It is easy to check that for R = 3 this is just another formulation of formula (1.3). 1.11–1.12. These things as well as the monographs and textbooks which pay due attention to the theory of Fredholm operators are well known. In German and Russian Fredholm operators are (more correctly) called Noether operators, since Friedrich Noether [370] was the ﬁrst to discover that a singular integral operator with nonvanishing continuous symbol is normally solvable and has ﬁnite kernel and cokernel dimensions; he also computed the index of such operators. The term Fredholm operator is in German and Russian used for operators with index zero. 1.13–1.15. Results of the type of Theorem 1.14 were ﬁrst obtained by N. Ya. Krupnik (see Kesler and Krupnik [311]). Theorem 1.14 itself was ﬁrst established by K¨ ohler and Silbermann [314], [315] and independently also by Markus and Feldman [347]. Theorem 1.15 is due to Markus and Feldman [346]. The proofs of these two theorems are in the original papers and also in Chapter 1 of Krupnik’s book [329]. (We also recommend to have a glance at Math. Reviews 87a:15006). 1.16–1.29. This material is well known and a major part of it is in almost each ˙ [587], [588]. book on Banach or C ∗ algebras. The result 1.20(c) is Zelazko’s Theorem 1.22 was established by Glicksberg [216] and is a quantitative version of Corollary 1.23, which had already been known from earlier work of Shilov and Bishop. The proof given in the text is Ransford’s [415]. Later we shall also need vectorvalued analogs of Glicksberg’s theorem (Theorem 3.6). For this topic see Burckel [127], Machado [342], Szyma´ nski [512], Ransford [415]. For 1.28 we refer to Glicksberg [216] and Gamelin [209]. All other results are explicitly proved, e.g., in Gelfand, Raikov, Shilov [212], Naimark [362], ˙ Douglas [162], Zelazko [588], Dixmier [156]. 1.30–1.35. The simplest local principle is the Gelfand theory of commutative Banach algebras with identity: if A is such an algebra and a ∈ A, then a ∈ GA ⇐⇒ a ∈ /N
∀ N ∈ M (A) ⇐⇒ a(N ) = 0 ∀ N ∈ M (A).
Thus, in a sense, invertibility in commutative Banach algebras is local in nature. Theorems 1.32 and 1.35 are what we need from the arsenal of “local techniques” for noncommutative algebras. We want not to go into the history of this topic, but only mention that Simonenko [493], [494], [495] was indisputably the ﬁrst who both realized the local nature of Fredholmness of convolution and related operators and at the same time created a powerful machinery (his local principle) for tackling successfully a whole series of problems. This local principle is presented in
1.11 Notes and Comments
43
Simonenko, Chin Ngok Min [497]. Simonenko’s local principle was generalized by Kozak [318], [319], [320] to arbitrary Banach algebras. A reader who wishes to become acquainted with Simonenko’s principle is therefore recommended to consult also Kozak’s works [318], [319], [320]. Also see Chapter XV of Mikhlin, Pr¨ ossdorf [361] and B¨ ottcher, Krupnik, Silbermann [98]. Lemma 1.31 and Theorem 1.32 are due to Gohberg and Krupnik [232] (except for 1.32(c), which is due to the authors). This local principle is a modiﬁcation of Simonenko’s principle, and it is distinguished for its simplicity on the one hand and for its wide area of applications on the other hand. An application of 1.32(c) is in the proof of Theorem 5.57. It is a delicate problem to say by whom and where Lemma 1.34 and Theorem 1.35 were established for the ﬁrst time. As far as we know, the earliest references where these two results appeared explicitly are Allan [5], [6]. Independently, Douglas [162] stated Theorem 1.35 for the case of C ∗ algebras and he was the ﬁrst to realize its importance for the investigation of Toeplitz operators. The example in the remark in 1.35 is taken from Krupnik [329]. The comparison of several known local principles (theories) should be the subject of further investigations. Steps in this direction have been made by Clancey, Gosselin [139], Clancey, Gohberg [137], and B¨ ottcher, Krupnik, Silbermann [98]; also see B¨ottcher, Roch, Silbermann [100]. An extension of the AllanDouglas local principle which can be applied to the study of C ∗ algebras of operators of “nonlocal type” was developed by Karlovich [304] (full proofs of the results of that note are contained in Karlovich [305]). Concrete applications of this socalled localtrajectory method to C ∗ algebras of convolution type operators with shifts and/or oscillating symbols are in B¨ ottcher, Karlovich, Silbermann [95], B¨ ottcher, Karlovich, Spitkovsky [97], Bastos, Fernandes, Karlovich [46]. Another extension of the local principle by Allan and Douglas, again for the investigation of operators of “nonlocal type,” but now for operators acting on Banach spaces, is in Karlovich, Silbermann [306]. The latter local method is illustrated by an application in Karlovich, Silbermann [307]. 1.36–1.46. See, e.g., Hoﬀman [284], Duren [178], Koosis [316], Garnett [211], Rudin [443], [444], Douglas [162], Sarason [457], Nikolski [368], [369], Rosenblum and Rovnyak [442]. 1.47–1.48. Excellent presentations of the BM O and V M O theory are in Sarason [457], Garnett [211], and Koosis [316]. See also the nicely written original works Sarason [454] and Stegenga [507]. 1.49. The discrete HuntMuckenhouptWheeden condition is in these authors’ work [289, Theorem 10]. 1.50. These facts are taken from Peller and Khrushchev [390] and Peller [389, Appendix 2.3]. For periodic Besov spaces see Schmeisser and Triebel [462]. There one can also explicitly ﬁnd 1.50(a) and (b) in Remark 3 and the Corollary of 3.5.5 and in Theorem 1 of 3.5.1, respectively.
2 Basic Theory
2.1 Multiplication Operators 2.1. Deﬁnition. If a ∈ L∞ and 1 < p < ∞, then the operator M (a) : Lp → Lp ,
f → af
(2.1)
is obviously bounded and M (a)L(Lp ) ≤ a∞ . It is called the multiplication p q operator on Lp generated by * the function a. For f ∈ L and g ∈ L (1/p + 1 f g dm. It is clear that (M (a)χ , χ ) is equal to 1/q = 1), write (f, g) := 2π j k T the (k − j)th Fourier coeﬃcient of a. The following proposition shows that every bounded operator with such a property is a multiplication operator. 2.2. Proposition. Let A ∈ L(Lp ) (1 < p < ∞) and suppose there is a sequence {an }n∈Z of complex numbers such that (Aχj , χk ) = ak−j . Then there is an a ∈ L∞ such that A = M (a) and {an } is the Fourier coeﬃcient sequence of a. Moreover, M (a)L(Lp ) = a∞ . Proof. Put a = Aχ0 . Then a ∈ Lp and the nth Fourier coeﬃcient of a is (a, χn ) = (Aχ0 , χn ) = an . If f ∈ L∞ , then both Af and af are in Lp . We claim that (2.2) Af = af ∀ f ∈ L∞ . coeﬃcient sequence of f . Then the jth Let {fn }n∈Z denote the Fourier
a Fourier coeﬃcient of af is k∈Z j−k fk . On the other hand, since the se
f χ converges to f in the Lp norm, we deduce that the series ries k∈Z k k
j ) for each j ∈ Z. This shows that the k∈Z fk (Aχk , χj ) converges to (Af, χ
jth Fourier coeﬃcient of Af equals k∈Z aj−k fk , too. Thus, Af = af . We now prove that a ∈ L∞ . Let E be a measurable subset of T with positive measure on which a > A and let χE denote the characteristic function of E. Then, by (2.2), ( AχE p = aχE p = ap dm = Ap χE p . E
46
2 Basic Theory
But this is impossible and so a ≤ A a.e. on T. Hence a ∈ L∞ and a∞ ≤ A. Thus, since in view of (2.2) the operators A and M (a) coincide on a dense subset of Lp and both operators are bounded, it follows that A = M (a). The norm equalities are now obvious. 2.3. Deﬁnitions. Let a ∈ L1 have Fourier coeﬃcients sequence {an }n∈Z . Given ϕ = {ϕj }j∈Z ∈ 0 (Z) deﬁne the sequence a ∗ ϕ by (a ∗ ϕ)j := aj−k ϕk (j ∈ Z). k∈Z
For 1 ≤ p < ∞, let M p denote the collection of all a ∈ L1 for which a ∗ ϕ belongs to p (Z) whenever ϕ ∈ 0 (Z) and sup
a ∗ ϕ
p
ϕp
: ϕ ∈ 0 (Z), ϕ = 0 < ∞.
If a ∈ M p then the operator 0 (Z) → p (Z), ϕ → a ∗ ϕ extends to a bounded operator a ∗ ϕ, (2.3) M (a) : p (Z) → p (Z), ϕ → which is referred to as the multiplication operator on p (Z) generated by the function a. The M (a) we have just deﬁned on p (Z) is sometimes also called the p q Laurent operator
generated by a. For ϕ ∈ (Z) and ψ ∈ (Z) (1/p+1/q = 1) put (ϕ, ψ) := n∈Z ϕn ψn . It is easy to verify that (M (a)ej , ek ) = ak−j for every a ∈ M p . Here is a converse of this. 2.4. Proposition. Let A ∈ L(p (Z)) (1 ≤ p < ∞) and suppose (Aej , ek ) = ak−j for some sequence {an }n∈Z of complex numbers. Then there is an a ∈ M p such that A = M (a) and {an } is the Fourier coeﬃcient sequence of a. Proof. First let 1 ≤ p ≤ 2. Put α := Ae0 ∈ p (Z). Then α = {an }n∈Z . Since p (Z) ⊂ 2 (Z), there is an a ∈ L2 whose Fourier coeﬃcients sequence is {an }. Clearly, Aϕ = a ∗ ϕ for all ϕ ∈ 0 (Z). Therefore, A = sup = sup
Aϕ
p
ϕp a ∗ ϕ
: ϕ ∈ 0 (Z), ϕ = 0
p
ϕp
: ϕ ∈ 0 (Z), ϕ = 0
and it follows that a ∈ M p and A = M (a). Now let p ≥ 2. The adjoint A∗ ∈ L(q (Z)) (1/p + 1/q = 1) of A satisﬁes (A∗ ej , ek ) = (ej , Aek ) = aj−k . By what has been proved above, we have A∗ = M (b) for some b ∈ M q whose nth Fourier coeﬃcient bn equals a−n . Deﬁne a ∈ L1 by a(t) = b(t) (t ∈ T). Then the nth Fourier coeﬃcient of a is an and therefore (a ∗ ϕ, ψ) = (ϕ, b ∗ ψ) for all ϕ, ψ ∈ 0 (Z). It follows that a ∗ ϕ ∈ (q (Z))∗ = p (Z) for all ϕ ∈ 0 (Z) and that
2.1 Multiplication Operators
47
sup a ∗ ϕp : ϕ ∈ 0 (Z), ϕp ≤ 1 = sup (a ∗ ϕ, ψ) : ϕ, ψ ∈ 0 (Z), ϕp ≤ 1, ψq ≤ 1 = M (b)L(q (Z)) < ∞, whence a ∈ M p , which completes the proof.
2.5. Basic properties of M p . Incidentally, in the preceding proof we established that M p ⊂ L2 . Thus, if a and b are in M p , then ab ∈ L1 and now it is easily seen that actually ab ∈ M p . The conclusion is that M p is an algebra (under pointwise operations). For p = 2, the multiplication operators deﬁned on L2 and 2 (Z) by (2.1) and (2.3), respectively, are unitarily equivalent through the isomorphism L2 → 2 (Z), ϕn χn → {ϕn }n∈Z . n∈Z
This shows that M 2 = L∞ . It is also easy to see that M 1 = W := a ∈ L1 : aW := an  < ∞ . n∈Z
Indeed, if a ∈ M 1 , then {an } = M (a)e0 ∈ 1 (Z), whence M 1 ⊂ W , and that 1 W is contained in M
follows from the fact that, for a ∈ W , M (a) can be written as M (a) = n∈Z an M (χn ) with an  M (χn )L(1 (Z)) = an  < ∞. n∈Z
n∈Z
Moreover, the preceding argument also shows that M (a)L(1 (Z)) = aW . Here are some more properties of the algebras M p . In what follows let 1 < p < ∞ and 1/p + 1/q = 1. Put [p, q] := [min{p, q}, max{p, q}], let · p denote the norm in L(p (Z)) and for a ∈ L1 deﬁne a ∈ L1 by a(t) = a(t) (t ∈ T). (a) If a ∈ M p , then a ∈ M q and the adjoint M ∗ (a) ∈ L(q (Z)) equals M (a). Proof. If a ∈ M p , then M ∗ (a) ∈ L(q (Z)), and from the equality (ei , M (a)ej ) = (M (a)ei , ej ) we deduce that M ∗ (a) = M (a). (b) M p = M q . If a ∈ M p , then a ∈ M p and M (a)p = M (a)p = M (a)q = M (a)q .
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2 Basic Theory
Proof. It is easy to see that M (a) = J M (a)J , where J is the isometry on p (Z) given by J : {ϕi }i∈Z → {ϕ−i }i∈Z . Thus, if a ∈ M p then a ∈ M p . From (a) we then deduce that a ∈ M q . The norm equalities follow from (a) and the fact that J is an isometry. (c) If a ∈ M p , then a ∈ M r for all r ∈ [p, q]. Proof. This results from (b) and the RieszThorin interpolation theorem. (d) If 1 ≤ p ≤ r ≤ 2, then W = M 1 ⊂ M p ⊂ M r ⊂ M 2 = L∞ and, moreover, a∞ ≤ M (a)r ≤ M (a)p ≤ aW .
Proof. The representation M (a) = an M (χn ) shows that M 1 ⊂ M r and that M (a)r ≤ M (a)1 . If p ≤ r ≤ 2, then, by the RieszThorin interpolation theorem, M (a)r ≤ M (a)tp M (a)1−t q , where 1/r = t/p + (1 − t)/q and 0 ≤ t ≤ 1, and in view of (b) the righthand side equals M (a)tp M (a)1−t = M (a)p . p (e) If a ∈ M p and r ∈ [p, q], then γ M (a)r ≤ a1−γ ∞ M (a)p
with
γ=
pr − 2 . rp − 2
Proof. Again apply the RieszThorin interpolation theorem. (f) If a ∈ L∞ is of bounded variation, then a ∈ M p and M (a)p ≤ sp a∞ + V1 (a) , where sp is a constant independent of a and V1 (a) denotes the total variation of a. Proof. By virtue of (b) there is no loss of generality in assuming that 2 < p < ∞. For α ∈ (0, 2π) let χα (eiθ ) be 1 on (α, 2π) and 0 on (0, α). The nth Fourier coeﬃcient of χα is (i/(2πn))(1 − e−inα ) for n = 0. Hence, if ϕ ∈ 0 (Z) then (M (χα )ϕ)j − (χα )0 ϕj equals i −ijα eikα ϕk i ϕk − e . 2π j − k 2π j−k k =j
k =j
From the discrete HuntMuckenhouptWheeden theorem in 1.49 we deduce that χα ∈ M p and that there is a constant sp such that M (χα )p ≤ sp for all α ∈ (0, 2π). Here and in the following sp denotes a constant depending only on p but not necessarily the same at each occurrence. If a is a nondecreasing simple function with a(eiθ ) = αk for θ ∈ (θk−1 , θk ) (0 = θ0 < θ1 < . . . < θn = 2π), then a can be written in the form
2.2 Toeplitz Operators
a=
49
n−1
(αk+1 − αk )χθk + α1 ,
k=1
and from what has just been proved it follows that a ∈ M p and M (a)p ≤ sp V1 (a) + a∞ .
(2.4)
To obtain the assertion for general a it suﬃces to assume that a is a realvalued monotonically nondecreasing function. Then there exists a sequence {a(m) }m∈Z+ of nondecreasing simple functions converging to a uniformly on [0, 2π) as m → ∞. Since p > 2, we have " " " M (a) − M a(m) ϕ" ≤ M (a − a(m) )ϕ2 ≤ a − a(m) ∞ ϕ2 = o(1) p as m → ∞ for every ϕ ∈ 0 (Z). Because of (2.4), M (a(m) )p ≤ sp V1 (a(m) ) + a(m) ∞ ≤ sp V1 (a) + a∞ . Thus, M (a) ∈ L(p (Z)) and (2.4) holds for general a. (g) M p is a Banach algebra with respect to the norm ap := M (a)L(p (Z)) . Proof. It remains to show that M p is complete. Let {a(m) }m∈Z+ be a Cauchy sequence in M p . By virtue of (d), {a(m) }m∈Z+ is a Cauchy sequence in L∞ and, consequently, there is an a ∈ L∞ such that a(m) −a∞ → 0 as m → ∞. Since L(p (Z)) is complete, there is an A ∈ L(p (Z)) such that M (a(m) ) − Ap → 0 as m → ∞. Because M (a)ϕ = Aϕ for ϕ ∈ 0 (Z), we ﬁnally conclude that A = M (a).
2.2 Toeplitz Operators 2.6. Deﬁnition. The operator T (a) deﬁned for a ∈ L∞ and 1 < p < ∞ by T (a) : H p → H p ,
f → P (af )
(2.5)
is obviously bounded and T (a)L(H p ) ≤ cp a∞ , where cp is the norm of the Riesz projection on Lp . This operator is called the Toeplitz operator on H p generated by the function a. If a ∈ M p , then the operator T (a) given on p = p (Z+ ) (1 ≤ p < ∞) as T (a) : p → p ,
ϕ → P (a ∗ ϕ)
(2.6)
is clearly bounded and T (a)L(p ) ≤ M (a)L(p (Z)) . Here P denotes the discrete Riesz projection. The operator deﬁned by (2.6) is called the Toeplitz operator on p generated by the function a.
50
2 Basic Theory
The function a generating the Toeplitz operators (2.5) and (2.6) is usually referred to as the symbol of the corresponding operator. Toeplitz operators on p are sometimes also called discrete WienerHopf operators. For p = 2, the Toeplitz operators deﬁned on H 2 and 2 by (2.5) and (2.6), respectively, are unitarily equivalent through the isomorphism H 2 → 2 , ϕn χn → {ϕn }n∈Z+ . (2.7) n∈Z+
Therefore we shall frequently identify these operators without mentioning this explicitly. For f ∈ H p , g ∈ H q , ϕ ∈ p , ψ ∈ q (1/p + 1/q = 1) let ( 1 (f, g) := f g dm, (ϕ, ψ) := ϕn ψn . 2π T n∈Z+
It is clear that the operators (2.5) and (2.6) satisfy (T (a)χj , χk ) = ak−j ,
(T (a)ej , ek ) = ak−j
∀ j, k ∈ Z+ .
The following theorem states that every bounded operator on H p resp. p with this property is a Toeplitz operator and, moreover, relates the norm of a Toeplitz operator with the norm of the multiplication operator generated by the same function. 2.7. Theorem (Brown/Halmos). (a) Let A ∈ L(H p ) (1 < p < ∞) and suppose there is a sequence {an }n∈Z of complex numbers such that (Aχj , χk ) = ak−j for all k, j ∈ Z+ . Then there exists an a ∈ L∞ such that A = T (a) and {an } is the Fourier coeﬃcient sequence of a. Moreover, a∞ ≤ T (a)L(H p ) ≤ cp a∞ ,
(2.8)
where cp is the norm of P on Lp . (b) Let a ∈ L(p ) (1 ≤ p < ∞) and suppose there is a sequence {an }n∈Z of complex numbers such that (Aej , ek ) = ak−j for all k, j ∈ Z+ . Then there exists an a ∈ M p such that A = T (a) and {an } is the Fourier coeﬃcient sequence of a. Moreover, T (a)L(p (Z+ )) = M (a)L(p (Z)) .
(2.9)
Proof. (a) For n ≥ 0, deﬁne bn ∈ Lp as bn := χ−n (Aχn ). Then bn p ≤ A. Since Lp = (Lq )∗ , the BanachAlaoglu theorem implies that there is a b ∈ Lp such that bp ≤ A and some subsequence {bnk } of {bn } converges to b in the weak topology on Lp . In particular, (bnk , χj ) → (b, χj ) for all j ∈ Z, and because (bnk , χj ) = (Aχnk , χnk +j ) = aj whenever nk + j ≥ 0, it follows that (b, χj ) = aj
∀ j ∈ Z.
(2.10)
2.2 Toeplitz Operators
51
Now deﬁne the mapping B by B : P → Lp ,
f → bf.
(2.11)
If f, g ∈ P, then, by virtue of (2.10), (Bf, g) is equal to (M (χ−n )AM (χn )f, g) whenever n is chosen large enough. Hence (Bf, g) ≤ lim sup (M (χ−n )AM (χn )f, g) ≤ A f p gq n→∞
and thus Bf p = sup (Bf, g) : g ∈ P, gq ≤ 1 ≤ A f p for all f ∈ P. This shows that the linear mapping (2.11) extends to an operator B ∈ L(Lp ) with B ≤ A. Again from (2.10) we deduce that (Bχj , χk ) = (b, χk−j ) = ak−j for all j, k ∈ Z. Now Proposition 2.2 gives the existence of an a ∈ L∞ such that B = M (a) and {an } is the Fourier coeﬃcient sequence of a. Since both (T (a)χj , χk ) and (Aχj , χk ) equal ak−j , it follows that A = T (a). Finally, because M (a) = B ≤ A = T (a), the norm equality in Proposition 2.2 gives the ﬁrst “≤” in (2.8). The second “≤” in (2.8) is trivial. (b) Since Aen ∈ p for all n ∈ Z+ , it is obvious that the sequence {an }n∈Z belong to p (Z). After deﬁning B as ! # aj−k ϕk B : 0 (Z) → p (Z), {ϕj }j∈Z → j∈Z
k∈Z
the rest of the proof is completely analogous to (a).
2.8. Corollary. (a) If a ∈ L∞ , then T (a)L(H 2 ) = T (a)L(2 ) = a∞ . (b) If a ∈ M p , then T (a)L(p ) = aM p . (c) If 1 < p < ∞, 1/p + 1/q = 1, and if a ∈ M p , then T (a) ∈ L(q ) and T (a)L(q ) = T (a)L(p ) . The adjoint T ∗ (a) ∈ L(q ) is equal to T (a). (d) If 1 < p < ∞, 1/p + 1/q = 1, and if a ∈ M p , then T (a) ∈ L(r ) for all r ∈ [p, q] and γ a∞ ≤ T (a)L(r ) ≤ a1−γ ∞ aM p ≤ aW ,
where γ = pr − 2/(rp − 2). Proof. Combine 2.4, 2.5, and 2.7.
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2 Basic Theory
2.9. Shift operators. Recall that χn is given by χn (t) = tn (t ∈ T). The operators M (χ1 ) and T (χ1 ) are usually referred to as the bilateral and unilateral shift, respectively, because they act on p (Z) resp. p (Z+ ) (1 ≤ p < ∞) by the rules M (χ1 ) : {ϕi }i∈Z → {ϕi−1 }i∈Z , T (χ1 ) : {ϕ0 , ϕ1 , . . .} → {0, ϕ0 , ϕ1 , . . .}. Also as usual, we put U := M (χ1 ) and V := T (χ1 ), and, for n ∈ Z+ , we let U n = M (χn ),
U −n = M (χ−n ),
V n = T (χn ),
V (−n) = T (χ−n ).
Then obviously U ±n = (U ±1 )n , V n = (V 1 )n , V (−n) = (V (−1) )n , and U −n U n = U n U −n = I,
U ∗ = U −1 ,
V (−n) V n = I,
V ∗ = V (−1) .
Note that V n V (−n) is not the identity operator. It is clear that U ±n are isometries on Lp and p (Z) and that the operators V n are isometries on H p (1 < p < ∞) and p (Z+ ) (1 ≤ p < ∞). In particular, the range of V n is always closed. It is easy to see that V (−n) is onto, dim Ker V (−n) = n, dim Coker V n = n, V n is onetoone. Thus, Ind T (χk ) = −k for all k ∈ Z.
2.3 Hankel Operators 2.10. Deﬁnitions. The deﬁnition of Hankel operators is slightly complicated by the circumstance that there is neither a deﬁnition in general use nor a unique notation for them in the literature and that there is in fact no compelling reason for adopting such a unique deﬁnition or notation. Besides the projections P and Q = I − P , we now need a third operator J, the socalled ﬂip operator. This is the (obviously isometric) operator acting on Lp (1 < p < ∞) by the formula 1 1 (Jf )(t) = f fn t−n−1 (t ∈ T) = t t n∈Z
and, accordingly, acting on p (Z) (1 ≤ p < ∞) by the rule (Jϕ)n = ϕ−n−1
(n ∈ Z).
For a ∈ L∞ , we deﬁne the Hankel operator H(a) on H p (1 < p < ∞) by H(a) : H p → H p ,
f → P M (a)QJf.
We let H() a) denote the operator given by
2.3 Hankel Operators
H() a) : H p → H p ,
53
f → JQM (a)P f
and refer to H() a) also as a Hankel operator (see 2.15 below). It is clear that both H(a) and H() a) are bounded whenever a ∈ L∞ and 1 < p < ∞. p Given a ∈ M (1 ≤ p < ∞) with Fourier coeﬃcients sequence {an }n∈Z we analogously deﬁne H(a) : p → p , H() a) : p → p ,
ϕ → P M (a)QJϕ, ϕ → JQM (a)P ϕ
and call H(a) and H() a) the Hankel operators on p generated by the function a. It is easily seen that H(a) and H() a) are bounded on p . Moreover, it is p readily veriﬁed that their action on can be given by the following formulas: ! # H(a) : p → p , {ϕj }j∈Z+ → aj+k+1 ϕk , j∈Z+
k∈Z+
H() a) : p → p ,
{ϕj }j∈Z+ →
!
# a−j−k−1 ϕk
k∈Z+
. j∈Z+
If p = 2, the Hankel operators deﬁned on H 2 and 2 are again unitarily equivalent through the isomorphism (2.7). The operators resulting from Hankel operators by omitting the ﬂip operator, that is, the four operators ◦
p → H p, P M (a)Q : H− ◦
p QM (a)P : H p → H− ,
P M (a)Q : Qp (Z) → P p (Z), QM (a)P : P p (Z) → Qp (Z)
will occasionally also be referred to as Hankel operators. If a ∈ L∞ , then (H(a)χj , χk ) = aj+k+1 for all j, k ∈ Z+ . The following theorem describes the bounded operators on H p with this property and provides an important norm estimate for Hankel operators on H p . 2.11. Theorem (Nehari). Let A ∈ L(H p ) (1 < p < ∞) and suppose there is a sequence {an }n∈N of complex numbers such that (Aχj , χk ) = aj+k+1 for all j, k ≥ 0. Then there is a function b ∈ L∞ such that A = H(b) and the nth Fourier coeﬃcient bn of b is equal to an for all n ≥ 1. Moreover, distL∞ (b, H ∞ ) ≤ H(b)L(H p ) ≤ cp distL∞ (b, H ∞ ), where cp = P L(Lp ) . In particular, for p = 2 we have H(b)L(H 2 ) = distL∞ (b, H ∞ ).
(2.12)
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2 Basic Theory
Proof. Note that AL(H p ) equals sup{G(Af ) : f H p ≤ 1, G(H p )∗ ≤ 1}. So we deduce from (1.15) that 1 AL(H p ) ≤ Φ(A) ≤ AL(H p ) , cp
(2.13)
where Φ(A) := sup{(Af, g) : f H p ≤ 1, gH q ≤ 1} and 1/p + 1/q = 1. The hypothesis implies that (Af, g) = (Aχ0 , f c g)
∀ f, g ∈ PA ,
where f c (t) = f (1/t) (t ∈ T). It is obvious that f c H p = f H p . Thus, Φ(A) = sup (Aχ0 , f g) : f H p ≤ 1, gH q ≤ 1 . We claim that f g : f H p ≤ 1, gH q ≤ 1 = h ∈ H 1 : hH 1 ≤ 1 .
(2.14)
H¨ older’s inequality immediately gives the inclusion “⊂”. To get the reverse inclusion note ﬁrst that, by 1.41(a), every h ∈ H 1 factors as h = ϕhe , where 1/p ϕ ∈ H ∞ is inner and he ∈ H 1 is outer. From 1.41(e) we see that he ∈ H p 1/q 1/p 1/q and he ∈ H q . Thus, if we let f = ϕhe and g = he , then h = f g and f H p = gH q = hH 1 . This completes the proof of (2.14). Taking into account (2.14) we obtain Φ(A) = sup{(Aχ0 , h) : hH 1 ≤ 1}. Hence, the mapping ( 1 1 (Aχ0 )h dm C : H → C, h → 2π T ∗ is a linear functional belonging to H 1 and we have C 1 ∗ = Φ(A). Again from 1.42 we conclude that there is a c ∈ L∞ such that ( ( (Aχ0 )h dm = ch dm ∀ h ∈ H 1 . T
H
T
Letting h = χn (n ≥ 0), we get (Aχ0 , χn ) = an+1 = cn , and thus the function b = χ1 c ∈ L∞ has the desired property: bn = an for n ≥ 1. We are left with the norm estimate (2.12). Any extension of C to a bounded ) on L1 is given by linear functional C ( ) : L1 → C, f → C ϕf dm T
with some ϕ ∈ L∞ and we have C
H1
∗ ≤ C ) (L1 )∗ = ϕ∞ . Due to
)0 ∈ (L1 )∗ of C such that the HahnBanach theorem there is an extension C )0 . Thus, C = C
2.3 Hankel Operators
Φ(A) = C
H1
55
∗ = inf C ) (L1 )∗ = inf ϕ∞
) is an extension of C (where, in fact, the inf can be replaced by min). But C if and only if ( (ϕ − Aχ0 )h dm = 0 ∀ h ∈ H1 T
⇐⇒
(ϕ − Aχ0 )n = 0
⇐⇒ ⇐⇒
(χ1 ϕ)n − an = 0 χ1 ϕ − b ∈ H ∞ ,
∀n≥0 ∀n≥1
therefore, Φ(A) = inf ϕ∞ : χ1 ϕ − b ∈ H ∞ = inf χ1 ϕ∞ : χ1 ϕ − b ∈ H ∞ = distL∞ (b, H ∞ ) and now (2.13) gives (2.12). Remark. Let the hypothesis of the preceding theorem be satisﬁed. Then Aχ0 ∈ H p and so there exists a function a ∈ H p whose nth Fourier coefﬁcient is an (n ∈ N). One can now formulate the following criterion for the boundedness of A: A ∈ L(H p ) if and only if a ∈ BM O. Indeed, if A ∈ L(H p ), then, by the above theorem, a = P b for some b ∈ L∞ and thus a ∈ BM O by 1.48(k); on the other hand, if a ∈ BM O, then, by virtue of 1.48(l), there are u, v ∈ L∞ such that a = u + P v, whence u = P u, so a = P b with b = u + v ∈ L∞ , and the above theorem implies the boundedness of A. 2.12. Open problem. Establish a Nehari theorem for p . The conjecture is that for this case in the preceding theorem “b ∈ L∞ ” must be replaced by “b ∈ M p ” and the norm H(b)L(p ) is equivalent (or, maybe, even equal) to distM p (b, H ∞ ∩ M p ). The following two sections are intended to give a ﬁrst idea of the connection between multiplication, Toeplitz, and Hankel operators. Moreover, the formulas stated in 2.14, thought being very simple, will be of extreme importance for all what follows. 2.13. Decomposition of the multiplication operator. L2 decomposes ◦
into the orthogonal sum of H 2− and H 2 . Accordingly, every operator in L(L2 ) can be represented as a 2 × 2 operator matrix, ◦ ◦ 2 2 AB H − → H− . : 2 CD H H2 This applies, in particular, to the multiplication operator M (a) ∈ L(L2 ). In the corresponding matrix representation we meet operators closely related to Toeplitz and Hankel operators:
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2 Basic Theory
M (a) =
QM (a)Q P M (a)Q
⎛ ◦ ⎞ ⎛ ◦ ⎞ 2 H H2 : ⎝ −⎠ → ⎝ −⎠. P M (a)P H2 H2
QM (a)P
(2.15)
Clearly, the P M (a)P in the right lower corner is nothing else than the Toeplitz operator T (a). The operators QM (a)P and P M (a)Q diﬀer from H() a) and H(a), respectively, only by the ﬂip operator. Finally, if we deﬁne T () a) on H 2 as T () a) = JQM (a)QJ, then the QM (a)Q in the left upper corner is equal to JT () a)J. In terms of Fourier coeﬃcients we have ! # aj−i ϕj . T () a) : 2 → 2 , {ϕi }i∈Z+ → i∈Z+
j∈Z+
Thus, the operator matrix in (2.15) can be written as ⎛ ◦ ⎞ ⎛ ◦ ⎞ 2 JT () a)J JH() a) H H2 : ⎝ −⎠ → ⎝ −⎠. M (a) = H(a)J T (a) H2 H2 In more detail, if we express things via Fourier coeﬃcients, the 6 × 6 matrix in the center of M (a) equals ⎛ ⎞ a0 a−1 a−2 a−3 a−4 a−5 ⎜ a1 a0 a−1 a−2 a−3 a−4 ⎟ ⎜ ⎟ ⎜ a2 a1 a0 a−1 a−2 a−3 ⎟ ⎜ ⎟ ⎜a a a a a a ⎟. 1 0 −1 −2 ⎟ ⎜ 3 2 ⎝ a4 a3 a2 a1 a0 a−1 ⎠ a5 a4 a3 a2 a1 a0 ◦
·
·
p Since Lp =H− + H p (1 < p < ∞) and p (Z) = Qp (Z) + P p (Z) (1 ≤ p < ∞), all what has been said above applies to the case p = 2 as well. In particular, a) deﬁned by if a ∈ L∞ or a ∈ M p , then the operators T ()
T () a) : H p → H p ,
f → JQM (a)QJf
and T () a) : → , p
p
{ϕi }i∈Z+ →
! j∈Z+
# aj−i ϕj
(2.16) i∈Z+
will be bounded on H p and p , respectively. Finally, notice the obvious identity M (a) = P M (a)P + P M (a)Q + QM (a)P + QM (a)Q.
