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. Böttcher, S. M. Grudsky, and B. Silbermann

Abstract. This paper is concerned with Wiener-Hopf integral operators on L^{p} and with Toeplitz operators (or matrices) on l^{p}. The symbols of the operators 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 Förderpreis für 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).

\begin{matrix}c \\ ◯ \end{matrix}1997 State University of New York ISSN 1076-9803/97

1. Introduction

Given a complex number c ∈ C and a function k ∈ L^{1}(R), we consider the Wiener-Hopf integral operator W defined by

(W φ)(x) = cφ(x) + ∫_{0}^{∞}k(x − t)φ(t) dt (0 < x < ∞) (1)

on L^{p}(0, ∞) (1 ≤ p ≤ ∞). Let F k be the Fourier transform of k,

(F k)(ξ) = ∫_{−∞}^{∞} k(x)e^{iξx}dx (ξ ∈ R).

Many properties of the operator (1) can be read off from the function a = c + F k. 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 + F k : c ∈ C, k ∈ L^{1}(R)} is denoted by C + F L^{1}(R) and called the Wiener algebra.

For τ ∈ (0, ∞), the truncated Wiener-Hopf integral operator W_{τ} (a) is the compression of W (a) to L^{p}(0, τ). Thus, W_{τ} (a) acts by the rule

(W_{τ} (a)φ)(x) = cφ(x) + ∫_{0}^{τ} k(x − t)φ(t) dt (0 < x < τ). (2)

The norm and the spectrum of an operator A on L^{p} will be denoted by ∥A∥_{p} and σ_{p}(A), respectively. We write ∥A^{−1}∥_{p} = ∞ if A is not invertible on L^{p}. For ε ∈ (0, ∞), the ε-pseudospectrum of an operator A on L^{p} is defined as the set

σ_{p}^{ε}(A) = {λ ∈ C : ∥(A − λI)^{−1}∥_{p} ≥ 1/ε}; (3)

of course, the points of

σ_{p}(A) = {λ ∈ C : ∥(A − λI)^{−1}∥_{p} = ∞}

all belong to σ_{p}^{ε}(A). In case W (a) or W_{τ} (a) is invertible, we denote the inverse by W^{−1}(a) and W_{τ}^{−1}(a), respectively.

This paper concerns the limits of ∥W_{τ}^{−1}(a)∥_{p}, σ_{p}(W_{τ} (a)), σ_{p}^{ε}(W_{τ} (a)) as τ goes to infinity. We also study ∥A_{τ}^{−1}∥ and σ_{p}^{ε}(A_{τ}) in case {A_{τ} } is a more complicated family of operators, say

A_{τ} = ∑_{j} ∏_{k} W_{τ} (a_{jk}) or A_{τ} = P_{τ} ∑_{j} ∏_{k} W (a_{jk})P_{τ} , (4)

where P_{τ} is given by

(P_{τ} φ)(x) = \{ \begin{matrix}φ(x) & if & 0 < x < τ, \\ 0 & if & τ < x; \end{matrix} (5)

here and in what follows we freely identify L^{p}(0, τ) and the image of P_{τ} on L^{p}(0, ∞).

Let a ∈ C + F L^{1}(R). The set a(Ṙ) := {c} ∪ {a(ξ) : ξ ∈ R} is a closed continuous curve in the complex plane. We give this curve the orientation induced by the change of ξ from −∞ to +∞. For λ ∉ a(Ṙ), we denote by wind (a, λ) the winding number of a(Ṙ) about λ. A classical result says that

σ_{p}(W (a)) = a(Ṙ) ∪ {λ ∈ C \ a(Ṙ) : wind (a, λ) ≠ 0} (6)

for 1 ≤ p ≤ ∞ (see [19] and [14]).

Now assume a ∈ C+F L^{1}(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

lim sup_{τ→∞} ∥W_{τ}^{−1}(a)∥_{p} < ∞ (7)

if and only if W (a) is invertible on L^{p}(0, ∞). Note that (7) includes the requirement that W_{τ} (a) is invertible on L^{p}(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 + F k ∈ C + F L^{1}(R), the associated symbol ã ∈ C + F L^{1}(R) is defined by ã(ξ) := a(−ξ). Thus, W (ã) and W_{τ} (ã) may be given by (1) and (2) with k(x − t) replaced by k(t − x). Since ã(Ṙ) differs from a(Ṙ) only in the orientation, we infer from (6) that σ_{p}(W (a)) = σ_{p}(W (ã)). In particular, W (ã) is invertible if and only if W (a) is invertible.

In this paper we prove the following results.

Theorem 1.1. Let a ∈ C + F L^{1}(R) and 1 ≤ p < ∞. If (7) holds then the limit lim_{τ→∞} ∥W_{τ}^{−1}(a)∥_{p} exists and

lim_{τ→∞} ∥W_{τ}^{−1}(a)∥_{p} = max {∥W^{−1}(a)∥_{p}, ∥W^{−1}(ã)∥_{p}}. (8)

Theorem 1.2. There exist a ∈ C + F L^{1}(R) and p ∈ (1, ∞) such that

∥W^{−1}(ã)∥_{p} > ∥W^{−1}(a)∥_{p}. (9)

Given a family {M_{τ} }_{τ∈(0,∞)} of subsets M_{τ} of C, we define the limiting set lim_{τ→∞} M_{τ} as the set of all λ ∈ C for which there are τ_{1}, τ_{2}, … and λ_{1}, λ_{2}, … such that

0 < τ_{1} < τ_{2} < …, τ_{n} → ∞, λ_{n} ∈ M_{τ_{n}}, λ_{n} → λ.

Here is what we will prove about the limiting sets of σ_{p}(W_{τ} (a)) and σ_{p}^{ε}(W_{τ} (a)).

Theorem 1.3. Let a ∈ C + F L^{1}(R) and 1 ≤ p ≤ ∞. Then for every τ > 0 the spectrum σ_{p}(W_{τ} (a)) is independent of p. In particular,

lim_{τ→∞} σ_{p}(W_{τ} (a)) (10)

does not depend on p. There exist a ∈ C + F L^{1}(R) such that (10) is not equal to the spectrum of W (a).

Theorem 1.4. If a ∈ C + F L^{1}(R) and 1 < p < ∞ then for each ε > 0,

lim_{τ→∞} σ_{p}^{ε}(W_{τ} (a)) = σ_{p}^{ε}(W (a)) ∪ σ_{p}^{ε}(W (ã)). (11)

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 W_{τ} (a)W_{τ} (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 P_{τ} 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 compact (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 ↦ F^{−1}aF φ is bounded on L^{2}(R). The set of all a ∈ L^{∞}(R) for which there exists a constant C = C(a, p) < ∞ such that

∥F^{−1}aF φ∥_{p} ≤ C∥φ∥_{p} for all φ ∈ L^{2}(R) ∩ L^{p}(R)

is denoted by M_{p}(R). The set M_{p}(R) is a Banach algebra with pointwise algebraic operations and the norm

∥a∥_{M_{p}(R)} = sup \{\frac{∥F^{−1}aF φ∥_{p}}{∥φ∥_{p}} : φ ∈ L^{2}(R) ∩ L^{p}(R), φ ≠ 0\}.

We have M_{1}(R) = C + F L^{1}(R) and M_{2}(R) = L^{∞}(R). If p ∈ (1, 2) ∪ (2, ∞) and 1/p + 1/q = 1, then

C + F L^{1}(R) ⊂ M_{p}(R) = M_{q}(R) ⊂ L^{∞}(R),

both inclusions being proper. For a ∈ M_{p}(R), the Wiener-Hopf integral operator W (a) is defined by

W (a)φ = χ_{+}(F^{−1}aF)φ (φ ∈ L^{p}(0, ∞)),

where (F^{−1}aF) is the continuous extension of F^{−1}aF from L^{2} ∩ L^{p} to all of L^{p} and χ_{+} denotes the characteristic function of (0, ∞). Of course, if a ∈ C + F L^{1}(R) then W (a)φ is given by (1).

One can also associate a Hankel operator H(a) ∈ L(L^{p}(0, ∞)) with every a ∈ M_{p}(R). We confine ourselves to stating that H(a) can be given by the formula

(H(a)f)(x) = ∫_{0}^{∞}k(x + t)φ(t) dt (0 < x < ∞)

if a = c + F k ∈ C + F L^{1}(R).

Let C_{p}(Ṙ) denote the closure of C + F L^{1}(R) in M_{p}(R). We have

C_{1}(Ṙ) = C + F L^{1}(R), C_{2}(Ṙ) = C(Ṙ),

where C(Ṙ) stands for the continuous functions on R which have finite and equal limits at +∞ and −∞. If p ∈ (1, 2) ∪ (2, ∞) and 1/p + 1/q = 1, then

C + F L^{1}(R) ⊂ C_{p}(Ṙ) = C_{q}(Ṙ) ⊂ C(Ṙ) ∩ M_{p}(R);

again both inclusions being proper. One can show that if a ∈ C_{p}(Ṙ) and a(ξ) ≠ 0 for all ξ ∈ Ṙ = R ∪ {∞}, then a^{−1} ∈ C_{p}(Ṙ).

If a ∈ M_{p}(R) then ∥W (a)∥_{p} = |W (a)|_{p} = ∥a∥_{M_{p}(R)}, where |W (a)|_{p} is the essential norm of W (a) on L^{p}(0, ∞),

|W (a)|_{p} := inf {∥W (a) + K∥_{p} : K ∈ K(L^{p}(0, ∞))}.

Further, if a ∈ C_{p}(Ṙ) and K ∈ K(L^{p}(0, ∞)), then σ_{p}(W (a)) is the set (6) and σ_{p}(W (a)) ⊂ σ_{p}(W (a) + K). In particular, W (a) and a are always invertible in case W (a) + K is invertible.

We denote by A_{p} the smallest closed subalgebra of L(L^{p}(0, ∞)) containing the set {W (a) : a ∈ C + F L^{1}(R)}. If a, b ∈ C_{p}(Ṙ), then

W (a)W (b) = W (ab) − H(a)H(b̃) (12)

and H(a) and H(b̃) (where b̃(ξ) = b(−ξ)) are compact. This and the equality |W (a)|_{p} = ∥a∥_{M_{p}(R)} imply that every operator in A_{p} is of the form W (a) + K with a ∈ C_{p}(Ṙ) and K ∈ K(L^{p}(0, ∞)). It is well known that in fact

A_{p} = {W (a) + K : a ∈ C_{p}(Ṙ), K ∈ K(L^{p}(0, ∞))}. (13)

Suppose a ∈ C_{p}(Ṙ), K ∈ K(L^{p}(0, ∞)), and W (a) + K is invertible. Then a^{−1} ∈ C_{p}(Ṙ), and (12) with b = a^{−1} gives

(W (a) + K)W (a^{−1}) = I − H(a)H(ã^{−1}) + KW (a^{−1}),

which shows that (W (a) + K)^{−1} equals

W (a^{−1}) + (W (a) + K)^{−1}H(a)H(ã^{−1}) − (W (a) + K)^{−1}KW (a^{−1}).

