞ p/2 (43) In(r, ϕ, f)= C(r, x)+ B2nl(x, r, ϕ) dµ(x). X l=1 We now apply the inequality |a + b|p/2 + |a − b|p/2 (44) ≥|a|p/2 2 Truncated Wiener-Hopf Operators and Toeplitz Matrices 19 to X∞ X∞ a = C(r, x)+ B2l(r, x, ϕ),b= B2l−1(r, x, ϕ) l=0 l=1 and integrate the result over X. Taking into account that Z p/2 |a − b| dµ(x)=I0(r, ϕ + π, f), X we get I (r, ϕ, f)+I (r, ϕ + π, f) (45) 0 0 ≥ I (r, ϕ, f). 2 1 Letting X∞ X∞ a = C(r, x)+ B4l(r, x, ϕ),b= B4l−2(r, x, ϕ) l=1 l=1 in (44), we analogously obtain after integration that I (r, ϕ, f)+I (r, ϕ + π/2,f) (46) 1 1 ≥I (r, ϕ, f). 2 2 Combining (45) and (46) we arrive at the inequality I (r, ϕ, f)+I (r, ϕ + π/2,f)+I (r, ϕ + π, f)+I (r, ϕ +3π/2,f) (47) 0 0 0 0 4 I (r, ϕ, f)+I (r, ϕ + π/2,f) ≥ 1 1 ≥I (r, ϕ, f). 2 2 In the same way as above we see that I (r, ϕ, f)+I (r, ϕ + π/4,f) 2 2 ≥I (r, ϕ, f), 2 3 which together with (47) gives 1 X7 mπ I r, ϕ + ,f ≥I (r, ϕ, f). 8 0 4 3 m=0 Obviously, continuing this process we obtain n 1 2X−1 mπ (48) I r, ϕ + ,f ≥I (r, ϕ, f) 2n 0 2n−1 n m=0 for every n ≥ 0. Now put ϕ = 0 in (48). From (43) we infer that the right-hand side of (48) converges to Z |C(r, x)|p/2dµ(x) X as n →∞. The left-hand side of (48) is an integral sum and hence passage to the limit n →∞gives n Z2π 1 π 2X−1 mπ 1 lim I0 r, ,f = I0(r, ϕ, f) dϕ. n→∞ 2π 2n−1 2n−1 2π m=0 0 20 A. B¨ottcher, S. M. Grudsky, and B. Silbermann iϕ −1 p As I0(r, ϕ, f)=k(A−re I) fkp by (42), it follows that Z2π Z 1 (49) k(A − reiϕI)−1fkpdϕ ≥ |C(r, x)|p/2dµ(x). 2π p 0 X −1 Assume kfkp = 1. Then our hypothesis k(A−λI) kp ≤ M and (49) imply that Z2π Z 1 M p ≥ k(A − reiϕI)−1fkpdϕ ≥ |C(r, x)|p/2dµ(x) 2π p Z 0 X p/2 ≥ |(A−1f)(x)|2 + r2|(A−2f)(x)|2 dµ(x). X Since (|a| + |b|)p/2 ≥|a|p/2 + |b|p/2,weget Z Z (50) Mp ≥ |(A−1f)(x)|pdµ(x)+rp |(A−2f)(x)|pdµ(x) X X −1 p p −2 p = kA fkp + r kA fkp. Let ε>0 be an arbitrary number. By virtue of (41), we can find an f ∈ Lp(X, dµ) −1 p p such that kfkp = 1 and kA fkp ≥ M − ε. Since −2 2 −1 2 −1 kA fkp ≥kA kp kfkp =kA kp , we finally obtain from (50) that p p p 2 −p M ≥ M − ε + r kA kp for all r ∈ (0,r0], which is impossible if ε is sufficiently small. This contradiction completes the proof. Theorem 5.2. Let 1 Proof. Put A = W (a)+K and B = W (˜a)+L. We first show the inclusion ε ε (52) σp(A) ⊂ lim σp(Aτ ). τ→∞ Pick λ ∈ σp(A). We claim that −1 (53) lim sup k(Aτ − λI) kp = ∞. τ→∞ −1 Assume the contrary, i.e., let k(Aτ − λI) kp ≤ m<∞for all τ>τ0. Then kPτ ϕkp ≤ mk(Aτ − λI)Pτ ϕkp p for every ϕ ∈ L (0, ∞) and all τ>τ0, whence kϕkp ≤ mk(A − λI)ϕkp for every ϕ ∈ Lp(0, ∞). This shows that A − λI has a closed range and a trivial kernel. Our ∗ −1 assumption implies that k(Aτ + λI) kq ≤ m<∞where 1/p +1/q = 1. Arguing as above, we see that A∗ − λI has a trivial kernel on Lq(0, ∞), which implies that the range of A − λI is dense in Lp(0, ∞). In summary, A − λI must be invertible