(2.17)
This is merely a translation of the representation (2.15) into another language. The four operators on the right of (2.17) are the building stones of the operators
2.4 Invertibility of Toeplitz Operators on H 2
M (a)P + M (b)Q,
57
P M (a) + QM (b),
which are usually called singular integral operators when considered as acting on Lp and paired convolution operators when considered as acting on p (Z). 2.14. Proposition. Let a, b ∈ L∞ resp. a, b ∈ M p . Then T (ab) = T (a)T (b) + H(a)H()b), H(ab) = T (a)H(b) + H(a)T ()b).
(2.18) (2.19)
In particular, if the positive Fourier coeﬃcients of a = a− and the negative Fourier coeﬃcients of b = b+ vanish, then for every c ∈ L∞ resp. c ∈ M p , T (a− cb+ ) = T (a− )T (c)T (b+ ).
(2.20)
Proof. We have T (ab) = P M (ab)P = P M (a)M (b)P = P M (a)P M (b)P + P M (a)QM (b)P = P M (a)P · P M (b)P + P M (a)QJ · JQM (b)P and this is (2.18). Similarly, H(ab) = P M (ab)QJ = P M (a)M (b)QJ = P M (a)P · P M (b)QJ + P M (a)QJ · JQM (b)QJ, which is (2.19). To complete the proof, note that the conditions imposed upon a− and b+ imply that H(a− ) = 0 and H(b. + ) = 0. 2.15. Important remark. The ) a used in 2.10, 2.13 and 2.14 has nothing to do with the conjugate function of a (in the sense of 1.43). Given a measurable function a on T we now deﬁne ) a by ) a(t) = a(1/t) (t ∈ T). Moreover, this point of view eliminates any confusion that might arise in connection with 2.10 and deﬁnitions in 2.13. For instance, H() a) may be thought of as both JQM (a)P and as P M () a)QJ, and T () a) may be interpreted as both JQM (a)QJ and P M () a)P . In either case, both is the same. The notation ) a is in general use for the conjugate function of a as well as for the function given by ) a(t) = a(1/t) (t ∈ T). As a rule, henceforth ) a will always refer to the function ) a(t) = a(1/t) unless it is explicitly indicated that ) a means the conjugate function.
2.4 Invertibility of Toeplitz Operators on H 2 We ﬁrst show how the expressions for the norms T (a)L(H 2 ) and H(a)L(H 2 ) obtained in Theorems 2.7 and 2.11 can be used to derive results on the invertibility of Toeplitz operators on H 2 .
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2 Basic Theory
2.16. Deﬁnition. Let a ∈ L∞ . If there exist a real number ε > 0 and a complex number c of modulus 1 such that Re (ca) ≥ ε a.e. on T, then a is called sectorial. It is obvious that a sectorial function is necessarily invertible in L∞ . It is also easy to see that a is sectorial if and only if so is a/a. Thus, sectoriality is a matter of the argument. Also notice that a ∈ GL∞ is sectorial if and only if a/a can be written as ei(v+c) where c ∈ R and v ∈ L∞ is a realvalued function with v∞ < π/2. Finally, it is easily veriﬁed that a function a ∈ GL∞ is sectorial if and only if distL∞ (a/a, C) < 1, where distL∞ (g, C) := inf{g − c∞ : c ∈ C}. 2.17. Theorem (Brown/Halmos). If a ∈ L∞ is sectorial then T (a) is invertible on H 2 . Proof. It is readily seen that there are a suﬃciently small real number δ > 0 and a complex number c of modulus 1 such that 1 − δca∞ < 1. From Theorem 2.7 (or, more explicitly, from its Corollary 2.8(a)) we get I − δcT (a)L(H 2 ) < 1. This implies the invertibility of δcT (a) and thus also that of T (a). In order to state the consequence of Nehari’s theorem we have promised above, we need two propositions which are interesting on their own and will often be applied in the following. 2.18. Proposition (Wintner). Let h ∈ H ∞ . Then T (h) ∈ GL(H 2 ) ⇐⇒ h ∈ GH ∞ . Proof. If h−1 ∈ H ∞ , then, by Proposition 2.14, T (h−1 ) ∈ L(H 2 ) is the inverse of T (h). Conversely, if T (h) ∈ GL(H 2 ), then the equation T (h)f = P (hf ) = hf = g must have a solution f ∈ H 2 for every g ∈ H 2 . Thus hH 2 = H 2 and 1.41(g) completes the proof. 2.19. Proposition. Let a ∈ GL∞ . Then T (a) ∈ GL(H 2 ) ⇐⇒ T (a/a) ∈ GL(H 2 ). Proof. Since a−1 ∈ L∞ , we deduce from 1.41(d) that there is an outer function h ∈ GH ∞ such that a−1 1/2 = h. So a/a = hah and, by Proposition 2.14, T (a/a) = T (h)T (a)T (h). Due to the preceding proposition T (h) and T (h) = T ∗ (h) are invertible on H 2 and this gives the assertion at once. Remark. This proposition reduces the invertibility problem for Toeplitz operators on H 2 to the case of unimodular symbols. In other words, the invertibility of a Toeplitz operator on H 2 is exclusively dictated by the behavior of the argument of its symbol.
2.4 Invertibility of Toeplitz Operators on H 2
59
2.20. Theorem (Widom/Devinatz). Let ϕ ∈ L∞ be a unimodular function, i.e., ϕ = 1 a.e. on T. Then (a) T (ϕ) is leftinvertible on H 2 ⇐⇒ distL∞ (ϕ, H ∞ ) < 1, (b) T (ϕ) is rightinvertible on H 2 ⇐⇒ distL∞ (ϕ, H ∞ ) < 1, (c) T (ϕ) is invertible on H 2 ⇐⇒ distL∞ (ϕ, GH ∞ ) < 1. Proof. (a) We have M (ϕ)P = P M (ϕ)P + QM (ϕ)P and since ϕ = 1, it follows that ) 2 f 2 = ϕf 2 = P (ϕf )2 + Q(ϕf )2 = T (ϕ)f 2 + H(ϕ)f for every f ∈ H 2 . The operator T (ϕ) is leftinvertible on H 2 if and only if there exists an ε > 0 such that εf 2 ≤ T (ϕ)f 2 for all f ∈ H 2 (recall 1.12(g)). ) < 1, which, Consequently, T (ϕ) is leftinvertible on H 2 if and only if H(ϕ) by Nehari’s theorem 2.11 for p = 2, is the same as " " ) − h"∞ : h ∈ H ∞ 1 > dist(ϕ, ) H ∞ ) = inf "ϕ " " )−) h"∞ : h ∈ H ∞ = inf "ϕ = inf ϕ − h∞ : h ∈ H ∞ = dist(ϕ, H ∞ ). (b) Since T ∗ (ϕ) = T (ϕ), this is immediate from (a). (c) Suppose T (ϕ) ∈ GL(H 2 ). Then, by (a), there is an h ∈ H ∞ such that ϕ − h∞ < 1 and it remains to show that h ∈ GH ∞ . We have (Corollary 2.8(a)) (2.21) I − T (ϕh) = 1 − ϕh∞ = ϕ − h∞ < 1, this implies the invertibility of T (ϕh) = T (ϕ)T (h) = T ∗ (ϕ)T (h),
(2.22)
and because T ∗ (ϕ) is invertible, so also is T (h). From Proposition 2.18 we deduce that h ∈ GH ∞ . Now suppose h ∈ GH ∞ and ϕ−h∞ < 1. Then (2.21) holds and therefore the operator (2.22) is invertible. By Proposition 2.18, T (h) ∈ GL(H 2 ), hence T ∗ (ϕ) ∈ GL(H 2 ) and thus T (ϕ) ∈ GL(H 2 ). 2.21. Lemma. Suppose B is a subset of L∞ with the property that cb ∈ B whenever c ∈ C \ {0} and b ∈ B. Let ϕ ∈ L∞ be a unimodular function. Then distL∞ (ϕ, B) < 1 if and only if there are a function b ∈ B and a sectorial function s ∈ GL∞ such that ϕ = bs. Proof. If distL∞ (ϕ, B) < 1, then 1 − ϕ−1 b∞ = ϕ − b∞ < 1 for some b ∈ B. Hence ϕ−1 b is equal to a function s whose (essential) range is contained in some disk with center 1 and radius less than 1. Thus, ϕ−1 b = s with sectorial s, so ϕ = bs−1 and it remains to observe s−1 is sectorial whenever s is so.
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2 Basic Theory
Let ϕ = bs, where b ∈ B and s is sectorial. There is a c ∈ C \ {0} such that the (essential) range of cs−1 is contained in some disk with center 1 and radius less than 1. This implies that ϕ − cb∞ = 1 − ϕ−1 cb∞ = 1 − cs−1 ∞ < 1, whence distL∞ (ϕ, B) < 1.
2.22. Corollary. Let a ∈ GL∞ . Then the operator T (a) is left (right, resp. twosided ) invertible on H 2 if and only if a/a = hs, where h ∈ H ∞ ∩ GL∞ (h ∈ H ∞ ∩ GL∞ , resp. h ∈ GH ∞ ) and s ∈ GL∞ is sectorial. Proof. Combine Theorem 2.20, Proposition 2.19, and Lemma 2.21. For several applications the following restatement of Theorem 2.20(c) is useful. 2.23. Theorem (Widom/Devinatz). Let a ∈ GL∞ . Then T (a) is invertible on H 2 if and only if a/a = ei()u+v+c)
a.e. on
T,
(2.23)
∞
where c ∈ R, u and v are realvalued functions in L , and v∞ < π/2. Here u ) refers to the conjugate function of u. Proof. Put ϕ = a/a. By virtue of Proposition 2.19, T (a) is invertible if and only if T (ϕ) is invertible. Let T (ϕ) be invertible. Due to Corollary 2.22 there is an h ∈ GH ∞ such that ϕh is sectorial. Thus, ϕh = he−iv with some realvalued function v ∈ L∞ for which v∞ < π/2. Hence ϕ = (h/h)eiv = (h/h)eiv .
(2.24)
Since h is outer, there is an analytic logarithm log h in D (see 1.41(g), (ii)). The real part of log h is u(z) := log h(z) (z ∈ D) and log h can be written as log h(z) = u(z) + i) u(z) + ic
(z ∈ D)
with some c ∈ R. Consequently, h(z) = eu(z) ei()u(z)+c)
(z ∈ D).
∞
Because h ∈ H , the nontangential limit of eu(z) = h(z) exists a.e. on T and equals h(t). Therefore the nontangential limit of ei()u(z)+c) also exists a.e. on T and is equal to h(t)/h(t). In other words, h/h = ei()u+c) , and (2.24) gives (2.23) with u = log h, which is clearly in L∞ . Now let ϕ = a/a be of the form (2.23). Put ψ = ei(v+c) ,
h = ei(u+i)u)/2 .
It is obvious that h ∈ GH ∞ and that ϕ = (1/h)ψh, whence T (ϕ) = T (1/h)T (ψ)T (h). But the operators T (1/h) and T (h) are invertible by Proposition 2.18, while T (ψ) is invertible due to Theorem 2.17. Thus, T (ϕ) is invertible, too.
2.4 Invertibility of Toeplitz Operators on H 2
61
2.24. Remark. The WidomDevinatz theorems solve the invertibility problem in H 2 for Toeplitz operators (with symbols in GL∞ ) completely. However, given an a ∈ GL∞ it is, in general, by no means easy to decide whether there is an outer function h ∈ GL∞ such that a − h∞ < 1 or to check whether a can be represented in the form (2.23). This is the reason for a great part of all further investigations devoted to the invertibility of Toeplitz operators. The main goal of these investigations is to obtain invertibility criteria, or, equivalently, descriptions of the spectrum, in terms of geometric data of the symbol. The WidomDevinatz theorems answer the question in an analytic language. Nevertheless, there are situations in which Theorem 2.23 can be almost directly applied to decide whether a given Toeplitz operator is invertible or not. It can also be used to produce interesting examples of invertible Toeplitz operators. We shall demonstrate this in Proposition 2.26 below. 2.25. The class C(T◦ ). Let T◦ denote the punctured circle T \ {−1} and let C(T◦ ) denote the collection of all functions on T which are continuous at every point t ∈ T◦ . We denote by CU (T◦ ) the unimodular and by CR(T◦ ) the realvalued functions in C(T◦ ). Every a ∈ CU (T◦ ) can be written as a = eib with b ∈ CR(T◦ ) and b is uniquely determined by a up to an additive constant of the form 2kπ, k ∈ Z. Given a ∈ CU (T◦ ) choose any b ∈ CR(T◦ ) for which a = eib and deﬁne the realvalued function a# ∈ C(R) as i−x a# (x) = b (x ∈ R). i+x The behavior of a# (x) as x → ±∞ provides a good picture of the argument of a near the possible discontinuity of a at −1. We write a# (+∞) = −∞ if lim a# (x) = −∞; the notations a# (+∞) = +∞, a# (−∞) = +∞, x→+∞
and a# (−∞) = −∞ are deﬁned analogously. The function a# is said to be bounded from above (below) at −∞ if there exist both M ∈ R and x0 ∈ R such that a# (x) ≤ M
∀ x < x0
(a# (x) ≥ M
∀ x < x0 ).
It is clear that these deﬁnitions express properties of the function a that do not depend on the particular choice of the function b ∈ CR(T◦ ) which deﬁnes a# . 2.26. Proposition. (a) There are a ∈ CU (T◦ ) such that a# (+∞) = +∞, a# (−∞) = +∞ and T (a) is invertible on H 2 . (b) There exist functions a ∈ CU (T◦ ) such that a# (+∞) = +∞, a# is not bounded neither from above nor from below at −∞ and T (a) is invertible on H 2 . (c) If a ∈ CU (T◦ ) and if a# (+∞) = +∞ and a# is bounded from above at −∞, then T (a) is not invertible on H 2 .
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2 Basic Theory
(d) Let a ∈ CU (T◦ ) and suppose a# = ψ + δ, where δ ∈ L∞ (R), ψ is monotonous on (−∞, 0) and (0, ∞), and ψ(±∞) = +∞. Then if T (a) is invertible, we have a# (x) = O(log x)
as
x → +∞.
Proof. (a) Let w be a conformal mapping of D onto the region π π . Ω1 = z = x + iy ∈ C : y >  tan x, − < x < 2 2 There is a point t0 ∈ T such that w(z) → ∞ as z → t0 , z ∈ D. Without loss of generality assume t0 = −1. Deﬁne (2.25) w(t) := lim w(z) t ∈ T◦ z→t,z∈D
and, for t ∈ T◦ , put a(t) := eiIm w(t) . Then a ∈ CU (T◦ ), a# (±∞) = +∞, and, of course, a ∈ GL∞ . Since a is of the form (2.23) with u = Re w ∈ L∞ (so u ) = Im w + const) and v = 0, we deduce that T (a) is invertible on H 2 . (b) Now let Ω2 be the region Ω2 = z = x + iy ∈ C : y > − cot x, 0 < x < 2π and let S be the countable union of vertical halflines given by S=
∞ / 1 z = x + iy ∈ C : x = , y ≤ n . n n=1
Then Ω3 := Ω2 \ S is a simply connected region. Let w denote a conformal mapping of D onto Ω3 and without loss of generality suppose w(z) → ∞ as z → −1, z ∈ D. Deﬁne w on T◦ as in (2.25) and put a := eiIm w on T◦ . Then a has all the required properties and T (a) is invertible by Theorem 2.23. (c) Assume the contrary, that is, assume T (a) ∈ GL(H 2 ). Then a can be written in the form (2.23) and it follows from the Feﬀerman theorem 1.48(g) that the argument of a is in BM O. From 1.48(m) we deduce that a# is in BM O(R). But a function a# with the properties required in the hypothesis cannot be in BM O(R). This can be seen, for instance, as follows. Assume a# ∈ BM O(R). Then the function g(ξ) := (1/2) a# (ξ)−a# (−ξ) , ξ ∈ R, is also in BM O(R). Since g is odd, gI must be zero for every I of the form I = (−x, x), and therefore we have ( 1 x g(ξ) dξ =: N < ∞. (2.26) sup x>0 x 0 *x Put G(x) := 0 g(ξ) dξ. Obviously, G(0) = 0, G(x) ≥ 0 for x ≥ 0, and (2.26) says that
2.4 Invertibility of Toeplitz Operators on H 2
G(x) ≤ N x
for
x > 0.
63
(2.27)
It is precisely the conditions that a# (+∞) = +∞ and that a# be bounded from above at −∞ which imply that g(ξ) → +∞ as ξ → +∞. This in turn ensures the existence of a (suﬃciently large) x0 > 0 such that ( 2x0 G(2x0 ) − G(x0 ) = g(ξ) dξ ≥ (2N + 1)x0 x0
(apply the meanvalue theorem). Hence, G(2x0 ) G(x0 ) + (2N + 1)x0 (2N + 1)x0 1 = ≥ =N+ , 2x0 2x0 2x0 2 which contradicts (2.27) and completes the proof. (d) As in the proof of part (c) we deduce that a# ∈ BM O(R), whence ψ = a# − δ ∈ BM O(R). Put g(x) := ψ(1/x) (x = 0). It is not diﬃcult to see from 1.48(m) that g is also in BM O(R). The assertion can now be derived from the JohnNirenberg theorem 1.48(n) as follows. There is an x0 > 0 such that g(x) > 0 for x ∈ (−x0 , x0 ). Deﬁne ( x0 1 g(x) dx. g0 = 2x0 −x0 We now conclude from 1.48(n) that, for λ > 0, + + + x ∈ (−x0 , x0 ) : g(x) − g0  > λ + ≤ Ce−c0 λ with some constants C and c0 independent of λ. Hence, if we deﬁne x1 (λ) ∈ (0, x0 ) and x2 (λ) ∈ (0, x0 ) by g(−x1 (λ)) = g(x2 (λ)) = g0 + λ (note that g is monotonous on (−x0 , 0) and (0, x0 )), then x1 (λ) + x2 (λ) ≤ Ce−c0 λ (again use monotonicity). So xi (λ) ≤ Ce−c0 λ , whence log xi (λ) ≤ log C − c0 λ, and therefore g(±xi (λ)) = g0 + λ ≤ g0 +
log C log xi (λ) − c0 c0
(i = 1, 2).
On replacing xi (λ) by x we get g(x) ≤ A log(1/x) for all x ∈ (−x3 , x3 ) with some x3 > 0 (once more take into account monotonicity). Thus, ψ(x) = O(log x) as x → ∞, and consequently, a# (x) = ψ(x) + δ(x) = O(log x)
as x → ∞.
Remark. This proposition, though being a simple consequence of the WidomDevinatz theorem 2.23 obtained by invoking some deep BM O results in a luxurious way, is already concerned with geometric data of the symbol. It says, roughly speaking, that if a ∈ CU (T◦ ) has a discontinuity of oscillating type at −1 then
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2 Basic Theory
(a), (b) T (a) may be invertible if a has the possibility of changing the orientation of the oscillation (=rotation ) into the opposite direction when passing through −1; (c) T (a) cannot be invertible if the orientation of the oscillation is preserved when passing through −1; (d) T (a) cannot be invertible if the oscillation is allowed to alter its orientation into the opposite direction when passing through −1 but is, in addition, required to be “monotonous” and “suﬃciently fast”. Let us still dwell a bit on symbols a ∈ CU (T◦ ) for which a# is an even function. We saw that if a# (x) tends monotonically to inﬁnity as x → ±∞, then T (a) is not invertible unless a# increases suﬃciently slowly. We shall soon be in a position to decide whether T (a) is invertible if the limits a# (±∞) exist and are ﬁnite (this corresponds to the situation in which a ∈ CU (T◦ ) is continuous or has a jump discontinuity at −1). Much more diﬃculties arise for the “intermediate case,” e.g., for the cases where a# approaches +∞ sufﬁciently slowly or where the limits a# (±∞) do not exist at all. For instance, if a# (x) = cos x,
a# (x) = log log x,
or a# (x) = log log x + cos x
(x ∈ R, x large) we have situations of that kind. The spectral inclusion theorems we are now going to derive can be viewed as a ﬁrst step towards the description of invertibility of Toeplitz operators in geometrical language.
2.5 Spectral Inclusion Theorems 2.27. Deﬁnitions. The essential range R(a) of a function a ∈ L∞ is the spectrum of a considered as an element of the C ∗ algebra L∞ . Equivalently, R(a) is the set of all λ ∈ C such that {t ∈ T : a(t) − λ < ε} has positive (Lebesgue) measure for every ε > 0. Let X be a Banach space and let π denote the canonical homomorphism of L(X) onto the Calkin algebra L(X)/C∞ (X). For A ∈ L(X), the spectrum sp A of A is deﬁned by sp A := spL(X) A = λ ∈ C : A − λI ∈ GL(X) and the essential spectrum spess A of A is deﬁned as spess A := spL(X)/C∞ (X) (πA) = λ ∈ C : A − λI ∈ Φ(X) . The essential norm of A is given by Aess := πAL(X)/C∞ (X) = inf A + K : K ∈ C∞ (X) .
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65
In order to avoid confusion, we shall sometimes write spΦ(X) A and AΦ(X) for spess A and Aess , respectively. Note that obviously spess A ⊂ sp A and Aess ≤ A for every A ∈ L(X). 2.28. Proposition. (a) If a ∈ L∞ and 1 < p < ∞, then M (a) ∈ GL(Lp ) if and only if a ∈ GL∞ . In other words, spL(Lp ) M (a) = R(a). (b) If a ∈ M p and 1 ≤ p < ∞, then M (a) ∈ GL(p (Z)) if and only if a ∈ GM p . Consequently, spL(p (Z)) M (a) = spM p a ⊃ R(a). Proof. (a) If a ∈ GL∞ and b ∈ L∞ is the inverse of a, then M (b) ∈ L(Lp ) is the inverse of M (a). Conversely, suppose M (a) ∈ GL(Lp ). Then the equation M (a)b = 1 has a solution b ∈ Lp and we have ab = 1. Let B ∈ L(Lp ) denote the inverse of M (a). So a·Bf = f for all f ∈ P, whence Bf = bf for f ∈ P, and this implies that (Bχj , χk ) equals the (k − j)th Fourier coeﬃcient of b. The assertion now follows from Proposition 2.2. (b) If a ∈ GM p and b ∈ M p is the inverse of a, then M (b) ∈ L(p (Z)) is the inverse of M (a). Now suppose M (a) ∈ GL(p (Z)). By virtue of 2.5(a) it suﬃces to consider the case 1 ≤ p ≤ 2. The invertibility of M (a) implies that the equation = {ϕn } ∈ p (Z). Since p (Z) ⊂ 2 (Z), we M (a)ϕ = e0 has a solution ϕ
conclude that the function b = n∈Z ϕn χn belongs to L2 and that ab = 1. Thus, for a sequence ψ = {ψi } ∈ 0 (Z) the inverse B of M (a) is given by bi−j ψj (i ∈ Z). (Bψ)i = j∈Z
Due to the boundedness of B we !"! # " " sup " bi−j ψj j∈Z
have " # " " : ψ ∈ 0 (Z), ψp ≤ 1 < ∞, "
i∈Z p
which implies that b ∈ M p by the deﬁnition of M p . 2.29. Proposition. (a) Suppose a ∈ L∞ and 1 < p < ∞. If M (a) ∈ Φ+ (Lp ) or M (a) ∈ Φ− (Lp ), then M (a) ∈ GL(Lp ). (b) Suppose a ∈ W . If M (a) ∈ Φ+ (1 (Z)) or M (a) ∈ Φ− (1 (Z)), then M (a) ∈ GL(1 (Z)). (c) If a ∈ M p , 2 ≤ p < ∞, and M (a) ∈ Φ+ (p (Z)), then M (a) belongs to GL(p (Z)). (d) If a ∈ M p , 1 ≤ p < ∞, and M (a) ∈ Φ(p (Z)), then M (a) ∈ GL(p (Z)). Open problem. We are embarrassed to report that we have not been able to prove (c) for 1 < p < 2, although there seems to be no reason that (c) be false in that case.
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Proof. (a) Suppose M (a) ∈ Φ+ (Lp ); otherwise pass to the adjoint operator and take into account 1.12(h). We ﬁrst show that Ker M (a) = {0}. Let af = 0 for some f ∈ Lp , f = 0. Then f χn ∈ Ker M (a) for all n ∈ Z and it is easily seen that the system {f χn }n∈Z is linearly independent in Lp (if f p = 0 for some p ∈ P, then f = 0 a.e. on T). It would follow that dim Ker M (a) = ∞, which is a contradiction. Thus Ker M (a) = {0}. Consequently, Im M ∗ (a) = Im M (a) = Lq (1/p + 1/q = 1) and there is a b ∈ Lq such that ab = 1 a.e. on T, and it follows that Ker M (a) = {0}. Thus M ∗ (a) ∈ GL(Lq ), whence M (a) ∈ GL(Lp ). (b) Suppose M (a) ∈ Φ+ (1 (Z)). As in the proof of part (a), one can see that then necessarily Ker M (a) = {0}. Thus, by 1.12(g), there is a δ > 0 / GL(1 (Z)). such that abW ≥ δbW for all b ∈ W . Now assume M (a) ∈ −1 Then a ∈ / GW , since otherwise M (a ) would be an inverse of M (a). But the maximal ideal space, T, of the Banach algebra W coincides with its Shilov boundary. Therefore, a is a topological divisor of zero, that is, there exists a sequence {bn }∞ n=1 of functions bn ∈ W such that bn W = 1 and abn W → 0 as n → ∞. We arrived at a contradiction. Now let M (a) ∈ Φ− (1 (Z)). Since c0 (Z) := ϕ = {ϕn }n∈Z : ϕn → 0 as n → ∞ is a predual of 1 (Z), we conclude from 1.12(h) that M (a) ∈ Φ+ (c0 (Z)). Assume M (a)ψ = 0, ψ ∈ c0 (Z), ψ = 0. Then M (a)(ψ ∗ en ) = 0 for all n ∈ Z, where (ψ ∗ en )i := ψn−i (i ∈ Z). We claim that the system {ψ ∗ en }n∈Z is linearly independent in c0 (Z). To see this, let π ∈ 0 (Z) and assume ψ ∗ π = 0. Let p ∈ PA denote the polynomial whose Fourier coeﬃcients sequence is π, assume p(t) = q(t)(t − α) (t ∈ T) with q ∈ PA and α ∈ C, and let ∈ 0 (Z) be the Fourier coeﬃcients sequence of q. Then ξ := ψ ∗ ∈ c0 (Z) and we have ξ ∗ (e1 − αe0 ) = 0, i.e., ξn−1 = αξn (n ∈ Z). If α = 0, then ξ = 0, and in case α = 0 we have ξ−n = αn ξ0 and ξn = (1/α)n ξ0 (n ∈ Z+ ), which also implies that ξ = 0. On repeating this argument with in place of π etc., we ﬁnally see that πn = 0 for n = 0. This proves the linear independence of the system {ψ ∗ en }n∈Z . Thus, what results is that Ker M (a) = {0} in c0 (Z). Consequently, Im M (a) = 1 (Z), hence there is a b ∈ W such that ab = 1, whence M (a) ∈ GL(1 (Z)). (c) As in the proof of the Φ− part of (b) we conclude that M (a) has a trivial kernel in p (Z) whenever M (a) ∈ Φ+ (p (Z)). Therefore M (a) is onto on q (Z) (1/p + 1/q = 1). In particular, there is a ψ ∈ q (Z) such that M (a)ψ = e0 . Since q (Z) ⊂ 2 (Z), the function whose Fourier coeﬃcients sequence is ψ belongs to L2 and we have af = 1. It follows that a = 0 a.e. on T. Thus, if M (a)ϕ = 0 for some ϕ ∈ q (Z) ⊂ 2 (Z), then, again by passing into L2 , we have ϕ = 0. So Ker M (a) = {0} in q (Z), hence M (a) ∈ GL(q (Z)), and thus M (a) ∈ GL(p (Z)).
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(d) For p = 1 and 2 ≤ p < ∞ this is immediate from (b) and (c), respectively. If 1 < p < 2 and M (a) ∈ Φ(p (Z)), then M (a) ∈ Φ− (p (Z)), whence M (a) ∈ Φ+ (q (Z)) (2 < q < ∞) and (c) applies again. 2.30. Theorem (Hartman/Wintner). (a) Let a ∈ L∞ and 1 < p < ∞. If T (a) ∈ Φ+ (H p ) or T (a) ∈ Φ− (H p ), then a ∈ GL∞ . Consequently, R(a) ⊂ spΦ(H p ) T (a). (b) If a ∈ M p , 1 ≤ p < ∞, and T (a) ∈ Φ(p ), then a ∈ GM p . Consequently, R(a) ⊂ spM p a ⊂ spΦ(p ) T (a). Proof. (a) Let T (a) ∈ Φ+ (H p ) and denote by K any (ﬁniterank) projection of H p onto Ker T (a). By 1.12(g), there is a δ > 0 such that T (a)f p + Kf p ≥ δf p
∀ f ∈ H p.
This implies that P M (a)P gp + P KP gp + δQgp ≥ δgp
∀ g ∈ Lp .
Hence, if we let U denote the bilateral shift, then P M (a)P U n gp + P KP U n gp + δQU n gp ≥ δU n gp
∀ g ∈ Lp ,
and since U ±n are isometries, we obtain for all g ∈ Lp , U −n P M (a)P U n gp + P KP U n p + δU −n QU n gp ≥ δgp .
(2.28)
The operators U −n P U n are uniformly bounded on Lp , and because, obviously, U −n P U n f converges in Lp to f for every f ∈ P, we deduce from 1.1(d) that U −n P U n converges strongly to the identity operator. Thus, U −n QU n → 0 strongly on Lp , U −n P M (a)P U n = U −n P U n M (a)U −n P U n → M (a) strongly on Lp . Because U n converges weakly to zero on Lp , we get, by 1.1(f), P KP U n → 0
strongly on Lp .
Thus, (2.28) gives that M (a)gp ≥ δgp for all g ∈ Lp . So M (a) ∈ Φ+ (Lp ) and Propositions 2.29(a) and 2.28(a) imply that a ∈ GL∞ . For T (a) ∈ Φ− (H p ) passage to the adjoint yields the desired result. (b) The proof is the same as that of part (a).
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Remark 1. Propositions 2.29 and 2.28 also imply that the following implications hold: a ∈ W, T (a) ∈ Φ+ (1 ) ∪ Φ− (1 ) =⇒ a ∈ GW ; a ∈ M p , 2 ≤ p < ∞, T (a) ∈ Φ+ (p ) =⇒ a ∈ GM p ; a ∈ M p , 1 ≤ p ≤ 2, T (a) ∈ Φ− (p ) =⇒ a ∈ GM p . To prove that T (a) ∈ Φ− (1 ) =⇒ a ∈ GW pass ﬁrst to the predual c0 of 1 and notice that U −n QU n → 0 strongly on c0 . Remark 2. The HartmanWintner theorem shows that in Proposition 2.19 and Theorem 2.23 the hypothesis that a be invertible is redundant. Both results can be stated in the form “Let a ∈ L∞ . Then T (a) is invertible on H 2 if and only if a ∈ GL∞ and ....” Remark 3. Suppose a, b ∈ L∞ and 1 < p < ∞. If M (a)P + M (b)Q ∈ Φ+ (Lp ) or M (a)P + M (b)Q ∈ Φ− (Lp ), then a ∈ GL∞ and b ∈ GL∞ . Proof. Indeed, if, for instance, M (a)P + M (b)Q ∈ Φ+ (Lp ), then there are a K ∈ C0 (Lp ) and a δ > 0 such that " −n " "U M (a)P + M (b)Q U n g "p + KU n gp ≥ δgp , " " n "U M (a)P + M (b)Q U −n g " + KU −n gp ≥ δgp p for all n ≥ 0 and g ∈ Lp , and because U ±n → 0 weakly on Lp as n → ∞ and U −n P U n → I,
U −n QU n → 0,
U n P U −n → 0,
U n QU −n → I
strongly on Lp as n → ∞, we have M (a)gp ≥ δgp ,
M (b)gp ≥ δgp
for all g ∈ Lp and the assertion follows as above.