Thus, (W (a)+K)^{−1} is W (a^{−1}) plus a compact operator. Note that this observation in particular implies that A_{p} is inverse closed in L(L^{p}(0, ∞)).

The finite section analogue of (12) is Widom’s formula

W_{τ} (a)W_{τ} (b) = W_{τ} (ab) − P_{τ} H(a)H(b̃)P_{τ} − R_{τ} H(ã)H(b)R_{τ} (14)

where P_{τ} is as in (5) and R_{τ} : L^{p}(0, ∞) → L^{p}(0, ∞) is defined by

(R_{τ} φ)(x) = \{ \begin{matrix}φ(τ − x) & if & 0 < x < τ \\ 0 & if & τ < x \end{matrix} (15)

(see [34] and [8, 9.43(d)]). Thus, if a ∈ C_{p}(Ṙ), N ∈ K(L^{p}(0, ∞)), and W (a) + N is invertible, then (14) with b = a^{−1} yields

(W_{τ} (a) + P_{τ} NP_{τ} )W_{τ} (a^{−1})

= P_{τ} − P_{τ} H(a)H(ã^{−1})P_{τ} − R_{τ} H(a^{−1})H(ã)R_{τ} + P_{τ} NP_{τ} W_{τ} (a^{−1})

= P_{τ} − P_{τ} Y P_{τ} − R_{τ} ZR_{τ} + P_{τ} NP_{τ} W_{τ} (a^{−1}),

where Y and Z are compact on L^{p}(0, ∞). Our assumptions on a and N imply that W_{τ} (a) + P_{τ} NP_{τ} is invertible for all sufficiently large τ and that

(W_{τ} (a) + P_{τ} NP_{τ} )^{−1}P_{τ} → (W (a) + N)^{−1} (16) strongly on L^{p}(0, ∞). Hence

(W_{τ} (a) + P_{τ} NP_{τ} )^{−1} = W_{τ} (a^{−1}) + (W_{τ} (a) + P_{τ} NP_{τ} )^{−1}P_{τ} Y P_{τ}

+ (W_{τ} (a) + P_{τ} NP_{τ} )^{−1}R_{τ} ZR_{τ} + (W_{τ} (a) + P_{τ} NP_{τ} )^{−1}P_{τ} NP_{τ} W_{τ} (a^{−1}).

The space L^{p}(0, ∞) is the dual space of L^{q}(0, ∞) (1/p+1/q = 1) in case 1 < p < ∞ and the dual space of

L_{0}^{∞}(0, ∞) = {φ ∈ L^{∞}(0, ∞) : lim_{T →∞} ess sup_{x>T} |φ(x)| = 0}

in case p = 1. Since P_{τ} → I strongly on L^{r}(0, ∞) (1 ≤ r < ∞) and on L_{0}^{∞}(0, ∞) and since Y is compact, we obtain with the help of (16) that

(W_{τ} (a) + P_{τ} NP_{τ} )^{−1}P_{τ} Y P_{τ} = P_{τ} (W (a) + N)^{−1}Y P_{τ} + C_{τ}^{(1)}

where ∥C_{τ}^{(1)}∥_{p} → 0 as τ → ∞. Further, as R_{τ} W_{τ} (a)R_{τ} = W_{τ} (ã) and R_{τ} NR_{τ} → 0 strongly, we get analogously that

(W_{τ} (a) + P_{τ} NP_{τ} )^{−1}R_{τ} ZR_{τ} = R_{τ} (W_{τ} (ã) + R_{τ} NR_{τ} )^{−1}P_{τ} ZR_{τ}
= R_{τ} W^{−1}(ã)ZR_{τ} + C_{τ}^{(2)}

with ∥C_{τ}^{(2)}∥_{p} → 0 as τ → ∞. Finally, in the same way we obtain

(W_{τ} (a) + P_{τ} NP_{τ} )^{−1}P_{τ} NP_{τ} W_{τ} (a^{−1}) = P_{τ} (W_{τ} (a) + N)^{−1}NW (a^{−1})P_{τ} + C_{τ}^{(3)}

where ∥C_{τ}^{(3)}∥_{p} → 0 as τ → ∞. In summary, we have shown the following result.

Proposition 2.1. Let 1 ≤ p < ∞, a ∈ C_{p}(Ṙ), N ∈ K(L^{p}(0, ∞)). If W (a) + N is invertible, then for all sufficiently large τ

(W_{τ} (a) + P_{τ} NP_{τ} )^{−1} = W_{τ} (a^{−1}) + P_{τ} KP_{τ} + R_{τ} LR_{τ} + C_{τ} (17)

where K, L ∈ K(L^{p}(0, ∞)) and ∥C_{τ} ∥_{p} → 0 as τ → ∞.

Formula (17) is well known to workers in the field and extraordinarily useful in connection with all questions concerning the behavior of W_{τ}^{−1}(a) for large τ. Such formulas were first established and employed in [34] and [30] (also see [18]).

Let S_{p} denote the set of all families {A_{τ} } = {A_{τ} }_{τ∈(0,∞)} of operators A_{τ} ∈ L(L^{p}(0, τ)) with sup_{τ>0} ∥A_{τ} ∥_{p} < ∞. With obvious algebraic operations and the norm

∥{A_{τ} }∥_{p} = sup_{τ>0} ∥A_{τ} ∥_{p},

the set S_{p} is a Banach algebra. We let C_{p} stand for the collection of all elements {C_{τ} } ∈ S_{p} such that ∥C_{τ} ∥_{p} → 0 as τ → ∞. Obviously, C_{p} is a closed two-sided ideal of S_{p} and if {A_{τ} } ∈ S_{p}, then the norm of the coset {A_{τ} } + C_{p} in the quotient algebra S_{p}/C_{p} is

∥{A_{τ} } + C_{p}∥_{p} = lim sup_{τ→∞} ∥A_{τ} ∥_{p}.

It is clear that {W_{τ} (a)} ∈ S_{p} for every a ∈ M_{p}(R). We denote by F_{p} the smallest closed subalgebra of S_{p} containing all families {W_{τ} (a)} with a ∈ C + F L^{1}(R). If a, b ∈ C_{p}(Ṙ) then all Hankel operators in (14) are compact and hence

{W_{τ} (a)}{W_{τ} (b)} = {W_{τ} (ab) + P_{τ} KP_{τ} + R_{τ} LR_{τ} }

with certain compact operators K and L. This simple observation anticipates the following result.

Proposition 2.2. If 1 ≤ p < ∞ then

(18) F_{p} = {{W_{τ} (a) + P_{τ} KP_{τ} + R_{τ} LR_{τ} + C_{τ} } :

a ∈ C_{p}(Ṙ), K, L ∈ K(L^{p}(0, ∞)), {C_{τ} } ∈ C_{p}}

and C_{p} is a closed two-sided ideal of F_{p}.

The discrete analogue of Proposition 2.2 is proved in [6] and [8, Proposition 7.27]. On first approximating arbitrary families {C_{τ} } ∈ C_{p} by piecewise constant families, Proposition 2.2 can be proved in the same way as its discrete version.

Remark. For 1 ≤ p ≤ ∞, let Q(L^{p}(0, ∞)) be the so-called quasi-commutator ideal of L^{p}(0, ∞), i.e., let Q(L^{p}(0, ∞)) stand for the smallest closed two-sided ideal of L(L^{p}(0, ∞)) containing the set

{W (ab) − W (a)W (b) : a, b ∈ C + F L^{1}(R)}.

Proposition 2.1 easily implies that Q(L^{p}(0, ∞)) = K(L^{p}(0, ∞)) if 1 ≤ p < ∞. One can show that Q(L^{∞}(0, ∞)) is a proper subset of K(L^{∞}(0, ∞)) and that, although P_{τ} does not converge strongly on L^{∞}(0, ∞), one has ∥K −P_{τ} K∥ → 0 and ∥K −KP_{τ} ∥ → 0 as τ → ∞ for every K ∈ Q(L^{∞}(0, ∞)) (see [24, Proposition 4.55]). Furthermore, if K ∈ Q(L^{∞}(0, ∞)), then R_{τ} KR_{τ} goes to zero in the sense of Pconvergence (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(L^{p}(0, ∞)) is replaced by Q(L^{p}(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(L_{m}^{∞}(0, ∞)) and Theorem 7.2 with N ∈ Q(l_{m}^{∞})).

3. Norms of Inverses

Let A_{p} (1 ≤ p < ∞) be the algebra (13). We equip the direct sum A_{p} ⊕ A_{p} with the norm

∥(A, B)∥_{p} = max {∥A∥_{p}, ∥B∥_{p}}.

By virtue of Proposition 2.2, we may consider the quotient algebra F_{p}/C_{p}. Here is the key result of this paper.

Theorem 3.1. If 1 ≤ p < ∞, then the map Sym_{p} : F_{p}/C_{p} → A_{p} ⊕ A_{p} given by

{W_{τ} (a) + P_{τ} KP_{τ} + R_{τ} LR_{τ} + C_{τ} } + C_{p} ↦ (W (a) + K, W (ã) + L)

is an isometric Banach algebra homomorphism. Moreover, if {A_{τ} } ∈ F_{p} then

lim sup_{τ→∞} ∥A_{τ} ∥_{p} = lim_{τ→∞} ∥A_{τ} ∥_{p}.

Before giving a proof, a few comments are in order. We know from Proposition 2.2 that every family {A_{τ} } ∈ F_{p} is of the form

A_{τ} = W_{τ} (a) + P_{τ} KP_{τ} + R_{τ} LR_{τ} + C_{τ}

with K, L ∈ K(L^{p}(0, ∞)) and {C_{τ} } ∈ C_{p}. Clearly, W (a) + K and W (ã) + L are nothing but the strong limits of A_{τ} and R_{τ} A_{τ} R_{τ}, respectively. Thus, we could equivalently also define

Sym_{p} : {A_{τ} } + C_{p} ↦ (s−lim_{τ→∞} A_{τ} , s−lim_{τ→∞} R_{τ} A_{τ} R_{τ} ).

This identification of Sym_{p} shows in particular that Sym_{p} is a well-defined continuous Banach algebra homomorphism and that

(19) ∥Sym_{p}({A_{τ} } + C_{p})∥_{p} ≤ lim inf_{τ→∞} ∥A_{τ} ∥_{p}.

Since the only compact Wiener-Hopf operator is the zero operator, it follows that Sym_{p} is injective. Thus, Sym_{p} is a Banach algebra homomorphism and the only thing we must show is that Sym_{p} is an isometry.

If p = 2, then F_{2}/C_{2} and A_{2} ⊕ A_{2} are C^{∗}-algebras and Sym_{2} is easily seen to be a ∗-homomorphism. Since injective ∗-homomorphisms of C^{∗}-algebras are always isometric, the proof is complete. In the p ≠ 2 case, the latter conclusion requires hand-work.