p on L (0, ∞). As this is impossible if λ ∈ σp(A), we see that (53) is true. Truncated Wiener-Hopf Operators and Toeplitz Matrices 21 From (53) we deduce the existence of τ1,τ2,... such that τn →∞and −1 k(Aτn − λI) kp ≥ 1/ε. This shows that λ belongs to the set on the right of (52). ε −1 Now suppose λ ∈ σp(A) \ σp(A). Then k(A − λI) kp ≥ 1/ε. Let U ⊂ C be any open neighborhood of λ. From Theorem 5.1 we see that there is a point µ ∈ U −1 such that k(A − µI) k > 1/ε. Hence, we can find an integer n0 > 0 such that 1 k(A − µI)−1k ≥ for all n ≥ n . p ε − 1/n 0 Because U was arbitrary, it follows that there exists a sequence λ1,λ2,... such that ε−1/n λn ∈ σp (A) and λn → λ. For every invertible operator T ∈L(Lp(0, ∞)) we have −1 −1 −1 kT ψkp kϕkp kTϕkp (54) kT kp = sup = sup = inf . ψ6=0 kψkp ϕ6=0 kTϕkp ϕ6=0 kϕkp −1 Since k(A − λnI) kp ≥ 1/(ε − 1/n), we obtain from (54) that inf k(A − λnI)ϕkp ≤ ε − 1/n, kϕkp=1 p implying the existence of a ϕn ∈ L (0, ∞) such that kϕnkp = 1 and k(A − λnI)ϕnkp <ε−1/(2n). Because k(Aτ −λnI)Pτ ϕnkp →k(A−λI)ϕkp and kPτ ϕnkp →kϕnkp =1asτ→∞, it follows that k(Aτ − λnI)Pτ ϕnkp/kPτ ϕnkp <ε−1/(3n) for all τ → τ0(n). Again invoking (54) we see that −1 −1 k(Aτ − λnI) kp > ε − 1/(3n) > 1/ε ε and thus λn ∈ σp(Aτ ) for all τ>τ0(n). This implies that λ = lim λn lies in the set on the right of (52). At this point the proof of (52) is complete. Repeating the above reasoning with Rτ Aτ Rτ and B in place of Aτ and A, respectively, we get the inclusion ε ε (55) σp(B) ⊂ lim σp(Rτ Aτ Rτ ). τ→∞ p Because Rτ is an isometry on L (0,τ) and Rτ Aτ Rτ − λPτ = Rτ (Aτ − λI)Rτ , ε ε it is clear that σp(Rτ Aτ Rτ )=σp(Aτ). Thus, in (55) we may replace Rτ Aτ Rτ by Aτ , which in conjunction with (52) proves the inclusion “⊃”in(51). ε ε We are left with proving the inclusion “⊂”of(51). So let λ 6∈ σp(A) ∪ σp(B). Then −1 −1 k(A − λI) kp < 1/ε, k(B − λI) kp < 1/ε, whence −1 (56) k(Aτ − λI) kp < 1/ε − δ<1/ε for all τ>τ0 22 A. B¨ottcher, S. M. Grudsky, and B. Silbermann with some δ>0 due to Corollary 3.2.If|µ−λ|is sufficiently small, then Aτ − µI is invertible together with Aτ − λI, and we have k(A − λI)−1k (57) k(A − µI)−1k ≤ τ p . τ p −1 1 −|µ−λ|k(Aτ −λI) kp Let |µ − λ| <εδ/(1/ε − δ). In this case (56) and (57) give 1/ε − δ 1 k(A − µI)−1k < = . τ p 1 − εδ(1/ε − δ)/(1/ε − δ) ε ε ε Thus, µ 6∈ σp(Aτ ) for τ>τ0. This shows that λ cannot belong to lim σp(Aτ ). τ→∞ Obviously, Theorem 1.4 is a special case of Theorem 5.2. We remark that in the p = 2 case the equalities (34) and (35) imply that ε ε (58) lim σ2 Wτ (a) = σ2 W (a) τ→∞ for each ε>0. With the notation as in Corollary 3.4, we have for each ε>0, X Y ε ε ε lim σp Wτ (ajk = σp(A) ∪ σp(B1), τ→∞ j k X Y ε ε ε lim σp Pτ W (ajk)Pτ = σp(A) ∪ σp(B2). τ→∞ j k ε Finally, note that Theorems 5.2 and 1.2 imply that σp(Wτ (a)) in general depends on p. 6. Matrix Case p Given