Thus, when investigating Fredholmness or invertibility of singular integral operators (over the unit circle) we may a priori assume that the coeﬃcients are in GL∞ . Moreover, we then have M (a)P + M (b)Q = M (b) M (b−1 a)P + Q = M (b) P M (b−1 a)P + Q QM (b−1 a)P + I (2.29) and since QM (b−1 a)P +I is always invertible (the inverse is I −QM (b−1 a)P ), we arrive at the following conclusion. Let a, b ∈ L∞ . Then M (a)P + M (b)Q is in Φ(Lp ) (resp. GL(Lp )) if and only if a, b ∈ GL∞ and T (b−1 a) is in Φ(H p ) (resp. GL(H p )). In the case of Fredholmness, Ind M (a)P + M (b)Q = Ind T (b−1 a).
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69
This shows in what a sense the study of Fredholmness and invertibility for singular integral operators over the unit circle (and thus over smooth curves) is equivalent to the study of the corresponding problems for Toeplitz operators. We now extend Proposition 2.18 to the case p = 2. Note that passage to adjoints yields results for antianalytic symbols, that is, for h ∈ H ∞ . 2.31. Proposition. (a) If 1 < p < ∞ and h ∈ H ∞ , then T (h) ∈ GL(H p ) ⇐⇒ h ∈ GH ∞ . (b) If 1 ≤ p < ∞ and h ∈ M p ∩ H ∞ , then T (h) ∈ GL(p ) ⇐⇒ h ∈ GM p and h is outer. Proof. The implications “⇐=” follow as in the proof of Proposition 2.18. So we are left with the reverse implications. Theorem 2.30 gives that h ∈ GL∞ resp. h ∈ GM p . Thus, by 1.41(g) it remains to show that h−1 ∈ H ∞ . The identity (2.19) implies that −1 −1 −1 H ) h +H ) h T (h). H ) h ) h =T ) h −1 −1 But H ) h ) T (h) = 0, and h = H(1) = 0 and H ) h) = 0, whence H ) h −1 ) = 0. Because ) h−1 = (h−1 )), since T (h) is invertible, it results that H h −1 ∞ we conclude that h ∈ H . 2.32. Proposition. Let a ∈ L∞ and 1 < p < ∞. Then T (a) is in GL(H p ) (Φ± (H p ) resp. Φ(H p )) if and only if a ∈ GL∞ and T (a/a) is in GL(H p ) (Φ± (H p ) resp. Φ(H p )). Moreover, if a ∈ GL∞ , then dim Ker T (a) = dim Ker T (a/a),
dim Coker T (a) = dim Coker T (a/a).
Proof. It follows from Theorem 2.30(a) that a may be assumed to belong to GL∞ . As in the proof of Proposition 2.19 we see that a/a = hah for some h ∈ GH ∞ . Since T (a/a) = T (h)T (a)T (h), the preceding proposition implies all assertions. 2.33. Theorem (Brown/Halmos). If a ∈ L∞ , then spL(H 2 ) T (a) ⊂ conv R(a),
(2.30)
where conv R(a) is the closed convex hull of R(a). Proof. Immediate from Theorem 2.17.
Remark. We shall see later that if E is any subarc of T and χE is the characteristic function of E, neither spL(H p ) T (χE ) nor spL(p ) T (χE ) is contained in conv R(χE ) = [0, 1] for 1 < p < ∞ and p = 2.
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2.34. Realvalued continuous functions. For p = 2, Theorems 2.30 and 2.33 together give that R(a) ⊂ spess T (a) ⊂ sp T (a) ⊂ conv R(a).
(2.31)
This is all what is needed to derive the following: if a is a realvalued continuous function, then T (a) ∈ GL(H 2 ) ⇐⇒ T (a) ∈ Φ(H 2 ) ⇐⇒ a(t) = 0 ∀ t ∈ T and
spΦ(H 2 ) T (a) = spL(H 2 ) T (a) = min a(t), max a(t) . t∈T
t∈T
Note that both the spectrum and the essential spectrum are completely described via geometric data of the symbol. 2.35. Connectedness of the spectrum. A powerful tool for obtaining information about the spectra of Toeplitz operators are the following results. (a) (Widom). If a ∈ L∞ then spL(H p ) T (a) is connected. (b) (Douglas). If a ∈ L∞ then spΦ(H 2 ) T (a) is connected. Corollary 2.40 below implies that the boundary of spL(H p ) T (a) is contained in spL(H p ) T (a). Using this it is easy to derive the connectedness of the spectrum of a Toeplitz operator from the connectedness of the essential spectrum. Open problems. Is spΦ(H p ) T (a) always connected? We conjecture that the answer is yes and that a check of the proof in Widom [563] and Douglas [162] will indicate the modiﬁcation needed to obtain the desired result. The following problem seems to us to lie essentially deeper: what can be said about the connectedness of the spectra of a Toeplitz operator on p ? We do not know any symbol in M p generating a Toeplitz operator whose spectra are disconnected. 2.36. Realvalued symbols. If a ∈ L∞ is realvalued, then spΦ(H 2 ) T (a) = spL(H 2 ) = ess inf a(t), ess sup a(t) . t∈T
t∈T
This result is due to Hartman and Wintner, too. Proof. Combine (2.31) and 2.35(b). There is a simple direct proof, which goes as follows. Let λ ∈ R and put b = a − λ. We must show that sign b = const whenever T (b) ∈ Φ(H 2 ). If Ind T (b) = κ, then Ind T (b) = Ind T (b) = Ind T ∗ (b) = −κ, whence κ = 0. Coburn’s theorem, which will be proved below (Corollary 2.40 for p = 2), therefore shows that we may assume that T (b) is invertible. Then the equation
2.6 The Connection Between Fredholmness and Invertibility
71
◦
2 T (b)f = 1 has a solution f ∈ H 2 . So bf = 1 + g with g ∈H− , and we obtain, for n ≥ 1, ( ( ( bf 2 χn dm = bf f χn dm = (1 + g)f χn dm = 0. T
T
*
T
Since bf 2 is realvalued, it follows that T bf 2 χn dm = 0 for all n ∈ Z \ {0}, so bf 2 = const, that is, sign b = const. 2.37. The boundary of conv R(a). For a ∈ L∞ , denote by a the harmonic extension of a into D. The following result of Wolﬀ is sometimes very useful to get further information about the spectrum of a Toeplitz operator. Let a ∈ L∞ and let λ belong to the boundary of conv R(a). Then λ ∈ spL(H 2 ) T (a) ⇐⇒ λ ∈ clos a(D). An application of this result will be given in 4.75 and 4.78.
2.6 The Connection Between Fredholmness and Invertibility 2.38. Theorem (Coburn). A nonzero bounded Toeplitz operator has a trivial kernel or a dense range. The precise statement is as follows. (a) If a ∈ L∞ and if a does not vanish identically, then the kernel of T (a) in H p (1 < p < ∞) or the kernel of T (a) in H q (1/p + 1/q = 1) is trivial. (b) If a ∈ M p and if a does not vanish identically, then the kernel of T (a) in p (1 ≤ p < ∞) or the kernel of T (a) in q (1/p + 1/q = 1) is trivial. Proof. (a) Assume there are f+ ∈ H p , g+ ∈ H q , f+ = 0, g+ = 0 such that T (a)f+ = 0, T (a)g+ = 0. The F. and M. Riesz theorem 1.40(b) implies that ◦
f+ = 0 and g+ = 0 a.e. on T. Put f− := af+ and g− := ag+ . Then f− ∈H p , ◦
◦
◦
1 g− ∈H q , and so g− f+ ∈H 1 , g+ f− ∈H− . But g− f+ = ag+ f− , whence g+ f− = g− f+ = 0. Since f+ = 0 a.e. on T, we conclude that g− = 0 a.e. on T, and since g− = ag+ and g+ = 0 a.e. on T, it follows that a = 0 a.e. on T, which contradicts the hypothesis of the theorem.
(b) Let ﬁrst 1 < p < ∞. Since the assertion is symmetric in p and q (recall 2.5(a), (b)), we may assume that 1 < p ≤ 2. Let T (a)ϕ+ = 0, T (a)ψ+ = 0, where ϕ+ ∈ p , ψ ∈ q , ϕ+ = 0, ψ+ = 0. Put ϕ− := M (a)ϕ+ and ψ− := M (a)ψ+ . Then ϕ− ∈ p (Z), ψ− ∈ q (Z), (ϕ− )n = (ψ− )n = 0 for all n ≥ 0. s (Z) (1 ≤ r ≤ ∞, 1/r + 1/s = 1) the convolution ϕ ∗ ψ For ϕ ∈ r (Z) and ψ ∈
deﬁned by (ϕ ∗ ψ)i := j∈Z ϕi−j ψj belongs to ∞ (Z). For ϕ ∈ r (Z) deﬁne ϕ ∈ r (Z) by (ϕ)n = ϕn . Thus, we have
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(ψ− ∗ ϕ+ )n = 0 ∀ n ≤ 0,
(ψ+ ∗ ϕ− )n = 0 ∀ n ≤ 0
and because ψ− ∗ ϕ+ = (M (a)ψ+ ) ∗ ϕ+ = ψ+ ∗ (M (a)ϕ+ ) = ψ+ ∗ ϕ− , it follows that (ψ− ∗ ϕ+ )n = (ψ+ ∗ ϕ− )n = 0 for n ∈ Z. Since ψ+ = 0, we have (ϕ− )n = 0 for all n ∈ Z. Thus, M (a)ϕ+ = 0, and since ϕ+ ∈ p (Z) ⊂ 2 (Z), we deduce that af+ = 0 a.e. on T, where f+ ∈ H 2 is the function whose Fourier coeﬃcients sequence is ϕ+ . The function f+ has a nonvanishing Fourier coeﬃcient and therefore, by the F. and M. Riesz theorem 1.40(b), f+ = 0 a.e on T. This gives a = 0 a.e. on T and we arrived at a contradiction. Since M 1 is the Wiener algebra W , minor modiﬁcations of the proof also yield the result for p = 1. Notice that Toeplitz operators with symbols in W are obviously bounded on ∞ . Recall that, for a ∈ L1 , the function a is deﬁned by a(t) = a(t) (t ∈ T). 2.39. Lemma. Let a ∈ L∞ , 1 < p < ∞, 1/p + 1/q = 1. Then T (a) is Fredholm (invertible ) on H p if and only if T (a) is Fredholm (invertible ) on H q . In the case of Fredholmness one has dim Ker T (a) = dim Coker T (a),
dim Ker T (a) = dim Coker T (a). ◦
Remark. Some care is in order, since the dual of H p is Lq / H q− and not H q . Nevertheless, all is easy. Proof. The hypothesis that a be in L∞ ensures that all operators occurring ◦
·
◦
·
p q + H p and Lq =H− + H q , we have are bounded. Since Lp =H−
T (a) ∈ Φ(H p ) ⇐⇒ P M (a)P + Q ∈ Φ(Lp ), T (a) ∈ Φ(H q ) ⇐⇒ P M (a)P + Q ∈ Φ(Lq ) and this is true with Φ replaced by GL. But (Lp )∗ is Lq and (P M (a)P + Q)∗ is easily seen to be P M (a)P + Q. This implies all assertions of the lemma. 2.40. Corollary. A Toeplitz operator is invertible if and only if it is Fredholm and has index zero. More explicitly: if a ∈ L∞ and 1 < p < ∞, then T (a) ∈ GL(H p ) ⇐⇒ T (a) ∈ Φ(H p ) and Ind T (a) = 0; if a ∈ M p and 1 ≤ p < ∞, then T (a) ∈ GL(p ) ⇐⇒ T (a) ∈ Φ(p ) and Ind T (a) = 0.
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73
Proof. The previous lemma and Theorem 2.38 imply that Ind T (a) = dim Ker T (a) − dim Ker T (a) = 0 if and only if dim Ker T (a) = dim Ker T (a) = 0. 2.41. The index of a continuous function. Let a ∈ C and suppose a has no zeros on T. Then there is a b ∈ CR(T◦ ) (recall 2.25) such that a = ae2πib . The increment of b as the result of a circuit around T counterclockwise is an integer and depends only on a, i.e., it does not depend on the particular choice of b. This integer is referred to as the index (or winding number ) of a and is denoted by ind a. If a ∈ C has no zeros on T, then a/a is a continuous function belonging to CU (T◦ ). Therefore the limits lim (a/a)# (x) =: (a/a)# (±∞) exist, are x→±∞
ﬁnite, and its diﬀerence is an integral multiple of 2π. It is easily seen that ind a is nothing else than 1 1 0 (a/a)# (+∞) − (a/a)# (−∞) . 2π Note that ind χn = n, where χn (t) = tn (t ∈ T). Here are two important properties of the index. (a) If a, b ∈ C and a(t)b(t) = 0 for all t ∈ T, then ind (ab) = ind a + ind b. (b) If a, d ∈ C, a(t) = 0 for all t ∈ T, and d/a∞ < 1, then a(t)+d(t) = 0 for all t ∈ T and ind (a + d) = ind a. If a is continuously diﬀerentiable and does not vanish on T, then ( ( 2π iθ 1 a (t) a (e ) iθ 1 dt = e dθ. ind a = 2πi T a(t) 2π 0 a(eiθ ) Thus, if a ∈ C has no zeros on T, then by the above property (b) and by 1.38(b), ( 1 (hr a) (t) ind a = lim ind hr a = lim dt. r→1−0 r→1−0 2πi T (hr a)(t) Also notice the following. (c) If a is a rational function without poles and zeros on the unit circle T, then ind a = z − p, where z and p are the numbers of zeros and poles (counted up to multiplicity) of a in D, respectively. In the language of Banach algebras we have the following. (d) If a ∈ GC, then ind a = 0 if and only if a belongs to the connected component of GC containing the identity. Given a Banach algebra A of continuous functions on T that contains the constants we shall say that the maximal ideal space of A is T if
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(i) the general form of a multiplicative linear functional on A is given by ϕ : A → C, ϕ(a) = a(τ ), where τ ranges over T; (ii) the Gelfand topology on T coincides with the usual topology on T. The notion of the index allows us to specialize a result of Shilov as follows. (e) If A is a Banach algebra of continuous functions on T that contains the constants and whose maximal ideal is T, then every a ∈ GA of index zero has a logarithm log a ∈ A and, consequently, by 1.16(a), belongs to the connected component of GA containing the identity. We are now in a position to establish criteria for Fredholmness and invertibility of Toeplitz operators with continuous symbols on H p and p . 2.42. Theorem. Let a ∈ C and 1 < p < ∞. Then (a) H(a) ∈ C∞ (H p ); (b) T (a) ∈ Φ(H p ) if and only if a(t) = 0 for all t ∈ T; if T (a) is Fredholm on H p , then T (a−1 ) is a regularizer of T (a) and Ind T (a) = −ind a; (c) T (a) ∈ GL(H p ) if and only if a(t) = 0 for all t ∈ T and ind a = 0. Proof. (a) There are an ∈ P (for example, the Fej´er means of a) such that a − an ∞ → 0 as n → ∞. Then H(an ) has ﬁnite rank and since H(a) − H(an )p = P M (a − an )QJp ≤ c2p a − an ∞ , H(a) is compact on H p . (b) The implication “=⇒” follows from Theorem 2.30. So suppose a = 0 on T. By Proposition 2.14, −1 T (a−1 )T (a) = I − H(a−1 )H() (2.32) a), T (a)T (a−1 ) = I − H(a)H ) a and since, by (a), all Hankel operators occurring are compact, it follows that T (a) ∈ Φ(H p ) and that T (a−1 ) is a regularizer of T (a). So we are left with the index formula. Let T (a) ∈ Φ(H p ) and ind a = n. Then, by 2.41(a), ind (χ−n a) = 0. Hence, by 2.41(d), χ−n a belongs to the connected component of GC containing the identity. As T (f )L(H p ) ≤ cp f ∞ , the mapping T : GC → Φ(H p ), f → T (f ) is continuous. Consequently, the operator T (χ−n a) must be in the connected component of Φ(H p ) containing I and 1.12(d) gives Ind T (χ−n a) = 0. Because T (χ−n a) equals T (χ−n )T (a) or T (a)T (χ−n ), we deduce from Atkinson’s theorem 1.12(c) and from 2.9 that 0 = Ind T (χ−n a) = Ind T (χ−n ) + Ind T (a) = n + Ind T (a). (c) Immediate from (b) and Corollary 2.40. Remark. Theorem 2.30 even implies that a(t) = 0 for all t ∈ T provided T (a) ∈ Φ+ (H p ) or T (a) ∈ Φ− (H p ).
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75
2.43. The classes Cp and M p . For 1 ≤ p < ∞, let Cp denote the closure in M p of the Laurent polynomials: Cp = closM p P. Clearly, C1 = W and C2 = C. Note that Cp is a closed subalgebra of M p . For p ∈ (1, 2) ∪ (2, ∞), let M p denote the collection of all functions a ∈ L∞ which belong to M p) for all p) in some neighborhood of p, i.e., $ (M p+ε ∩ M p−ε ). M p := ε>0
Finally, let M 1 = M 1 = W and M 2 = M 2 = L∞ . The following Proposition 2.45 intends to give an alternative description of Cp and to provide a better understanding of which functions belong to Cp . However, neither the deﬁnition of M p nor that proposition are needed to prove Propositions 2.46 and Theorem 2.47. 2.44. Lemma. Let a ∈ M p and let σn a denote the nth Fej´er mean of a, n j (σn a)(t) = 1− a j tj n + 1 j=−n
(t ∈ T).
Then σn aM p ≤ aM p for all n ≥ 0. Proof. For θ ∈ (−π, π], let sin2 (n + 1)θ/2 1 Kn (θ) = 2π(n + 1) sin2 (θ/2) denote the nth Fej´er kernel and deﬁne the function ax by ax (eiθ ) := a(ei(θ−x) ). Thus, ( π
(σn a)(eiθ ) =
−π
ax (eiθ )Kn (x) dx.
(2.33)
It is easy to see that M (ax ) = D−x M (a)Dx , where Dx is the isometry Dx : p → p ,
{ϕj }j∈Z+ → {eijx ϕj }j∈Z+ .
Therefore, if ϕ ∈ 0 (Z), then the function (−π, π] → p ,
x → Kn (x)M (ax )ϕ
is continuous. This and (2.33) enable us to write M (σn a)ϕ as a Bochner integral: ( π M (ax )ϕKn (x) dx. M (σn a)ϕ = −π
Hence,
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2 Basic Theory
( M (σn a)ϕp ≤
π −π
M (ax )ϕp Kn (x) dx
≤ D−x p M (a)p Dx p ϕp
(
π −π
Kn (x) dx
= M (a)p ϕp , for all ϕ ∈ P, and consequently, M (σn a)p ≤ M (a)p . 2.45. Proposition. If 1 ≤ p < ∞, then Cp = closM p (C ∩ M p ). Proof. There is nothing to prove for p = 1 or p = 2. Thus let p ∈ (1, 2)∪(2, ∞). We ﬁrst show that C ∩ M p ⊂ Cp . By virtue of 2.5(b), we may without loss of generality assume that 2 < p < ∞. Then a ∈ C ∩ M p+ε for some ε > 0. Let σn a denote the nth Fej´er mean of a. From 2.5(e) we get γ M (a − σn a)p ≤ a − σn a1−γ ∞ M (a − σn a)p+ε ,
where 1/p = γ/(p + ε) + (1 − γ)/2. Since a ∈ C, we know that a − σn a∞ → 0 as n → ∞, and Lemma 2.44 applied to a ∈ M p+ε shows that M (a−σn a)p+ε remains bounded as n → ∞. Thus, the inclusion C ∩ M p ⊂ Cp is proved. Now it is easy to see that the asserted equality holds: Cp = closM p P ⊂ closM p (C ∩ M p ) ⊂ closM p Cp = Cp . 2.46. Proposition. Let 1 ≤ p < ∞. (a) The maximal ideal space M (Cp ) of Cp is T. Thus, if a ∈ Cp , then a ∈ GCp ⇐⇒ a(t) = 0 ∀ t ∈ T.
(2.34)
(b) The connected component of GCp containing the identity coincides with the functions in GCp of index zero. Proof. (a) By virtue of 2.5(d), we have a(τ ) ≤ a∞ aM p for all τ ∈ T and hence m : Cp → C, a → a(τ ) deﬁnes a multiplicative linear functional on Cp . Conversely, let m : Cp → C be a multiplicative linear functional. Put τ := m(χ1 ). It is easy to see that spM p (χ1 ) = T, whence τ ∈ T. This implies that m(f ) = f (τ ) for every f ∈ P and thus m(a) = a(τ ) for every a ∈ Cp , the closure of P. That the Gelfand topology on T coincides with the usual one on T can be checked in a standard way. (b) This follows from Shilov’s theorem 2.41(e). A proof which does not invoke that theorem is as follows. If a ∈ GCp has index zero, then, by 2.41(b), c ∈ GCp and ind c = 0 whenever c ∈ Cp and a − cM p is suﬃciently small. Since P is dense in Cp , among these c’s there is a c ∈ P. Using 2.41(a) and 2.41(c) it is not diﬃcult to see that c factors into a product of functions of the form χ1 − α and 1 − βχ−1 , where α > 1 and β < 1. But such functions are in the connected component of GCp containing the identity, hence so is c and therefore a, too.
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77
2.47. Theorem. Let a ∈ Cp and 1 ≤ p < ∞. Then (a) H(a) ∈ C∞ (p ); (b) T (a) ∈ GL(p ) if and only if a(t) = 0 for all t ∈ T; if T (a) is Fredholm on p , then T (a−1 ) is a regularizer of T (a) and Ind T (a) = −ind a; (c) T (a) ∈ GL(p ) if and only if a(t) = 0 for all t ∈ T and ind a = 0. Proof. (a) By the deﬁnition of Cp , there are an ∈ P such that H(a) − H(an )p ≤ a − an M p = o(1)
as n → ∞,
which gives the compactness of H(a) on p . (b) The implication “=⇒” follows from Theorem 2.30. Taking into account (2.34), one can obtain the implication “⇐=” as in the proof of Theorem 2.42. Proposition 2.46(b) shows that the argument applied in the proof of Theorem 2.42 can be used to verify the index formula in the case at hand, too. (c) This is immediate from (b) and Corollary 2.40. Remark 1. The index formula can also be proved with the help of the argument that will be used in the proof of Theorem 2.66 below. Remark 2. One can show, e.g., as in the proof of 2.29(b) or by using a perturbation argument (see the proof of Theorem 2.74 below), that a ∈ GCp whenever a ∈ Cp and T (a) ∈ Φ+ (p ) or T (a) ∈ Φ− (p ).
2.7 Compactness of Hankel Operators and C + H ∞ Symbols 2.48. Deﬁnition. For 1 < p < ∞, let Ap := a ∈ L∞ : H(a) ∈ C∞ (H p ) ,
Bp := a ∈ M p : H(a) ∈ C∞ (p ) .
2.49. Lemma. Ap and Bp are closed subalgebras of L∞ and M p , respectively. Proof. It is clear that Ap and Bp are linear spaces. From Proposition 2.14 we have H(ab) = T (a)H(b) + H(a)T ()b), which shows that ab ∈ Ap resp. ab ∈ Bp whenever a, b ∈ Ap resp. a, b ∈ Bp . Thus Ap and Bp are algebras. If an ∈ Bp , b ∈ M p , an − bp → 0 as n → ∞, then H(b) − H(an )L(p ) ≤ b − an p = o(1) as n → ∞, whence H(b) ∈ C∞ (p ). This shows that Bp is closed. The closedness of Ap can be proved analogously. 2.50. Theorem. (a) If a ∈ Ap , then T (a) ∈ Φ(H p ) ⇐⇒ a ∈ GAp . (b) If a ∈ Bp , then T (a) ∈ Φ(p ) ⇐⇒ a ∈ GBp .
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Proof. (a) Let a ∈ Ap . Then a, a−1 ∈ Ap and, by (2.18), T (a−1 )T (a) − I = −H(a−1 )H() a) ∈ C∞ (H p ), T (a)T (a−1 ) − I = −H(a)H ) a−1 ) ∈ C∞ (H p ), whence T (a) ∈ Φ(H p ). Conversely, suppose T (a) ∈ Φ(H p ). From Theorem 2.30 we know that then a ∈ GL∞ and it remains to show that a−1 ∈ Ap . Let RT (a) = I + K, where R ∈ L(H p ) is a regularizer of T (a) and K ∈ C∞ (H p ). Formula (2.19) gives −1 0 = H(aa−1 ) = T (a)H(a−1 ) + H(a)T ) , a hence, by acting with R from the left, −1 0 = H(a−1 ) + KH(a−1 ) + RH(a)T ) , a and it follows that H(a−1 ) ∈ C∞ (H p ), as desired. (b) The proof is the same. 2.51. Deﬁnition. For 1 < p < ∞, deﬁne Cp as in 2.43 and let Hp∞ := H ∞ ∩M p . It is obvious that Hp∞ is a closed subalgebra of M p . Let alg (Cp , Hp∞ ) denote the smallest closed subalgebra of M p containing Cp and Hp∞ . It is clear that alg (Cp , Hp∞ ) coincides with alg (χ1 , Hp∞ ), the smallest closed subalgebra of M p containing the set Hp∞ and the function χ1 . The discontinuous function (1 − χ1 )iβ (β ∈ R \ {0}) can be shown to belong to Hp∞ for all 1 < p < ∞ (Theorem 6.45). This shows that alg (Cp , Hp∞ ) is strictly larger than Cp . Of course, alg (Cp , Hp∞ ) contains Cp + Hp∞ = f + g : f ∈ Cp , g ∈ Hp∞ . It is an absolutely unexpected fact, the discovery of which goes back to Sarason, that alg (Cp , Hp∞ ) is actually equal to Cp + Hp∞ . Our next goal is to prove this. The proof will be based on the following interesting lemma. 2.52. Lemma (Zalcman/Rudin). Let X be a Banach space and let E and F be closed subspaces of X. Suppose {Sn }n∈Z+ is a sequence of operators Sn ∈ L(X) with the following properties: (i) Sn L(X) ≤ M for all n ∈ Z+ ; (ii) Sn (X) ⊂ E for all n ∈ Z+ ; (iii) Sn (F ) ⊂ F for all n ∈ Z+ ; (iv) Sn u − u → 0 as n → ∞ for all u ∈ E. Then E + F is a closed subspace of X.
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79
Proof. Let x ∈ closX (E + F ). Then
there are uk ∈ E, vk ∈ F such that ∞ uk + vk ≤ 1/2k for k ≥ 2 and x = k=1 (uk + vk ). For each k ≥ 2 choose k nk so that uk − Snk uk ≤ 1/2 . If we let xk := uk + vk , then obviously xk = (uk − Snk uk + Snk xk ) + (vk − Snk vk ). We have u )k := uk − Snk uk + Snk xk ∈ E and ) uk ≤
1 1+M + Snk xk ≤ k 2 2k
(note that xk ≤ 1/2k ). Furthermore, v)k := vk − Snk vk ∈ F and ) vk ≤ ) uk + xk ≤
2+M . 2k
∞
∞ )k and k=1 v)k are absolutely convergent. Let Consequently, the series k=1 u u ∈ E and v ∈ F denote their sums. What results is that x=
∞ k=1
xk =
∞
() uk + v)k ) = u + v ∈ E + F.
k=1
2.53. Theorem. Cp + Hp∞ is a closed subalgebra of M p and Cp + Hp∞ = alg (Cp , Hp∞ ) = alg (χ1 , Hp∞ ).
(2.35)
Proof. We apply the preceding lemma with X = M p , E = Cp , F = Hp∞ , and Sn ∈ L(M p ) given by Sn a = σn a, where σn a denotes the nth Fej´er mean of a. It is clear that (ii) and (iii) are satisﬁed. Lemma 2.44 shows that (i) is fulﬁlled. Finally, given a ∈ Cp and ε > 0 choose f ∈ P so that a − f M p < ε/3. Then σn a − aM p ≤ σn (a − f )M p + σn f − f M p + f − aM p ≤ a − f M p + σn f − f W + f − aM p 2ε + σn f − f W < 3 (Lemma 2.44 and 2.5(d)) and σn f − f W < ε/3 whenever n is large enough. Thus, the requirement (iv) is also met and it follows that Cp + Hp∞ is closed. Now let a, b ∈ Cp + Hp∞ . There are an , bn ∈ P + Hp∞ such that a − an M p → 0,
b − bn M p → 0 as
n → ∞.
It is obvious that an bn ∈ P +Hp∞ , and since an bn −abM p → 0 as n → ∞ and Cp + Hp∞ is closed, we conclude that ab ∈ Cp + Hp∞ . Consequently, Cp + Hp∞ is an algebra. Once we know that Cp + Hp∞ is a closed subalgebra of M p , the equalities (2.35) are obvious.
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2.54. Theorem (Hartman/Adamyan/Arov/Krein). Let a ∈ L∞ and 1 < p < ∞. Then distL∞ (a, C + H ∞ ) ≤ H(a)Φ(H p ) ≤ cp distL∞ (a, C + H ∞ ), where cp = P L(Lp ) . In particular, Ap = C + H ∞ , that is, H(a) ∈ C∞ (H p ) ⇐⇒ a ∈ C + H ∞ . Proof. We have distL∞ (a, C + H ∞ ) = inf a − f − h∞ : f ∈ C, h ∈ H ∞ = inf
inf a − f − h∞ = inf distL∞ (a − f, H ∞ )
f ∈C h∈H ∞
c∈C
≥
1 inf H(a − f )L(H p ) cp c∈C
=
1 inf H(a) − H(f )L(H p ) cp f ∈C
=
1 H(a)Φ(H p ) cp
(Theorem 2.11)
(Theorem 2.42(a)).
Now let V = T (χ1 ). Since (V n )∗ = T (χ−n ) converges strongly to zero on H p as n → ∞, we conclude that KV n → 0 as n → ∞ for every K ∈ C∞ (H p ) (see 1.3(d)). Thus, if K ∈ C∞ (H p ) then H(a) − K ≥ (H(a) − K)V n ≥ H(a)V n − KV n = H(χ−n a) − KV n ≥ distL∞ (χ−n a, H ∞ ) − KV n (Theorem 2.11) = distL∞ (a, χn H ∞ ) − KV n = distL∞ (a, C + H ∞ ) − KV n , whence H(a)Φ(H p ) ≥ distL∞ (a, C + H ∞ ). Since C + H ∞ is closed, we have distL∞ (a, C + H ∞ ) = 0 if and only if a ∈ C + H ∞. Remark. The above theorem implies the following compactness criterion for Hankel operators: if a ∈ L∞ and 1 < p < ∞, then H(a) ∈ C∞ (H p ) if and only if P a ∈ V M O. Indeed, if a ∈ C + H ∞ , then P a ∈ P (C) ⊂ V M O by 1.48(k), and if P a ∈ V M O, then P a = u + P v with u, v ∈ C by virtue of 1.48(l), which shows that a = u + v + Q(a − v) is in C + H ∞ . 2.55. Corollary. If a ∈ C + H ∞ , then T (a) ∈ Φ(H p ) ⇐⇒ a ∈ G(C + H ∞ ). If T (a) ∈ Φ(H p ), then T (a−1 ) is a regularizer of T (a).