Proof. Fix ε > 0. Since a ∈ C_{p}(Ṙ), there is a number c ∈ C and a function k ∈ L^{1}(R) with finite support, say supp k ⊂ (−s, s), such that

∥{W_{τ} (a − c − F k)}∥_{p} ≤ ∥W (a − c − F k)∥_{p} < ε.

Furthermore, if τ_{0} is large enough, then ∥P_{τ_{0}}KP_{τ_{0}} − K∥_{p} < ε, ∥P_{τ_{0}}LP_{τ_{0}} − L∥_{p} < ε. As ε > 0 may be chosen as small as desired, it follows that we are left with proving that

lim_{τ→∞} ∥B_{τ} ∥_{p} = max {∥W (b) + K∥_{p}, ∥W (b̃) + L∥_{p}} (20)

where

B_{τ} := W_{τ} (b) + P_{τ} KP_{τ} + R_{τ} LR_{τ} ,

b := c + F k ∈ C + F L^{1}(R), supp k ⊂ (−s, s), and the compact operators K, L are subject to the condition

P_{τ_{0}}KP_{τ_{0}} = K, P_{τ_{0}}LP_{τ_{0}} = L. (21)

Here s > 0 and τ_{0} > 0 are certain fixed numbers.

Pick l > max{τ_{0}, s} and put τ = 4l. Let φ ∈ L^{p}(0, τ) be any function such that ∥φ∥_{p} = 1. We claim that there exists a number

β ∈ (l + s, 3l − s)

such that

∫_{β−s}^{β+s}|φ(t)|^{p}dt ≤ [\frac{l}{s}]^{−1} (22)

where [l/s] stands for the integral part of l/s. Indeed, since

1 = ∥φ∥_{p}^{p} ≥ ∫_{l}^{3l}|φ(t)|^{p}dt ≥ ∑_{d=0}^{d_{0}−1} ∫_{l+2ds}^{l+2(d+1)s}|φ(t)|^{p}dt

with d_{0} := [l/s], there exists a d_{1} such that 0 ≤ d_{1} ≤ d_{0} − 1 and

∫_{l+2d_{1}s}^{l+2(d_{1}+1)s}|φ(t)|^{p}dt ≤ [\frac{l}{s}]^{−1},

which proves our claim with β := l + (2d_{1} + 1)s.

Given 0 < τ_{1} < τ_{2}, we put P_{(τ_{1},τ_{2})} := P_{τ_{2}} − P_{τ_{1}}. With β as above, we have

B_{τ} = P_{β}B_{τ} P_{β} + P_{(β,τ)}B_{τ} P_{(β,τ)} + P_{β}B_{τ} P_{(β,τ)} + P_{(β,τ)}B_{τ} P_{β}.

Since β + τ_{0} < 3l − s + l < 4l = τ and P_{τ_{0}}LP_{τ_{0}} = L, the equality P_{β}R_{τ} LR_{τ} P_{β} = 0 holds. Hence,

P_{β}B_{τ} P_{β} = P_{β}(W (b) + K)P_{β},

which imples that ∥P_{β}B_{τ} P_{β}∥_{p} ≤ ∥W (b) + K∥_{p} =: M_{1}. (23) As β > τ_{0} and P_{τ_{0}}KP_{τ_{0}} = K, we see that P_{(β,τ)}P_{τ} KP_{τ} P_{(β,τ)} = 0. Therefore,

P_{(β,τ)}B_{τ} P_{(β,τ)} = P_{(β,τ)}(W_{τ} (b) + R_{τ} LR_{τ} )P_{(β,τ)}
= P_{(β,τ)}R_{τ} (W_{τ} (b̃) + L)R_{τ} P_{(β,τ)}
= R_{τ} P_{τ−β}(W_{τ} (b̃) + L)P_{τ−β}R_{τ}
= R_{τ} P_{τ−β}(W (b̃) + L)P_{τ−β}R_{τ} ,

whence ∥P_{(β,τ)}B_{τ} P_{(β,τ)}∥_{p} ≤ ∥W (b̃) + L∥_{p} =: M_{2} (24) Again using (21) we get

P_{(β,τ)}B_{τ} P_{β} + P_{β}B_{τ} P_{(β,τ)} = P_{(β,τ)}W_{τ} (b)P_{β} + P_{β}W_{τ} (b)P_{(β,τ)}.

Since supp k ⊂ (−s, s), we have

(W_{τ} (b)P_{β}φ)(x) = ∫_{0}^{β} k(x − t)φ(t) dt = ∫_{β−s}^{β} k(x − t)φ(t) dt

10 A. Böttcher, S. M. Grudsky, and B. Silbermann for x ∈ (β, τ) and

(W_{τ} (b)P_{(β,τ)}φ)(x) = ∫_{β}^{τ} k(x − t)φ(t) dt = ∫_{β}^{β+s}k(x − t)φ(t) dt

for x ∈ (0, β). Consequently,

(25)  ∥P_{(β,τ)}B_{τ} P_{β}φ + P_{β}B_{τ} P_{(β,τ)}φ∥_{p}^{p}

= ∫_{β}^{τ} | ∫_{β−s}^{β} k(x − t)φ(t) dt|^{p}dx + ∫_{0}^{β} | ∫_{β}^{β+s}k(x − t)φ(t) dt|^{p}dx

≤ ∥k∥_{1}^{p}( ∫_{β−s}^{β} |φ(t)|^{p}dt + ∫_{β}^{β+s}|φ(t)|^{p}dt)

= ∥k∥_{1}^{p} ∫_{β−s}^{β+s}|φ(t)|^{p}dt ≤ ∥k∥_{1}^{p}[\frac{l}{s}]^{−1},

the last two estimates resulting from the inequality ∥k ∗ φ∥_{p} ≤ ∥k∥_{1}∥φ∥_{p} and from (22), respectively. Put

f_{1} = P_{β}B_{τ} P_{β}φ, f_{2} = P_{(β,τ)}B_{τ} P_{(β,τ)}φ,
f_{3} = (P_{(β,τ)}B_{τ} P_{β} + P_{β}B_{τ} P_{(β,τ)})φ.

Then B_{τ} φ = f_{1} + f_{2} + f_{3} and thus,

∥B_{τ} φ∥_{p} ≤ ∥f_{1} + f_{2}∥_{p} + ∥f_{3}∥_{p} = ( ∫_{0}^{τ} |f_{1}(t) + f_{2}(t)|^{p}dt)^{1/p} + ∥f_{3}∥_{p}

= ( ∫_{0}^{β} |f_{1}(t)|^{p}dt + ∫_{β}^{τ} |f_{2}(t)|^{p}dt)^{1/p} + ∥f_{3}∥_{p} = (∥f_{1}∥_{p}^{p} + ∥f_{2}∥_{p}^{p})^{1/p} + ∥f_{3}∥_{p}.

From (23), (24), (25) we therefore get

∥B_{τ} φ∥_{p} ≤ (M_{1}^{p}∥P_{β}φ∥_{p}^{p} + M_{2}^{p}∥P_{(β,τ)}φ∥_{p}^{p})^{1/p} + ∥f_{3}∥_{p}
≤ max{M_{1}, M_{2}}(∥P_{β}φ∥_{p}^{p} + ∥P_{(β,τ)}φ∥_{p}^{p})^{1/p} + ∥f_{3}∥_{p}
= max{M_{1}, M_{2}}∥φ∥_{p} + ∥f_{3}∥_{p}
≤ max{M_{1}, M_{2}} + ∥k∥_{1}[l/s]^{−1}.

If τ → ∞ then [l/s]^{−1} = [τ/(4s)]^{−1} → 0, which gives the estimate

(26) lim sup_{τ→∞} ∥B_{τ} ∥_{p} ≤ max{M_{1}, M_{2}}.

Since max{M_{1}, M_{2}} = ∥Sym_{p}({B_{τ} } + C_{p})∥_{p}, we finally obtain (20) from (26) and (19). □

Corollary 3.2. Let 1 ≤ p < ∞ and {A_{τ} } ∈ F_{p}. Put

A = s−lim_{τ→∞} A_{τ} , B = s−lim_{τ→∞} R_{τ} A_{τ} R_{τ} .

and suppose A − λI and B − λI are invertible on L^{p}(0, ∞). Then A_{τ} − λI (= A_{τ} − λP_{τ}) is invertible on L^{p}(0, τ) for all sufficiently large τ and

lim_{τ→∞} ∥(A_{τ} − λI)^{−1}∥_{p} = max {∥(A − λI)^{−1}∥_{p}, ∥(B − λI)^{−1}∥_{p}}.

Proof. Since (A−λI, B−λI) = Sym_{p}({A_{τ} −λI}+C_{p}), we deduce from Theorem 3.1 and the inverse closedness of A_{p} in L(L^{p}(0, ∞)) that {A_{τ} − λI} + C_{p} is invertible in F_{p}/C_{p}. Let {D_{τ} } + C_{p} ∈ F_{p}/C_{p} be the inverse. Then

Sym_{p}({D_{τ} } + C_{p}) = ((A − λI)^{−1}, (B − λI)^{−1}),
∥{D_{τ} } + C_{p}∥_{p} = lim sup_{τ→∞} ∥(A_{τ} − λI)^{−1}∥_{p},

and therefore the assertion follows from Theorem 3.1. □

The following corollary of Theorem 3.1 clearly contains Theorem 1.1.

Corollary 3.3. Let 1 ≤ p < ∞, a ∈ C_{p}(Ṙ), N ∈ K(L^{p}(0, ∞)). If W (a) + N is invertible on L^{p}(0, ∞) then the limit

lim_{τ→∞} ∥(W_{τ} (a) + P_{τ} NP_{τ} )^{−1}∥_{p} (27)

exists and is equal to

max {∥(W (a) + N)^{−1}∥_{p}, ∥W^{−1}(ã)∥_{p}}. (28)

Proof. Theorem 3.1 and Proposition 2.1 show that the limit (27) exists and equals

max {∥W (a^{−1}) + K∥_{p}, ∥W (ã^{−1}) + L∥_{p}}.

Passing in (17) to the strong limit τ → ∞ we get (W (a) + N)^{−1} = W (a^{−1}) + K, and multiplying (17) from both sides by R_{τ} and then passing to the strong limit τ → ∞ we see that W^{−1}(ã) = W (ã^{−1}) + L. □

Corollary 3.4. Let 1 ≤ p < ∞ and let a_{jk} be a finite collection of functions in C_{p}(Ṙ). Put

A = ∑_{j} ∏_{k} W (a_{jk}), B_{1} = ∑_{j} ∏_{k} W (ã_{jk}), B_{2} = W ( ∑_{j} ∏_{k} ã_{jk}).

and suppose A is invertible on L^{p}(0, ∞)

(a) If B_{1} is invertible on L^{p}(0, ∞) then

lim_{τ→∞} ∥( ∑_{j} ∏_{k} W_{τ} (a_{jk}))^{−1}∥_{p} = max {∥A^{−1}∥_{p}, ∥B_{1}^{−1}∥_{p}}, (29)

while if B_{1} is not invertible on L^{p}(0, ∞) then

lim_{τ→∞} ∥( ∑_{j} ∏_{k} W_{τ} (a_{jk}))^{−1}∥_{p} = ∞. (30)

(b) The operator B_{2} is invertible and

lim_{τ→∞} ∥(P_{τ} ∑_{j} ∏_{k} W (a_{jk})P_{τ} )^{−1}∥ = max {∥A^{−1}∥_{p}, ∥B_{2}^{−1}∥_{p}}. (31)

Proof. Let h = ∑_{j} ∏_{k} a_{jk}.