a positive integer m, we denote by Lm(0, ∞) the direct sum of m copies p p of L (0, ∞) and we think of the elements of Lm(0, ∞) as columns. We define the p norm on Lm(0, ∞)by ∞ Z Xm p/2 T p 2 k(f1,... ,fm) kp := |fj(x)| dx. 0 j=1 m×m ˙ Let Cp (R) stand for the collection of all m × m matrix functions with entries in ˙ m m×m ˙ Cp(R). For a =(ajk)j,k=1 ∈ Cp (R) the (block) Wiener-Hopf integral operator p m W (a) is defined on Lm(0, ∞) by the operator matrix (W (ajk))j,k=1. The direct sum p p p Lm(0,τ)ofmcopies of L (0,τ) may be identified with a subspace of Lm(0, ∞)in p a natural manner. The operators on Lm(0, ∞) given by the diagonal matrices diag (Pτ ,... ,Pτ) and diag (Rτ ,... ,Rτ) are also denoted by Pτ and Rτ .Ifa= m m×m ˙ p (ajk)j,k=1 ∈ Cp (R), then the compression Wτ (a)ofW(a)toLm(0,τ) is nothing m m×m m×m m×m but (Wτ (ajk))j,k=1. Finally, we give Fp , Cp , Ap the natural meaning. The results proved above extend to the matrix case with obvious adjustments. We confine ourselves to formulating the basic theorems. Truncated Wiener-Hopf Operators and Toeplitz Matrices 23 Theorem 6.1. Let 1 ≤ p<∞. Then n o m×m m×m ˙ p Ap = Wτ (a)+K:a∈Cp (R),K∈K Lm(0, ∞) , n m×m m×m ˙ Fp = Wτ (a)+PτKPτ + RτLRτ + Cτ : a ∈ Cp (R), o p K, L ∈K Lm(0, ∞) , kCτ kp → 0 as τ →∞ , m×m m×m m×m m×m the map Symp : Fp /Cp →Ap ⊕Ap sending m×m {Wτ (a)+PτKPτ + RτLRτ + Cτ } + Cp to W (a)+K, W(˜a)+L is an isometric Banach algebra homomorphism, and lim sup kAτ kp = lim kAτ kp τ→∞ τ→∞ m×m for every family {Aτ }∈Fp . m×m ˙ p Theorem 6.2. Let 1 ≤ p<∞,a∈Cp (R), and N ∈K(Lm(0, ∞)).If p W(a)+N and W (˜a)(where a˜(ξ):=a(−ξ)) are invertible on Lm(0, ∞) then the limit (27) exists and is equal to (28). If one of the operators W (a)+N and W (˜a) p is not invertible on Lm(0, ∞), then −1 lim sup k(Wτ (a)+PτNPτ) kp = ∞. τ→∞ Note that in the scalar case (m = 1) the operator W (˜a) is automatically invertible if W (a)+N is invertible. This is no longer true in the matrix case (see e.g., [14]). Also notice that in the matrix case the equality n o −1 −1 −1 lim kWτ (a)k2 = max kW (a)k2, kW (˜a)k2 τ→∞ cannot be simplified to (36). Theorem 6.3. Let 1 We emphasize again that the equality ε ε ε lim σ2 Wτ (a) = σ2 W (a) ∪ σ2 W (˜a) τ→∞ cannot be reduced to (58)ifm>1. 7. Block Toeplitz Matrices Let T be the complex unit circle. Given a function a ∈ L∞(T), we denote by {an}n∈Z the sequence of its Fourier coefficients, Z2π 1 a = a(eiθ)e−inθdθ, n 2π 0 24 A. B¨ottcher, S. M. Grudsky, and B. Silbermann and we let W (T) stand for the Wiener algebra of all functions a with absolutely p p p convergent Fourier series. We think of l = l (Z+)=l({0,1,2,...