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81
Proof. Immediate from Theorems 2.50 and 2.54 and formula (2.18). 2.56. Open problem. Establish the analogue of Theorem 2.54 for p . In this connection recall 2.12. It is clear that Cp + Hp∞ ⊂ Bp , but we have not been able to prove that Bp ⊂ Cp + Hp∞ . Nevertheless we shall show that the p version of Corollary 2.55 holds (see Theorem 2.60 below). 2.57. Deﬁnition. Put R = p/q : p ∈ PA , q ∈ PA , q(t) = 0 ∀t ∈ T . Note that R is the restriction to the unit circle T of the set of all rational functions deﬁned on the whole plane C and having no poles on T. 2.58. Theorem (Kronecker). (a) Let 1 < p < ∞ and a ∈ L∞ . Then H(a) ∈ C0 (H p ) ⇐⇒ a ∈ R + H ∞ . (b) Let 1 < p < ∞ and a ∈ M p . Then H(a) ∈ C0 (p ) ⇐⇒ a ∈ R + Hp∞ . Proof. We ﬁrst prove that the operator H(a) has ﬁnite rank for a ∈ R (and thus for a ∈ R+H ∞ resp. a ∈ R+Hp∞ ). This is obvious if a is a polynomial. If a = 1/(χ1 −λ) with some λ ∈ C, then H(a) = 0 for λ < 1 and rank H(a) = 1 for λ > 1. So application of the formula H(bc) = T (b)H(c) + H(b)T () c) shows that H(a) has ﬁnite rank if a is of the form polynomial /(χ1 − λ)n (n ∈ Z+ , λ ∈ C), and decomposition into partial fractions gives the assertion for all a ∈ R. Now suppose rank H(a) = r < ∞. This implies that the ﬁrst r + 1 columns of the matrix (aj+k+1 )∞ j,k=0 are linearly dependent, where an denotes the nth Fourier coeﬃcient of a. Hence, if we let b :=
∞
ar+k+1 χk
(∈ H 2 ),
k=0
then there exist complex numbers λ0 , λ1 , . . . , λr such that at least one of them is nonzero and λ0 (a1 + a2 χ1 + . . . + ar χr−1 + χr b) +λ1 (a2 + a3 χ1 + . . . + ar χr−2 + χr−1 b) + . . . + λr b = 0. It follows that (λ0 χr + λ1 χr−1 + . . . + λr )b is a polynomial and therefore b must be a rational function. Since b ∈ H 2 , b cannot have poles on T, whence b ∈ R. If a ∈ L∞ resp. a ∈ M p , then χ−r−1 a − b belongs to H ∞ resp. M p ∩ H ∞ = Hp∞ , which shows that a ∈ R + H ∞ resp. a ∈ R + Hp∞ .
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2.59. Corollary. (a) R + Hp∞ is an algebra. (b) If a ∈ R + H ∞ , then T (a) ∈ Φ(H p ) ⇐⇒ a ∈ GL∞ and a−1 ∈ R + H ∞ . (c) If a ∈ R + Hp∞ , then T (a) ∈ Φ(p ) ⇐⇒ a ∈ GL∞ and a−1 ∈ R + Hp∞ . Proof. (a) This follows from Theorem 2.58 together with the formula (2.19). (b), (c) Combine the reasoning in the proof of Theorem 2.50 with Theorem 2.58 (also take into account 1.12(a), (iii)). 2.60. Theorem. Let 1 < p < ∞ and a ∈ Cp + Hp∞ . Then T (a) ∈ Φ(p ) ⇐⇒ a ∈ G(Cp + Hp∞ ). If T (a) ∈ Φ(p ), then T (a−1 ) is a regularizer of T (a). Proof. If a and a−1 are in Cp + Hp∞ , then H(a) and H(a−1 ) are compact, and so the argument used in the proof of Theorem 2.50 gives the implication “⇐=” and shows that T (a−1 ) is a regularizer of T (a). Now suppose T (a) ∈ Φ(p ). Then, by Theorem 2.30(b), a ∈ GM p and it remains to prove that a−1 ∈ Cp + Hp∞ . By the deﬁnition of Cp , there are functions bn ∈ R+Hp∞ such that a−bn M p → 0 as n → ∞. If n is suﬃciently large, then T (bn ) ∈ Φ(p ) (by 1.12(d)) and so Corollary 2.59 implies that p p ∞ b−1 n ∈ R+Hp . But if bn converges to an element a ∈ GM in the norm of M , −1 −1 p −1 then bn converges to a in the norm of M . Hence a ∈ closM p (R + Hp∞ ) and since Cp + Hp∞ is closed, it follows that a−1 ∈ Cp + Hp∞ . Our next concern is an invertibility criterion for C + H ∞ . We ﬁrst need a property of the harmonic extension. Recall that, for f ∈ L∞ and 0 < r < 1, the function fr ∈ C is deﬁned by fr (eiθ ) := f(reiθ ). 2.61. Lemma. If c ∈ C and a ∈ L∞ , then (ca)r − cr ar ∞ → 0 as r → 1 − 0. Proof. Since cr − c∞ → 0 it suﬃces to show that (ca)r − car ∞ → 0. Let kr denote the Poisson kernel (see 1.37). Then ( 2π 1 (ca)r (eiθ ) − c(eiθ )ar (eiθ ) = [c(eit ) − c(eiθ )]a(eit )kr (θ − t) dt. 2π 0 Given ε > 0 there is a δ > 0 such that c(eit ) − c(eiθ ) < ε whenever t − θ < δ and so + +( ( 2π + + + + it iθ it ≤ ε [c(e ) − c(e )]a(e )k (θ − t) dt a(eit )kr (θ − t) dt + + r + + t−θ<δ 0 ≤ 2πεa∞ .
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83
If r is suﬃciently close to 1, then +( + ( + + + + it iθ it [c(e ) − c(e )]a(e )kr (θ − t) dt+ ≤ 2c∞ a∞ kr (τ ) dτ + + t−θ>δ + τ >δ ≤ 2c∞ a∞ ε. This completes the proof.
2.62. Theorem (Douglas). (a) If a, b ∈ C + H ∞ then (ab)r − ar br ∞ → 0 as
r → 1 − 0.
a is bounded (b) Let a ∈ C + H ∞ . Then a ∈ G(C + H ∞ ) if and only if away from zero in some annulus near T, i.e., if and only if there exist δ > 0 and ε > 0 such that  a(z) > ε for 1 − δ < z < 1. Proof. (a) Let a = c + h, b = d + g, where c, d ∈ C and h, g ∈ H ∞ . It is clear that (hg)r = hr gr . From the preceding lemma we deduce that lim (cd)r −cr dr ∞ = lim (cg)r −cr gr ∞ = lim (hd)r −hr dr ∞ = 0,
r→1−0
r→1−0
r→1−0
and this gives the assertion at once. (b) Let a ∈ G(C + H ∞ ) and b = a−1 . Then, by (a), ar br − 1∞ → 0 as r → 1 − 0, and since br ∞ is bounded from above (by b∞ ), it follows that ar  must be bounded away from zero if r is close enough to 1. a(z) > ε for 1 − δ < z < 1. Then Now let a ∈ C + H ∞ and assume  a ≥ ε a.e. on T and so a ∈ GL∞ . Because a ∈ C + H ∞ , there are hn ∈ H ∞ such that χ−n hn → a as n → ∞ in the norm of L∞ . Part (a) and the fact 2n is bounded that a is bounded away from zero near T imply that each h away from zero in some annulus 1 − δn < z < 1. So 1/hn is bounded and 2n can there be written as analytic in 1 − δn < z < 1. Consequently, 1/h the sum of a function which extends to be bounded and analytic in D and a function which extends to be bounded and analytic in z > 1 − δn . This decomposition yields a representation of 1/hn as the sum of a function in H ∞ ∞ ∞ and a function in C. Thus h−1 and therefore χn h−1 n ∈ C +H n ∈ C +H . ∞ ∞ −1 −1 But if χ−n hn → a ∈ GL in the norm of L , then χn hn → a in the L∞ norm. Since C + H ∞ is closed, it follows that a−1 ∈ C + H ∞ . We ﬁnally compute the index of Toeplitz operators with C + H ∞ and Cp + Hp∞ symbols. By virtue of Corollary 2.40 this solves the invertibility problem for these operators. 2.63. Deﬁnition. Let a ∈ L∞ and suppose a is bounded away from zero in some annulus near T. Then there are ε > 0 and δ > 0 such that ar (eiθ ) ≥ ε for all r ∈ (1 − δ, 1) and θ ∈ [0, 2π). By 2.41(b), the mapping (1 − δ, 1) → Z, r → ind ar is continuous, and because (1 − δ, 1) is connected, ind ar must be constant for r ∈ (1 − δ, 1). That constant value of ind ar will be denoted by ind {ar }.
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2.64. Theorem. Let h ∈ H ∞ and 1 < p < ∞. Then T (h) ∈ Φ(H p ) if and only if h = bg where b is a ﬁnite Blaschke product and g ∈ GH ∞ . If T (h) is Fredholm, then h is bounded away from zero in some annulus near T and Ind T (h) = −ind {hr }. Proof. If h = bg with b ∈ GC and g ∈ GH ∞ , then T (h) = T (b)T (g) ∈ Φ(H p ) by Theorem 2.42 and Proposition 2.31. Conversely, suppose T (h) ∈ Φ(H p ). Then h ∈ G(C + H ∞ ) due to Corollary 2.55. In view of 1.41(a), 1.41(b), we have h = bSg, where b is a Blaschke product, S is a singular inner function, and g ∈ GH ∞ . Thus bS ∈ G(C + H ∞ ) and by virtue of Theorem 2.62(b), b(z)S(z) must be bounded away from zero in some annulus near T. Since the radial limit of S(z) vanishes at the points in the support of the singular measure deﬁning S, it follows that S = 1, and b(z) is bounded away from zero in an annulus near T only if b is a ﬁnite Blaschke product. The index formula can be derived as follows: Ind T (h) = Ind T (b) + Ind T (g) (Atkinson) = Ind T (b) (Proposition 2.31) = −ind b (Theorem 2.42) = −ind {br } (1.38(b) and 2.41(b)) = −ind {br } − ind {gr } (1.41(g)) = −ind {(bg)r } (Lemma 2.61 and 2.41(b)).
2.65. Theorem (Douglas). Let a ∈ C + H ∞ and 1 < p < ∞. Then T (a) is a is bounded away from zero in some annulus near T, in Φ(H p ) if and only if and in that case Ind T (a) = −ind {ar }. Proof. The Fredholm criterion follows by combining Corollary 2.55 and Theorem 2.62(b). So it remains to prove the index formula. There is a number ε > 0 with the following property: if b − a∞ < ε, then T (b) ∈ Φ(H p ), Ind T (b) = Ind T (a), b is bounded away from zero in some annulus near T, and ind {br } = ind {ar } (recall 1.12(d), 1.38(b), 2.41(b)). Among these b’s we can ﬁnd a b ∈ C + H ∞ of the form b = χ−n h, n ∈ Z+ , h ∈ H ∞ . Because T (b) = T (χ−n )T (h) ∈ Φ(H p ), it follows that T (h) ∈ Φ(H p ) and the preceding theorem gives (2.36) Ind T (h) = −ind {hr }. The desired index formula can now be veriﬁed as follows: Ind T (a) = Ind T (χ−n h) = Ind T (χ−n ) + Ind T (h) (Atkinson) = n + Ind T (h) (2.9) = n − ind {hr } (equality (2.36)) = −ind {(χ−n )r } − ind {hr }
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85
= −ind {(χ−n )r hr } (2.41(a)) = −ind {(χ−n h)r } (Lemma 2.61 and 2.41(b)) = −ind {ar }. a is 2.66. Theorem. Let a ∈ Cp + Hp∞ and 1 < p < ∞. If T (a) ∈ Φ(p ), then bounded away from zero in some annulus near T and Ind T (a) = −ind {ar }. Proof. Suppose 1 < p < 2 and let 1/p + 1/q = 1. From Theorem 2.60 we know that a ∈ G(Cp + Hp∞ ). Hence, due to 2.5(c), (d), a ∈ G(Cs + Hs∞ ) for all s ∈ [p, q] and thus, again by Theorem 2.60, T (a) ∈ Φ(s ) for all s ∈ [p, q]. Given an operator A ∈ Φ(s ) denote by αs (A) the dimension of the kernel of A in s and by Inds A the index of A considered as operator on s . Since p ⊂ 2 ⊂ q , we have αp (T (a)) ≤ α2 (T (a)),
α2 (T (a)) ≤ αq (T (a)),
hence Indp T (a) = αp (T (a)) − αq (T (a)) ≤ α2 (T (a)) − α2 (T (a)) = Ind2 T (a) = −ind {ar }. On repeating this argument with a−1 in place of a we arrive at the inequality Indp T (a−1 ) ≤ −ind {(a−1 )r } = ind {ar } (recall Theorem 2.62(a)). But T (a−1 ) is a regularizer of T (a). So Indp T (a) = −Indp T (a−1 ) ≥ −ind {ar }, and we ﬁnally get Indp T (a) = −ind {ar }. If 2 < p < ∞, then 1 < q < 2 and from the equality Indp T (a) = −Indq T (a) = ind {ar } = −ind {ar } we get the desired formula.
2.8 Local Methods for Scalar Toeplitz Operators 2.67. The local distance at a point. Given a ∈ L∞ and an open subarc U of T denote by aU the restriction of a to U regarded as an element of L∞ (U ). For τ ∈ T, let Uτ denote the family of all open subarcs of T containing the point τ . The local distance of a, b ∈ L∞ at τ ∈ T is deﬁned as distτ (a, b) = inf aU − bU L∞ (U ) . U ⊂Uτ
Clearly, if aU = bU for some neighborhood U of τ , then distτ (a, b) = 0. If the ﬁnite limits a(τ ±0) exist, then distτ (a, b) = 0 if and only if a(τ −0) = b(τ −0)
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and a(τ + 0) = b(τ + 0). In particular, if a and b are continuous at τ , then distτ (a, b) = 0 if and only if a(τ ) = b(τ ). Let Rτ denote the collection of all functions f ∈ C such that 0 ≤ f ≤ 1 on T and f is identically 1 in some neighborhood of τ (depending on f ). It is not diﬃcult to verify that distτ (a, b) = inf (a − b)c∞ . c∈Rτ
(2.37)
2.68. Theorem. Let 1 < p < ∞ and a ∈ L∞ . Assume that for each τ ∈ T there is an aτ ∈ L∞ such that distτ (a, aτ ) = 0 and T (aτ ) ∈ Φ(H p ). Then T (a) ∈ Φ(H p ). Proof. This theorem provides a good occasion of giving an application of Theorem 1.32. Put A = L(H p )/C∞ (H p ) and, for a ∈ L∞ , let T π (a) denote the coset in A containing T (a). If f ∈ C and b ∈ L∞ , then, by Proposition 2.14, T (f )T (b) = T (f b) − H(f )H()b) = T (b)T (f ) + H(b)H(f)) − H(f )H()b) and since H(f ) and H(f)) are compact on H p , we get T π (f )T π (b) = T π (f b) = T π (b)T π (f ).
(2.38)
For τ ∈ T, deﬁne Rτ as in 2.67 and put Mπτ := T π (f ) ∈ A : f ∈ Rτ . Using (2.38) it is easy to see that Mπτ is a localizing class in A. Given a of T we can choose a family {T π (fτ )}τ ∈T (fτ ∈ Rτ ), due to the compactness
ﬁnite subfamily {T π (fτj )}nj=1 such that g := j fτj ≥ 1 and Theorem 2.42(b) shows that T π (g) is invertible in A. Hence, {Mπτ }τ ∈T is a covering system of localizing classes in A. Also from (2.38) we deduce that T π (a) commutes with every T π (f ) in the union of all Mπτ . We ﬁnally have, again making use of (2.38), " " " " inf " T π (a) − T π (aτ ) T π (f )"A = inf "T π (a − aτ )f "A f ∈Rτ f ∈Rτ " " ≤ inf "T (a − aτ )f "L(H p ) ≤ cp inf (a − aτ )f ∞ = 0 f ∈Rτ
and analogously
f ∈Rτ
" " inf "T π (f ) T π (a) − T π (aτ ) "A = 0.
f ∈Rτ
In other words, T π (a) and T π (aτ ) are Mπτ equivalent from the left and from the right at each τ ∈ T. Since T π (aτ ) is invertible in A, it is of course Mπτ invertible from the left and from the right. Thus, we have collected together all the things allowing us to apply Theorem 1.32. The conclusion is that T π (a) is invertible in A and this yields the assertion.
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2.69. Theorem. Let 1 ≤ p < ∞ and a ∈ M p . Suppose for each τ ∈ T there exists an aτ ∈ M p such that distτ (a, aτ ) = 0 and T (aτ ) ∈ Φ(p ). Then T (a) ∈ Φ(p ). Proof. Since M 1 = M 1 = W , the case p = 1 is covered by Theorem 2.47. The case p = 2 is contained in the preceding theorem. Thus let 1 < p < 2; for 2 < p < ∞ the assertion can be proved analogously or can be obtained immediately from the case 1 < p < 2 by taking adjoints. Put A = L(p )/C∞ (p ) and for a ∈ M p let T π (a) := T (a) + C∞ (p ). It is clear that (2.38) holds for every b ∈ M p and every f ∈ Cp . Now, for τ = eiθ0 ∈ T, let Rτ := f ∈ C ∞ : 0 ≤ f ≤ 1, there is an ε > 0 (depending on f ) such that f (eiθ ) = 1 for θ − θ0  < ε, f (eiθ ) = 0 for θ − θ0  > 2ε, f is monotonically increasing for θ0 − 2ε < θ < θ0 − ε, f is monotonically decreasing for θ0 + ε < θ < θ0 + 2ε , Mπτ := T π (f ) ∈ A : f ∈ Rτ . As in the preceding proof it is readily seen that {Mπτ }τ ∈T is a covering system of localizing classes in A (note that C ∞ ∈ M p ) and that T π (a)T π (f ) = T π (f )T π (a) for every f ∈ Rτ . Since 1 < p < 2 and a, aτ ∈ M p , there is an r = rτ ∈ (1, p) such that a, aτ ∈ M r . Hence, " " " " inf " T π (a) − T π (aτ ) T π (f )" = inf "T π (a − aτ )f " A
f ∈Rτ
≤ inf (a − aτ )f M p ≤ inf (a − aτ )f 1−γ ∞ (a − f ∈Rτ
f ∈Rτ
A
f ∈Rτ
aτ )f γM r ,
where γ = rp−2/(pr−2) and the last estimate results from 2.5(e). To prove that T π (a) is Mπτ equivalent from the left to T π (aτ ) it therefore remains to show that (a − aτ )f M r is bounded by a constant, K, as f varies over Rτ . But from 2.5(f) we obtain (a − aτ )f M r ≤ a − aτ M r f M r ≤ aM r + aτ M r sp f ∞ + V1 (f ) ≤ aM r + aτ M r sp · 3 =: K, since f ∞ = 1 and V1 (f ) = 2 for every f ∈ Rτ . Theorem 1.32 now completes the proof. 2.70. Remark. Let X be a Banach space and A ∈ L(X). Then A is said to be leftFredholm (resp rightFredholm) if A + C∞ (X) is leftinvertible (resp. rightinvertible) in L(X)/C∞ (X). A look at Theorem 1.32 shows that in the preceding two theorems the requirement that T (aτ ) be Fredholm for each τ ∈ T can be replaced by the requirement that T (aτ ) be leftFredholm (resp. rightFredholm) in order to deduce that T (a) be leftFredholm (resp. rightFredholm).
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The relation between left and rightFredholm operators and Φ± operators is clariﬁed by the following result of Yood [584]. (a) A ∈ L(X) is leftFredholm if and only if A ∈ Φ+ (X) and if Im A is a complemented subspace of X. (b) A ∈ L(X) is rightFredholm if and only if A ∈ Φ− (X) and if Ker A is a complemented subspace of X. Thus, in general, a Φ+ (Φ− )operator need not be leftFredholm (rightFredholm). However, for a bounded Hilbert space operator to be a Φ+ (Φ− )operator is equivalent to being leftFredholm (rightFredholm). 2.71. Deﬁnition. A function a ∈ L∞ is called sectorial on a subarc U of T if there is an ε > 0 and a c ∈ C of modulus 1 such that Re (ca) ≥ ε a.e. on U . A function a ∈ L∞ is said to be locally arcwise sectorial if for each τ ∈ T there is a subarc Uτ ∈ Uτ such that a is sectorial on Uτ . Since T is compact, a function a ∈ L∞ is locally arcwise sectorial if and only if T can be covered by a ﬁnite number of open subarcs Ui such that a is sectorial on each Ui . 2.72. Theorem. If a ∈ L∞ is locally arcwise sectorial then T (a) is Fredholm on H 2 . Proof. Immediate from Theorems 2.68 and 2.17. Remark. The index of a Toeplitz operator generated by a locally arcwise sectorial symbol is, loosely speaking, minus the winding number of the “sectorial cloud” associated with the symbol. We do not make precise what this means, but shall later provide another way of computing the index, namely, via the harmonic extension of the symbol. 2.73. The algebra P C. Let P C0 denote the collection of all piecewise continuous functions on T which have at most ﬁnitely many jumps. The closure of P C0 in L∞ is denoted by P C. A function a ∈ P C possesses ﬁnite limits a(t ± 0) everywhere on T and there are at most countably many t ∈ T such that a(t − 0) = a(t + 0). Note that P C is a C ∗ algebra of L∞ . Given a ∈ P C0 deﬁne a function a2 : T × [0, 1] → C by the formula a2 (t, µ) = (1 − µ)a(t − 0) + µa(t + 0)
(t ∈ T,
µ ∈ [0, 1]).
(2.39)
The range of a2 is a continuous closed curve with a natural orientation; it is obtained from the (essential) range of a by ﬁlling in the straight line segment [a(t − 0), a(t + 0)] for each t ∈ T at which a has a jump. If this curve does not pass through the origin, we let ind a2 denote its winding number with respect to the origin. A more precise deﬁnition of ind a2 is as follows. With each ﬁnite subset S of T we associate a function ωS : T → T × [0, 1]. To construct this function let S = {eiθ1 , . . . , eiθR }, 0 ≤ θ1 < . . . < θR < 2π, put θR+1 := θR + 2π, and for j = 1, . . . , R let
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θj + θj+1 θj + ϕj ϕj + θj+1 , ϕj = , ϕj = . 2 2 2 Then deﬁne, for j = 1, . . . , R and 0 ≤ λ ≤ 1, 1 i(θj +λ(ϕj −θj )) iθj 1 )= e , + λ , ωS (e 2 2 ϕj =
ωS (ei(ϕj +λ(ϕj −ϕj )) ) = (ei(θj +λ(θj+1 −θj )) , 1 − λ), i(ϕ +λ(θj+1 −ϕ )) iθj+1 1 j j ωS (e )= e , λ . 2 Given a ∈ P C0 denote by S(a) the ﬁnite subset of T formed by the points at which a has a jump. It is easily seen that a2 (t, µ) = 0 for all (t, µ) ∈ T×[0, 1] if and only if the continuous function a2 ◦ωS(a) : T → C does not vanish on T. In that case ind a2 is deﬁned as ind (a2 ◦ ωS(a) ), where the latter “ind ” refers to the index as it was deﬁned in 2.41. For a ∈ P C deﬁne a2 : T × [0, 1] → C again by (2.39). It is not diﬃcult to see that the origin belongs to the range of a2 if and only if there is a sequence of functions an ∈ P C0 such that a−an ∞ → 0 and dist(0, range(an )2 ) → 0 as n → ∞. If a2 (t, µ) = 0 for all (t, µ) ∈ T × [0, 1], we choose any sequence of functions an ∈ P C0 with a − an ∞ → 0 as n → ∞ and deﬁne ind a2 as lim ind (an )2 . It can be easily seen that this limit always n→∞ exists and that it does not depend on the particular choice of the sequence {an }. 2.74. Theorem. Let a ∈ P C. Then T (a) ∈ Φ(H 2 ) ⇐⇒ a2 (t, µ) = 0
∀ (t, µ) ∈ T × [0, 1].
If T (a) is Fredholm, then Ind T (a) = −ind a2 . Proof. If a2 (t, µ) = 0 for all (t, µ) ∈ T×[0, 1], then a is locally arcwise sectorial and therefore T (a) ∈ Φ(H 2 ) due to Theorem 2.72. Our next objective is to prove the index formula. Thus, let a ∈ P C and a2 (t, µ) = 0 for all (t, µ) ∈ T × [0, 1]. If b ∈ P C0 is suﬃciently close to a in the L∞ norm, then Ind T (a) = Ind T (b), b2 (t, µ) = 0 for all (t, µ) ∈ T × [0, 1] and ind a2 = ind b2 (the latter fact per deﬁnitionem!). So it remains to show that Ind T (b) = −ind b2 . Let t1 , . . . , tn denote the points on T at which b has jumps. Choose suﬃciently small neighborhoods U1 , . . . , Un ⊂ T of the points t1 , . . . , tn and put U = U1 ∪ . . . ∪ Un . Then deﬁne c ∈ C as follows: let c = b on T \ U and on Ui let c be any continuous function such that c(Ui ) = b(Ui ) ∪ [b(ti − 0), b(ti + 0)]. The function d = b/c equals 1 on T \ U and it is easy to see that d is sectorial on U if only the neighborhoods U1 , . . . , Un have been chosen suﬃciently small. Thus, by (2.18), ) T (b) = T (cd) = T (c)T (d) + H(c)H(d) with H(c) compact (Theorem 2.42(a)) and T (d) invertible (Theorem 2.17). It follows that Ind T (b) = Ind T (c), and since ind c = ind b2 , Theorem 2.42(b) completes the proof of the index formula.
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We now prove the implication “=⇒”. Let T (a) ∈ Φ(H 2 ) but assume there is a (t0 , µ0 ) ∈ T × [0, 1] such that a2 (t0 , µ0 ) = 0. Then T (b) ∈ Φ(H 2 ) whenever a − b∞ is suﬃciently small and among these b’s there is a b ∈ P C0 such that b2 (t, µ) = 0. If b − c∞ and b − d∞ are small enough, then T (c) and T (d) are Fredholm and Ind T (b) = Ind T (c) = Ind T (d),
(2.40)
but it is easily seen that one can ﬁnd such functions c and d in P C0 which satisfy c2 (t, µ) = 0, d2 (t, µ) = 0 ∀ (t, µ) ∈ T × [0, 1] and ind c2 − ind d2 = 1. From the index formula proved above we get Ind T (d) = Ind T (c) + 1 which contradicts (2.40). Remark. The perturbation argument used in this proof also applies to show that a2 (t, µ) = 0 for all (t, µ) ∈ T × [0, 1] if T (a) is a Φ+  or Φ− operator on the space H 2 . The following theorem is the “essentialization” of Theorem 2.20 and forms the basis for another local approach. 2.75. Theorem (Douglas/Sarason). Let ϕ ∈ L∞ be a unimodular function. Then (a) T (ϕ) ∈ Φ+ (H 2 ) ⇐⇒ distL∞ (ϕ, C + H ∞ ) < 1; (b) T (ϕ) ∈ Φ− (H 2 ) ⇐⇒ distL∞ (ϕ, C + H ∞ ) < 1; (c) T (ϕ) ∈ Φ(H 2 ) ⇐⇒ distL∞ (ϕ, G(C + H ∞ )) < 1. Proof. We abbreviate distL∞ to dist. (a) Let T (ϕ) ∈ Φ+ (H 2 ). If dim Ker T (ϕ) = 0, then T (ϕ) is leftinvertible and so Theorem 2.20(a) gives dist(ϕ, H ∞ ) < 1. If dim Ker T (ϕ) = n > 0, then dim Ker T ∗ (ϕ) = 0 by Theorem 2.38. Thus, T (ϕ) is Fredholm of index n > 0. It follows that T (ϕχn ) is invertible, whence dist(ϕχn , H ∞ ) < 1 by Theorem 2.20(a), and thus dist(ϕ, χ−n H ∞ ) < 1. The proof of the implication “=⇒” is complete. Now suppose dist(ϕ, C + H ∞ ) < 1. Then dist(ϕχn , H ∞ ) < 1 for some n ≥ 0, and Theorem 2.20(a) yields the leftinvertibility of T (ϕ)T (χn ). Since T (χn ) is Fredholm, it results that T (ϕ) is leftFredholm. This proves the implication “⇐=”. (b) Take adjoints and apply (a). (c) If T (ϕ) is Fredholm, then T (χn ϕ) is invertible for some n ∈ Z (Corollary 2.40), hence dist(χn ϕ, GL∞ ) < 1 (Theorem 2.20(c)), and thus dist(ϕ, G(C + H ∞ )) < 1. On the other hand, if dist(ϕ, G(C +H ∞ )) < 1, then there are an n ≥ 0 and an h ∈ H ∞ ∩ G(C + H ∞ ) such that ϕ − χ−n h∞ < 1. Since T (h) ∈ Φ(H 2 )
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(Corollary 2.55), we have h = bg, where b is a ﬁnite Blaschke product and g is in GH ∞ (Theorem 2.64). Consequently, dist(b−1 ϕχn , GH ∞ ) < 1, and so Theorem 2.20(c) implies that T (b−1 )T (ϕ)T (χn ) is invertible. Because T (b−1 ) and T (χn ) are Fredholm, it follows that T (ϕ) must also be Fredholm. 2.76. Corollary. Let a ∈ GL∞ . Then T (a) is in Φ+ (H 2 ) (Φ− (H 2 ) resp. Φ(H 2 )) if and only if a = bs, where b ∈ C + H ∞ (b ∈ C + H ∞ resp. b ∈ G(C + H ∞ )) and s ∈ GL∞ is sectorial. Proof. Combine Theorem 2.75, Proposition 2.32, and Lemma 2.21. Theorem 2.75 and its Corollary 2.76 do not answer the question on the Fredholmness of Toeplitz operators in terms of the geometric data of the symbol. The purpose of what follows in the next sections is to combine Theorem 2.75 with Glicksberg’s theorem 1.22 in order to make the things little bit more geometrical. Before doing this we need a few pieces of information about the maximal ideal space of L∞ and its decompositions. 2.77. M (L∞ ). The maximal ideal space M (L∞ ) of the Banach algebra L∞ will be denoted by X. Since L∞ is a C ∗ algebra with respect to the involution a → a, where a(t) = a(t) (t ∈ T), L∞ is starisometrically isomorphic to C(X). The Gelfand transform of a function a ∈ L∞ will also be denoted by a. Thus, if a ∈ L∞ and x ∈ X, then a(x) = x(a). The topological space X is totally disconnected in the following sense: the closure of every open set is again open. 2.78. L∞ ﬁbers over M (C). The maximal ideal space of C is T: the general form of a functional in M (C) is given by vτ : C → C, where τ ∈ T. Let
f → f (τ ),
Xτ := Mτ (L∞ ) := x ∈ X : xC = vτ .
It is easy to see that the ﬁbers Xτ are homeomorphic to each other. Because Xτ , it follows that Xτ = ∅. This is also a consequence of 1.20(b) X = τ ∈T
(recall also 1.27(b)). Given a ∈ L∞ and an open subarc U of T denote by R(aU ) the spectrum of the restriction of a to U regarded as an element of L∞ (U ). Equivalently, R(aU ) is the set of all µ ∈ C such that {t ∈ U : a(t) − µ < ε} has positive (Lebesgue) measure for each ε > 0. Finally, recall that according to 2.67 distτ (a, b) = inf aU − bU L∞ (U ) U ∈Uτ
while in accordance with 1.21 distXτ (a, b) = max a(x) − b(x). x∈Xτ
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2.79. Proposition. (a) If a ∈ L∞ and τ ∈ T, then $ R(aU ). a(Xτ ) = U ∈Uτ ∞
(b) If a, b ∈ L
and τ ∈ T, then distτ (a, b) = distXτ (a, b). In particular, distτ (a, b) = 0 ⇐⇒ aXτ = bXτ .
Proof. (a) A little thought shows that µ ∈ /
U ∈Uτ
R(aU ) if and only if
∃ b, c ∈ L∞ : (a − µ)b + (χ1 − τ )c = 1.
(2.41)
If (2.41) holds, then (a(x) − µ)b(x) = 1 for all x ∈ Xτ , whence µ ∈ / a(Xτ ). On the other hand, if µ ∈ / a(Xτ ) then there is no x ∈ X such that a(x) = µ and χ1 (x) = τ . Thus, the closed ideal (a − µ)b + (χ1 − τ )c : b, c ∈ L∞ is not contained in any maximal ideal of L∞ , which gives (2.41). (b) Since distτ (a, b) = distτ (a − b, 0), it suﬃces to prove that max f (x) = distτ (f, 0)
x∈Xτ
for every f ∈ L∞ . By virtue of part (a), c(Xτ ) = {1} for every c ∈ Rτ (see 2.67). So max f (x) = max f (x)c(x) x∈Xτ
x∈Xτ
for every c ∈ Rτ , whence, by (2.37), max f (x) ≤ inf max f (x)c(x) = distτ (f, 0).
x∈Xτ
c∈Rτ x∈X
To establish the reverse inequality we need the following well known fact: if K1 ⊃ K2 ⊃ K3 ⊃ ... are compact nonempty subsets of a Hausdorﬀ space, if ∞
n=1
Kn ⊂ Ω, and if Ω is open, then there is an n0 such that Kn0 ⊂ Ω.
Now put M = max f (x). Given any ε > 0 we have, due to part (a), x∈Xτ ∞ $
R(f Un ) ⊂ z ∈ C : z < M + ε ,
n=1
where Un = {t ∈ T : t − τ  < 1/n}, and since each set R(f Un ) is compact and nonempty (as the spectrum of f Un ∈ L∞ (Un )), it follows that there is an U0 ∈ Uτ such that R(f U0 ) ⊂ {z ∈ C : z < M + ε}. Now it is clear that there exists a c0 ∈ Rτ such that f c0 ∞ < M + ε, and therefore distτ (f, 0) = inf max f (x)c0 (x) = max f (x)c0 (x) = f c0 ∞ < M + ε. c∈Rτ x∈X
x∈X
Since ε > 0 can be chosen arbitrarily, we get distτ (f, 0) ≤ M .