(a) We know from Section 2 that A = W (h) + M and B_{1} = W (h̃) + N with compact operators M and N. Obviously,

(32) A_{τ} := ∑_{j} ∏_{k} W_{τ} (a_{jk}) → A strongly

and

(33) R_{τ} A_{τ} R_{τ} = ∑_{j} ∏_{k} W_{τ} (ã_{jk}) → B_{1} strongly.

Suppose the limit in (30) does not exist or is finite. Then there are τ_{k} → ∞ such that ∥(R_{τ_{k}}A_{τ_{k}}R_{τ_{k}})^{−1}∥_{p} = ∥A_{τ_{k}}^{−1}∥_{p} ≤ m < ∞. This implies that ∥P_{τ_{k}}φ∥_{p} ≤ m∥R_{τ_{k}}A_{τ_{k}}R_{τ_{k}}φ∥_{p} and thus ∥φ∥_{p} ≤ m∥B_{1}φ∥_{p} for all φ ∈ L^{p}(0, ∞). Consequently, B_{1} is injective. Since A = W (h) + M is assumed to be invertible, the operator B_{1} = W (h̃) + N must be Fredholm of index zero (see [14]). This in conjunction with the injectivity of B_{1} shows that B_{1} is actually invertible. Thus, if B_{1} is not invertible, then (30) holds.

Now suppose B_{1} is invertible. From (32) and (33) we infer that Sym_{p}({A_{τ} }+C_{p}) equals (A, B_{1}), so that (29) is immediate from Theorem 3.1.

(b) If A = W (h)+M is invertible, then so also is B_{2} = W (h̃) 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 appeared 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 A_{p} and F_{p}/C_{p} do not admit such simple descriptions 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 ≠ 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

lim sup_{τ→∞} ∥W_{τ}^{−1}(a)∥_{2} = ∥W^{−1}(a)∥_{2}

cannot be proved by bare hands”, which is on p. 274 of [4]. Finally, it should be mentioned that the l^{p} version of Theorem 3.1 was de facto predicted already in [27, 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 L^{p}(0, ∞) by (Rφ)(x) = \overline{φ(x)}. Obviously,

R^{2} = I and W (ã) = RW (\overline{a})R

where \overline{a}(ξ) = \overline{a(ξ)}. Notice that if a = c + F k ∈ C + F L^{1}(R), then W (\overline{a}) and W_{τ} (\overline{a}) are given by (1) and (2) with \overline{k(t − x)} in place of k(x − t). Since R is an isometry, we have

∥W^{−1}(ã)∥_{p} = ∥W^{−1}(\overline{a})∥_{p} (34)

for 1 ≤ p ≤ ∞. On identifying the dual space of L^{p} (1 < p < ∞) with L^{q} (1/p + 1/q = 1), we may think of W (\overline{a}) as the adjoint operator of W (a). Thus,

∥W^{−1}(\overline{a})∥_{p} = ∥W^{−1}(a)∥_{q} (35)

for 1 < p < ∞, and in Theorem 1.1 and Corollary 3.3 we can replace ∥W^{−1}(ã)∥_{p} by ∥W^{−1}(a)∥_{q} in case 1 < p < ∞. If p = 2, then Corollary 3.3 implies that

lim_{τ→∞} ∥W^{−1}(a)∥_{2} = ∥W^{−1}(a)∥_{2}. (36)

Theorem 1.2 shows that in the above results the maximum cannot be removed. We remark that the previous results extend to certain spaces with weights. Given

a real number μ and a measurable function f on (0, ∞), we define

(Λ_{μ}f)(x) := (1 + |x|)^{μ}f(x).

Let L^{p,μ}(0, ∞) stand for the measurable functions f on (0, ∞) satisfying

∥f∥_{p,μ} := ∥Λ_{μ}f∥_{p} < ∞,

and let L^{1,μ}(R) be the space of all measurable functions on R for which Λ_{μ}f is in L^{1}(R). If a ∈ C + F L^{1,|μ|}(R), then W (a) and W_{τ} (a) are bounded on L^{p,μ}(0, ∞) and L^{p,μ}(0, τ) for all p ∈ [1, ∞) and all μ ∈ R. The algebras A_{p,μ} and F_{p,μ} are defined in the natural manner. Taking into account [24, Remark 4.51, 2^{∘}] one can construct an isometric Banach algebra isomorphism of F_{p,μ}/C_{p,μ} onto the direct sum A_{p,μ} ⊕ A_{p}. It follows in particular that if a ∈ C + F L^{1,|μ|}(R), a(ξ) ≠ 0 for all ξ ∈ Ṙ, and wind (a, 0) = 0, then

lim sup_{τ→∞} ∥W_{τ}^{−1}(a)∥_{p,μ} = lim_{τ→∞} ∥W_{τ}^{−1}(a)∥_{p,μ} = max {∥W^{−1}(a)∥_{p,μ}, ∥W^{−1}(ã)∥_{p}}.

Proof of Theorem 1.2. For ξ ∈ R, put

a_{−}(ξ) = 1 + \frac{i}{2(ξ − i)}, a_{+}(ξ) = 1 + \frac{2i}{ξ + i} − \frac{6}{(ξ + i)^{2}}

and consider a = a_{−}^{−1}a_{+}^{−1}. Since a_{±} ∈ C+F L^{1}(R) and these functions have no zeros on R, the symbol a belongs to C + F L^{1}(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 L^{p}(0, ∞) for 1 ≤ p ≤ ∞ and that W^{−1}(a) = W (a_{+})W (a_{−}). Consequently,

(W^{−1}(a)φ)(x) = φ(x) + ∫_{0}^{∞}γ(x, t)φ(t) dt (0 < x < ∞) (37)

where

γ(x, t) = γ_{+}(x − t) + γ_{−}(x − t) + ∫_{0}^{min{t,x}}γ_{+}(x − r)γ_{−}(r − t) dr

and γ_{±} are the kernels defined by a_{±} = 1 + F γ_{±}. Clearly,

γ_{+}(x) = \frac{1}{2π} ∫_{−∞}^{∞} (\frac{2i}{ξ + i} − \frac{6}{(ξ + i)^{2}} )e^{−iξx}dξ = \{ \begin{matrix}2e^{−x} + 6xe^{−x} & if & x > 0, \\ 0 & if & x < 0, \end{matrix}

γ_{−}(x) = \frac{1}{2π} ∫_{−∞}^{∞} \frac{i}{2(ξ − i)}e^{−iξx}dξ = \{ \begin{matrix}−\frac{1}{2}e^{x} & if & x < 0, \\ 0 & if & x > 0, \end{matrix}

whence

γ(x, t) = \frac{1}{4} \{ \begin{matrix}(3 + 18(x − t))e^{−(x−t)} + (5 + 6x)e^{−(x+t)} & if & x > t, \\ −7e^{x−t} + (5 + 6x)e^{−(x+t)} & if & x < t. \end{matrix}

From the representation (37) we infer that

∥W^{−1}(a)∥_{1} = 1 + sup_{t>0} ∫_{0}^{∞}|γ(x, t)| dx, ∥W^{−1}(a)∥_{∞} = 1 + sup_{x>0} ∫_{0}^{∞}|γ(x, t)| dt.

It is obvious that γ(x, t) > 0 for x > t, and taking into account that f(x) = (5 + 6x)e^{−2x} is monotonically decreasing on (0, ∞) it is easily seen that γ(x, t) < 0 for x < t. Thus, a straightforward computation gives

S_{1}(t) := ∫_{0}^{∞}|γ(x, t)| dt = 7 − \frac{9}{2}e^{−t} + (\frac{11}{2} + 3t)e^{−2t},

S_{∞}(x) := ∫_{0}^{∞}|γ(x, t)| dt = 7 − (4 + 3x)e^{−x} − (\frac{5}{2} + 3x)e^{−2x}.

We have S_{1}(0) = 8, S_{1}(∞) = 7, and S′_{1}(t) = e^{−2t}((9/2)e^{t} − (8 + 6t)). Let t_{0} > 0 be the unique solution of the equation (9/2)e^{t} = 8 + 6t. Then S_{1}(t) is monotonically decreasing on (0, t_{0}) and monotonically increasing on (t_{0}, ∞), which implies that

∥W^{−1}(a)∥_{1} = 1 + sup_{t>0} S_{1}(t) = 1 + S_{1}(0) = 9. (38)

Since S′_{∞}(x) = (1 + 3x)e^{−x} + (2 + 6x)e^{−2x} > 0 for all x ∈ (0, ∞), it follows that

∥W^{−1}(a)∥_{∞} = 1 + sup_{x>0} S_{∞}(x) = 1 + S_{∞}(∞) = 8. (39)

We now show that if p ∈ (1, ∞) is sufficiently large and 1/p + 1/q = 1, then

∥W^{−1}(a)∥_{p} < ∥W^{−1}(a)∥_{q}. (40)

This together with (34) and (35) gives the desired inequality (9).

Abbreviate W^{−1}(a) to A. By (38), there is a function f ∈ L^{∞}(R) with finite support, say supp f ⊂ (0, T), such that ∥Af∥_{1}/∥f∥_{1} > 9 − ε. Since

∥f∥_{1+δ}^{1+δ} = ∫_{0}^{T} |f(x)|^{1+δ}dx

depends continuously on δ ∈ [0, ∞), we see that ∥f∥_{1+δ}^{1+δ} → ∥f∥_{1} as δ → 0, whence ∥f∥_{1+δ} → ∥f∥_{1} as δ → 0. Further, we know that

(Af)(x) = f(x) + ∫_{0}^{T} γ(x, t)f(t) dt.

If x > T, then ∫_{0}^{T} γ(x, t)f(t) dt equals

∫_{0}^{T} (3 + 18(x − t))e^{−x}e^{t}f(t) dt + ∫_{0}^{T}(5 + 6x)e^{−x}e^{t}f(t) dt,

which is of the form αe^{−x} + βxe^{−x} with α, β ∈ C. Consequently,

(Af)(x) = g(x) + (α + βx)e^{−x}

where g ∈ L^{∞}(0, ∞) and supp g ⊂ (0, T). This easily implies that ∥Af∥_{1+δ}^{1+δ} → ∥Af∥_{1} and thus ∥Af∥_{1+δ} → ∥Af∥_{1} as δ → 0.