}) as a space of infinite columns. If a ∈ L∞(T), then the Toeplitz operator T (a) given by the Toeplitz matrix ∞ (59) (aj−k)j,k=0 2 ∞ is bounded on l . For 1 ≤ p<∞, let Mp(T) be the set of all functions a ∈ L (T) satisfying 2 p kT (a)ϕkp ≤ Ckϕkp for all ϕ ∈ l ∩ l with some C = C(a, p) < ∞ independent of p. The set Mp(T) is a Banach algebra with the norm n o kak = sup kT (a)ϕk /kϕk : ϕ ∈ l2 ∩ lp,ϕ6=0 . Mp(T) p p It is easily seen that W (T) ⊂ Mp(T) for all p ∈ [1, ∞). The closure of W (T)in Mp(T) is denoted by Cp(T). One can show that C1(T)=W(T),C2(T)=C(T), and that if p ∈ (1, 2) ∪ (2, ∞), then Cp(T)=Cq(T)⊂C(T)∩Mp(T), the inclusion being proper (see, e.g., [7], [8]). If a ∈ Cp(T), then the matrix (59) induces a bounded operator on lp. This operator is denoted by T (a) and called the Toeplitz operator (or the Toeplitz matrix) with the symbol a. For a ∈ Cp (T) and n ∈ Z+ we denote by Tn(a) the (n +1)×(n+ 1) Toeplitz n p matrix (aj−k)j,k=0. On defining Pn on l by ϕ if 0 ≤ j ≤ n, (P ϕ) = j n j 0if j>n, p p we can identify Tn(a) with the compression of T (a)tol(Zn)=l ({0,1,... ,n}). The discrete analogue of the operator Rτ given by (15) is the operator Rn defined by ϕ if 0 ≤ j ≤ n, (R ϕ) = n−j n j 0if j>n. p p p p We regard l (Zn) as a subspace of l = l (Z+). This specifies the norms in l (Zn) p p p and L(l (Zn)). The norm and the spectrum of an operator A on l (Zn)orl (Z+) ε are denoted by kAkp and σp(A), respectively. The ε-pseudospectrum σp(A)is defined as the set (3). p p m×m Finally, given a positive integer m, we define lm,lm(Zn),Cp (T),Pn,Rn m×m as well as T (a) and Tn(a) for a ∈ Cp (T) in the natural fashion. The norm of T p (f1,... ,fm) ∈lm is defined by XXm p/2 T p 2 k(f1,... ,fm) kp = |(fk)j| . j k=1 All results established above for Wiener-Hopf integral operators have analogues for Toeplitz operators. The proofs in the Toeplitz case are completely analogous to (and sometimes even simpler than) the proofs for Wiener-Hopf integral operators. We therefore only state the results. Truncated Wiener-Hopf Operators and Toeplitz Matrices 25 m×m p Let Ap be the smallest closed subalgebra of L(lm) containing {T (a):a∈ W(T)}. One can show that n o m×m m×m p Ap = T (a)+K:a∈Cp (T),K∈K(lm) m×m p m×m and that Ap is inverse closed in L(lm). Denote by Sp the Banach algebra ∞ p of all sequences {An} = {An}n=0 of operators (matrices) An ∈L(lm) such that k{An}kp := sup kAnkp < ∞, n∈Z+ m×m m×m let Cp be the closed two-sided ideal of all sequences {An}∈Sp for which m×m kAnkp → 0asn→∞, and let Fp stand for the smallest closed subalgebra of m×m ∞ Sp containing all sequences {Tn(a)}n=0 with a ∈ W (T). Theorem 7.1. Let 1 ≤ p<∞. Then n m×m m×m Fp = Tn(a)+PnKPn + RnLRn + Cn : a ∈ Cp (T), o p K, L ∈K(lm), kCnkp →0 as n →∞ . The map m×m m×m m×m m×m Symp : Fp /Cp →Ap ⊕Ap sending n o m×m m×m {An} + Cp = Tn(a)+PnKPn + RnLRn + Cn + Cp to s−lim An, s−lim RnAnRn = T (a)+K, T(˜a)+L n→∞ n→∞ (˜a(t):=a(1/t) for t ∈ T) is an isometric Banach algebra homomorphism. Further- more, lim sup kAnkp = lim kAnkp n→∞ n→∞ m×m for every {An}∈Fp . m×m p Theorem 7.2. Let 