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93
Remark. Thus, a function a ∈ L∞ is continuous at a point τ ∈ T if and only if a(Xτ ) is a singleton. If a has a jump discontinuity at τ , then a(Xτ ) is a doubleton, but if a(Xτ ) is known to be a doubleton, then all one can say is that a has two essential cluster points at τ , which does, in general, not imply that a has a jump at τ . 2.80. QC. The largest C ∗ subalgebra of C + H ∞ is denoted by QC and is referred to as the algebra of quasicontinuous functions. Thus QC = (C + H ∞ ) ∩ (C + H ∞ ). Although H ∞ ∩ H ∞ is the set of constant functions, QC is strictly larger than C. Indeed, let 1 Ω = z = x + iy ∈ C : 0 < x < 1, −2 < y < sin x and let ω map D conformally onto Ω. Then ω ∈ H ∞ and Re ω ∈ C, whence Im ω = iRe ω − iω ∈ C + H ∞ . Since Im ω is a realvalued function, Im ω ∈ C + H ∞ . But Im ω is obviously discontinuous and therefore Im ω ∈ QC \ C. Since QC is a C ∗ algebra, we have, for c ∈ QC, c ∈ GQC ⇐⇒ c ∈ GL∞ . 2.81. L∞ ﬁbers over M (QC). If ξ ∈ M (QC), then by virtue of 1.20(b) (or 1.27(b)) the ﬁber Xξ = Mξ (L∞ ) is not empty. To every ξ ∈ M (QC) there corresponds a τ ∈ M (C) = T such that ξ ∈ Mτ (QC), and it is clear that Mξ (L∞ ) ⊂ Mτ (L∞ ). We have / Mτ (L∞ ) = Mξ (L∞ ). ξ∈Mτ (QC)
Since QC = C, the partition
/
M (L∞ ) =
Mξ (L∞ )
ξ∈M (QC)
is a proper reﬁnement of the partition / Mτ (L∞ ). M (L∞ ) = τ ∈T
Because the restriction of a function in C to a ﬁber Xτ (τ ∈ T) is constant, we have C + H ∞ Xτ = H ∞ Xτ
and
C + H ∞ Xξ = H ∞ Xξ .
(2.42)
We know from 1.27(c), (d) (in the setting Y = X, A = C +H ∞ , B = QC) that each maximal antisymmetric set for C + H ∞ is contained in some ﬁber Xξ , where ξ ∈ M (QC). Consequently, Corollary 1.23 implies that, for a ∈ L∞ ,
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2 Basic Theory
a ∈ C + H ∞ ⇐⇒ aXξ ∈ H ∞ Xξ
∀ ξ ∈ M (QC).
This in turn gives that, for a ∈ L∞ , a ∈ QC ⇐⇒ aXξ = const ∀ ξ ∈ M (QC).
(2.43)
Note that the implications “=⇒” are trivial. Let B be a C ∗ subalgebra of L∞ containing the constant functions. Then for each β0 ∈ M (B) the ﬁber Xβ0 = Mβ0 (L∞ ) is a peak set for B (recall 1.28). Indeed, because B is isometrically isomorphic to C(M (B)), there is an f ∈ B with f (β0 ) = 1 and 0 < f (β) < 1 for β = β0 , whence f Xβ0 = 1 and 0 < f < 1 on X \ Xβ0 . In particular, for τ ∈ T, Xτ is a peak set for C and therefore for C + H ∞ , and for ξ ∈ M (QC), Xξ is a peak set for QC and thus also for C + H ∞ . So we deduce from 1.28 that C + H ∞ Xτ and C + H ∞ Xξ are closed subalgebras of L∞ Xτ and L∞ Xξ , respectively, for every τ ∈ T and ξ ∈ M (QC). Taking into account (2.42) we arrive at the conclusion that the algebras H ∞ Xτ and H ∞ Xξ are closed. It also follows that the algebras QCXτ and QCXξ are closed. Clearly, QCXξ is the complex ﬁeld C, while 1.27(b) shows that M (QCXτ ) can be identiﬁed with Mτ (QC). We ﬁnally mention that both M (H ∞ Xτ ) (which can be identiﬁed with the ﬁber Mτ (H ∞ ) of H ∞ over τ ∈ T = M (CA ) \ D) and M (H ∞ Xξ ) are connected (τ ∈ T, ξ ∈ M (QC)). The connectedness of the ﬁrst space is shown in Hoﬀman’s book [284] and that the second space is connected was proved by Gorkin [239, Corollary 2.9]. 2.82. Deﬁnition. Let a ∈ L∞ and let F be a closed subset of X = M (L∞ ). The Toeplitz operator T (a) will be said to be F restricted invertible (left resp. rightinvertible) if there is a b ∈ L∞ such that aF = bF and T (b) is invertible (left resp. rightinvertible) on H 2 . If F is contained in some ﬁber Xτ (τ ∈ T), then T (a) is F restricted invertible (left resp. rightinvertible) if and only if there is a b ∈ L∞ such that aF = bF and T (b) is Fredholm (left resp. rightFredholm) on H 2 (recall Remark 2.70). This follows from Corollary 2.40 together with the fact that continuous functions restricted to Xτ are constants. Proposition 2.79 shows that Theorem 2.68 for p = 2 may also be stated as follows: if T (a) is Xτ restricted invertible for each τ ∈ T, then T (a) is Fredholm on H 2 . In this form the theorem was established by Douglas and Sarason [164] using a method which actually applies to prove the following much “more local” result. 2.83. Theorem (Axler). Let a ∈ L∞ and let B ⊂ C + H ∞ be a closed subalgebra of L∞ containing the constants. If T (a) is Srestricted invertible (left resp. rightinvertible) for each maximal antisymmetric set S for B, then T (a) is Fredholm (left resp. rightFredholm) on H 2 . Proof. Let T (a) be Srestricted leftinvertible for some S ⊂ X. Then there is a b ∈ L∞ such that aS = bS and T (b) is leftinvertible. By Theorem 2.30(a),
2.8 Local Methods for Scalar Toeplitz Operators
95
b ∈ GL∞ and hence a(x) = 0 for all x ∈ S. Proposition 2.32 implies that T (b/b) is leftinvertible and so Theorem 2.75(a) gives distX (b/b, C + H ∞ ) = distL∞ (b/b, C + H ∞ ) < 1. Because b/b equals a/a on S, we have distS (a/a, C + H ∞ ) < 1.
(2.44)
If (2.44) holds for each maximal antisymmetric set S for B, then it also holds for each maximal antisymmetric set S for C + H ∞ , since the latter ones are contained in the former ones (see 1.27(d)). Thus, Theorem 1.22 gives that distL∞ (a/a, C + H ∞ ) = distX (a/a, C + H ∞ ) < 1, from Theorem 2.75(a) we deduce that T (a/a) ∈ Φ+ (H 2 ) and once more applying Proposition 2.32 we see that T (a) ∈ Φ+ (H 2 ). The proof for the rightFredholmness is analogous. Finally, if T (a) is Srestricted invertible for each maximal antisymmetric set S for B, then, by what has already been proved, T (a) is in both Φ+ (H 2 ) and Φ− (H 2 ), and hence in Φ(H 2 ). 2.84. Deﬁnitions. Let F be a closed subset of X = M (L∞ ). A function a ∈ L∞ is said to be sectorial on F if there are an ε > 0 and a c ∈ C of modulus 1 such that Re (ca(x)) ≥ ε for all x ∈ F . If a is sectorial on F , then aF is obviously invertible in L∞ F , and it is easy to see that a is sectorial on F if and only if so is a/a. Moreover, for a ∈ L∞ to be sectorial on F it is necessary and suﬃcient that a(x) = 0 for all x ∈ F and distF (a/a, C) < 1. Now let B be a closed subalgebra of C + H ∞ containing the constants. A function a ∈ L∞ will be called locally sectorial over B if it is sectorial on each maximal antisymmetric set for B. The most important special cases are B = C + H ∞ , B = QC, B = C, and B = C. So, by virtue of 1.27(c), a ∈ L∞ is locally sectorial over QC (resp. C) if and only if it is sectorial on each ﬁber Xξ , ξ ∈ M (QC) (resp. Xτ , τ ∈ T). The functions that are sectorial in the sense of Deﬁnition 2.16 are just the functions which are sectorial on X = M (L∞ ) or, equivalently, locally sectorial over C. Finally, from 1.27(d) (with Y = X) we deduce that if B ⊂ A, then a locally sectorial over B =⇒ a locally sectorial over A. 2.85. Theorem. If a ∈ L∞ is locally sectorial over a closed subalgebra B of C + H ∞ containing the constants then T (a) is Fredholm on H 2 . Proof. The hypothesis implies that a ∈ GL∞ and that a is locally sectorial over C + H ∞ . Hence
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distS (a/a, C + H ∞ ) ≤ distS (a/a, C) < 1 for each maximal antisymmetric set S for C + H ∞ . Since the maximal antisymmetric sets for C + H ∞ are the same as those for C + H ∞ , Theorem 1.22 and Theorem 2.75(a), (b) can be combined to obtain that T (a/a) ∈ Φ+ (H 2 ) ∩ Φ− (H 2 ) = Φ(H 2 ) and Proposition 2.32 completes the proof.
The following proposition provides an idea of what the diﬀerent notions of local sectoriality involve. 2.86. Proposition. (a) If a ∈ L∞ is locally sectorial over C + H ∞ then a can be written as f s with f ∈ G(C + H ∞ ) and s ∈ GL∞ sectorial (on T). (b) Let B be a C ∗ subalgebra of L∞ between C and QC. Then a is locally sectorial over B if and only if a can be represented as a = bs with b ∈ GB and s ∈ GL∞ sectorial (on T). (c) For a ∈ L∞ the following are equivalent: (i) a is locally sectorial over C; (ii) a = cs with c ∈ GC and s is sectorial; (iii) a is locally arcwise sectorial. Proof. (a) Theorem 1.22 gives distL∞ (a/a, C + H ∞ ) < 1,
distL∞ (a/a, C + H ∞ ) < 1,
so Theorem 2.75 shows that distL∞ (a/a, G(C + H ∞ )) < 1 and Lemma 2.21 ends the proof. (b) One half of the assertion can be proved as in (a). On the other hand, if a = bs with b ∈ GB and s sectorial, then aS = (bS)(sS) for each maximal antisymmetric set S for B, and since these sets are just the ﬁbers Xβ , β ∈ M (B) (1.27(c)), bS is a nonzero constant and hence a is sectorial on S. (c) The implication (i) =⇒ (ii) follows from part (b) and the implication (iii) =⇒ (i) results from Proposition 2.79(a). Finally, if (ii) holds, then Re (γs(t)) ≥ ε for some ε > 0, some γ ∈ C, and for almost all t ∈ T. Hence, if τ ∈ T, Re γ/c(τ ) c(τ )s(t) ≥ ε > 0 for almost all t ∈ T and since c is continuous, Re
γ/c(τ ) c(t)s(t) ≥ ε > 0
for almost all t in some neighborhood of τ , which gives (iii).
2.8 Local Methods for Scalar Toeplitz Operators
97
Remark. Thus, Theorem 2.85 can also be proved as follows: if a ∈ L∞ is locally sectorial over C+H ∞ , then a = f s with f ∈ G(C+H ∞ ) and s sectorial, so T (a) = T (s)T (f )+ compact operator ((2.18) and Theorem 2.54), and since T (s) is invertible (Theorem 2.17) and T (f ) is Fredholm (Corollary 2.55), we conclude that T (a) is Fredholm. Our next concern is the index computation (and thus the solution of the invertibility problem) for Toeplitz operators whose symbol is locally sectorial over QC (or over any closed subalgebra of QC containing the constants). The key observations are Propositions 2.86(b) and the following generalization of Lemma 2.61. 2.87. Lemma (Sarason). If b ∈ QC and a ∈ L∞ , then (ba)r − br ar ∞ → 0 as
r → 1 − 0.
Proof. Let kr denote the Poisson kernel. Then + + ( 2π + 1 + [b(eit ) − br (eiθ )]a(eit )kr (θ − t) dt++ (ba)r (eiθ ) − br (eiθ )ar (eiθ ) = ++ 2π 0 ( a∞ 2π ≤ b(eit ) − br (eiθ )kr (θ − t) dt 2π 0 1/2 ( 2π a∞ it iθ 2 ≤ b(e ) − br (e ) kr (θ − t) dt 2π 0 1/2 a∞ (bb)r (eiθ ) − br (eiθ )br (eiθ ) = . 2π But if b ∈ QC, then b ∈ C + H ∞ and b ∈ C + H ∞ , whence, by virtue of Theorem 2.62(a), (bb)r − br br ∞ → 0 as r → 1 − 0. 2.88. Theorem. If a ∈ L∞ is locally sectorial over QC, then T (a) ∈ Φ(H 2 ), the harmonic extension a is bounded away from zero in some annulus near T, and Ind T (a) = −ind {ar }. Proof. Due to Proposition 2.86(b) we have a = bs, where b ∈ GQC and s ∈ GL∞ is sectorial (on T). So T (a) = T (b)T (s)+ compact operator ((2.18) and Theorem 2.54), and since T (b−1 ) is a regularizer of T (b) ((2.18) and Corollary 2.55) and T (s) is invertible (Theorem 2.17), it follows that T (a) ∈ Φ(H 2 ). Of course, the same conclusion might be also drawn from Theorem 2.85. Lemma 2.87 shows that ar − br sr ∞ → 0 as r → 1 − 0. If Re s ≥ ε > 0 a.e. on T, then Re s ≥ ε > 0 in D, because the Poisson kernel is positive. Hence, if s is sectorial then s is bounded away from zero in D and ind {sr } = 0.
(2.45)
If b ∈ GQC, then b is bounded away from zero in some annulus near T by Theorem 2.62. Thus, under our hypothesis, a is bounded away from zero in some annulus near T.
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The index formula can now be veriﬁed as follows: Ind T (a) = Ind T (b) + Ind T (s) = Ind T (b) = −ind {br } (Theorem 2.65) = −ind {br } − ind {sr } (by (2.45)) = −ind {(bs)r } (2.41(a), Lemma 2.87, 2.41(b)).
We ﬁnally show that for a relatively large class of symbols local sectoriality is not only suﬃcient but also necessary for the Fredholmness of the corresponding Toeplitz operator. 2.89. Deﬁnition. Let B be a C ∗ subalgebra of QC containing the constants. We denote by P2 B the collection of all functions a ∈ L∞ which take at most two values on each ﬁber Xξ , ξ ∈ M (B). For instance, P2 C contains P C, and if E any measurable subset of T, then the functions aχE + b (χE being the characteristic function of E, a and b being in C) belong to P2 C. Further, P2 QC contains all functions of the form a=
n
pi qi ,
pi ∈ P2 C,
qi ∈ QC.
i=1
We shall see later that P QC, the closed subalgebra of L∞ generated by P C and QC, is also a subset of P2 QC (see Remark 1 of 3.36). 2.90. Lemma. Let B = C or B = QC and let ξ ∈ M (B). If ϕ ∈ L∞ is unimodular and ϕ(Xξ ) is a pair of antipodal points, then distXξ (ϕ, H ∞ ) = 1. Proof. Without loss of generality suppose ϕ(Xξ ) is the doubleton {−1, 1}. Assume there is an h ∈ H ∞ such that max ϕ(x) − h(x) ≤ 1 − δ < 1. Put E± := {x ∈ Xξ : ϕ(x) = ±1}. So 1 − h(x) ≤ 1 − δ
∀ x ∈ E+ ,
x∈Xξ
1 + h(x) ≤ 1 − δ
∀ x ∈ E− .
(2.46)
Let B denote the restriction algebra L∞ Xξ , which is closed by 1.28. From (2.46) we see that the spectrum of hXξ in B is contained in the union of two disks with center at +1 and −1 and radius 1 − δ, and, moreover, that each of these two disks contains a point of that spectrum, i.e., that there are z1 , z2 ∈ C such that Re z1 < 0, Re z2 > 0, z1 ∈ spB (hXξ ), z2 ∈ spB (hXξ ). Now put A := H ∞ Xξ . From 2.81 we know that A is closed and M (A) is connected (Hoﬀman for B = C and Gorkin for B = QC). Consequently, spA (hXξ ) = h(M (A)) is a connected subset of C. By virtue of 1.16(b), spA (hXξ ) is the union of spB (hXξ ) and a (possibly empty) collection of bounded connected
2.9 Matrix Symbols
99
components of spB (hXξ ). However, the set {z ∈ C : Re z < δ/2} is contained in the unbounded complement of spB (hXξ ), hence {z ∈ C : Re z < δ/2} ∩ spA (hXξ ) = ∅. But this is a contradiction, since together with z1 and z2 some points of the stripe {Re z < δ/2} must belong to the (connected!) set spA (hXξ ). 2.91. Theorem. Let B = C or B = QC and let a ∈ P2 B. Then T (a) ∈ Φ(H 2 ) ⇐⇒ a is locally sectorial over B. Proof. The implication “⇐=” is immediate from Theorem 2.85 (or can be established as in the proof of Theorem 2.88). So we are left with the reverse implication. Let T (a) ∈ Φ(H 2 ). Then T (a/a) ∈ Φ(H 2 ) by Proposition 2.32, and hence distL∞ (a/a, C + H ∞ ) < 1 by Theorem 2.75(a). It follows that distXξ (a/a, H ∞ ) < 1 for each ξ ∈ M (B) (recall (2.42)). The preceding lemma shows that the singleton or doubleton (a/a)(Xξ ) cannot be a doubleton consisting of two antipodal points and this is equivalent to saying that a/a (and thus a itself) is sectorial on Xξ .
2.9 Matrix Symbols We conclude this chapter by stating some facts on Toeplitz operators with matrix symbols. We here conﬁne ourselves to settling a few problems the solution of which merely requires minor modiﬁcations of the methods developed above for scalar Toeplitz operators. The more delicate questions on block Toeplitz operators will be deferred to the forthcoming chapters. ∞ 2.92. Deﬁnitions. Given a matrix function a = (ajk )N j,k=1 ∈ LN ×N the mulp tiplication operator M (a) is deﬁned on LN (1 < p < ∞) by
M (a) :
LpN
→
LpN ,
(fk )N k=1
→
N
N M (akj )fj
j=1
k=1
p (recall the notations introduced in 1.13). For a ∈ MN ×N the operator M (a) p p is deﬁned on N (Z) := ( (Z))N analogously. Here LpN and pN (Z) can be regarded as being equipped with the norms
f LpN :=
N
fj Lp ,
f = (fj )N j=1 ,
j=1
ϕpN (Z) :=
N
ϕj p (Z) ,
j=1
or with any norms equivalent to those ones.
ϕ = (ϕj )N j=1 ,
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Similarly, if a ∈ L∞ N ×N , the Toeplitz operator T (a) and the Hankel operap (1 < p < ∞) by tor H(a) are given on HN p p T (a) : HN → HN ,
(fk )N k=1 →
N
N T (akj )fj
p p H(a) : HN → HN ,
(fk )N k=1 →
N
, k=1
j=1
N H(akj )fj
, k=1
j=1
p p and for a ∈ MN ×N an analogous deﬁnition is made for T (a) and H(a) on N (1 ≤ p < ∞). p and pN can be viewed as subspaces of LpN and pN (Z), The spaces HN respectively. Thus, whenever a norm in the latter two spaces is speciﬁed it will always be clear what the norm in the ﬁrst two spaces is. Both the Riesz projection P : Lp → H p and the canonical projection P : p (Z) → p extend in a natural way to LpN and pN (Z). We denote these projections again by P . Thus, P = diag (P, . . . , P ). If the norm on pN (Z) is given by 1/p N N p ϕj p (Z) or ϕj p (Z) , j=1
j=1
then obviously P L(pN (Z)) = 1. In the same fashion the projection Q and the ﬂip operator J are deﬁned on LpN and pN (Z). We then have T (a) = P M (a)P Im P,
H(a) = P M (a)QJIm P,
etc. In particular, formulas (2.18)–(2.20) remain true for the matrix case without any changes. The matrix function a will always be referred to as the symbol of the corresponding operator. With every ϕ ∈ pN we may associate a CN valued sequence ψ ∈ p (Z+ , CN ) as follows: p j ∞ p if ϕ = (ϕk )N k=1 ∈ N where ϕk = {ϕk }j=0 ∈ , j N p then ψ = {ψj }∞ j=0 ∈ (Z+ , CN ) where ψj = (ϕk )k=1 ∈ CN .
So the Toeplitz operator on pN can also be thought of as acting on p (Z+ , CN ) by the rule ! #∞ ∞ → a ψ , T (a) : {ψj }∞ j−k k j=0 k=0
j=0
where an (n ∈ Z) denotes the N × N matrix ((ajk )n )N j,k=1 formed by the p N Fourier coeﬃcients of a = (ajk )j,k=1 ∈ MN ×N . Therefore Toeplitz operators on pN are sometimes also called block Toeplitz operators. It is clear that M (a)
2.9 Matrix Symbols
101
and H(a) can be viewed as acting on p (Z, CN ) and p (Z+ , CN ), respectively, in a similar manner. p We deﬁne norms on L∞ N ×N and MN ×N by := M (a)L(L2N ) , aL∞ N ×N
aMNp ×N := M (a)L(pN ) .
p Provided with these norms L∞ N ×N and MN ×N (1 ≤ p < ∞) are (noncommutative) Banach algebras with identity I. Clearly, a ∈ GL∞ N ×N resp. p p ∞ a ∈ GMN ×N if and only if there is a b ∈ LN ×N resp. b ∈ MN ×N such that ∞ and ab = ba = I. It is also obvious that a ∈ GL∞ N ×N ⇐⇒ det a ∈ GL p p a ∈ GMN ×N ⇐⇒ det a ∈ GM . By virtue of 1.29(a),
= ess sup a(t)L(CN ) = max a(x)L(CN ) . aL∞ N ×N t∈T
x∈X
2.93. Theorem. (a) If 1 < p < ∞ and a ∈ L∞ N ×N , then M (a) ∈ Φ± (LpN ) ⇐⇒ M (a) ∈ GL(LpN ) ⇐⇒ a ∈ GL∞ N ×N , p T (a) ∈ Φ(HN ) =⇒ a ∈ GL∞ . N ×N p (b) If 1 ≤ p < ∞ and a ∈ MN ×N , then p M (a) ∈ Φ(pN (Z)) ⇐⇒ M (a) ∈ GL(pN (Z)) ⇐⇒ a ∈ GMN ×N , p p ∞ T (a) ∈ Φ(N ) =⇒ a ∈ GMN ×N =⇒ a ∈ GLN ×N .
Proof. The assertions about the multiplication operators can be proved by the same argument as in the scalar case (2.28, 2.29). After deﬁning the bilateral shift U on LpN or pN (Z) as U = M (χ1 I) = diag (U, . . . , U ), the proof of Theorem 2.30 also works in the matrix case. 1 Note that the implication T (a) ∈ Φ(1N ) =⇒ a ∈ GMN ×N can also be veriﬁed by invoking Theorem 1.14(c). Indeed, we have, by (2.18), T (f )T (g) − T (g)T (f ) = −H(f )H() g ) + H(g)H(f))
(2.47)
for all f, g ∈ M 1 , and since M 1 = W , Theorem 2.47(a) shows that the occurring Hankel operators are compact. Important remark. A decisive distinction between the scalar case (N = 1) and the matrix case (N > 1) is that a Fredholm block Toeplitz operator of index zero is not necessarily invertible (compare Corollary 2.40). For instance, if a = diag (χ1 , χ−1 ), then obviously T (a) ∈ Φ(H22 ) although both dim Ker T (a) end dim Coker T (a) equal 1. 2.94. Theorem. (a) Let a ∈ (C + H ∞ )N ×N and 1 < p < ∞. Then H(a) is p ) and in C∞ (HN p ) ⇐⇒ det a ∈ G(C + H ∞ ); T (a) ∈ Φ(HN
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2 Basic Theory
p if T (a) is Fredholm on HN , then T (a−1 ) is a regularizer of T (a) and
Ind T (a) = Ind T (det a) = −ind {(det a)r }. (b) Let a ∈ (Cp + Hp∞ )N ×N and 1 < p < ∞. Then H(a) ∈ C∞ (pN ) and T (a) ∈ Φ(pN ) ⇐⇒ det a ∈ G(Cp + Hp∞ ); if T (a) is Fredholm on pN , then T (a−1 ) is a regularizer of T (a) and Ind T (a) = Ind T (det a) = −ind {(det a)r }. Proof. The compactness of the Hankel operators can be shown as in the scalar case. This and identity (2.47) allow us the application of Theorem 1.14(c). p resp. pN if and only if T (det a) What results is that T (a) is Fredholm on HN is so on H p resp. p , which, by Corollary 2.55 and Theorem 2.60, is equivalent to the invertibility of det a in C + H ∞ resp. Cp + Hp∞ . That T (a−1 ) is a regularizer of T (a) follows from (2.18). In view of Theorems 2.65 and 2.66 it remains to show that Ind T (a) = Ind T (det a). To this end approximate a suﬃciently close in the norm of L∞ N ×N p ∞) ∞) resp. MN by b ∈ (R+H resp. b ∈ (R+H . Then, by 1.12(d), N ×N N ×N p ×N Ind T (a) = Ind T (b),
Ind T (det a) = Ind T (det b).
Taking into account identity (2.47) and Theorem 2.58 we see that the entries of T (b) commute modulo ﬁniterank operators. So Theorem 1.15(b) gives Ind T (b) = Ind T (det b). Remark. Obvious modiﬁcations of the proof show that part (b) is true for p = 1 if only Cp + Hp∞ is replaced by W = M 1 . p
2.95. Theorem. Let a ∈ MN ×N and 1 ≤ p < ∞. Assume for each τ ∈ T p
there exists an aτ ∈ MN ×N such that aτ Xτ = aXτ and T (aτ ) ∈ Φ(pN ). Then T (a) ∈ Φ(pN ). Proof. This follows from applying Theorem 1.32. Put A = L(pN )/C∞ (pN ) and, p for a ∈ MN ×N , denote the coset of A containing T (a) by T π (a). For τ ∈ T, deﬁne Rτ as in the proof of Theorem 2.69 and put Fτ = {diag (f, . . . , f ) : f ∈ Rτ },
Mπτ = {T π (ϕ) : ϕ ∈ Fτ }.
It is readily seen that {Mπτ }τ ∈T is a covering system of localizing classes in A and that (2.48) T π (a)T π (ϕ) = T π (aϕ) = T π (ϕa) = T π (ϕ)T π (a) for every ϕ ∈ Fτ . As in the proof of Theorem 2.69 one can show that T π (a) τ
and T π (aτ ) are Mπτ equivalent from the left and from the right for each τ ∈ T. Theorem 1.32 then gives the assertion.
2.10 Notes and Comments
103
∗ 2.96. Theorem. Let a ∈ L∞ N ×N and 1 < p < ∞, and let B be a C subalgebra of QC containing the constants. Suppose for each ξ ∈ M (B) there is an aξ in p p L∞ N ×N such that aξ Xξ = aXξ and T (aξ ) ∈ Φ(HN ). Then T (a) ∈ Φ(HN ). p p Proof. We shall derive this from Theorem 1.32. Put A = L(HN )/C∞ (HN ) p ∞ π and, for a ∈ LN ×N , let T (a) := T (a) + C∞ (HN ). For ξ ∈ M (B), deﬁne Rξ as the collection of all f ∈ B such that 0 ≤ f ≤ 1 and f is identically 1 in some neighborhood Uξ ⊂ M (B) of ξ. Then let
Fξ = {diag (f, . . . , f ) : f ∈ Rξ }, Mπξ = {T π (ϕ) : ϕ ∈ Fξ }. It is clear that (2.48) holds for ϕ ∈ Fξ and it is easy to see that {Mπξ }ξ∈M (B) ξ
is a covering system of localizing classes in A. We now show that T π (a) and T π (aξ ) are Mπξ equivalent from the left. Choose ε > 0, set b = a − aξ , and let U = η ∈ M (B) : b(x)L(CN ) < ε ∀ x ∈ Xη . Assume U is not an open subset of M (B). Then there is an η ∈ U and a net ηi in M (B) such that ηi → η and such that for each i, there exists xi ∈ Xηi with b(xi ) ≥ ε. Taking a subnet, we can suppose that there is an x ∈ X such that xi → x. Since the mapping (y ∈ X) → (yB ∈ M (B)) is continuous, it follows that x ∈ Xη . But b(x) ≥ ε, which is impossible for x ∈ Xη and η ∈ U . This contradiction shows that U is open. It is clear that ξ ∈ U , and hence there is a closed subset S of M (B) such that ξ ∈ S ⊂ U . So there exists an f ∈ B satisfying 0 ≤ f ≤ 1, f S = 1, f (M (B) \ U ) = 0. If we let ϕ = diag (f, . . . , f ), then ϕ ∈ Fξ and " " π " T (a) − T π (aξ ) T π (ϕ)" = T π (bϕ) ≤ cp bϕL∞ , N ×N where cp = P L(LpN ) . Since b(x)ϕ(x) < ε for all x ∈ X, and as ε > 0 can be chosen arbitrarily, it results that T π (a) and T π (aξ ) are Mπξ equivalent from the left, as desired. It can be shown analogously that they are Mπξ equivalent from the right. Now Theorem 1.32 completes the proof.
2.10 Notes and Comments 2.2–2.4. These facts are well known. The proof of Theorem 2.2 is patterned after Halmos [265, Problems 50 and 193]. 2.5. These results form only a little part of what is known about multipliers on p and similar spaces, and signiﬁcant contributions to this topic have been made by many people. For more about this see, e.g., H¨ ormander [288], Hirschman [280], Zygmund [591], Nikolski [365], Gohberg, Krupnik [232], Duduchava [169], [173], Verbitsky [538]. Inequality 2.5(f) goes back to S. B. Stechkin.
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2.6. Commemorative articles on life and work of Otto Toeplitz may be found in Gohberg et al. [235]. 2.7. See Brown, Halmos [125] or Halmos [265] for H 2 and Duduchava [169] for p . Let a ∈ C. Then neither the spectral radius of T (a) ∈ L(H p ) nor the spectral radius of T (a) + C∞ (H p ) ∈ L(H p )/C∞ (H p ) depend on p (see Theorem 2.42). However, one can show that T (a)L(H p ) depends on p. In particular, T (χ−1 )L(H p ) > 1 for all p = 2. Does the essential norm T (a)Φ(H p ) depend on p (see 2.27)? This problem is equivalent to the following question: Is T (χ−1 )Φ(H p ) equal 1 for all p ∈ (1, ∞)? For more about these things see B¨ottcher, Krupnik, Silbermann [98]. 2.9. We recorded these trivialities mainly to ﬁx some notation and to have a reference. It is well known that shift operators have many remarkable and nontrivial properties. We only mention the following. Given any Hilbert space E let 2 (E) refer to the Hilbert space of all Evalued sequences {xn }n∈Z+ such
(−1) that xn 2E < ∞ and deﬁne VE as (−1)
VE
: 2 (E) → 2 (E),
{x0 , x1 , x2 , . . .} → {x1 , x2 , x3 , . . .}.
Then if A is any bounded linear operator on a Hilbert space H such that A ≤ 1 and An → 0 strongly as n → ∞, there exists a Hilbert space E (−1) with dim E = dim(I − A∗ A)H, an invariant subspace K ⊂ 2 (E) of VE , (−1) and a unitary operator W : H → K such that A = W −1 (VE K)W . Thus, shifts turn out to be “universal operators.” If I − A∗ A has rank one, we may take E = C and identify 2 (E) with H 2 . The above result is then completed by Beurling’s theorem, which states that every (closed) invariant subspace K ⊂ H 2 of V (−1) , other than H 2 , is of the form K = H 2 θH 2 with some inner function θ. The beginner should consult Rudin [443] and Halmos [265] for these things; excellent presentations of this topic and of related questions are Nikolski [368] and Rosenblum, Rovnyak [442]. 2.10. Good discussions of the main facts about Hankel operators are Nikolski [369], Partington [375], Peetre [384] (this reference also contains an outline of the life of Hermann Hankel), Peller [389], Peller, Khrushchev [390], Power [403], [404]. 2.11. For p = 2, this theorem was established by Nehari [363]. The proof given in the text is due to Sarason [457]. It is this proof which makes the extension of Nehari’s theorem to the spaces H p to a relatively simple matter (this has also been observed by Peetre [384]; also see Peller [388] and Tolokonnikov [519]). Sarason’s proof was adopted in A. Karlovich [299] to extend the Nehari theorem to the case of Hardy type subspaces or rearrangementinvariant spaces. The proof of an analogue of (2.14) in A. Karlovich [299] is based on the innerouter factorization 1.41 and the socalled Lozanovsky factorization of L1 functions.