In summary, we have shown that ∥Af∥_{1+δ}/∥f∥_{1+δ} > 9 − 2ε and thus ∥A∥_{1+δ} > 9 − 2ε whenever δ > 0 is sufficiently small. From the Riesz-Thorin interpolation theorem and from (38), (39) we obtain that

∥A∥_{p} ≤ ∥A∥_{1}^{1/p}∥A∥_{∞}^{1−1/p} = 9^{1/p}8^{1−1/p} < 8 + ε

if only p is sufficiently large. But if p is large enough, then q = 1+δ with δ as small as desired. As 8 + ε < 9 − 2ε for ε < 1/3, we arrive at (40). □

4. Spectra

The comparision of the spectra of W (a) and W_{τ} (a) as well as the discrete analogue 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 Szegö-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 discussed 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 ∈ C_{p}(Ṙ) and shows that σ_{p}(W (a)) is independent 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.

Theorem 4.1. If a ∈ C + F L^{1}(R) and τ > 0, then σ_{p}(W_{τ} (a)) does not depend on p ∈ [1, ∞].

Proof. Define the shift operators V_{τ} and V_{τ}^{(−1)} on L^{p}(0, ∞) by

(V_{τ} φ)(x) = \{ \begin{matrix}φ(x − τ) & if & τ < x \\ 0 & if & 0 < x < τ \end{matrix} , (V_{τ}^{(−1)}φ)(x) = φ(x + τ).

We think of the elements of the direct sum L^{p}(0, ∞) ⊕ L^{p}(0, ∞) =: L_{2}^{p}(0, ∞) as column vectors and consider the operator B given on L_{2}^{p}(0, ∞) by the operator matrix

B = ( \begin{matrix}V_{τ} & W (a) \\ 0 & V_{τ}^{(−1)} \end{matrix} ) .

It is easily seen that W_{τ} (a) is invertible on L^{p}(0, τ) if and only if B is invertible on L_{2}^{p}(0, ∞).

Let a = c + F k with c ∈ C and k ∈ L^{1}(R). The operator B may be identified with the block Wiener-Hopf operator W (b) whose symbol is

b(ξ) = ( \begin{matrix}e^{iτξ} & a(ξ) \\ 0 & e^{−iτξ} \end{matrix} )

(see [14]). Letting

m_{−}(ξ) := ( \begin{matrix}−c & 0 \\ e^{iτξ} & 1 \end{matrix} ) , m_{+}(ξ) := ( \begin{matrix}1 & 0 \\ e^{−iτξ} & −c \end{matrix} ) ,

we can write W (m_{−})W (b)W (m_{+}) = W (d) where

d(ξ) = m_{−}(ξ)b(ξ)m_{+}(ξ)

= ( \begin{matrix}e^{iτξ}(F k)(ξ) & c + (F k)(ξ) \\ −c + (F k)(ξ) & e^{−iτξ}(F k)(ξ) \end{matrix} )

= ( \begin{matrix}(F k_{τ})(ξ) & c + (F k)(ξ) \\ −c + (F k)(ξ) & (F k_{−τ})(ξ) \end{matrix} )

with k_{τ} (x) = k(x − τ), k_{−τ} (x) = k(x + τ). The operators W (m_{±}) are invertible, the inverses being W (m_{±}^{−1}). 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 + F L^{1}(R). We have det d(ξ) = c^{2}, and hence W (d) is invertible if and only if c ≠ 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 of our original operator W_{τ} (a) on L^{p}(0, ∞) is independent of p.

Since W_{τ} (a) − λI = W_{τ} (a − λ), we finally see that the spectrum σ_{p}(W_{τ} (a)) is the same for all p ∈ [1, ∞]. □

Thus, when considering the limiting set

Λ(a) := lim_{τ→∞} σ_{p}(W_{τ} (a))

for a ∈ C + F L^{1}(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 Wiener-Hopf determinants contained in [3]. We only remark that if a ∈ C+F L^{1}(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 spectrum of W (a). For example, if a(ξ) = (3 + iξ)/(1 + ξ^{2}), then σ_{p}(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−\sqrt{2}, 3/2+\sqrt{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 σ_{p}(W_{τ} (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

(W (a)φ)(x) = \frac{1}{πi} ∫_{0}^{∞} \frac{φ(t)}{t − x} dt (0 < x < ∞).

Of course, W_{τ} (a) is given by the same formula with ∞ replaced by τ. It is well known (see, e.g., [13]) that if 1 < p < ∞, then

σ_{p}(W_{τ} (a)) = σ_{p}(W (a)) = O_{p}(−1, 1) for all τ > 0,

where O_{p}(−1, 1) is the set of all λ ∈ C at which the line segment [−1, 1] is seen at an angle θ satisfying

max{2π/p, 2π/q} ≤ θ ≤ π (1/p + 1/q = 1).

Thus, we are in the best of all possible cases. In a sense, it is continuous and non-rational (or non-analytic) symbols which cause real problems. We refer to [36] and [2] for a discussion of this phenomenon.

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 L^{p} 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 < p < ∞. Suppose A is a bounded linear operator on L^{p}(X, dμ) and A−λI is invertible for all λ in some open subset U of C. If ∥(A − λI)^{−1}∥_{p} ≤ M for all λ ∈ U, then ∥(A − λI)^{−1}∥_{p} < M for all λ ∈ U.

Remark. It should be noted that such a result is not true for general analytic operator-valued functions. To see this, equip C^{2} with the norm ∥( \begin{matrix}z \\ ω \end{matrix} )∥_{p} =

\sqrt[p]{|z|^{p} + |ω|^{p}} and consider the function

A : C → L(C^{2}), λ ↦ ( \begin{matrix}λ & 0 \\ 0 & 1 \end{matrix} ) .

Clearly, ∥A(λ)∥_{p} = max{|λ|, 1} and thus ∥A(λ)∥_{p} = 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 subset of C containing the origin and ∥(A − λI)^{−1}∥_{p} ≤ M for all λ ∈ U, then ∥A^{−1}∥_{p} < M. Assume the contrary, i.e., let

∥A^{−1}∥_{p} = M. (41)

We have

(A − λI)^{−1} = ∑_{j=0}^{∞} λ^{j}A^{−(j+1)}

for all λ = re^{iφ} in some disk |λ| = r ≤ r_{0}. Hence, for every f ∈ L^{p}(X, dμ),

(42)  ∥(A − λI)^{−1}f∥_{p}^{p} = ∫_{X} | ∑_{j=0}^{∞} λ^{j}(A^{−(j+1)}f)(x)|^{p}dμ(x)

= ∫_{X} | ∑_{j=0}^{∞} r^{j}e^{ijφ}(A^{−(j+1)}f)(x)|^{2p/2}dμ(x)

= ∫_{X} |C(r, x) + ∑_{l=1}^{∞} B_{l}(r, x, φ)|^{p/2}dμ(x)

with

C(r, x) = ∑_{j=0}^{∞} r^{2j}|(A^{−(j+1)}f)(x)|^{2},

B_{l}(r, x, φ) = 2 ∑_{k=0}^{∞} r^{l+2k}Re (e^{ilφ}(A^{−(l+k+1)}f)(x)\overline{(A^{−(k+1)}f)(x)} ).

For n = 0, 1, 2, …, put

I_{n}(r, φ, f) = ∫_{X} |C(r, x) + ∑_{l=1}^{∞} B_{2^{n}l}(x, r, φ)|^{p/2}dμ(x). (43)

We now apply the inequality

\frac{|a + b|^{p/2} + |a − b|^{p/2}}{2} ≥ |a|^{p/2} (44)

to

a = C(r, x) + ∑_{l=0}^{∞} B_{2l}(r, x, φ), b = ∑_{l=1}^{∞} B_{2l−1}(r, x, φ)

and integrate the result over X. Taking into account that

∫_{X} |a − b|^{p/2}dμ(x) = I_{0}(r, φ + π, f),

we get

\frac{I_{0}(r, φ, f) + I_{0}(r, φ + π, f)}{2} ≥ I_{1}(r, φ, f). (45)

Letting

a = C(r, x) + ∑_{l=1}^{∞} B_{4l}(r, x, φ), b = ∑_{l=1}^{∞} B_{4l−2}(r, x, φ)

in (44), we analogously obtain after integration that

\frac{I_{1}(r, φ, f) + I_{1}(r, φ + π/2, f)}{2} ≥ I_{2}(r, φ, f). (46)

Combining (45) and (46) we arrive at the inequality

(47) \frac{I_{0}(r, φ, f) + I_{0}(r, φ + π/2, f) + I_{0}(r, φ + π, f) + I_{0}(r, φ + 3π/2, f)}{4}

≥ \frac{I_{1}(r, φ, f) + I_{1}(r, φ + π/2, f)}{2} ≥ I_{2}(r, φ, f).

In the same way as above we see that

\frac{I_{2}(r, φ, f) + I_{2}(r, φ + π/4, f)}{2} ≥ I_{3}(r, φ, f),

which together with (47) gives

\frac{1}{8} ∑_{m=0}^{7} I_{0}(r, φ + \frac{mπ}{4} , f) ≥ I_{3}(r, φ, f).

Obviously, continuing this process we obtain

\frac{1}{2^{n}} ∑_{m=0}^{2^{n}−1} I_{0}(r, φ + \frac{mπ}{2^{n−1}} , f) ≥ I_{n}(r, φ, f) (48)

for every n ≥ 0.

Now put φ = 0 in (48). From (43) we infer that the right-hand side of (48) converges to

∫_{X} |C(r, x)|^{p/2}dμ(x)

as n → ∞. The left-hand side of (48) is an integral sum and hence passage to the limit n → ∞ gives

lim_{n→∞} \frac{1}{2π} \frac{π}{2^{n−1}} ∑_{m=0}^{2^{n}−1} I_{0}(r, \frac{mπ}{2^{n−1}} , f) = \frac{1}{2π} ∫_{0}^{2π}I_{0}(r, φ, f) dφ.

As I_{0}(r, φ, f) = ∥(A − re^{iφ}I)^{−1}f∥_{p}^{p} by (42), it follows that

\frac{1}{2π} ∫_{0}^{2π}∥(A − re^{iφ}I)^{−1}f∥_{p}^{p}dφ ≥ ∫_{X} |C(r, x)|^{p/2}dμ(x). (49)

Assume ∥f∥_{p} = 1. Then our hypothesis ∥(A−λI)^{−1}∥_{p} ≤ M and (49) imply that

M^{p} ≥ \frac{1}{2π} ∫_{0}^{2π}∥(A − re^{iφ}I)^{−1}f∥_{p}^{p}dφ ≥ ∫_{X} |C(r, x)|^{p/2}dμ(x)

≥ ∫_{X} (|(A^{−1}f)(x)|^{2} + r^{2}|(A^{−2}f)(x)|^{2})^{p/2}dμ(x).