1 ≤ p<∞,a∈Cp (T), and N ∈K(lm).IfT(a)+N and p T (˜a) are invertible on lm then the limit −1 lim k(Tn(a)+PnNPn) kp n→∞ exists and equals n o −1 −1 max k(T (a)+N) kp, kT (˜a)kp . p If one of the operators T (a)+N and T (˜a) is not invertible on lm then −1 lim sup k(Tn(a)+PnNPn) kp = ∞. n→∞ 26 A. B¨ottcher, S. M. Grudsky, and B. Silbermann Theorem 7.3. Let 1 ε In Theorem 7.3, limn→∞ σp(An) is defined as the set of all λ ∈ C for which there exist n1,n2,... and λ1,λ2,... such that n −1 lim k(Pn(aP + bQ + K)Pn) kp n→∞ n o −1 −1 −1 ˜ = max k(aP + bQ + K) kp, kT (˜a)kp, kT (b)kp and, for ε>0, ε ε ε ε ˜ lim σp Pn(aP + bQ + K)Pn = σp aP + bQ + K ∪ σp T (˜a) ∪ σp T(b) . n→∞ Truncated Wiener-Hopf Operators and Toeplitz Matrices 27 Of course, systems of paired Wiener-Hopf integral operators may be tackled in the same way. We remark that all results can also be carried over to block Toeplitz operators on the Hardy spaces Hp(T)(1 8. Pseudospectra of Infinite Toeplitz Matrices From (6) we know the spectrum of the Wiener-Hopf operator W (a)onLp(0, ∞) (1 ≤ p ≤∞) in case a ∈ C + FL1(R). An analogous result is true for Toeplitz operators on lp (1 ≤ p ≤∞): if b ∈ W (T), then n o (62) σp T (b) = b(T) ∪ λ ∈ C \ b(T) : wind (b, λ) 6=0 . Theorems 5.2 and 7.3 do not pertain to spectra but to pseudospectra. Thus, it is ε ε desirable to know more about the pseudospectra σp(W (a)) and σp(T (b)). Given a Banach space X and a bounded linear operator A on X, we define the ε spectrum σX (A) and the ε-pseudospectrum σX (A) in the natural manner: n o σX (A)= λ∈C:A−λI is not invertible , n o ε −1 σX (A)= λ∈C:k(A−λI) k≥1/ε . One always has ε (63) σX (A)+∆ε ⊂σX(A) ε −1 −1 where ∆ε = {λ ∈ C : |λ|≤ε}. Indeed, if λ 6∈ σX (A) then ε Theorem 8.1. If ε ≥ 0,a∈L∞(R),b∈L∞(T), then ε (64) σ2 W (a) +∆ε ⊂σ2 W(a) ⊂conv R(a)+∆ε, ε (65) σ2 T (b) +∆ε ⊂σ2 T(b) ⊂conv R(b)+∆ε. Proof. The left inclusions in (64) and (65) are immediate from (63). For ε =0, the right inclusions of (64) and (65) are known as the Brown-Halmos theorem (see, e.g., [8, Theorem 2.33]). For ε>0, the proof of the latter inclusions is similar to the proof of the Brown-Halmos theorem. Here it is. Let λ 6∈ conv R(a)+∆ε. Then 0 6∈ conv R(a − λ)+∆ε, and hence we can rotate conv R(a − λ)+∆ε into the right open half-plane, i.e., there is a γ ∈ T such that conv R γ(a − λ) +∆ε ⊂{z∈C:Rez>0}. 28 A. B¨ottcher, S. M. Grudsky, and B. Silbermann Then we can find a (sufficiently large) disk {z ∈ C : |z − r − ε| γ(a(ξ) − λ) r − 1 < r + ε r + ε for almost all ξ ∈ R. Since γ γ(a−λ) W (a − λ)=I+W −1 r + ε r+ε and the norm of the Wiener-Hopf operator on the right is less than r/(r + ε) < 1, it follows that (γ/(r + ε))W (a − λ) is invertible and that r + ε 1 kW −1(a − λ)k < , |γ| 2 1 − r/(r + ε) −1 ε whence kW (a − λ)k2 < |γ|/ε =1/ε. Thus, λ 6∈ σp(W (a)). The proof is the same for the Toeplitz operator T (b). It is readily seen that the right inclusions of (64) and (65) may be proper. Indeed, suppose a ∈ C + FL1(R) and R(a) is the half-circle {z ∈ T :Imz≥0}. From (6) we infer that W (a) is invertible and hence, if ε>0 is small enough then −1 ε kW (a)k2 < 1/ε. Thus, 0 6∈ σ2(W (a)) although 0 ∈ conv R(a)+∆ε. The following result tells us that the left inclusions of (64) and (65) may also be proper. Theorem 8.2. Given ε>0, there exist a ∈ C + FL1(R) and b ∈ W (T) such that ε ε σ2 W (a) +∆ε 6=σ2 W(a) ,σ2 T(b)+∆ε 6=σ2 T(b) . Proof. Put e2iθ for 0 ≤ θ<π, b(eiθ)= e−2iθ for π ≤ θ<2π. If eiθ traverses T, then b(eiθ) twice traces out the unit circle, once in the positive and once in the negative direction. Thus, σ2(T (b)) = T due to (62). We claim that 3/4 (66) σ2 T (b) = {λ ∈ C : |λ|≤7/4}, which is clearly properly larger than T +∆3/4. From Theorem 8.1 we deduce that 3/4 {λ ∈ C :1/4≤|λ|≤7/4}⊂σ2 T(b) ⊂{λ∈C:|λ|≤7/4}. Consequently, (66) will follow as soon as we have shown that −1 (67) kT (b − λ)k2 ≥ 4/3 whenever |λ| < 1/4. So assume |λ| < 1/4. The Fourier coefficients of b can be easily computed: b0 =0,b2=b−2=1/2, bn =0 if n6=2iseven, 2 bn=4/ πi(n − 4) if n is odd. Truncated Wiener-Hopf Operators and Toeplitz Matrices 29 It follows in particular that b ∈ W (T). Writing the elements of l2 in the form (x0,x1,x2,...) and taking into account (54)weget −1 −1 kT (b − λ)ϕk2 kT (b−λ)k2 = inf ϕ6=0 kϕk2 −1 ≥kT(b−λ)(1, 0, 0,...)k2 −1 =k(b0 −λ, b1,b2,b3,...)k2 X −1/2 2 2 = |b0 −λ| + |bn| . n≥1 Since b0 = 0 and |bn| = |b−n| for all n, we obtain X 1 X |b − λ|2 + |b |2 = |λ|2+ |b |2 0 n 2 n n≥1 n∈Z Z2π 1 1 = |λ|2 + |b(eiθ)|2dθ (Parseval’s equality) 2 2π 0 = |λ|2 +1/2 (note that b is unimodular) < (1/4)2 +1/2=9/16. −1 −1/2 Thus, kT (b − λ)k2 > (9/16) =4/3, which gives (67) and completes the proof of (66). Replacing b by bε := (4ε/3)b we easily conclude from (66) that ε σ2 T (bε) = {λ ∈ C : |λ|≤7ε/3}, which is strictly larger than σ2 T (bε) +∆ε ={λ∈C:ε/3 ≤|λ|≤7/ε}. At this point we have proved the assertion for Toeplitz matrices. To prove the theorem for Wiener-Hopf operators, notice√ first that an orthonormal 2 ∞ −x basis in L (0, ∞) is given by {en}n=0 where en(x)= 2e Πn(2x) and Πn is the normalized nth Laguerre polynomial. The map X∞ 2 2 U : l → L (0, ∞), (ϕ0,ϕ1,ϕ2,...)7→ ϕnen n=0 is an isometric isomorphism, and if b ∈ W (T) has the Fourier coefficients {bn}n∈Z then UT(b)U−1 is the Wiener-Hopf operator W (a) with the symbol X ξ−i n a(ξ)= b (ξ∈R); n ξ+i n∈Z 1 see, e.g., [14, Chap. III, Sec. 3]. Obviously, a ∈ C+FL (R),σ2(W(a)) = σ2(T (b)), and since −1 −1 −1 −1 kW (a − λ)k2 = kUT (b−λ)U k2 = kT (b−λ)k2, ε ε it results that σ2(W (a)) = σ2(T (b)) for each ε>0. This reduces the proof in the Wiener-Hopf case to the proof in the Toeplitz case. 30 A. B¨ottcher, S. M. Grudsky, and B. 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