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105
There are at least three other proofs of Theorem 2.11: Nehari’s original one (which is quite complicated), the proof using the commutant lifting theorem (see Page [374], Clancey, Gohberg [138, Chap. VIII, Theorem 5.2], and Peller [389, Appendix 1.5]), and Parrott’s proof [376]. The latter two proofs also work in the matrix case; Parrott’s proof will be given in 4.32 and 4.33. Also see Bonsall [55]. Finally notice the following characterization of bounded positive Hankel operators due to Widom [564]: if µ is a positive Borel measure on [−1, 1] and if we let ( 1 xn dµ(x) µn = −1
and H[µ] = (µi+j )∞ i,j=0 , then H[µ] ∈ L(2 ) ⇐⇒ µn = O(1/n) (n → ∞) ⇐⇒ µ((−1, −x) ∪ (x, 1)) = O(1 − x) (x → 1). 2.14. Although similar (end equivalent) formulas had been used for a long time, identities (2.18), (2.19) appeared in Widom [569] for the ﬁrst time. In connection with (2.20) we mention the following result of Brown, Halmos [125]: T (a)T (b) is a Toeplitz operator if and only if a or b is analytic. 2.17. Brown, Halmos [125] and Devinatz [151]. Corollary 4.2 generalizes this result to the matrix case. 2.18. Wintner [579]. Extensions to H p and p are in 2.31. It is clear from the proof in 2.31 that these results extend to the matrix case. 2.19–2.23. Widom [557], Devinatz [151]. There are diﬃculties in the matrix case, but see 4.35–4.38 and Devinatz, Shinbrot [155] or Speck [499]. For extensions to the spaces H p see 2.32, 5.3, 5.20, 5.22. We do not know a WidomDevinatz criterion for the spaces p . It should be noted here that the general invertibility problem for Toeplitz operators can be reduced to the special case that the symbol is of the form ω1 ω2 , where ω1 and ω2 are inner functions. Using certain factorization theorems of S. Axler, Th. H. Wolﬀ, and D. Sarason one can show that for every function a ∈ GL∞ there exist inner functions ω1 and ω2 , an outer function h ∈ H ∞ , and a continuous function c such that a = ω1 ω2 hc and that T (a) ∈ Φ(H 2 ) (resp. GL(H 2 )) if and only if T (ω1 ω2 ) ∈ Φ(H 2 ) (resp. GL(H 2 )). However, it is by no means easy to decide whether T (ω1 ω2 ) is invertible or Fredholm. For a nice discussion of this topic see Nikolski [366], [367]. Let us also mention the following result of Lee and Sarason [335]. For an inner function ω let supp ω := {τ ∈ T : 0 ∈ ClH (ω, τ )} (see 3.72). If ω1 and ω2 are inner and supp ω1 = supp ω2 , then sp T (ω1 ω2 ) = clos D. 2.25–2.26. This material is taken from B¨ottcher [68]. See also 4.72, 4.73, 9.18. Parts (a) and (b) of Proposition 2.26 are well known. It may be that parts (c) and (d) of this proposition are known to specialists, but we have not found any reference. It should also be noted that Kats [310] considered the Riemann
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2 Basic Theory
boundary value problem f + = af − +g on T with a coeﬃcient a ∈ C(T◦ ) in the class of all functions holomorphic in z < 1 resp. z > 1, bounded in z ≤ 1 resp. z ≥ 1, and continuous in {z ≤ 1, z = −1} resp. {z ≥ 1, z = −1}. He obtained necessary and suﬃcient conditions for the homogeneous problem to have a ﬁnite number of linearly independent solutions, computed this number, and studied the solvability of the inhomogeneous problem. 2.28–2.29. These results are well known, but apart from the case p = 2 (Halmos [265, Problem 52], Douglas [162, 4.24]) we do not know any reference. 2.30. Hartman and Wintner [267] showed that R(a) is contained in the spectrum spL(H 2 ) T (a), and Simonenko [496] proved that a ∈ GL∞ if T (a) is a Φ+  or Φ− operator on H p . The proof given here is based on arguments of Widom [556]. Also see 2.93. Finally, note that the normal solvability of T (a) on H p (1 < p < ∞) implies that either a ∈ GL∞ or a ≡ 0. This was proved by Leiterer [336] for p = 2 and by Heunemann [275] for general p. 2.31–2.33. See the notes to 2.17–2.23. 2.35. In 1963, Halmos posed the following as a test question for any theory of invertibility of Toeplitz operators: Is the spectrum of a Toeplitz operator necessarily connected? Widom showed that the answer is yes (in [562] for p = 2 using the HelsonSzeg˝o theorem 1.45 and in [563] for general p without using this theorem). The connectedness of spΦ(H 2 ) T (a) was ﬁrst proved by Douglas [162, 7.45]. See also 4.68. 2.36. Hartman, Wintner [267]. It has been open for a long time whether an analogous result holds for quarterplane Toeplitz operators (see Chapter 8); in B¨ottcher [71], we observed that an argument used by McDonald and Sundberg [354] in the context of Toeplitz operators on the disk also applies to halfplane Toeplitz operators and so, by 8.13 and 8.14, proves the connectedness of both the spectrum and the essential spectrum of quarterplane Toeplitz operators with realvalued symbols. Open problems: Is the (essential) spectrum of T 2 (a) on H 2 (T2 ) connected for all a ∈ L∞ (T2 )? As we know that this is so for realvalued symbols, are there interesting applications of Theorem 4.100 to quarterplane Toeplitz operators? Selfadjoint operators live in another world than the generically nonselfadjoint Toeplitz operators. However, if a Toeplitz operator is selfadjoint, one is faced with the same speciﬁc questions as for general selfadjoint operators (such as diagonalization or spectral projections). These questions are treated in the pioneering papers by Rosenblum [439], [440], [441] and Ismagilov [292] and in the book Rosenblum, Rovnyak [442]. A recent remarkable contribution to the matter was made by Vreugdenhil [552]. He showed, for example, that if a(t) = t − 12 (t ∈ T), then the positive square root of T (a) is 3 4∞ 4 1 1 − . π 4(j + k + 2)2 + 1 4(j − k)2 + 1 j,k=0 See also Chapter 1 of B¨ ottcher, Grudsky [86].
2.10 Notes and Comments
107
2.37. Wolﬀ [581]. See also 4.76. 2.38–2.40. Coburn [141] for p = 2, Simonenko [496] for H p , Duduchava [169] for p . For generalizations of Theorem 2.38 and for still another proof of 2.38(b) see Volberg, Tolokonnikov [551]. Vukoti´c [553] proved a more explicit form of 2.38(a): If T (a) ∈ L(H 2 ) is a nonzero Toeplitz operator which is not onetoone, then T (a)(lin{P Ker T (a)}) = P, where lin S denotes the linear span of the set S and P Ker T (a) is deﬁned as {pf : p ∈ P, f ∈ Ker T (a)}. 2.41. For (e) see Gelfand, Raikov, Shilov [212]. 2.42. This theorem is the culmination of several authors including Noether [370], Mikhlin [360], Gohberg [217], Simonenko [491], Krein [322], Calder´ on, Spitzer, Widom [129], Devinatz [151]. Note that Simonenko’s 1960 work [491] actually contains Theorems 2.68 and 2.72! Stegenga [507] proved that a Toeplitz operator T (a) is bounded on H 1 if and only if a ∈ L∞ ∩ BM O1/ log t . Recently Virtanen [546] obtained an analogue of Theorem 2.42(b) for Toeplitz operators T (a) on H 1 provided a ∈ C ∩ V M O1/ log t . Here BM O1/ log t and V M O1/ log t are certain generalizations of BM O and V M O. 2.43–2.47. Theorem 2.47 was established by Krein [322] (for symbols in W ) and by Gohberg, Feldman [220] (for symbols in Cp ). Lemma 2.44 (and its proof given here) as well as Proposition 2.46 are due to Nikolski [365]. 2.50. This is an extract of arguments due to Krein [323] and B¨ ottcher, Silbermann [106, Chap. IV]. 2.51–2.53. Sarason [451] was the ﬁrst to observe that C + H ∞ is closed and thus an algebra. This (at the ﬁrst glance unpretentious and rather curious) discovery is certainly one of the most signiﬁcant achievements in mathematical analysis during the last decades, and it has stimulated and determined subsequent developments in various ﬁelds (in particular in the theory of Toeplitz operators) essentially. See also Sarason [452], [457], Douglas [162], Koosis [316], Garnett [211]. For the ZalcmanRudin lemma see Koosis [316, Chap. VII, 3◦ ]. The observation that Cp + Hp∞ is a closed subalgebra of M p (p = 2) is due to the authors (B¨ ottcher, Silbermann [113]). 2.54. Hartman [266] showed that the Hankel operator H(a) is compact on H 2 if and only if a ∈ C + H ∞ , and Adamyan, Arov, Krein [2] established the equality H(a)Φ(H 2 ) = distL∞ (a, C + H ∞ ). The proof given here follows Axler, Berg, Jewell, Shields [14]. It makes use of the fact that C + H ∞ is closed. There are proofs of Hartman’s result (e.g. Hartman’s original one) which do not use the closedness of C + H ∞ . Thus the fact that C + H ∞ is closed is also a consequence of Hartman’s result (see Sarason [457, p. 102]). For more about compactness of Hankel operators (snumbers, trace class criteria, vectorvalued versions of Hartman’s theorem etc.) see Page [374], Peller [385] – [388], Peller, Khrushchev [390], Nikolski [366] – [369], Rochberg
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[434], Power [403], [404], Peetre [384], Treil [524], Havin, Khrushchev, Nikolski [269]. Widom [564] showed that for positive Hankel operators the following are equivalent (recall the notes to 2.11): (i) H[µ] ∈ C∞ (2 ), (ii) µn = o(1/n) (n → ∞), (iii) µ((−1, −x) ∪ (x, 1)) = o(1 − x) (x → 1). If a ∈ P C, then H(a)Φ(H 2 ) = distL∞ (a, C + H ∞ ) = distL∞ (a, C), which in particular implies that P C + H ∞ is a closed subset of L∞ . This was shown by Bonsall and Gillespie [56]. They also pointed out that P C + H ∞ is not an algebra. 2.55. This result (for p = 2) was obtained by Douglas [159], [162]. The proof presented here is essentially simpler than that of Douglas. 2.58. The theorem is Kronecker’s [325] and the proof in the text is after Gantmacher [210, Chap. XV, Section 10]. An alternative proof of this result (due to Axler) is contained in Sarason [457] and Peller [389, Chap. 1, Section 3]. See also the references listed in the notes to 2.54. Notice that the rank of H(a) is equal to the number of poles of the rational function P a (counted up to multiplicity). 2.59–2.60. These results are due to the authors. Corollary 2.59 appeared ﬁrst in B¨ ottcher, Silbermann [106, 4.6] and Theorem 2.60 was ﬁrst published in B¨ottcher, Silbermann [113]. 2.61–2.66. The results of 2.61–2.65 were established by Douglas [159], [162, 7.36] (for p = 2). Theorem 2.66 is new and due to the authors. Our presentation also relies on Sarason [452]. 2.67–2.69. Simonenko [491], [496] and Gohberg, Krupnik [232]. In the case p = 2 these results were also obtained by Douglas and Sarason [164], who used Glicksberg’s theorem 1.22 (see 2.83). 2.71–2.72. The notion of local sectoriality was introduced by Simonenko [496] and Douglas, Widom [166]. Theorem 2.72 is in Simonenko [491], Devinatz [151], Douglas, Widom [166]. 2.74. Widom [556], Simonenko [491], Devinatz [151], Gohberg [218]. 2.75. Douglas, Sarason [164]. 2.77–2.81. See Hoﬀman [284], Gamelin [209], Garnett [211], Gelfand, Raikov, Shilov [212]. The algebra QC was introduced by Douglas. 2.82. This deﬁnition is from Clancey, Gosselin [139]. 2.83. This theorem was established by Axler [12] using transﬁnite localization (Axler’s method will be described in Chapter 4). For B = C, the result goes over into Theorem 2.67 (p = 2). The proof presented here ﬁrst appeared in B¨ottcher [69] and is new in the following sense: on the one hand it is not terribly new, since it mimics the argument used by Douglas and Sarason [164] to prove this theorem for B = C, and on the other hand it is strange that
2.10 Notes and Comments
109
Axler (who wrote his dissertation under the supervision of Sarason) did not at the very least mention this possibility of proving the theorem in [12]. 2.84–2.85. The term “locally sectorial over QC” had already been used by Douglas [161], a systematic study of symbols which are locally sectorial over C ∗ algebras between C and QC has begun in Silbermann [483]. There Theorem 2.85 was established for B = QC. For B = C + H ∞ , Theorem 2.85 is due to B¨ottcher [69]. Note that Theorem 2.85 is neither an immediate consequence of Theorem 2.83 nor of Theorems 4.63 and 4.64. 2.86. Part (c) goes back to Simonenko [491] and Douglas, Widom [166], parts (a) and (b) were probably ﬁrst proved in B¨ ottcher [69]. The fact that every function which is locally sectorial over QC can be written as a product of a function in GQC and a sectorial function is nontrivial and is a key result, which simpliﬁes the theory of Toeplitz operators with symbols that are locally sectorial over QC substantially. We remark that it was the search for a proof of this result which led us to Glicksberg’s theorem 1.22 and, subsequently, to the proofs of Theorems 2.83 and 2.85 given here. For the matrix case see 3.7, 3.8, 4.31, and the remark after 4.49. 2.87. Sarason [456]. 2.88. The result was established in Silbermann [483], the proof presented here is from B¨ ottcher [69]. 2.90. For B = C, both the result and the proof are taken from Clancey [135] (see also Clancey, Morrel [140]). Once Gorkin [239] had shown that M (H ∞ Xξ ) (ξ ∈ M (QC)) is connected, Silbermann [483] stated this lemma for B = QC. 2.91. See Silbermann [483] for B = QC. The present proof is taken from B¨ottcher [69]. 2.93. Simonenko [492]. Also see Devinatz, Shinbrot [155]. 2.94. Douglas [160] for p = 2 and Spitkovsky [502] for H p . 2.95–2.96. Both theorems are well known. Theorem 2.96 for B = C goes back to Simonenko [492]. It is clear that 1.32 is the appropriate tool to prove 2.95. That 2.96 (in the case B = QC) can be proved with the help of 1.32 is less obvious and requires an argument which was also used by Axler [12, pp. 39–40].
3 Symbol Analysis
3.1 Local Sectoriality 3.1. Deﬁnitions. Let F be a closed subset of X = M (L∞ ) and let a be in L∞ N ×N . The matrix function a is called analytically sectorial on F if there exist a real number ε > 0 and two invertible matrices b, c ∈ CN ×N such that Re (ba(x)c) ≥ ε for all x ∈ F , that is, Re (ba(x)cz, z) ≥ εz2
∀x∈F
∀ z ∈ CN ,
and is said to be geometrically sectorial on F if conv a(F ) ⊂ GCN ×N , that is, if each matrix in the closed convex hull of a(F ) is invertible. It is easy to see that a scalarvalued function (N = 1) is analytically sectorial on F if and only if it is geometrically sectorial on F . In this case the function is simply called sectorial on F , which is in accordance with 2.84. Functions which are (analytically or geometrically) sectorial on the whole maximal ideal space X will be called (analytically or geometrically) sectorial. In the scalar case this agrees with 2.16. Let B be a closed subalgebra of C + H ∞ containing the constants. A function a ∈ L∞ N ×N will be called (analytically or geometrically) locally sectorial over B if it is (analytically or geometrically) sectorial on each maximal antisymmetric set for B. In case B is a C ∗ subalgebra (of QC), the ﬁbers Xβ , β ∈ M (B), occupy the place of the maximal antisymmetric sets (see 1.27(c)). 3.2. Proposition. If a ∈ L∞ N ×N is analytically sectorial on a closed subset F of X, then a is geometrically sectorial on F .
)c) ≥ ε, then Re b λ a(x )c ≥ ε whenever λi ≥ 0 and Proof. If Re (ba(x i i i i
λ = 1. But if Re d ≥ ε > 0 for a matrix d ∈ C N ×N , then d ∈ GCN ×N . So, i i
since b, c ∈ GCN ×N , we conclude that i λi a(xi ) is invertible for all λi ≥ 0 such that i λi = 1.
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3 Symbol Analysis
Remark. Azoﬀ and Clancey [16] constructed an example of a matrix function a ∈ L∞ 2×2 which is geometrically sectorial but not analytically sectorial. The following lemma is needed to prove Theorem 3.4, which represents an important special case in which geometrical sectoriality implies analytic sectoriality. 3.3. Lemma. (a) Let Jλ be the m × m ⎛ λ 1 ⎜ 0 λ ⎜ Jλ = ⎜ ⎜ 0 0 ⎝... ... 0 0
Jordan cell ⎞ 0 ... 0 1 ... 0 ⎟ ⎟ λ ... 0 ⎟ ⎟ ... ... ...⎠ 0 ... λ
and suppose the origin does not belong to the line segment [λ, 1]. Then there are B, C ∈ GCm×m and δ > 0 such that Re (BC) ≥ δ > 0,
Re (BJλ C) ≥ δ > 0.
(b) Let E, F ∈ Cm×m and suppose det µE + (1 − µ)F = 0
∀µ ∈ [0, 1].
Then there are B, C ∈ GCm×m and δ > 0 such that Re (BEC) ≥ δ > 0, Proof. (a) Put ⎛ 0 ⎜ 0 ⎜ N =⎜ ⎜... ⎝ 0 0
1 0 ... 0 0
0 1 ... 0 0
... ... ... ... ...
⎞ 0 0 ⎟ ⎟ ...⎟ ⎟, 1 ⎠ 0
Re (BF C) ≥ δ > 0. ⎛
β m−1 β m−2 ⎜ 0 β m−2 ⎜ 0 Vβ = ⎜ ⎜ 0 ⎝ ... ... 0 0
β m−3 β m−3 β m−3 ... 0
... ... ... ... ...
β β β ... 0
⎞ 1 1 ⎟ ⎟ 1 ⎟ ⎟. ...⎠ 1
/ [λ, 1], there If β = 0, then Vβ N = βN Vβ , whence Vβ N Vβ−1 = βN . Since 0 ∈ are ν ∈ T and α > 0 such that Re ν ≥ α > 0,
Re (νλ) ≥ α > 0.
Consequently, if we let U = diag (ν, . . . , ν), then Re (U I) ≥ α > 0,
Re (U λI) ≥ α > 0.
Thus, Re (U Vβ IVβ−1 ) = Re (U I) ≥ α > 0, Re (U Vβ Jλ Vβ−1 ) = Re (U Vβ (λI + N )Vβ−1 ) = Re (U λI) + Re (βU N ) ≥
α >0 2
3.1 Local Sectoriality
113
if only β is suﬃciently small. It follows that then B = U Vβ , C = Vβ−1 , δ = α/2 have the desired properties. (b) The hypothesis implies that E and F are invertible. There is a D in GCm×m such that D−1 E −1 F D = J is in the Jordan canonical form. We have det(µI + (1 − µ)J) = det(µD−1 E −1 ED + (1 − µ)D−1 E −1 F D) = det(D−1 E −1 ) det(µE + (1 − µ)F ) det D = 0 for all µ ∈ [0, 1]. The matrix J is block diagonal, J = block diag (Jλk ) with each Jλk of the form as in part (a). Because det(µI + (1 − µ)Jλk ) ∀ µ ∈ [0, 1], 0 = det(µI − (1 − µ)J) = k
we conclude that 0 ∈ / [λk , 1] for each k. So part (a) ensures the existence of matrices Bk , Ck and of numbers δk > 0 such that Re (Bk Ck ) ≥ δk > 0,
Re (Bk Jλk Ck ) ≥ δk > 0.
If we let B = block diag (Bk ), C = block diag (Ck ), then Re (B C ) ≥ δ > 0,
Re (B JC ) ≥ δ > 0
with some δ > 0. Now it is easily seen that the matrices B = B D−1 E −1 and C = DC have the desired properties. 3.4. Theorem (Clancey). Let F be a closed subset of X and let a ∈ L∞ N ×N be geometrically sectorial on F . If conv a(F ) is a line segment (i.e., a set of the form [z, w] = (1 − λ)z + λw : λ ∈ [0, 1] , where z, w ∈ CN ×N ), then a is analytically sectorial on F . Proof. Since F is compact and a is continuous on X, there are x1 , x2 ∈ F such that conv a(F ) = [a(x1 ), a(x2 )]. Because a is geometrically sectorial on F , the line segment [a(x1 ), a(x2 )] consists of invertible matrices only. So Lemma 3.3(b) can be applied to see that there are b, c ∈ GCN ×N and ε > 0 such that Re (ba(x1 )c) ≥ ε > 0, Re (ba(x2 )c) ≥ ε > 0. This implies that Re b λa(x1 ) + (1 − λ)a(x2 ) c ≥ ε > 0 ∀ λ ∈ [0, 1], whence Re (ba(x)c) ≥ ε > 0 for all x ∈ F .
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3 Symbol Analysis
Important convention. In what follows we shall mainly deal with matrix functions that are analytically sectorial on closed subsets of X. Therefore a matrix function which is analytically sectorial on a set F or over an algebra B will henceforth be simply called sectorial on F or over B, i.e., in the following “sectorial” always means “analytically sectorial.” The representations stated in Proposition 2.86 for scalarvalued locally sectorial functions played a crucial role in the local theory of scalar Toeplitz operators and it is therefore desirable to have analogous representations for locally sectorial matrix functions. We ﬁrst show how the arguments used to prove Propositions 2.86(a), (b) can be extended to the matrix case. 3.5. Theorem (Machado/Szyma´ nski). Let Y be a compact Hausdorﬀ space and B a closed subalgebra of C(Y ) containing the constants. Let S denote the family of maximal antisymmetric sets for B. Then, for every a ∈ [C(Y )]N ×N , dist(a, BN ×N ) = max distS (a, BN ×N ), S∈S
where for a closed subset F of Y , distF (a, BN ×N ) :=
inf
max a(y) − b(y)L(CN ) ,
b∈BN ×N y∈F
dist(a, BN ×N ) refers to distY (a, BN ×N ), and the norm on CN is the one given by (1.9). Proof. For a proof see Machado [342], Szyma´ nski [512], Burckel [127], Ransford [415], or P´erez Carreras and Bonet [391]. 3.6. Lemma. Let X = M (L∞ ) and let F be a closed subset of X. (a) A matrix function a ∈ L∞ N ×N is sectorial on F if and only if there are a d ∈ GCN ×N and an ε > 0 such that Re (a(x)d) ≥ ε for all x ∈ F . (b) A matrix function a ∈ L∞ N ×N is sectorial on F if and only if there is a d ∈ GCN ×N such that I − a(x)dL(CN ) < 1 for all x ∈ F . −1 (c) Let u ∈ GL∞ (x) = u∗ (x) for all N ×N be unitaryvalued on F , i.e., u x ∈ F . Then u is sectorial on F if and only if distF (u, CN ×N ) < 1. ∞ (d) Let B be any subset of L∞ N ×N , let u ∈ GLN ×N be unitaryvalued on −1 ∗ X (u (x) = u (x) for all x ∈ X), and suppose distX (u, B) < 1. Then there are a b ∈ B and a sectorial matrix function s ∈ GL∞ N ×N such that u = sb.
Proof. (a) If Re (ba(x)cz, z) ≥ δz2 for all x ∈ F with some b, c ∈ GCN ×N and δ > 0, then Re (a(x)c(b∗ )−1 b∗ z, b∗ z) = Re (a(x)cz, b∗ z) = Re (ba(x)cz, z) ≥ δz2 ≥ δb∗ −2 b∗ z2 ∀ z ∈ CN
3.1 Local Sectoriality
115
and so d = c(b∗ )−1 and ε = δb∗ −2 have the desired property. (b) The “if” portion follows from the observation that Re (a(x)d) = I − Re (I − a(x)d) ≥ I − I − a(x)d. On the other hand, if a is sectorial on F , then, by part (a), there are c ∈ GCN ×N and δ > 0 such that Re (a(x)c) ≥ δ for all x ∈ F . Put α := max a(x)c (> 0) and ε := δ/α2 . Then, for zCN = 1, x∈F
(I − a(x)c)z2 = 1 − 2εRe (a(x)cz, z) + ε2 a(x)cz2 δ2 ≤ 1 − 2εδ + ε2 α2 = 1 − 2 < 1, α which gives the assertion with d = εc. (c) From part (b) we deduce that there is a d ∈ CN ×N such that I − u(x)d ≤ 1 for all x ∈ F . Consequently, u(x) − d∗ = u∗ (x) − d = u(x)(u∗ (x) − d) = I − u(x)d < 1 for all x ∈ F , i.e., distF (u, CN ×N ) < 1. Conversely, if there is a c ∈ CN ×N with u(x) − c < 1 − δ < 1 for all x ∈ F , then I − u(x)c∗ = u(x)(u∗ (x) − c∗ ) = u∗ (x) − c∗ < 1 − δ for x ∈ F , which implies that, for zCN = 1, 2Re (u(x)c∗ z, z) = 1 + u(x)c∗ z2 − (I − u(x)c∗ )z2 ≥ 1 − I − u(x)c∗ 2 ≥ δ > 0 ∀ x ∈ F. (d) If there is a b ∈ B with u − b < 1 − δ < 1, then I − bu−1 = (u − b)u∗ < 1 − δ, hence, for zCN = 1, 2Re (bu−1 z, z) = 1 + bu−1 z2 − (I − bu−1 )z2 ≥ 1 − I − bu−1 2 > δ > 0 and thus bu−1 = s with s satisfying Re s(x) ≥ δ/2 for all x ∈ X. It follows that u = s−1 b, and because Re (s−1 z, z) = Re (s−1 sy, sy) = Re (y, sy) ≥ =
δ −1 2 δ s z ≥ s−2 z2 2 2
s−1 must be sectorial (on X).
δ y2 2
∀ z ∈ CN ,
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3 Symbol Analysis
3.7. Proposition. Let B be a C ∗ subalgebra of L∞ containing the constants −1 (x) = u∗ (x) for all x ∈ X. and let u ∈ GL∞ N ×N be unitaryvalued, that is, u Then for u to be locally sectorial over B it is necessary and suﬃcient that u be of the form u = sb with b ∈ GBN ×N and s ∈ GL∞ N ×N being sectorial. Proof. If u is locally sectorial over B, then, by Lemma 3.6(c), distXξ (u, BN ×N ) = distXξ (u, CN ×N ) < 1 ∀ ξ ∈ M (B), and Theorem 3.5 in conjunction with 1.27(c) implies that dist(u, BN ×N ) < 1. Now Lemma 3.6(d) gives that there are a b ∈ BN ×N and a sectorial ∞ s ∈ GL∞ N ×N such that u = sb. Since obviously b ∈ GLN ×N , we deduce from 1.26(d) that actually b ∈ GBN ×N . This proves the necessity portion. The suﬃciency part is trivial. We now remove the restriction to unitaryvalued matrixfunctions by using other (even more elementary) techniques. 3.8. Theorem. Let B be a C ∗ algebra between C and L∞ and let a ∈ L∞ N ×N . Then a is locally sectorial over B if and only if a is of the form a = sb where s ∈ GL∞ N ×N is sectorial and b ∈ GBN ×N . Proof. It suﬃces to prove the “only if” portion. So suppose a is locally sectorial over B. For ξ ∈ M (B), deﬁne Dξ := d ∈ CN ×N : I − a(x)dL(CN ) < 1 ∀ x ∈ Xξ . Lemma 3.6(b) implies that Dξ is nonempty. It is clear that Dξ is an open convex subset of CN ×N . If d ∈ Dξ , then d ∈ Dη for all η in some open neighborhood U (ξ) ⊂ M (B) of ξ; this follows from the upper semicontinuity of the mapping M (B) → R+ ,
ξ → max I − a(x)d, x∈Xξ
which, in turn, can be derived from Theorem 1.35(b) in the setting A = L∞ N ×N and B = {ϕIN ×N : ϕ ∈ B}. Associate with each ξ ∈ M (B) a matrix dξ ∈ Dξ and a neighborhood U (ξ) of ξ such that dξ ∈ Dη for all η ∈ U (ξ). Because M (B) is a compact Hausdorﬀ space, it is a normal space and hence there are open neighborhoods U (ξ) such that ξ ∈ U (ξ) ⊂ clos U (ξ) ⊂ U (ξ). By the compactness of M (B), there are ξ1 , . . . , ξn in M (B) such that n U (ξi ). Consider the constant functions M (B) = i=1
fi : clos U (ξi ) → CN ×N ,
ξ → dξi .
3.1 Local Sectoriality
117
Accept for a moment the validity of the following claim: If U and V are open subsets of M (B) such that U \ clos V = ∅, if g is a continuous function on clos U with g(ξ) ∈ Dξ for all ξ ∈ U , and if d is some matrix belonging to Dξ for all ξ in some open neighborhood W (clos V ) of clos V , then there is a continuous function h on clos (U ∪ V ) such that h(ξ) ∈ Dξ for all ξ ∈ U ∪ V . Hence, letting U = U (ξ1 ), V = U (ξ2 ), g = f1 , d = dξ2 , we get a continuous function h1 on clos [U (ξ1 ) ∪ U (ξ2 )] such that h1 (ξ) ∈ Dξ for all ξ ∈ U (ξ1 ) ∪ U (ξ2 ). Then the claim for U = U (ξ1 ) ∪ U (ξ2 ), V = U (ξ3 ), g = h1 , d = dξ3 gives a continuous function h2 on clos [U (ξ1 ) ∪ U (ξ2 ) ∪ U (ξ3 )] with h2 (ξ) ∈ Dξ for all ξ ∈ U (ξ1 ) ∪ U (ξ2 ) ∪ U (ξ3 ). Continuing, we ﬁnally arrive at a continuous function h on M (B) with h(ξ) ∈ Dξ for all ξ ∈ M (B), that is, we have an h ∈ BN ×N with I − a(x)h(x)L(CN ) < 1 ∀ x ∈ X.
(3.1)
From (3.1) we see that h ∈ GL∞ N ×N , whence h ∈ GBN ×N . Also by (3.1), the matrix function s = ah is sectorial. Hence, if we let b = h−1 , then a = sb is the desired factorization. It remains to prove the above claim. Since M (B) is a normal space, there is an open neighborhood W = W (clos V ) of clos V such that V ⊂ clos V ⊂ W (clos V ) ⊂ W (clos V ),
U := clos U \ W (clos V ) = ∅.
Put V = clos (V \ U ) and notice that U ∪ V ⊂ U ∪ V ∪ (W ∩ U ). The sets U and V are closed and U ∩ V = ∅. Hence, by Uryson’s extension theorem (see, e.g., Naimark [362, p. 43]), there exists a continuous function ϕ on clos (U ∪ V ) such that 0 ≤ ϕ ≤ 1, ϕU = 1, ϕV = 0. Extend g arbitrarily to a function on clos (U ∪ V ) and put h = gϕ + d(1 − ϕ). Because gϕ is continuous on clos U and vanishes identically on V = clos (V \ U ), it follows that h is continuous on clos (U ∪ V ). If ξ ∈ U , then h(ξ) = g(ξ) ∈ Dξ and if ξ ∈ V , then h(ξ) = d ∈ Dξ . Finally, if ξ ∈ W ∩ U , then h(ξ) is a convex linear combination of g(ξ) ∈ Dξ and d ∈ Dξ , which, by the convexity of Dξ , implies that h(ξ) ∈ Dξ . This completes the proof of our claim. We ﬁnally state a matrix analogue of 2.86(c). 3.9. Theorem. Let a ∈ L∞ N ×N . Then the following are equivalent: (i) a is locally sectorial over C; (ii) for each τ ∈ T there are open neighborhoods Uτ ⊂ T of τ , matrices bτ , cτ ∈ GCN ×N , and an ετ > 0 such that Re (bτ a(t)cτ ) ≥ ετ
for almost all
t ∈ Uτ ;
(iii) there are a ﬁnite number of open subarcs U1 , . . . , Un of T, matrices c1 , . . . , cn ∈ GCN ×N and an ε > 0 such that Re (a(t)ck ) ≥ ε
for almost all
t ∈ Uk ;
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3 Symbol Analysis
(iv) a = sψ with ψ ∈ GCN ×N and s ∈ GL∞ N ×N sectorial ; (v) a = ϕsψ with ϕ, ψ ∈ GCN ×N and s ∈ GL∞ N ×N sectorial. Proof. The implications (iv) =⇒ (v) =⇒ (i) are trivial and the implication (i) =⇒ (iv) results from the preceding theorem. The implication (iii) =⇒ (ii) is also trivial, while the implication (ii) =⇒ (iii) is a consequence of the compactness of T and of Lemma 3.6(a). Finally, the implication (iv) =⇒ (ii) can be veriﬁed as in the proof of Proposition 2.86(c) and the implication (ii) =⇒ (i) follows from Proposition 2.79(a). 3.10. Pn C. Let a ∈ L∞ N ×N and τ ∈ T. Each point of CN ×N which belongs to the (compact) set a(Xτ ) is called an essential cluster point of a at τ . (Pn C)N,N is deﬁned as the set of all functions a ∈ L∞ N ×N which have at most n essential cluster points at each point of T. The set (Pn C)1,1 will be abbreviated to Pn C. Example. Let τ ∈ T and let E1 , . . . , En (n ≥ 2) be pairwise disjoint measurn Ek and U ∩ Ek has positive measure for able subsets of T such that T = k=1
each k ∈ {1, . . . , n} and each neighborhood U ⊂ T of τ . Then
n let α1 , . . . , αn be any pairwise distinct complex numbers and put a = k=1 αk χEk . Due to Proposition 2.79(a) we have a(Xτ ) = {α1 , . . . , αn } and a belongs to Pn C \ Pn−1 C. Let Ak denote the set of all x ∈ Xτ for which a(x) = ατ . It is easy to see that χEk (x) = 1 for x ∈ Ak and χEk (x) = 0 for x ∈ Xτ \ Ak . Note that, for N > 1 and n > 1, (Pn C)N,N is properly contained in (Pn C)N ×N , the collection of all N × N matrix functions whose entries are in Pn C. Indeed, if we let E1 , E2 , E3 , E4 be as in the above example and if α, β are distinct complex numbers, then the entries of the diagonal matrix function diag (αχE1 + αχE2 + βχE3 + βχE4 , αχE1 + βχE2 + αχE3 + βχE4 ) belong to P2 C, while the matrix function itself takes the four values diag (α, α),
diag (α, β),
diag (β, α),
diag (β, β)
on A1 , A2 , A3 , A4 , respectively. It is clear that P CN ×N (see 2.73) is contained in (P2 C)N ×N and that there are functions in (P2 C)N,N that do not belong to P CN ×N . Finally, it is obvious that (P1 C)N ×N = CN ×N . The following proposition characterizes the matrix functions in (P2 C)N,N that are locally sectorial over C. 3.11. Proposition. Let a ∈ (P2 C)N,N , and for τ ∈ T denote by a1τ and a2τ the essential cluster points of a at τ (it may be that a1τ = a2τ ). Then the following are equivalent: (i) a is locally sectorial over C;
3.2 Asymptotic Multiplicativity
119
(ii) for each τ ∈ T, the (possibly degenerate) line segment [a1τ , a2τ ] consists of invertible matrices only; (iii) det[(1 − µ)a1τ + µa2τ ] = 0
∀ (τ, µ) ∈ T × [0, 1].