Since (|a| + |b|)^{p/2} ≥ |a|^{p/2} + |b|^{p/2}, we get

(50)  M^{p} ≥ ∫_{X} |(A^{−1}f)(x)|^{p}dμ(x) + r^{p} ∫_{X} |(A^{−2}f)(x)|^{p}dμ(x)
= ∥A^{−1}f∥_{p}^{p} + r^{p}∥A^{−2}f∥_{p}^{p}.

Let ε > 0 be an arbitrary number. By virtue of (41), we can find an f ∈ L^{p}(X, dμ) such that ∥f∥_{p} = 1 and ∥A^{−1}f∥_{p}^{p} ≥ M^{p} − ε. Since

∥A^{−2}f∥_{p} ≥ ∥A^{2}∥_{p}^{−1}∥f∥_{p} = ∥A^{2}∥_{p}^{−1},

we finally obtain from (50) that

M^{p} ≥ M^{p} − ε + r^{p}∥A^{2}∥_{p}^{−p} for all r ∈ (0, r_{0}],

which is impossible if ε is sufficiently small. This contradiction completes the proof. □

Theorem 5.2. Let 1 < p < ∞ and A_{τ} = W_{τ} (a) + P_{τ} KP_{τ} + R_{τ} LR_{τ} + C_{τ} with a in C_{p}(Ṙ), K, L ∈ K(L^{p}(0, ∞)), and ∥C_{τ} ∥_{p} → 0 as τ → ∞. Then for each ε > 0,

lim_{τ→∞} σ_{p}^{ε}(A_{τ}) = σ_{p}^{ε}(W (a) + K) ∪ σ_{p}^{ε}(W (ã) + L). (51)

Proof. Put A = W (a) + K and B = W (ã) + L. We first show the inclusion

σ_{p}^{ε}(A) ⊂ lim_{τ→∞} σ_{p}^{ε}(A_{τ}). (52)

Pick λ ∈ σ_{p}(A). We claim that

lim sup_{τ→∞} ∥(A_{τ} − λI)^{−1}∥_{p} = ∞. (53)

Assume the contrary, i.e., let ∥(A_{τ} − λI)^{−1}∥_{p} ≤ m < ∞ for all τ > τ_{0}. Then

∥P_{τ} φ∥_{p} ≤ m∥(A_{τ} − λI)P_{τ} φ∥_{p}

for every φ ∈ L^{p}(0, ∞) and all τ > τ_{0}, whence ∥φ∥_{p} ≤ m∥(A − λI)φ∥_{p} for every φ ∈ L^{p}(0, ∞). This shows that A − λI has a closed range and a trivial kernel. Our assumption implies that ∥(A_{τ}^{∗} + \overline{λ}I)^{−1}∥_{q} ≤ m < ∞ where 1/p + 1/q = 1. Arguing as above, we see that A^{∗} − \overline{λ}I has a trivial kernel on L^{q}(0, ∞), which implies that the range of A − λI is dense in L^{p}(0, ∞). In summary, A − λI must be invertible on L^{p}(0, ∞). As this is impossible if λ ∈ σ_{p}(A), we see that (53) is true.

From (53) we deduce the existence of τ_{1}, τ_{2}, … such that τ_{n} → ∞ and

∥(A_{τ_{n}} − λI)^{−1}∥_{p} ≥ 1/ε.

This shows that λ belongs to the set on the right of (52).

Now suppose λ ∈ σ_{p}^{ε}(A) \ σ_{p}(A). Then ∥(A − λI)^{−1}∥_{p} ≥ 1/ε. Let U ⊂ C be any open neighborhood of λ. From Theorem 5.1 we see that there is a point μ ∈ U such that ∥(A − μI)^{−1}∥ > 1/ε. Hence, we can find an integer n_{0} > 0 such that

∥(A − μI)^{−1}∥_{p} ≥ \frac{1}{ε − 1/n} for all n ≥ n_{0}.

Because U was arbitrary, it follows that there exists a sequence λ_{1}, λ_{2}, … such that λ_{n} ∈ σ_{p}^{ε−1/n}(A) and λ_{n} → λ.

For every invertible operator T ∈ L(L^{p}(0, ∞)) we have

(54) ∥T^{−1}∥_{p} = sup_{ψ≠0} \frac{∥T^{−1}ψ∥_{p}}{∥ψ∥_{p}} = sup_{φ≠0} \frac{∥φ∥_{p}}{∥T φ∥_{p}} = ( inf_{φ≠0} \frac{∥T φ∥_{p}}{∥φ∥_{p}} )^{−1}.

Since ∥(A − λ_{n}I)^{−1}∥_{p} ≥ 1/(ε − 1/n), we obtain from (54) that

inf_{∥φ∥_{p}=1} ∥(A − λ_{n}I)φ∥_{p} ≤ ε − 1/n,

implying the existence of a φ_{n} ∈ L^{p}(0, ∞) such that

∥φ_{n}∥_{p} = 1 and ∥(A − λ_{n}I)φ_{n}∥_{p} < ε − 1/(2n).

Because ∥(A_{τ} −λ_{n}I)P_{τ} φ_{n}∥_{p} → ∥(A−λI)φ∥_{p} and ∥P_{τ} φ_{n}∥_{p} → ∥φ_{n}∥_{p} = 1 as τ → ∞, it follows that

∥(A_{τ} − λ_{n}I)P_{τ} φ_{n}∥_{p}/∥P_{τ} φ_{n}∥_{p} < ε − 1/(3n)

for all τ → τ_{0}(n). Again invoking (54) we see that

∥(A_{τ} − λ_{n}I)^{−1}∥_{p} > (ε − 1/(3n))^{−1} > 1/ε

and thus λ_{n} ∈ σ_{p}^{ε}(A_{τ}) for all τ > τ_{0}(n). This implies that λ = lim λ_{n} lies in the set on the right of (52). At this point the proof of (52) is complete.

Repeating the above reasoning with R_{τ} A_{τ} R_{τ} and B in place of A_{τ} and A, respectively, we get the inclusion

σ_{p}^{ε}(B) ⊂ lim_{τ→∞} σ_{p}^{ε}(R_{τ} A_{τ} R_{τ}). (55)

Because R_{τ} is an isometry on L^{p}(0, τ) and

R_{τ} A_{τ} R_{τ} − λP_{τ} = R_{τ} (A_{τ} − λI)R_{τ} ,

it is clear that σ_{p}^{ε}(R_{τ} A_{τ} R_{τ}) = σ_{p}^{ε}(A_{τ}). Thus, in (55) we may replace R_{τ} A_{τ} R_{τ} by A_{τ}, which in conjunction with (52) proves the inclusion “⊃” in (51).

We are left with proving the inclusion “⊂” of (51). So let λ ∉ σ_{p}^{ε}(A) ∪ σ_{p}^{ε}(B). Then

∥(A − λI)^{−1}∥_{p} < 1/ε, ∥(B − λI)^{−1}∥_{p} < 1/ε,

whence

∥(A_{τ} − λI)^{−1}∥_{p} < 1/ε − δ < 1/ε for all τ > τ_{0} (56) with some δ > 0 due to Corollary 3.2. If |μ − λ| is sufficiently small, then A_{τ} − μI is invertible together with A_{τ} − λI, and we have

∥(A_{τ} − μI)^{−1}∥_{p} ≤ \frac{∥(A_{τ} − λI)^{−1}∥_{p}}{1 − |μ − λ| ∥(A_{τ} − λI)^{−1}∥_{p}} . (57)

Let |μ − λ| < εδ/(1/ε − δ). In this case (56) and (57) give

∥(A_{τ} − μI)^{−1}∥_{p} < \frac{1/ε − δ}{1 − εδ(1/ε − δ)/(1/ε − δ)} = \frac{1}{ε}.

Thus, μ ∉ σ_{p}^{ε}(A_{τ}) for τ > τ_{0}. This shows that λ cannot belong to lim_{τ→∞} σ_{p}^{ε}(A_{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

lim_{τ→∞} σ_{2}^{ε}(W_{τ} (a)) = σ_{2}^{ε}(W (a)) (58)

for each ε > 0. With the notation as in Corollary 3.4, we have for each ε > 0,

lim_{τ→∞} σ_{p}^{ε}( ∑_{j} ∏_{k} W_{τ} (a_{jk}) = σ_{p}^{ε}(A) ∪ σ_{p}^{ε}(B_{1}),

lim_{τ→∞} σ_{p}^{ε}(P_{τ} ∑_{j} ∏_{k} W (a_{jk})P_{τ} ) = σ_{p}^{ε}(A) ∪ σ_{p}^{ε}(B_{2}).

Finally, note that Theorems 5.2 and 1.2 imply that σ_{p}^{ε}(W_{τ} (a)) in general depends on p.

6. Matrix Case

Given a positive integer m, we denote by L_{m}^{p}(0, ∞) the direct sum of m copies of L^{p}(0, ∞) and we think of the elements of L_{m}^{p}(0, ∞) as columns. We define the norm on L_{m}^{p}(0, ∞) by

∥(f_{1}, …, f_{m})^{T} ∥_{p}^{p} := ∫_{0}^{∞}( ∑_{j=1}^{m} |f_{j}(x)|^{2})^{p/2}dx.

Let C_{p}^{m×m}(Ṙ) stand for the collection of all m × m matrix functions with entries in C_{p}(Ṙ). For a = (a_{jk})_{j,k=1}^{m} ∈ C_{p}^{m×m}(Ṙ) the (block) Wiener-Hopf integral operator W (a) is defined on L_{m}^{p}(0, ∞) by the operator matrix (W (a_{jk}))_{j,k=1}^{m}. The direct sum L_{m}^{p}(0, τ) of m copies of L^{p}(0, τ) may be identified with a subspace of L_{m}^{p}(0, ∞) in a natural manner. The operators on L_{m}^{p}(0, ∞) given by the diagonal matrices diag (P_{τ} , …, P_{τ}) and diag (R_{τ} , …, R_{τ}) are also denoted by P_{τ} and R_{τ}. If a = (a_{jk})_{j,k=1}^{m} ∈ C_{p}^{m×m}(Ṙ), then the compression W_{τ} (a) of W (a) to L_{m}^{p}(0, τ) is nothing but (W_{τ} (a_{jk}))_{j,k=1}^{m}. Finally, we give F_{p}^{m×m}, C_{p}^{m×m}, A_{p}^{m×m} the natural meaning.

The results proved above extend to the matrix case with obvious adjustments. We confine ourselves to formulating the basic theorems.

Theorem 6.1. Let 1 ≤ p < ∞. Then

A_{p}^{m×m} = {W_{τ} (a) + K : a ∈ C_{p}^{m×m}(Ṙ), K ∈ K(L_{m}^{p}(0, ∞))},
F_{p}^{m×m} = {W_{τ} (a) + P_{τ} KP_{τ} + R_{τ} LR_{τ} + C_{τ} : a ∈ C_{p}^{m×m}(Ṙ),

K, L ∈ K(L_{m}^{p}(0, ∞)), ∥C_{τ} ∥_{p} → 0 as τ → ∞},

the map Sym_{p} : F_{p}^{m×m}/C_{p}^{m×m} → A_{p}^{m×m} ⊕ A_{p}^{m×m} sending

{W_{τ} (a) + P_{τ} KP_{τ} + R_{τ} LR_{τ} + C_{τ} } + C_{p}^{m×m} to (W (a) + K, W (ã) + L)

is an isometric Banach algebra homomorphism, and

lim sup_{τ→∞} ∥A_{τ} ∥_{p} = lim_{τ→∞} ∥A_{τ} ∥_{p}

for every family {A_{τ} } ∈ F_{p}^{m×m}.