Proof. The equivalence (ii) ⇐⇒ (iii) is trivial. To establish the equivalence (i) ⇐⇒ (ii), notice ﬁrst that conv a(Xτ ) = [a1τ , a2τ ]. Therefore, by Proposition 3.2, (i) implies (ii). On the other hand, Theorem 3.4 (or Lemma 3.3(b)) shows that (i) is a consequence of (ii).
3.2 Asymptotic Multiplicativity 3.12. The Poisson kernels. For ϕ ∈ L1 , we deﬁned hr ϕ (0 < r < 1), the AbelPoisson means of the Fourier series (= harmonic extension), as rl ϕl eilx , x ∈ [0, 2π). (hr ϕ)(eix ) = l∈Z
Using the Poisson kernel, we have ( 2π kr (x − t)ϕ(eit ) dt, (hr ϕ)(eix ) =
x ∈ [0, 2π),
0
where kr (x) =
1 − r2 1 , 2π 1 − 2r cos x + r2
x ∈ R.
The slight change in notation (recall how kr was deﬁned in 1.37) should not cause confusion. If we extend ϕ to a function Φ ∈ L1loc (R) periodically, i.e., if we set Φ(x) = ϕ(eix ), x ∈ R, then hr ϕ can be written as ( ∞ (hr ϕ)(eix ) = λK(λ(x − t))Φ(t) dt, x ∈ R, −∞
where K(x) = 1/(π(1 + x2 )) and λ = −1/ log r ∈ (0, ∞) (see Ahiezer [3, Section 62]). 3.13. The Fej´ er kernels. The Fej´er (or Fej´erCesaro) means σn ϕ, n ∈ N, of a function ϕ ∈ L1 are deﬁned in terms of Fourier coeﬃcients by n (σn ϕ)(e ) = 1− ix
l=−n
l n+1
ϕl eilx ,
x ∈ [0, 2π).
We also have (
2π
kn (x − t)ϕ(eit ) dt,
(σn ϕ)(eix ) = 0
x ∈ [0, 2π),
120
3 Symbol Analysis
where the (Fej´er) kernel kn is given by kn (x) =
sin2 ((n + 1)x/2) 1 , 2π(n + 1) sin2 (x/2)
x ∈ R.
Let Φ ∈ L1loc (R) denote the periodic extension of ϕ, Φ(x) = ϕ(eix ) for x ∈ R. Then ( ∞ (n + 1)K (n + 1)(x − t) Φ(t) dt, x ∈ R, (σn ϕ)(eix ) = −∞
with
2 sin2 (x/2) , π x2 (again see Ahiezer [3, Section 62]). K(x) =
x∈R
3.14. Approximate identities. The Poisson and Fej´er kernels are typical examples of what is usually called an approximate identity. Let K be a function in L1 (R) which has the following properties: ( ∞ K(x) ≥ 0, K(x) = K(−x), K(x) dx = 1, (3.2) −∞
0 < ess inf K(x) ≤ ess sup K(x) < ∞, x∈(−π,π)
(3.3)
x∈(−π,π)
there is a constant M > 0 such that K(x) ≤
M for x ≥ 1. x2
(3.4)
The generalized sequence {Kλ }λ∈Λ , where Λ = {l0 , l0 + 1, l0 + 2, . . .} (l0 ∈ N)
or
Λ = (r0 , ∞)
(r0 ∈ R+ ),
and Kλ (x) = λK(λx),
x ∈ R,
will be called the approximate identity generated by K. Given ϕ ∈ L1 let Φ ∈ L1loc (R) denote the periodic extension of ϕ. Then deﬁne ( ∞ (kλ Φ)(x) :=
−∞
λK(λ(x − t))Φ(t) dt,
x ∈ R,
and put (kλ ϕ)(eix ) := (kλ Φ)(x),
x ∈ [0, 2π).
Thus, for λ ∈ Λ, kλ Φ is a 2πperiodic function in L1loc (R) and kλ ϕ is a function in L1 = L1 (T). Finally we shall sometimes write kλ,t ϕ := (kλ ϕ)(t) (t ∈ T),
kλ,x Φ := (kλ Φ)(x) (x ∈ R).
The following facts are well known and can be veriﬁed without substantial diﬃculty.
3.2 Asymptotic Multiplicativity
121
(a) If a ∈ L∞ , then kλ a ∈ C for all λ ∈ Λ. (b) If a ∈ L∞ , then sup kλ a∞ ≤ a∞ . λ∈Λ
(c) If a ∈ L , then kλ a − aL2 → 0 as λ → ∞.
Let ϕ(x) = l ϕl eilx be a Laurent polynomial. Then ( ∞ ix λK(λt) ϕl eil(x−t) dt (kλ ϕ)(e ) = 2
−∞
=
ϕl eilx
l
=
l
where K(y) =
(
∞
ϕl K
l
(
∞
λK(λt)e−ilt dt
−∞
l eilx λ
K(x)e−ixy dx =
−∞
(
(x ∈ R),
(3.5)
∞
K(x) cos(yx) dx.
(3.6)
−∞
∈ C(R), K(±∞) So K = 0, K(0) = 1 and K(y) < 1 for y = 0. This can be used to derive the following fact, which will be needed later. (d) If (kλn χ1 )(eiθn ) → χ1 (τ ) = τ as n → ∞ (τ ∈ T), then eiθn → τ and λn → ∞ as n → ∞. If K is the Poisson kernel, then (d) says that eiθn → τ and rn = e−1/λn → 1 whenever rn eiθn → τ ; thus, in that case (d) is trivial. Finally, note that K(y) = e−y if K is the Poisson kernel, ! 1 − y (y ≤ 1) K(y) = if K is the Fej´er kernel. 0 (y ≥ 1) 3.15. Asymptotic multiplicativity. Let A and B be subsets of L∞ and let {Kλ }λ∈Λ be an approximate identity. We say that {Kλ }λ∈Λ is asymptotically multiplicative on the pair (A, B) if kλ (ab) − (kλ a)(kλ b)∞ → 0 as
λ → ∞ ∀ a ∈ A,
∀ b ∈ B.
Thus, Lemma 2.61 says that the Poisson kernels are asymptotically multiplicative on the pair (C, L∞ ), Lemma 2.87 states that the Poisson kernels are even asymptotically multiplicative on the pair (QC, L∞ ), and the statement of Theorem 2.62(c) is the asymptotic multiplicativity of the Poisson kernels on the pair (C + H ∞ , C + H ∞ ). Minor and obvious modiﬁcations of the proof of Lemma 2.61 imply that every approximate identity is asymptotically multiplicative on the pair (C, L∞ ). The purpose of what follows is to show that this remains true for the pair (QC, L∞ ).
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3 Symbol Analysis
If an approximate identity is asymptotically multiplicative on the pair (H ∞ , H ∞ ), then it is so on the pair (C + H ∞ , C + H ∞ ). Thus, in that case the argument of the proof of Lemma 2.87 gives its asymptotic multiplicativity on the pair (QC, L∞ ). In this connection and as motivation for our further investigations notice the following. 3.16. Remark. In 1988, Wolf and Havin [580] showed that shifts and contractions of the Poisson kernel, i.e. kernels of the form K(x) =
ε 1 , 2 π ε + (h − x)2
are the only approximate identities which are asymptotically multiplicative on the pair (H ∞ , H ∞ ). Our starting point is the following characterization of QC. 3.17. Theorem (Sarason). QC = L∞ ∩ V M O. Proof. Let a ∈ L∞ ∩V M O. By 1.48(h), there are u, v ∈ C such that a = u+ v). v ∈ H ∞ . Therefore, Hence v) = a − u ∈ L∞ , and it follows that v + i) a = −i(v + i) v ) + (u + iv) ∈ H ∞ + C. If a ∈ L∞ ∩ V M O, then obviously a ∈ L∞ ∩ V M O, and the same reasoning gives a ∈ H ∞ + C. Thus a ∈ QC. Conversely, let a ∈ QC. Then a = b + ic, where b and c are realvalued u) + (v + iw) with functions in QC. Since b ∈ H ∞ + C, we have b = (u + i) ) = −w, whence u + i) u ∈ H ∞ and v + iw ∈ C. Because b is realvalued, u ) u = −u ) = w, ) and it results that b = u + v = w ) + v, where w ∈ C and v ∈ C. Again by 1.48(h), b ∈ V M O. The same argument gives that c ∈ V M O. Thus a ∈ V M O ∩ L∞ . 3.18. Lemma. Let ϕ ∈ L1 and Φ(x) := ϕ(eix ), x ∈ R. Then (a) ϕ ∈ BM O ⇐⇒ Φ ∈ BM O(R); (b) cΦ∗ ≤ ϕ∗ ≤ Φ∗ with some absolute constant c; (c) ϕ ∈ V M O ⇐⇒ Φ ∈ V M O(R). Proof. (a), (b) It is clear that ϕ ∈ BM O if Φ ∈ BM O(R) and that then ϕ∗ ≤ Φ∗ . So let ϕ ∈ BM O and let J be a ﬁnite interval of R. Without loss of generality assume J = [−δ, 2πn], where 0 < δ < 2π and n ∈ N. Put I = [−δ, 0]. Then ( ( ( 2π 1 1 Φ − Φ[0,2π]  dt = Φ − Φ[0,2π]  dt + n Φ − Φ[0,2π]  dt J J δ + 2πn I 0 and we have
3.2 Asymptotic Multiplicativity
n δ + 2πn
(
2π
Φ − Φ[0,2π]  dt ≤ 0
1 2π
(
123
2π
Φ − Φ[0,2π]  dt ≤ ϕ∗ . 0
Further, 1 δ + 2πn
( Φ − Φ[0,2π]  dt ≤ I
1 δ
( Φ − ΦI  dt + I
1 δ + 2πn
( Φ[0,2π] − ΦI  dt, I
and the ﬁrst term is clearly not greater than ϕ∗ . The second term equals 2π δ δ δ ≤ Φ[−2π,0] − ΦI  = const · log ϕ∗ Φ[−2π,0] − ΦI  δ + 2πn 2π δ 2π (1.48(r), (i), (ii)), which is not greater than const·ϕ∗ . Now 1.48(a) completes the proof. (c) This is immediate from (a), (b) together with 1.48(f), (p). 3.19. The moving average. This is the approximate identity generated by K(x) =
1 2π
for
x ∈ (−π, π),
K(x) = 0 for x ∈ / (−π, π).
Note that ( kλ,x Φ = (kλ Φ)(x) = =
∞ −∞
λ 2π
(
Φ(t)λK λ(x − t) dt x+π/λ
Φ(t) dt = Φ[x−π/λ,x+π/λ] . x−π/λ
Moreover, the “norm” Φ∗ on BM O(R) is nothing else than sup sup kλ,x (Φ − kλ,x Φ) λ>0 x∈R
and if Φ ∈ BM O(R), then, by deﬁnition, Φ ∈ V M O(R) ⇐⇒ lim sup kλ,x (Φ − kλ,x Φ) = 0. λ→∞ x∈R
(3.7)
The proof of the following proposition shows how (3.7) can be used to study asymptotic multiplicativity. 3.20. Proposition. The moving average is asymptotically multiplicative on the pair (QC, L∞ ). Proof. Let ϕ ∈ QC and a ∈ L∞ . Let Φ ∈ L∞ (R) and A ∈ L∞ (R) denote the periodic extensions of ϕ and a, respectively. Then with K as in 3.19,
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3 Symbol Analysis
(kλ ΦA)(x) − (kλ Φ)(x)(kλ A)(x) + +( ∞ + + + [Φ(t) − (kλ Φ)(x)]A(t)Kλ (x − t) dt++ =+ −∞ ( ∞ ≤ A∞ Φ(t) − (kλ Φ)(x)Kλ (x − t) dt −∞
= A∞ kλ,x (Φ − kλ,x Φ). Theorem 3.17 and Lemma 3.18 imply that Φ ∈ L∞ (R) ∩ V M O(R) and so the assertion follows from (3.7). 3.21. Theorem. Let {Kλ }λ∈Λ be an approximate identity with generating kernel K. For ϕ ∈ L1 , deﬁne ϕK := sup sup kλ,t (ϕ − kλ,t ϕ). λ∈Λ t∈T
Then, if ϕ ∈ L1 ,
ϕ ∈ BM O ⇐⇒ ϕK < ∞.
Moreover, there are constants c1 and c2 depending only on K (and Λ) such that c1 ϕK ≤ ϕ∗ ≤ c2 ϕK . Proof. Put Φ(x) = ϕ(eix ), x ∈ R, and let ΦK := sup sup kλ,x (Φ − kλ,x Φ). λ∈Λ x∈R
Since in this deﬁnition sup may be replaced by x∈R
sup , we have ΦK = ϕK . x∈[0,2π)
Recall that Λ is either {λ0 , λ0 + 1, λ0 + 2, . . .}, where λ0 ≥ 1 is an integer, or the interval (λ0 , ∞), where λ0 > 0. First suppose ΦK < ∞. Let I be any interval of R whose length is less than δ0 := min{π, π/λ0 }: I = (x − δ, x + δ), 0 < δ < δ0 /2. Due to (3.3) there is a constant m > 0 such that K(τ ) ≥ m for (almost all) τ ∈ (−π, π). Choose a λ1 ∈ Λ so that π/(4δ) < λ1 < π/δ (this is always possible). Then, if t ∈ I ⊂ (x − π/λ1 , x + π/λ1 ), we have Kλ1 (x − t) = λ1 K(λ1 (x − t)) ≥ λ1 m > hence Kλ1 (x − t) ≥
πm 1 χ(−δ,δ) (x − t) 2 I
πm , 4δ
∀ t ∈ R,
and it follows that ( ( ∞ 1 1 Φ(t) − kλ1 ,x Φ dt ≤ Φ(t) − kλ1 ,x Φχ(−δ,δ) (x − t) dt I I I −∞ ( ∞ 2 2 ΦK . ≤ Φ(t) − kλ1 ,x ΦKλ1 (x − t) dt ≤ πm −∞ πm
3.2 Asymptotic Multiplicativity
125
Thus, 1.48(a) implies that Mδ0 (ϕ) = const · ΦK and 1.48(b) shows that ϕ ∈ BM O and ϕ∗ ≤ c2 ΦK . Now suppose ϕ ∈ BM O. By virtue of Lemma 3.18, Φ ∈ BM O(R). Let x ∈ R, λ ∈ Λ, and put Ij = [x − 2j /λ, x + 2j /λ] for j ≥ 0. Note that Ij  = 2j+1 /λ. We have ( ∞ ( Φ(t) − ΦI0 Kλ (x − t) dt ≤ Φ(t) − ΦI0 Kλ (x − t) dt −∞
+
+
I0
∞ (
Φ(t) − ΦIj Kλ (x − t) dt
j=1 Ij \Ij−1 ∞ ( j=1
ΦIj − ΦI0 Kλ (x − t) dt.
Ij \Ij−1
(3.8)
For the ﬁrst term in (3.8) we get ( ( Φ(t) − ΦI0  Kλ (x − t) dt ≤ K∞ λ Φ(t) − ΦI0  dt I0 I0 ( 2 Φ(t) − ΦI0  dt ≤ const · Φ∗ . = K∞ I0  I0 In view of (3.4) we have max
2j−1 ≤λx−t≤2j
K λ(x − t) ≤
M (2j−1 )2
=
4M . 22j
(3.9)
Therefore, the second term in (3.8) is not greater than ( ∞ 4M λ j=1
22j
Φ(t) − ΦIj  dt = Ij
( ∞ 4M λ 2j+1 1 Φ(t) − ΦIj  dt 22j λ Ij  Ij j=1
≤ 8M Φ∗
∞ 1 = 8M Φ∗ . 2j j=1
Finally, again using (3.9) we see that the third term in (3.8) is not greater than ( ∞ ∞ 4M λ 8M Φ − Φ  dt = ΦIj − ΦI0  I I j 0 2j 2 2j Ij j=1 j=1 ≤
∞ 8M j=1
2j
const · jΦ∗
= const · Φ∗ . Thus, what we have shown is that
(by 1.48(r), (ii))
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3 Symbol Analysis
(
∞
sup sup λ∈Λ x∈R
−∞
Φ(t) − ΦI0 Kλ (x − t) dt ≤ const · Φ∗ ,
where I0 = [x − 1/λ, x + 1/λ]. But ( ∞ ( Φ(t) − kλ,x ΦKλ (x − t) dt ≤ −∞
∞
(3.10)
Φ(t) − ΦI0 Kλ (x − t) dt
−∞ ( ∞
+ −∞
ΦI0 − kλ,x ΦKλ (x − t) dt
and the ﬁrst integral herein admits estimate (3.10), while +( ∞ + + + + ΦI0 − kλ,x Φ = + ΦI0 − Φ(t) Kλ (x − t) dt++ −∞ ( ∞ ≤ ΦI0 − Φ(t)Kλ (x − t) dt ≤ const · Φ∗ , −∞
the last “≤” again by (3.10), whence ( ∞ ( ΦI0 − kλ,x ΦKλ (x − t) dt ≤ const · Φ∗ −∞
Thus,
(
∞
sup sup λ∈Λ x∈R
−∞
∞ −∞
Kλ (x − t) dt = const · Φ∗ .
Φ(t) − kλ,x ΦKλ (x − t) dt ≤ cΦ∗
with some constant c depending only on K and Λ. It follows that ϕK = ΦK < ∞ and Lemma 3.18(b) gives that c1 ϕK ≤ ϕ∗ . 3.22. Lemma. Let {Kλ }λ∈Λ be an approximate identity and let ϕ ∈ V M O. Then kλ ϕ ∈ V M O for all λ ∈ Λ and ϕ − kλ ϕ∗ → 0 as λ → ∞. Proof. Put Φ(x) := ϕ(eix ) for x ∈ R. Then Φ ∈ V M O(R) by Lemma 3.18. For y ∈ R, let Φy (x) := Φ(x − y). Since ( ∞ ( ∞ (kλ Φ)(x) = Φ(t)Kλ (x − t) dt = Φ(x − y)Kλ (y) dy −∞
−∞
and since, by 1.48(q), the mapping R → V M O(R), y → Φy is continuous, kλ Φ can be written as a Bochner integral: ( ∞ kλ Φ = Φy Kλ (y) dy. −∞
This implies that kλ Φ ∈ V M O(R), whence, by Lemma 3.18, kλ ϕ ∈ V M O. Furthermore, we also have ( ∞ Φ − kλ Φ = (Φ − Φy )Kλ (y) dy −∞
3.2 Asymptotic Multiplicativity
and thus
(
Φ − kλ Φ∗ ≤ (
∞ −∞
127
Φ − Φy ∗ Kλ (y) dy
= y<δ
( Φ − Φy ∗ Kλ (y) dy +
y>δ
Φ − Φy ∗ Kλ (y) dy.
Given any ε > 0 there is, by virtue of 1.48(q), a δ > 0 such that ( ( ε ∞ ε Φ − Φy ∗ Kλ (y) dy ≤ Kλ (y) dy = , 2 2 y<δ −∞ and having chosen this δ, there is a λ > 0 such that ( ( ε Φ − Φy ∗ Kλ (y) dy ≤ 2Φ∗ Kλ (y) dy ≤ 2 y>δ y>δ whenever λ > λ . Application of Lemma 3.18 completes the proof.
3.23. Theorem. Every approximate identity is asymptotically multiplicative on the pair (QC, L∞ ). Proof. Let ϕ ∈ QC and a ∈ L∞ , denote the periodic extensions by Φ and A, and note that Φ ∈ L∞ (R) ∩ V M O(R) (Theorem 3.17 and Lemma 3.18) and A ∈ L∞ (R). As in the proof of Proposition 3.20 we see that sup (kλ ΦA)(x) − (kλ Φ)(x)(kλ A)(x) ≤ A∞ sup kλ,x (Φ − kλ,x Φ). (3.11) x∈R
x∈R
We have sup kλ,x Φ − kλ,x Φ ≤ sup kλ,x Φ − kµ Φ − kλ,x (Φ − kµ Φ) x∈R x∈R + sup kλ,x kµ Φ − kµ,x Φ x∈R + sup kλ,x kµ,x Φ − kλ,x (kµ Φ) .
(3.12)
x∈R
The ﬁrst term in (3.12) is not greater than Φ − kµ ΦK ≤ (1/c1 )Φ − kµ Φ∗ (Theorem 3.21) and consequently, by Lemma 3.22, there is a µ0 ∈ Λ such that this term is smaller than ε/3 for µ = µ0 . For µ = µ0 , the second term in (3.12) is not greater than ( sup (kµ0 Φ)(t) − (kµ0 Φ)(x)Kλ (x − t) dt x∈R x−t<δ ( + sup (kµ0 Φ)(t) − (kµ0 Φ)(x)Kλ (x − t) dt. (3.13) x∈R
x−t>δ
Because kµ0 Φ ∈ L∞ (R) is uniformly continuous (by 3.14(a)), there is a δ > 0 such that the ﬁrst term in (3.13) is smaller than ε/6, and having chosen this
128
3 Symbol Analysis
δ, we can ﬁnd a λ1 ∈ Λ such that the second term in (3.13) becomes smaller than ε/6 for all λ > λ1 . Finally, for µ = µ0 , the third term in (3.12) is smaller than ε/3 for all λ > λ2 , since kλ f − f ∞ → 0 as λ → ∞ in case f ∈ C. Thus, the righthand side of (3.11) is smaller than A∞ times an arbitrarily given ε > 0 whenever λ > max{λ1 , λ2 }. But this is the assertion.
3.3 Piecewise Quasicontinuous Functions We ﬁrst state some results on the C ∗ algebra P C of all piecewise continuous functions on T (recall 2.73). 3.24. Proposition. The maximal ideal space of P C is T × {0, 1} and the Gelfand map Γ : P C → C(T × {0, 1}) is given by (Γ f )(τ, 0) = f (τ − 0),
(Γ f )(τ, 1) = f (τ + 0).
An open neighborhood base of (τ, 0) is formed by the sets −iε (τ e , τ ] × {0} ∪ (τ e−iε , τ ) × {1} , 0 < ε < π, and an open neighborhood base of (τ, 1) is formed by the sets [τ, τ eiε ) × {1} ∪ (τ, τ eiε ) × {0} , 0 < ε < π, where (a, b) denotes the open subarc of T whose endpoints are a and b and whose length is less than π, [a, b) := {a} ∪ (a, b), (a, b] := (a, b) ∪ {b}. Proof. It is clear that (τ, 0) and (τ, 1) are in M (P C). Conversely, let v be in M (P C). Then v belongs to some ﬁber Mτ (P C) of M (P C) over τ ∈ M (C) = T. Every function f ∈ P C can be written as f = cχτ +g, where c ∈ C, χτ is the characteristic function of the arc (τ, τ eiπ/2 ), and g is a function in P C which is continuous at τ . The spectrum spP C (χτ ) is obviously the doubleton {0, 1}. Therefore v(χτ ) must be equal either 0 or 1. If v(χτ ) = 0, then v(f ) = g(τ ) = f (τ − 0) for all f ∈ P C, and if v(χτ ) = 1, then v(f ) = c + g(τ ) = f (τ + 0) for all f ∈ P C. Thus, Mτ (P C) is the doubleton {(τ, 0), (τ, 1)}. The assertion concerning the Gelfand topology of M (P C) can be checked straightforwardly. 3.25. Proposition. Let τ ∈ T and let F be a closed subset of Xτ = Mτ (L∞ ). Then the restriction algebra P CF is either isometrically isomorphic to the complex ﬁeld C or is a singly generated C ∗ algebra whose maximal ideal space is the doubleton {0, 1} with the discrete topology. Proof. Every f ∈ P C can be written as f = cχτ + g, where c, χτ , g are as in the proof of the preceding proposition. Let χτF denote the restriction χτ F . Then f F = cχτF + g(τ ), and therefore P CF coincides with the algebra of all functions of the form cχτF + d, where c, d ∈ C. If χτF = const on F , then
3.3 Piecewise Quasicontinuous Functions
129
P CF ∼ = C. If χτF is not constant on F , then the range of χτF is {0, 1}, and it is easily seen that P CF = {cχτF + d : c, d ∈ C} is closed (hence, a C ∗ algebra), that it is generated by χτF , and that the spectrum of χτF in P CF is {0, 1}. It remains to recall 1.19. 3.26. Lemma. Let χU be the characteristic function of the upper halfcircle {eiθ : 0 < θ < π} and put H(x) := χU (eix ) (x ∈ R). Let K be an approximate identity. Then for every µ ∈ (0, 1) there is a ν ∈ R such that (kλ H)(ν/λ) → µ as λ → ∞. Proof. For deﬁniteness, assume µ ∈ [1/2, 1). Then there is a ν ≥ 0 such that *ν K(x) dx = µ, and we have, as λ → ∞, −∞ ν ν ( (2n+1)π ( ν/λ−2nπ − x dx = = Kλ Kλ (x) dx (kλ H) λ λ n∈Z 2nπ n∈Z ν/λ−(2n+1)π ( ν ( ν/λ Kλ (x) dx + o(1) = K(x) dx + o(1) = (
ν/λ−π ν
=
ν−λπ
K(x) dx + o(1) = µ + o(1). −∞
Our next concern is to provide some information about the C ∗ algebra QC of all quasicontinuous functions on T (see 2.80). 3.27. Lemma. Let {Kλ }λ∈Λ be an approximate identity and let ν ∈ R. Then, for ϕ ∈ QC, ( π/λ ν λ Φ(x) dx → 0 (kλ Φ) − λ 2π −π/λ
as
λ → ∞,
where Φ(x) = ϕ(eix ) (x ∈ R). Proof. Let ε > 0 be given arbitrarily. By virtue of Theorem 3.23 there is a λ0 ∈ Λ such that ( ∞+ ν +2 ν ν + ν +2 + + + + − x dx = (kλ Φ2 ) − +(kλ Φ) +Φ(x) − (kλ Φ) + Kλ + λ λ λ λ −∞ ≤ kλ (ϕϕ) − (kλ ϕ)(kλ ϕ)∞ < ε2 for all λ ∈ Λ, λ > λ0 . Hence, by the CauchySchwarz inequality, ( ∞+ ν + ν + + − x dx < ε (3.14) +Φ(x) − (kλ Φ) +Kλ λ λ −∞ for λ > λ0 . Let Iλ denote the interval λν − πλ , λν + πλ . Then (recall the proof of Theorem 3.21)
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3 Symbol Analysis
Kλ
ν
ν πm 1 −x ≥ χ(−π/λ,π/λ) −x ∀ x ∈ R. λ 2 Iλ  λ
Thus, we obtain from (3.14) that ( + ν + 1 2 + + ε + dx < +Φ(x) − (kλ Φ) Iλ  Iλ λ πm Because
∀ λ > λ0 .
( + + ν + ν + 1 + + + + +Φ(x) − (kλ Φ) +ΦIλ − (kλ Φ) +≤ + dx, λ Iλ  Iλ λ
ν (kλ Φ) − ΦIλ → 0 as λ → ∞. (3.15) λ Taking into account that Φ ∈ V M O(R) (Theorem 3.17 and Lemma 3.18) and using 1.48(r), (iii), it is not diﬃcult to see that we deduce that
Φ(−π/λ,π/λ) − ΦIλ → 0 as
λ → ∞.
(3.16)
On combining (3.15) and (3.16) we get the assertion. 3.28. M (QC). Let A be a C ∗ subalgebra of L∞ (note that QC is a C ∗ subalgebra of L∞ ). Clearly, M (A) can be regarded as a subset of A∗ , the dual of A. We shall always think of A∗ as being equipped with its weakstar topology. Let {Kλ }λ∈Λ be an approximate identity. Each µ = (λ, t) ∈ Λ × T induces a functional δµ ∈ A∗ given by δµ : A → C,
a → (kλ a)(t),
and therefore Λ × T may be viewed as a subset of A∗ . By virtue of 3.14(b), Λ × T is contained in the unit ball of A∗ . If Λ = (r0 , ∞), then Λ × T can be identiﬁed with a circular annulus or a punctured disk in a natural way, and if Λ = {l0 , l0 + 1, l0 + 2, . . .} (l0 ∈ N), then Λ × T is the countable disjoint union of copies of the circle T. The weakstar closure of a set S ⊂ A∗ will be denoted by closA∗ S. 3.29. Proposition. Let A be a C ∗ algebra of L∞ and let {Kλ }λ∈Λ be an approximate identity. Then (a) M (A) ⊂ closA∗ (Λ × T),
(b) if C ⊂ A ⊂ QC, one has M (A) = closA∗ (Λ × T) \ (Λ × T). Proof. (a) Let ξ0 ∈ M (A). Any A∗ neighborhood of ξ0 is of the form U = Uε;a1 ,...,an (ξ0 ) = ξ ∈ A∗ : ξ(ai ) − ξ0 (ai ) < ε ∀ i = 1, . . . , n ,
3.3 Piecewise Quasicontinuous Functions
131
where ε > 0 and a1 , . . . , an ∈ A. We must show that there is a µ ∈ Λ × T such that δµ ∈ U . Put a = a1 − ξ0 (a1 ) + . . . + an − ξ0 (an ). Since A is a / GA, C ∗ algebra, a belongs to A. By construction, ξ0 (a) = 0. Therefore a ∈ hence a ∈ / GL∞ , so ess inf a(t) = 0. It follows that there is a sequence t∈T
{µn } = {(λn , tn )} ⊂ Λ × T such that λn → ∞ and δµn a → 0 as n → ∞ (this is an immediate consequence of Corollary 3.57 below; this claim also results from the fact that, for a ∈ L∞ , (kλ a)(t) → a(t) a.e. on T as λ → ∞, for whose proof see, e.g., Ahiezer [4, pp. 133–137]). Since ai − ξ0 (ai ) ≤ a on T for each i, we have σµ ai − ξ0 (ai ) ≤ δµ a for all µ ∈ Λ × T and each i. The conclusion is that there exists a µ0 ∈ Λ × T such that δµ0 ai − ξ0 (ai ) ≤ δµ0 a < ε for each i. But this is the assertion. (b) It is clear that δµ ∈ / M (A) for µ ∈ Λ × T, since, by (3.5), 1 −1 K [kλ (χ−1 χ1 )](eix ) − (kλ χ−1 )(eix )(kλ χ1 )(eix ) = 1 − K > 0. λ λ Now let ξ0 ∈ closA∗ (Λ × T) \ (Λ × T), let a ∈ A, b ∈ A, and ε > 0. By Theorem 3.23, there is a λ0 ∈ Λ such that sup sup δλ,t (ab) − δλ,t (a)δλ,t (b) < ε. λ>λ0 t∈T
Then choose a µ1 = (λ1 , t1 ) ∈ Λ × T so that λ1 > λ0 and δµ1 ∈ Uε;a,b,ab (ξ0 ) (the proof of part (a) shows that this is always possible). We have ξ0 (ab) − ξ0 (a)ξ0 (b) ≤ ξ0 (ab) − δµ1 (ab) + δµ1 (ab) − δµ1 (a)δµ1 (b) +δµ1 (a) − ξ0 (a) δµ1 (b) + ξ0 (a) δµ1 (b) − ξ0 (b) ≤ ε + ε + εδµ1 (b) + ξ0 (a)ε and since ε > 0 can be chosen arbitrarily, we get ξ0 (ab) = ξ0 (a)ξ0 (b), that is, ξ0 ∈ M (A). Remark 1. In particular, taking the approximate identity generated by the Poisson kernel, we see that the open unit disk D can be naturally identiﬁed with a subset of A∗ and that, for C ⊂ A ⊂ QC, M (A) is the weakstar closure of D minus D. Remark 2. Notice how simple the things are for C ∗ algebras. The unit disk D can be identiﬁed with a subset M (H ∞ ) via the harmonic extension (which is multiplicative on H ∞ ). That M (H ∞ ) is contained in the weakstar closure of D (i.e., that M (H ∞ ) ⊂ clos(H ∞ )∗ D, whence, obviously, M (H ∞ ) = clos(H ∞ )∗ D) is Carleson’s corona theorem! 3.30. QC ﬁbers over M (C). For τ ∈ T = M (C), let Mτ+ (QC) (resp. Mτ− (QC)) denote the set of all ξ ∈ Mτ (QC) such that ϕ(ξ) = 0 whenever ϕ ∈ QC and lim sup ϕ(t) = 0 (resp. lim sup ϕ(t) = 0). For a function a ∈ L1 t→τ +0
t→τ −0
and numbers τ = eiθ ∈ T, λ ∈ (1, ∞) deﬁne
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3 Symbol Analysis
(mλ a)(τ ) =
λ 2π
(
θ+π/λ
a(eix ) dx.