Theorem 6.2. Let 1 ≤ p < ∞, a ∈ C_{p}^{m×m}(Ṙ), and N ∈ K(L_{m}^{p}(0, ∞)). If W (a) + N and W (ã) (where ã(ξ) := a(−ξ)) are invertible on L_{m}^{p}(0, ∞) then the limit (27) exists and is equal to (28). If one of the operators W (a) + N and W (ã) is not invertible on L_{m}^{p}(0, ∞), then

lim sup_{τ→∞} ∥(W_{τ} (a) + P_{τ} NP_{τ})^{−1}∥_{p} = ∞.

Note that in the scalar case (m = 1) the operator W (ã) 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

lim_{τ→∞} ∥W_{τ}^{−1}(a)∥_{2} = max {∥W^{−1}(a)∥_{2}, ∥W^{−1}(ã)∥_{2}}

cannot be simplified to (36).

Theorem 6.3. Let 1 < p < ∞ and A_{τ} = W_{τ} (a) + P_{τ} KP_{τ} + R_{τ} LR_{τ} + C_{τ} , where a ∈ C_{p}^{m×m}(Ṙ), K, L ∈ K(L_{m}^{p}(0, ∞)), and ∥C_{τ} ∥_{p} → 0 as τ → ∞. Then for each ε > 0,

lim_{τ→∞} σ_{p}^{ε}(A_{τ}) = σ_{p}^{ε}(W (a) + K) ∪ σ_{p}^{ε}(W (ã) + L).

We emphasize again that the equality

lim_{τ→∞} σ_{2}^{ε}(W_{τ} (a)) = σ_{2}^{ε}(W (a)) ∪ σ_{2}^{ε}(W (ã))

cannot be reduced to (58) if m > 1.

7. Block Toeplitz Matrices

Let T be the complex unit circle. Given a function a ∈ L^{∞}(T), we denote by {a_{n}}_{n∈Z} the sequence of its Fourier coefficients,

a_{n} = \frac{1}{2π} ∫_{0}^{2π}a(e^{iθ})e^{−inθ}dθ,

and we let W (T) stand for the Wiener algebra of all functions a with absolutely convergent Fourier series. We think of l^{p} = l^{p}(Z_{+}) = l^{p}({0, 1, 2, … }) as a space of infinite columns. If a ∈ L^{∞}(T), then the Toeplitz operator T (a) given by the Toeplitz matrix

(a_{j−k})_{j,k=0}^{∞} (59)

is bounded on l^{2}. For 1 ≤ p < ∞, let M_{p}(T) be the set of all functions a ∈ L^{∞}(T) satisfying

∥T (a)φ∥_{p} ≤ C∥φ∥_{p} for all φ ∈ l^{2} ∩ l^{p}

with some C = C(a, p) < ∞ independent of p. The set M_{p}(T) is a Banach algebra with the norm

∥a∥_{M_{p}(T)} = sup {∥T (a)φ∥_{p}/∥φ∥_{p} : φ ∈ l^{2} ∩ l^{p}, φ ≠ 0}.

It is easily seen that W (T) ⊂ M_{p}(T) for all p ∈ [1, ∞). The closure of W (T) in M_{p}(T) is denoted by C_{p}(T). One can show that C_{1}(T) = W (T), C_{2}(T) = C(T), and that if p ∈ (1, 2) ∪ (2, ∞), then

C_{p}(T) = C_{q}(T) ⊂ C(T) ∩ M_{p}(T),

the inclusion being proper (see, e.g., [7], [8]). If a ∈ C_{p}(T), then the matrix (59) induces a bounded operator on l^{p}. This operator is denoted by T (a) and called the Toeplitz operator (or the Toeplitz matrix) with the symbol a.

For a ∈ C_{p}(T) and n ∈ Z_{+} we denote by T_{n}(a) the (n + 1) × (n + 1) Toeplitz matrix (a_{j−k})_{j,k=0}^{n}. On defining P_{n} on l^{p} by

(P_{n}φ)_{j} = \{ \begin{matrix}φ_{j} & if & 0 ≤ j ≤ n, \\ 0 & if & j > n, \end{matrix}

we can identify T_{n}(a) with the compression of T (a) to l^{p}(Z_{n}) = l^{p}({0, 1, …, n}). The discrete analogue of the operator R_{τ} given by (15) is the operator R_{n} defined by

(R_{n}φ)_{j} = \{ \begin{matrix}φ_{n−j} & if & 0 ≤ j ≤ n, \\ 0 & if & j > n. \end{matrix}

We regard l^{p}(Z_{n}) as a subspace of l^{p} = l^{p}(Z_{+}). This specifies the norms in l^{p}(Z_{n}) and L(l^{p}(Z_{n})). The norm and the spectrum of an operator A on l^{p}(Z_{n}) or l^{p}(Z_{+}) are denoted by ∥A∥_{p} and σ_{p}(A), respectively. The ε-pseudospectrum σ_{p}^{ε}(A) is defined as the set (3).

Finally, given a positive integer m, we define l_{m}^{p}, l_{m}^{p}(Z_{n}), C_{p}^{m×m}(T), P_{n}, R_{n} as well as T (a) and T_{n}(a) for a ∈ C_{p}^{m×m}(T) in the natural fashion. The norm of (f_{1}, …, f_{m})^{T} ∈ l_{m}^{p} is defined by

∥(f_{1}, …, f_{m})^{T} ∥_{p}^{p} = ∑_{j} ( ∑_{k=1}^{m} |(f_{k})_{j}|^{2})^{p/2}.

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.

Let A_{p}^{m×m} be the smallest closed subalgebra of L(l_{m}^{p}) containing {T (a) : a ∈ W (T)}. One can show that

A_{p}^{m×m} = {T (a) + K : a ∈ C_{p}^{m×m}(T), K ∈ K(l_{m}^{p})}

and that A_{p}^{m×m} is inverse closed in L(l_{m}^{p}). Denote by S_{p}^{m×m} the Banach algebra of all sequences {A_{n}} = {A_{n}}_{n=0}^{∞} of operators (matrices) A_{n} ∈ L(l_{m}^{p}) such that

∥{A_{n}}∥_{p} := sup_{n∈Z_{+}} ∥A_{n}∥_{p} < ∞,

let C_{p}^{m×m} be the closed two-sided ideal of all sequences {A_{n}} ∈ S_{p}^{m×m} for which ∥A_{n}∥_{p} → 0 as n → ∞, and let F_{p}^{m×m} stand for the smallest closed subalgebra of S_{p}^{m×m} containing all sequences {T_{n}(a)}_{n=0}^{∞} with a ∈ W (T).

Theorem 7.1. Let 1 ≤ p < ∞. Then

F_{p}^{m×m} = {T_{n}(a) + P_{n}KP_{n} + R_{n}LR_{n} + C_{n} : a ∈ C_{p}^{m×m}(T),

K, L ∈ K(l_{m}^{p}), ∥C_{n}∥_{p} → 0 as n → ∞}.

The map

Sym_{p} : F_{p}^{m×m}/C_{p}^{m×m} → A_{p}^{m×m} ⊕ A_{p}^{m×m}

sending

{A_{n}} + C_{p}^{m×m} = {T_{n}(a) + P_{n}KP_{n} + R_{n}LR_{n} + C_{n}} + C_{p}^{m×m}

to

(s−lim_{n→∞} A_{n}, s−lim_{n→∞} R_{n}A_{n}R_{n}) = (T (a) + K, T (ã) + L)

(ã(t) := a(1/t) for t ∈ T) is an isometric Banach algebra homomorphism. Furthermore,

lim sup_{n→∞} ∥A_{n}∥_{p} = lim_{n→∞} ∥A_{n}∥_{p}

for every {A_{n}} ∈ F_{p}^{m×m}.

Theorem 7.2. Let 1 ≤ p < ∞, a ∈ C_{p}^{m×m}(T), and N ∈ K(l_{m}^{p}). If T (a) + N and T (ã) are invertible on l_{m}^{p} then the limit

lim_{n→∞} ∥(T_{n}(a) + P_{n}NP_{n})^{−1}∥_{p}

exists and equals

max {∥(T (a) + N)^{−1}∥_{p}, ∥T^{−1}(ã)∥_{p}}.

If one of the operators T (a) + N and T (ã) is not invertible on l_{m}^{p} then

lim sup_{n→∞} ∥(T_{n}(a) + P_{n}NP_{n})^{−1}∥_{p} = ∞.

Theorem 7.3. Let 1 < p < ∞ and A_{n} = T_{n}(a) + P_{n}KP_{n} + R_{n}LR_{n} + C_{n} where a ∈ C_{p}^{m×m}(T), K, L ∈ K(l_{m}^{p}), and ∥C_{N}∥_{p} → 0 as n → ∞. Then for each ε > 0,

lim_{n→∞} σ_{p}^{ε}(A_{n}) = σ_{p}^{ε}(T (a) + K) ∪ σ_{p}^{ε}(T (ã) + L).

In Theorem 7.3, lim_{n→∞} σ_{p}^{ε}(A_{n}) is defined as the set of all λ ∈ C for which there exist n_{1}, n_{2}, … and λ_{1}, λ_{2}, … such that

n_{1} < n_{2} < …, n_{k} → ∞, λ_{n} ∈ σ_{p}^{ε}(A_{n_{k}}), λ_{n} → λ.

For p = 2, the previous three theorems were established in [31] and [4] by means of C^{∗}-algebra techniques. For p = 1, m = 1, N = 0, Theorem 7.2 was first obtained in [16]. The paper [16] also contains an example of a function a ∈ W (T) such that

∥T^{−1}(a)∥_{1} > ∥T^{−1}(a)∥_{∞}; (60)

the proof of Theorem 1.2 given here is nothing but the continuous analogue of the proof of (60) presented in [16]. Of course, once (60) is known, one can proceed as in the proof of Theorem 1.2 to show that

∥T^{−1}(a)∥_{p} < ∥T^{−1}(ã)∥_{p}

if only p is sufficiently large. We remark that in [16] one can also find estimates for the convergence speed of ∥T_{n}^{−1}(a)∥_{1} to ∥T^{−1}(a)∥_{1} as n → ∞.

The discrete analogue of Theorem 4.1 amounts to the triviality that the eigenvalues of T_{n}(a) do not depend on the space T_{n}(a) is thought of as acting on.

Given a, b ∈ C_{p}^{m×m}(T), we put

aP + bQ := ( \begin{matrix}… & … & … & … & … & … & … \\ … & b_{0} & b_{−1} & a_{−2} & a_{−3} & a_{−4} & … \\ … & b_{1} & b_{0} & a_{−1} & a_{−2} & a_{−3} & … \\ … & b_{2} & b_{1} & a_{0} & a_{−1} & a_{−2} & … \\ … & b_{3} & b_{2} & a_{1} & a_{0} & a_{−1} & … \\ … & b_{4} & b_{3} & a_{2} & a_{1} & a_{0} & … \\ … & … & … & … & … & … & … \end{matrix} ) (61)

and call aP + bQ a paired matrix (see [14, Chapters V and VI]). Let l_{m}^{p}(Z) (1 ≤ p < ∞) denote the usual l_{m}^{p} spaces on the integers Z and define P_{n} on l_{m}^{p}(Z) by

P_{n} : (x_{k})_{k=−∞}^{∞} ↦ (…, 0, x_{−n}, …, x_{−1}, x_{0}, …, x_{n−1}, 0, …).

If K ∈ K(l_{m}^{p}(Z)), then P_{n}(aP + bQ + K)P_{n} may be identified with a 2nm × 2nm matrix. The hypothesis a, b ∈ C_{p}^{m×m}(T) implies that the north-east and southwest quarters of the matrix (61) represent compact operators. Thus (61) is in fact a compactly perturbed direct sum of two Toeplitz matrices. From Theorems 7.2 and 7.3 we therefore easily deduce that

lim_{n→∞} ∥(P_{n}(aP + bQ + K)P_{n})^{−1}∥_{p}

= max {∥(aP + bQ + K)^{−1}∥_{p}, ∥T^{−1}(ã)∥_{p}, ∥T^{−1}(b̃)∥_{p}}

and, for ε > 0,

lim_{n→∞} σ_{p}^{ε}(P_{n}(aP + bQ + K)P_{n}) = σ_{p}^{ε}(aP + bQ + K) ∪ σ_{p}^{ε}(T (ã)) ∪ σ_{p}^{ε}(T (b̃)).

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 H^{p}(T) (1 < p < ∞) and to Wiener-Hopf integral operators with matrix symbols on the Hardy spaces H^{p}(R) (1 < p < ∞).

8. Pseudospectra of Infinite Toeplitz Matrices

From (6) we know the spectrum of the Wiener-Hopf operator W (a) on L^{p}(0, ∞) (1 ≤ p ≤ ∞) in case a ∈ C + F L^{1}(R). An analogous result is true for Toeplitz operators on l^{p} (1 ≤ p ≤ ∞): if b ∈ W (T), then

σ_{p}(T (b)) = b(T) ∪ {λ ∈ C \ b(T) : wind (b, λ) ≠ 0}. (62)

Theorems 5.2 and 7.3 do not pertain to spectra but to pseudospectra. Thus, it is desirable to know more about the pseudospectra σ_{p}^{ε}(W (a)) and σ_{p}^{ε}(T (b)).

Given a Banach space X and a bounded linear operator A on X, we define the spectrum σ_{X}(A) and the ε-pseudospectrum σ_{X}^{ε}(A) in the natural manner:

σ_{X}(A) = {λ ∈ C : A − λI is not invertible},
σ_{X}^{ε}(A) = {λ ∈ C : ∥(A − λI)^{−1}∥ ≥ 1/ε}.

One always has

σ_{X}(A) + Δ_{ε} ⊂ σ_{X}^{ε}(A) (63)

where Δ_{ε} = {λ ∈ C : |λ| ≤ ε}. Indeed, if λ ∉ σ_{X}^{ε}(A) then ε < ∥(A − λI)^{−1}∥^{−1}, which implies that A − λI − δI is invertible whenever |δ| ≤ ε.

The following two theorems provide additional information about Hilbert space Wiener-Hopf and Toeplitz operators. For a function c in L^{∞} on R or T, we denote by R(c) its essential range and by conv R(c) the convex hull of R(c). Note that σ_{X}^{0}(A) is nothing but σ_{X}(A).

Theorem 8.1. If ε ≥ 0, a ∈ L^{∞}(R), b ∈ L^{∞}(T), then

(64)  σ_{2}(W (a)) + Δ_{ε} ⊂ σ_{2}^{ε}(W (a)) ⊂ conv R(a) + Δ_{ε},
(65)  σ_{2}(T (b)) + Δ_{ε} ⊂ σ_{2}^{ε}(T (b)) ⊂ conv R(b) + Δ_{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 ∉ conv R(a) + Δ_{ε}. Then 0 ∉ conv R(a − λ) + Δ_{ε}, and hence we can rotate conv R(a − λ) + Δ_{ε} into the right open half-plane, i.e., there is a γ ∈ T such that

conv R(γ(a − λ)) + Δ_{ε} ⊂ {z ∈ C : Re z > 0}.

Then we can find a (sufficiently large) disk {z ∈ C : |z − r − ε| < r} containing the set conv R(γ(a − λ)). We so have |γ(a(ξ) − λ) − r − ε| < r and therefore

|\frac{γ(a(ξ) − λ)}{r + ε} − 1| < \frac{r}{r + ε}

for almost all ξ ∈ R. Since

\frac{γ}{r + ε}W (a − λ) = I + W (\frac{γ(a − λ)}{r + ε} − 1)

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

\frac{r + ε}{|γ|}∥W^{−1}(a − λ)∥_{2} < \frac{1}{1 − r/(r + ε)},

whence ∥W^{−1}(a − λ)∥_{2} < |γ|/ε = 1/ε. Thus, λ ∉ σ_{p}^{ε}(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 + F L^{1}(R) and R(a) is the half-circle {z ∈ T : Im z ≥ 0}. From (6) we infer that W (a) is invertible and hence, if ε > 0 is small enough then ∥W^{−1}(a)∥_{2} < 1/ε. Thus, 0 ∉ σ_{2}^{ε}(W (a)) although 0 ∈ conv R(a) + Δ_{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 + F L^{1}(R) and b ∈ W (T) such that

σ_{2}(W (a)) + Δ_{ε} ≠ σ_{2}^{ε}(W (a)), σ_{2}(T (b)) + Δ_{ε} ≠ σ_{2}^{ε}(T (b)).

Proof. Put

b(e^{iθ}) = \{ \begin{matrix}e^{2iθ} & for & 0 ≤ θ < π, \\ e^{−2iθ} & for & π ≤ θ < 2π. \end{matrix}

If e^{iθ} traverses T, then b(e^{iθ}) twice traces out the unit circle, once in the positive and once in the negative direction. Thus, σ_{2}(T (b)) = T due to (62). We claim that

σ_{2}^{3/4}(T (b)) = {λ ∈ C : |λ| ≤ 7/4}, (66)

which is clearly properly larger than T + Δ_{3/4}. From Theorem 8.1 we deduce that

{λ ∈ C : 1/4 ≤ |λ| ≤ 7/4} ⊂ σ_{2}^{3/4}(T (b)) ⊂ {λ ∈ C : |λ| ≤ 7/4}.

Consequently, (66) will follow as soon as we have shown that

(67) ∥T^{−1}(b − λ)∥_{2} ≥ 4/3 whenever |λ| < 1/4.

So assume |λ| < 1/4. The Fourier coefficients of b can be easily computed:

b_{0} = 0, b_{2} = b_{−2} = 1/2,
b_{n} = 0 if n ≠ ±2 is even,
b_{n} = 4/(πi(n^{2} − 4)) if n is odd.

It follows in particular that b ∈ W (T). Writing the elements of l^{2} in the form (x_{0}, x_{1}, x_{2}, …) and taking into account (54) we get

∥T^{−1}(b − λ)∥_{2} = ( inf_{φ≠0} \frac{∥T (b − λ)φ∥_{2}}{∥φ∥_{2}} )^{−1}

≥ ∥T (b − λ)(1, 0, 0, …)∥_{2}^{−1}

= ∥(b_{0} − λ, b_{1}, b_{2}, b_{3}, …)∥_{2}^{−1}

= (|b_{0} − λ|^{2} + ∑_{n≥1} |b_{n}|^{2})^{−1/2}.

Since b_{0} = 0 and |b_{n}| = |b_{−n}| for all n, we obtain

|b_{0} − λ|^{2} + ∑_{n≥1} |b_{n}|^{2} = |λ|^{2}+\frac{1}{2} ∑_{n∈Z} |b_{n}|^{2}

= |λ|^{2} + \frac{1}{2} \frac{1}{2π} ∫_{0}^{2π}|b(e^{iθ})|^{2}dθ (Parseval’s equality)

= |λ|^{2} + 1/2 (note that b is unimodular)

< (1/4)^{2} + 1/2 = 9/16.

Thus, ∥T^{−1}(b − λ)∥_{2} > (9/16)^{−1/2} = 4/3, which gives (67) and completes the proof of (66).

Replacing b by b_{ε} := (4ε/3)b we easily conclude from (66) that

σ_{2}^{ε}(T (b_{ε})) = {λ ∈ C : |λ| ≤ 7ε/3},

which is strictly larger than

σ_{2}(T (b_{ε})) + Δ_{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 basis in L^{2}(0, ∞) is given by \{e_{n}\}_{n=0}^{∞} where e_{n}(x) = \sqrt{2}e^{−x}Π_{n}(2x) and Π_{n} is the normalized nth Laguerre polynomial. The map

U : l^{2} → L^{2}(0, ∞), (φ_{0}, φ_{1}, φ_{2}, …) ↦ ∑_{n=0}^{∞} φ_{n}e_{n}

is an isometric isomorphism, and if b ∈ W (T) has the Fourier coefficients {b_{n}}_{n∈Z} then UT (b)U^{−1} is the Wiener-Hopf operator W (a) with the symbol

a(ξ) = ∑_{n∈Z} b_{n}(\frac{ξ − i}{ξ + i})^{n} (ξ ∈ R);

see, e.g., [14, Chap. III, Sec. 3]. Obviously, a ∈ C+F L^{1}(R), σ_{2}(W (a)) = σ_{2}(T (b)), and since

∥W^{−1}(a − λ)∥_{2} = ∥UT^{−1}(b − λ)U^{−1}∥_{2} = ∥T^{−1}(b − λ)∥_{2},

it results that σ_{2}^{ε}(W (a)) = σ_{2}^{ε}(T (b)) for each ε > 0. This reduces the proof in the Wiener-Hopf case to the proof in the Toeplitz case. □

30 A. Böttcher, S. M. Grudsky, and B. Silbermann

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Faculty of Mathematics, TU Chemnitz-Zwickau, 09107 Chemnitz, Germany aboettch@mathematik.tu-chemnitz.de

Faculty of Mechanics and Mathematics, Rostov-on-Don State University, Bolshaya Sadovaya 105, 344 711 Rostov-on-Don, Russia

grudsk@ns.unird.ac.ru

Faculty of Mathematics, TU Chemnitz-Zwickau, 09107 Chemnitz, Germany silbermn@mathematik.tu-chemnitz.de