(3.17)
θ−π/λ
Thus, {mλ a}λ∈(1,∞) arises from a by applying the moving average. In accordance with 3.28, we may identify (1, ∞) × {τ } with a subset of QC ∗ : with (λ, τ ) ∈ (1, ∞) × {τ } we associate the functional δ(λ,τ ) : QC → C,
a → (mλ a)(τ ).
Let Mτ0 (QC) denote the points in M (QC) that lie in the weakstar closure of (1, ∞) × {τ }, i.e., Mτ0 (QC) = M (QC) ∩ closQC ∗ (1, ∞) × {τ } . It is clear that Mτ0 (QC) is a compact subset of the ﬁber Mτ (QC). Now let {Kλ }λ∈(1,∞) be any approximate identity. For ﬁxed ν ∈ R, the set Kν ⊂ (1, ∞) × R consisting, by deﬁnition, of all ordered pairs of the form (λ, ν/λ) (λ ∈ (1, ∞)) may be viewed as a subset of QC ∗ by identifying (λ, ν/λ) with the functional ν , δ(λ,ν/λ) : QC → C, a → (kλ A) λ where A(x) := a(eix ) (x ∈ R). Note that if {Kλ }λ∈(1,∞) is the moving average and ν = 0, then Kν is just the set (1, ∞) × {1} considered in the preceding paragraph. The following lemma shows that the points in M (QC) which lie in the weakstar closure of Kν are just the points in M10 (QC). 3.31. Lemma. M (QC) ∩ closQC ∗ Kν = M10 (QC). Proof. If ξ ∈ M10 (QC), then for each QC ∗ neighborhood U = Uε;ϕ1 ,...,ϕn (ξ) = η ∈ QC ∗ : η(ϕj ) − ϕj (ξ) < ε ∀ j = 1, . . . , n there is a λ1 ∈ (1, ∞) such that δ(λ1 ,1) ∈ U , i.e., + + ( + + λ1 π/λ1 + + Φj (x) dx+ < ε ∀ j = 1, . . . , n, +ϕj (ξ) − + + 2π −π/λ1
(3.18)
(3.19)
where Φj (x) = ϕj (eix ). So Lemma 3.27 implies that there is a λ2 ∈ (1, ∞) such that + ν + + + (3.20) +ϕj (ξ) − (kλ2 Φj ) + < 2ε ∀ j = 1, . . . , n. λ2 This shows that ξ is in the weakstar closure of Kν . Conversely, let ξ ∈ M (QC) be in the weakstar closure of Kν . Then obviously ξ ∈ M1 (QC). Given any neighborhood U of ξ of the form (3.18), there is a λ2 ∈ (1, ∞) satisfying (3.20) with ε in place of 2ε. Again by Lemma 3.27, there exists a λ1 ∈ (1, ∞) such that (3.19) holds with 2ε in place of ε. This implies that ξ ∈ M10 (QC).
3.3 Piecewise Quasicontinuous Functions
133
3.32. Deﬁnition. For ϕ ∈ L1 and τ = eiθ ∈ T, the integral gap γτ (ϕ) of ϕ at τ is deﬁned by + ( + ( θ + 1 θ+δ + 1 + + γτ (ϕ) := lim sup + ϕ(eix ) dx − ϕ(eix ) dx+ . + + δ δ δ→0+0 θ θ−δ If J is a subarc of T, we deﬁne V M O(J) in the natural manner (see 1.47 for J = T). As before, V M O := V M O(T). 3.33. Lemma. (a) If ϕ ∈ V M O, then γτ (ϕ) = 0 for each τ ∈ T. (b) If ϕ ∈ V M O(a, τ ) ∩ V M O(τ, b) and γτ (ϕ) = 0, then ϕ ∈ V M O(a, b). (c) If ϕ ∈ QC, ϕMτ0 (QC) = 0, and if p ∈ P C, then γτ (pϕ) = 0. Proof. (a) Let Φ(x) = ϕ(eix ) (x ∈ R). Without loss of generality assume τ = 1. Then, by Lemma 3.18 and 1.48(r), (iii), + ( + ( +1 δ + 1 0 + + Φ(x) dx − Φ(x) dx+ ≤ const · M2δ (Φ) = o(1) + +δ 0 + δ −δ as δ → 0, which is the assertion. (b) Let τ = eiθ . Fix ε > 0 and choose c > 0 so that ( 1 ϕ − ϕI  dm < ε I I whenever I is a subarc of (a, τ ) or of (τ, b) satisfying I < c, and so that + ( + ( + 1 θ+δ + 1 θ + + Φ(x) dx − Φ(x) dx+ < ε + +δ θ + δ θ−δ whenever δ < c. We show that if I is any subarc (a, b) such that I < c, then ( 1 ϕ − ϕI  dm < 6ε. I I It suﬃces to consider the case where τ ∈ I. First let τ be the center of I. Put I− = I ∩ (a, τ ) and I+ = I ∩ (τ, b). Then ϕI+ − ϕI−  < ε and ϕI = (1/2)(ϕI+ + ϕI− ), so that ϕI± − ϕI  < ε/2. Consequently, ( ( ( 1 1 1 ϕ − ϕI  dm = ϕ − ϕI  dm + ϕ − ϕI  dm I I 2I+  I+ 2I−  I− ( ( 1 1 ε ε ϕ − ϕI+  dm + ϕ − ϕI−  dm, < + + 4 4 2I+  I+ 2I−  I− which is less than 3ε/2. In fact this is also true for the case where I < 2c.
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3 Symbol Analysis
Now suppose I is any subarc of (a, b) containing τ , and let I0 be the smallest subarc of T that contains I and whose center is τ . By choosing c > 0 suﬃciently small we can guarantee that I0 ⊂ (a, b). So the preceding estimate applies to I0 , and therefore ( ( 1 1 ϕ − ϕI0  dm ≤ ϕ − ϕI0  dm < 3ε. I I 2I0  I0 Thus, + ( + ( ( + 1 + 1 1 + + ϕ dm − ϕI dm+ ≤ ϕ − ϕI0  dm < 3ε. ϕI − ϕI0  = + I I I I 0 I I * The two preceding inequalities give (1/I) I ϕ − ϕI  dm < 6ε, as desired. (c) Without loss of generality assume τ = 1. Every p ∈ P C can be written as p = cχ + g, where c ∈ C, χ is the characteristic function of the upper halfcircle, and g ∈ P C is continuous at τ = 1. By Theorem 3.17 and (a), γ1 (gϕ) = 0. It remains to show that γ1 (χϕ) = 0. We have + ( + +2 δ + + + Φ(x) dx+ 2γ1 (χϕ) = lim sup + + δ→0 + δ 0 + ( + ( +1 δ + 1 0 + + ≤ lim sup + Φ(x) dx − Φ(x) dx+ + + δ δ δ→0 0 −δ + ( + +1 + δ + + +2 lim sup + Φ(x) dx+ . (3.21) + δ→0 + 2δ −δ The ﬁrst term equals γ1 (ϕ), which is zero by (a). Let + ( + + + +1 + 1 ( δn + + δ + + + + Φ(x) dx+ = lim + Φ(x) dx+ . lim sup + n→∞ + + + + 2δ 2δ n −δn δ→0 −δ By the compactness of the unit ball of QC ∗ in the weakstar topology, there are a ξ ∈ M10 (QC) and a subsequence {δnk } of {δn } such that + + + 1 ( δnk + + + lim + Φ(x) dx+ = lim (mπ/δnk ϕ)(1) = ϕ(ξ) = 0, + k→∞ k→∞ + 2δnk −δ n k
i.e., the second term of (3.21) is zero, too.
3.34. Proposition (Sarason). If τ ∈ T, then Mτ+ (QC) ∩ Mτ− (QC) = Mτ0 (QC),
Mτ+ (QC) ∪ Mτ− (QC) = Mτ (QC).
3.3 Piecewise Quasicontinuous Functions
135
Proof. Without loss of generality assume τ = 1. Let ξ ∈ M10 (QC). If ϕ ∈ QC and lim sup ϕ(t) = 0, then t→1+0
+ ( + +1 δ + + + lim sup + Φ(x) dx+ = 0, + δ→0 + δ 0
whence, by Lemma 3.33(a) and Theorem 3.17, + ( 0 + +1 + + lim sup + Φ(x) dx++ = 0. δ −δ δ→0 + + ( + +1 δ + + lim sup + Φ(x) dx+ = 0, + + 2δ δ→0 −δ
So
and therefore
+ ( + +1 + δ + + ϕ(ξ) ≤ lim sup + Φ(x) dx+ = 0. + δ→0 + 2δ −δ
− It follows that ξ ∈ M1+ (QC). It can be shown similarly that ξ ∈ M 1 (QC). / M1− (QC) ∩ Now suppose ξ ∈ M1 (QC) \ M10 (QC). We show that then ξ ∈ M1+ (QC) ; this will give the ﬁrst equality in our proposition. There is a ϕ in QC such that ϕ(ξ) = 0 and ϕM10 (QC) = 0. Let p ∈ P C be continuous except for a jump at 1, with p(1 + 0) = 1 and p(1 − 0) = 0. Then pϕ ∈ V M O(T \ {1}) and Lemma 3.33(c) shows that γ1 (pϕ) = 0. Therefore, by Lemma 3.33(b) and Theorem 3.17, pϕ is in QC, and hence so also is (1 − p)ϕ. The function pϕ vanishes on M1− (QC) while (1 − p)ϕ vanishes on M1+ (QC). But since ϕ = pϕ + (1 − p)ϕ and ϕ(ξ) = 0, it is impossible for ξ to belong to both M1− (QC) and M1+ (QC). To establish the second equality, suppose ξ ∈ M1 (QC) \ M1+ (QC). Then there is a ϕ ∈ QC such that 0 ≤ ϕ ≤ 1, ϕ(ξ) = 0, ϕM1+ (QC) = 0. We claim that lim sup ϕ(t) = 0. To see this, let x := lim sup ϕ(t) and notice ﬁrst that t→1+0
t→1+0
x = lim sup ϕ(η) : η ∈ Mt (QC), t ∈ (1, eiθ ) . θ→0
It follows that there is a sequence {ηn } ∈ Mtn (QC) such that tn → 1 and ϕ(ηn ) → x. Due to the compactness of M (QC), there are a subsequence of {ηn }, again denoted by {ηn }, and an η ∈ M1 (QC) such that ηn → η in the weakstar topology. If ϕ ∈ QC and lim sup f (t) = 0, then f (η) = t→1+0
lim f (ηn ) = 0. Thus, η ∈ M1+ (QC). This shows that ϕ(η) = 0, and since
n→∞
both x and ϕ(η) are limits of ϕ(ηn ), we conclude that x = 0, as desired. Now let ψ be any function in QC with lim sup ψ(t) = 0. Then ϕψ is continuous at t→1−0
1 and takes the value 0 there. It follows that ϕψM1 (QC) = 0, in particular, ϕ(ξ)ψ(ξ) = 0. Because ϕ(ξ) = 0, we get ψ(ξ) = 0. Thus, ξ ∈ M1− (QC), and the proof is complete.
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3 Symbol Analysis
3.35. P QC. This is the smallest closed subalgebra of L∞ containing P C and QC, i.e., P QC = alg (P C, QC). Note that P QC is a C ∗ subalgebra of L∞ . The functions in PQC are referred to as piecewise quasicontinuous functions.
Let P QC0 denote the collection of all ﬁnite sums of the form i pi qi , where pi ∈ P C0 and qi ∈ QC. If ξ ∈ M (QC), then, by 1.27(b), the ﬁber Mξ (P QC) is not empty. Given y ∈ Mξ (P QC) put τ = yC and v = yP C (clearly, τ = ξC = vC). Since v belongs to Mτ (P C) and since Mτ (P C) is the doubleton {(τ, 0), (τ, 1)} (Proposition 3.24), we have either g(y) = pi (τ − 0)qi (ξ) ∀ g = pi qi ∈ P QC0 (3.22) i
or g(y) =
i
pi (τ + 0)qi (ξ)
∀g=
i
pi qi ∈ P QC0 .
(3.23)
i
Hence, if Mξ (P QC) would contain three distinct functionals, then two of them would coincide on P QC0 , and since P QC0 is dense in P QC, those two functionals would also coincide on P QC. The conclusion is that Mξ (P QC) contains at most two points. There is a natural mapping w : M (P QC) → M (QC) × {0, 1}, which is given as follows: for y ∈ M (P QC) let ξ = yQC, τ = yC, and v = yP C; if v = (τ, 0) (resp. v = (τ, 1)), i.e., if y satisﬁes (3.22) (resp. (3.23)), deﬁne w(y) = (ξ, 0) (resp. w(y) = (ξ, 1)). This mapping is clearly onetoone and therefore M (P QC) may be identiﬁed with a subset of the set M (QC)×{0, 1}. 3.36. Theorem (Sarason). Let ξ ∈ M (QC). Then (a) a(Mξ (L∞ )) = a(Mξ (P QC)) for all a ∈ P QC; (b) P QCMξ (L∞ ) = P CMξ (L∞ ); (c) Mξ (P QC) = {(ξ, 0)} for ξ ∈ Mτ− (QC) \ Mτ0 (QC), Mξ (P QC) = {(ξ, 1)} for ξ ∈ Mτ+ (QC) \ Mτ0 (QC); (d) Mξ (P QC) = {(ξ, 0), (ξ, 1)} for ξ ∈ M 0 (QC), and if {λn } ⊂ (1, ∞) is any sequence such that (λn , τ ) → ξ in the weakstar topology on QC ∗ , then for every a ∈ P QC the limits λn n→∞ π
(
θ0
lim
θ0 −π/λn
a(eix ) dx,
λn n→∞ π
(
θ0 +π/λn
lim
a(eix ) dx,
(3.24)
θ0
where τ = eiθ0 , exist and are equal to a(ξ, 0) and a(ξ, 1), respectively. Proof. (a) For x ∈ Mξ (L∞ ), put y = xP QC. It is clear that y ∈ Mξ (P QC). Therefore, if a ∈ P QC, then a(x) = x(a) = y(a) = a(y) and so a(Mξ (L∞ )) ⊂ a(Mξ (P QC)). On the other hand, by 1.20(b) and 1.26(c), each functional y ∈ Mξ (P QC) extends to a functional x ∈ Mξ (L∞ ). So the same argument as above shows that a(Mξ (P QC)) ⊂ a(Mξ (L∞ )).
3.3 Piecewise Quasicontinuous Functions
137
(b) If g = i pi qi ∈ P QC0 and x ∈ Mξ (L∞ ), then g(x) = i pi (x)qi (ξ), hence qi (ξ)(pi Mξ (L∞ )) ∈ P CMξ (L∞ ). gMξ (L∞ ) = i
If a ∈ P QC, then there are gn ∈ P QC0 such that a − gn ∞ → 0 as n → ∞, and since P CMξ (L∞ ) is closed (Proposition 3.25), it follows that aMξ (L∞ ) is in P CMξ (L∞ ). (c) Let y ∈ Mξ (P QC) and suppose (3.22) holds. We show that then ξ is in Mτ− (QC). Let ϕ ∈ QC and assume lim sup ϕ(t) = 0. If p ∈ P C is continuous t→τ −0
except for a jump at τ with p(τ − 0) = 1 and p(τ + 0) = 0, then obviously, γτ (pϕ) = 0, and so Lemma 3.33(b) and Theorem 3.17 imply that pϕ ∈ QC. Since pϕ is continuous at τ and takes the value 0 there, we have (pϕ)(ξ) = 0. Because of (3.22), (pϕ)(ξ) = (pϕ)(y) = p(τ − 0)ϕ(ξ) = ϕ(ξ), and it follows that ϕ(ξ) = 0. Thus ξ ∈ Mτ− (QC). / Mξ (P QC). This Consequently, if ξ ∈ Mτ (QC) \ Mτ− (QC), then (ξ, 0) ∈ and Proposition 3.34 prove the second equality of (c). The ﬁrst equality can be proved analogously. (d) Without loss of generality assume τ = 1. If a ∈ QC, then λn n→∞ 2π
(
π/λn
a(ξ) = lim hence
a(eix ) dx,
−π/λn
+ + + λ ( π/λn + + n + ix lim sup + a(e ) dx − a(ξ)+ + n→∞ + π 0 + + ( ( π/λn +λ + π/λn λ + n + n ≤ lim sup + a(eix ) dx − a(eix ) dx+ + + π 2π n→∞ 0 −π/λn + + + λ ( π/λn + + n + + lim sup + a(eix ) dx − a(ξ)+ + n→∞ + 2π −π/λn =
1 γ1 (a) + 0 = 0 (by 3.33(a)), 2
(3.25)
which implies that the second limit in (3.24) exists and equals a(ξ). If a ∈ P C0 , then obviously ( λn π/λn lim a(eix ) dx = a(τ + 0). (3.26) n→∞ π 0
To see that the second limit in (3.24) exists for any a = i pi qi ∈ P QC0 1 (pi ∈ P C0 , qi ∈ QC), note ﬁrst that, for any ϕ ∈ L ,
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3 Symbol Analysis
λn π
(
π/λn
ϕ(eix ) dx = (m2λn Φ)(δn ),
0
where δn := π/(2λn ), Φ denotes the periodic extension of ϕ, and we make use of notation (3.17). So + + + + + + pi (τ + 0)qi (ξ)+ lim sup +(m2λn A)(δn ) − + + n→∞ i ≤ lim sup (m2λn Pi Qi )(δn ) − (m2λn Pi )(δn )(m2λn Qi )(δn ) i
+
i
n→∞
lim sup (m2λn Pi )(δn )(m2λn Qi )(δn ) − pi (τ + 0)qi (ξ) n→∞
and the ﬁrst upper limit on the right is zero by virtue of the asymptotic multiplicativity of the moving average on the pair (L∞ , QC) (Proposition 3.20), while the second upper limit equals zero due to (3.25) and (3.26). It follows that the limit ( λn π/λn a(eix ) dx (3.27) y(a) := lim n→∞ π 0 exists for every a ∈ P QC0 and that (3.23) holds. This shows that y is linear and multiplicative on P QC0 . Since obviously y(a) ≤ a∞ for all a ∈ P QC0 , y extends to a linear and multiplicative (bounded) functional y) on P QC, and (3.23) implies that y) = (ξ, 1). Thus, (ξ, 1) ∈ Mξ (P QC). Finally, if a ∈ P QC and b ∈ P QC0 , then + + ( + + λn π/λn + + a(eix ) dx+ lim sup +y)(a) − + + π n→∞ 0 + + ( + + λn π/λn + + ix ≤ ) y (a) − y)(b) + lim sup +y)(b) − b(e ) dx+ + π 0 n→∞ + + + ( +λ + π/λn + n + + lim sup + [b(eix ) − a(eix )] dx+ + n→∞ + π 0 ≤ a − b∞ + 0 + b − a∞ = 2a − b∞ and a − b∞ can be made as small as desired. The conclusion is that the limit (3.27) exists for every a ∈ P QC and that it is equal to y)(a) = a(ξ, 1). This completes the proof for (ξ, 1). The proof for (ξ, 0) is analogous. Remark 1. We observed in 3.35 that Mξ (P QC) contains at most two points. This and part (a) of the above theorem imply that a function in P QC takes at most two values on each ﬁber Mξ (L∞ ). The same is true for functions in (P QC)N ×N .
3.3 Piecewise Quasicontinuous Functions
139
Remark 2. One can show that Mτ0 (QC) is a proper subset of Mτ (QC) (Sarason [456, p. 824]). This shows that the mapping w introduced in 3.35 is not onto. Remark 3. We saw in 3.35 that M (P QC) can be identiﬁed with a subset of M (QC)×{0, 1}. The preceding theorem shows which points of M (QC)×{0, 1} belong to M (P QC). The Gelfand topology on M (P QC) can now be described as follows. For ξ ∈ M (QC) let V(ξ) denote the family of open neighborhoods of ξ. For ξ ∈ Mτ (QC) and V ∈ V(ξ) let Vτ = V ∩ Mτ (QC) and let Vτ+ and Vτ− denote the sets of points in V that lie above the semicircles {eiθ : arg τ < θ < arg τ + π} and {eiθ : arg τ − π < θ < arg τ }, respectively. Then, if ξ ∈ Mτ+ (QC), the sets 0 1 (Vτ × {1}) ∪ (Vτ+ × {0, 1}) ∩ M (P QC), V ∈ V(ξ), form an open neighborhood base for (ξ, 1). If ξ ∈ Mτ− (QC), the sets 0 1 (Vτ × {0}) ∪ (Vτ− × {0, 1}) ∩ M (P QC), V ∈ V(ξ) form an open neighborhood base for (ξ, 0). 3.37. Pn QC. The collection of all matrix functions a ∈ L∞ N ×N which have the property that a(Mξ (L∞ )) contains at most n points for each ξ ∈ M (QC) will be denoted by (Pn QC)N,N . In the case N = 1 we shall write Pn QC in place of (Pn QC)1,1 . From (2.43) we know that (P1 QC)N,N = QCN ×N . It is obvious that P2 C ⊂ P2 QC and from Remark 1 in the preceding section it follows that P QC ⊂ P2 QC. Let a ∈ P QC and let m be any positive real number. We claim that the set {τ ∈ T : γτ (a) > m} is ﬁnite. Indeed, there is a function b ∈ P QC0 such that
a − b∞ < m/2, so γτ (b) > m/2 whenever γτ (a) > m. But b is a ﬁnite sum i pi qi , where pi ∈ P C0 and qi ∈ QC. This implies that γτ (b) = 0 for all τ at which all pi ’s are continuous and this proves our claim. Now let χE be the characteristic function of the set E=
∞ / n=1
En ,
En = eiθ ∈ T :
1 22n+1
< θ <
1 . 22n
The integral gap γτ (χE ) equals 1 at countably many points τ ∈ T. Hence, by what was said above, χE ∈ P2 C \ P QC. Thus, if q ∈ QC \ P4 C (the function Im ω constructed in 2.80 is in QC \P4 C), then χE +q ∈ P2 QC \(P2 C ∪P QC), and it follows that P2 QC is strictly larger than P2 C ∪ P QC. Finally, note that (P QC)N ×N ⊂ (P2 QC)N,N , by Remark 1 of the preceding section. For a ∈ (P2 QC)N,N and ξ ∈ M (QC) let a1ξ ∈ CN ×N and a2ξ ∈ CN ×N (possibly a1ξ = a2ξ ) denote the points lying in a(Mξ (L∞ )). The same reasoning as in the proof of Proposition 3.11 shows that a is locally sectorial over QC if
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3 Symbol Analysis
and only if the (possibly degenerate) line segment [a1ξ , a2ξ ] contains only invertible matrices. The disadvantage of this criterion is that it does not reﬂect the geometric data of the function a suﬃciently well. The purpose of what follows is to derive conditions for the local sectoriality of functions in (P2 QC)N,N (that will be necessary and suﬃcient ones for functions in (P QC)N,N ) which better correspond with their geometric properties. In order to attack that problem we shall develop a machinery which at the ﬁrst glance seems to be very heavy but is in fact very simple. Moreover, this machinery will become of decisive importance for a whole series of other problems we shall be concerned in the following.
3.4 Harmonic Approximation: Algebraization 3.38. Deﬁnitions. Let A and B be Banach algebras. A mapping i : A → B is called a quasiembedding if it is linear, continuous, has a closed image, and if its kernel Ker i is a (closed) twosided ideal of A. A quasiembedding whose kernel is trivial will be referred to as an embedding. If i : A → B is a quasiembedding, then the mapping ie deﬁned (correctly) by ie : A/Ker i → B,
a + Ker i → i(a)
is obviously an embedding. If i : A → B is linear and i(a)B = aA for every a ∈ A, then i is referred to as an isometry. Isometries are obviously embeddings. A mapping i : A → B is said to be submultiplicative if there is a constant γ > 0 such that " " " " n m m " " n " " "i " " ajk " ≤ γ " i(ajk )" (3.28) " " j=1 k=1
B
j=1 k=1
B
for every ﬁnite collection of elements ajk ∈ A. Any submultiplicative mapping i : A → B which satisﬁes (3.28) will also be called γsubmultiplicative. If i is a γsubmultiplicative quasiembedding then ie is obviously a γsubmultiplicative embedding. 3.39. Deﬁnitions. Let Λ be either of the sets {l0 , l0 + 1, l0 + 2, . . .} (l0 ∈ N) or (r0 , ∞) (r0 ∈ R+ ). We let A∞ N,N denote the collection of all (generalized) sequences {aλ }λ∈Λ of continuous matrix functions aλ ∈ CN ×N such that < ∞: sup aλ L∞ N ×N
λ∈Λ
A∞ N,N
! # ∞ := {aλ }λ∈Λ : aλ ∈ CN ×N , sup aλ LN ×N < ∞ . λ∈Λ
On deﬁning α{aλ } := {αaλ }, {aλ } + {bλ } := {aλ + bλ }, {aλ }{bλ } := {aλ bλ }, {aλ }∗ := {a∗λ },
3.4 Harmonic Approximation: Algebraization
{aλ } := sup aλ L∞ := sup M (aλ )L(L2N ) N ×N λ∈Λ
141
(3.29)
λ∈Λ
∗ we make A∞ N,N become a C algebra. Put
AN,N :=
∞ {aλ } ∈ A∞ N,N : there exists an a ∈ LN ×N such that
M (aλ ) → M (a), M ∗ (aλ ) → M ∗ (a) strongly on L2N as λ → ∞ .
Here the asterisk refers to the adjoint operator. It can be checked straightfor∞ ∞ wardly that AN,N is a C ∗ subalgebra of A∞ N,N . Denote A1,1 and A1,1 by A ∞ and A, respectively. Because (3.29) is an admissible norm in AN,N and AN,N ∞ (recall 1.29), we have A∞ N,N = AN ×N and AN,N = AN ×N . Therefore we shall ∞ henceforth AN,N and AN,N denote by A∞ N ×N and AN ×N , respectively. If Λ = (r0 , ∞), then A∞ N ×N and AN ×N can be thought of as algebras of matrix functions given on an annulus or a punctured disk. However, for certain reasons it will be more advantageous to work with algebras of generalized sequences, although for a moment this seems to be an unnecessary complication. Finally, given an approximate identity {Kλ }λ∈Λ deﬁne kλ a for a = ∞ N (ajk )N j,k=1 ∈ LN ×N as (kλ ajk )j,k=1 . 3.40. Proposition. Let {Kλ }λ∈Λ be an approximate identity. (a) If a ∈ L∞ N ×N , then {kλ a}λ∈Λ ∈ AN ×N . (b) The mapping K : L∞ N ×N → AN ×N , a → {kλ a}λ∈Λ is a 1submultiplicative isometry. Proof. (a) First let N = 1. From 3.14(a), (b) we deduce that {kλ a} ∈ A∞ and sup M (kλ a)L(L2 ) ≤ M (a)L(L2 )
(3.30)
λ
for every a ∈ L∞ . Since, by 3.14(c), ( M (kλ a)χn −
M (a)χn 2L2
= T
kλ a − a2 dm → 0 (λ → ∞)
for all n ∈ Z (χn (t) := tn ), we have M (kλ a)ϕ − M (a)ϕL2
as
λ → ∞ ∀ ϕ ∈ P.
(3.31)
But (3.30) and (3.31) in conjunction with 1.1(d) imply that M (kλ a) → M (a) strongly on L2 . The argument applies to a in place of a, which completes the proof for N = 1. The assertion for general N follows from the fact that the norm in A∞ N ×N is admissible. (b) If N = 1, then, by part (a), 1.1(e), 3.14(b),
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3 Symbol Analysis
a∞ = M (a) ≤ lim inf M (kλ a) ≤ sup M (kλ a) = sup kλ a ≤ a∞ , λ→∞
λ
which shows that {kλ a}A = a∞
λ
∀ a ∈ L∞ .
Now let N > 1. Then the norms in both AN ×N and in 1.29(b). Hence, for a = (ajk ) ∈ L∞ N ×N , " " " " " {k a }ϕ(I ) {kλ a}AN ×N = sup " λ jk jk " " ϕ
(3.32)
L∞ N ×N
can be written as
(by 1.29(b))
A
j,k
"! #" " " " k a ϕ(I ) = sup " λ jk jk " " ϕ A j,k " " " " = sup " ajk ϕ(Ijk )" (by (3.32)) " " ϕ
L∞
j,k
= aL∞ N ×N
(again by 1.29(b)).
(3.33)
Thus, K is an isometry. If aij is a ﬁnite collection of matrix functions in L∞ N ×N , then "! #" " " " " " " " " kλ " a = aij " (by (3.33)) ij " " ∞ " " AN ×N LN ×N i j i j " " " " " " " " " " " = "M aij " = " M (aij )" " i
j
i
j
i
j
" " " " " ≤ lim inf " M (kλ aij )" " (by part (a) and 1.1(e)) λ→∞ i j " " " " " " " " " " " ≤ sup " M (kλ aij )" = sup "M kλ aij " " λ
" ! #" " " " =" k a λ ij " " i
j
λ
i
j
. AN ×N
This proves that K is 1submultiplicative.
3.41. Deﬁnitions. Let A and B be Banach algebras and let i : A → B be a mapping of A into B. We denote by i(A) the image (range) of i; alg i(A) the closed subalgebra generated by i(A), that of B is, 5m n alg i(A) := closB j=1 k=1 i(ajk ) : ajk ∈ A ; Qi (A) the quasicommutator ideal of alg i(A), that is, the smallest closed twosided ideal of alg i(A) containing all elements of the form i(ab) − i(a)i(b), a ∈ A, b ∈ A. To avoid confusion, alg i(A) will be sometimes denoted by algB i(A).
3.4 Harmonic Approximation: Algebraization
143
3.42. Theorem. Let A and B be Banach algebras and suppose i : A → B is a submultiplicative quasiembedding. Then alg i(A) decomposes into the direct sum of i(A) and Qi (A): ·
alg i(A) = i(A) + Qi (A). Proof. The set F :=
! n m
i(ajk ) : n, m ∈ Z+ , ajk
# ∈A
j=1 k=1
is dense in alg i(A). Deﬁne the linear mapping S : F → F by S: i(ajk ) → i ajk . j
j
k
k
The hypothesis that i be implies that S is well deﬁned, i.e., 5
submultiplicative
5 if f = j k i(bjk ) = l m i(clm ) then i bjk = i clm , j
k
l
m
and that S is bounded on F . Therefore S extends to a bounded linear mapping on the whole algebra alg i(A) into itself. Since S(i(a)) = i(a) for every a ∈ A, it follows that S is a projection on alg i(A) and that i(A) ⊂ S(alg i(A)). The hypothesis that i(A) be closed in B implies that actually i(A) = S(alg i(A)). We claim that the closed set Ker S is a twosided ideal in alg i(A). Let b ∈ Ker S and c ∈ alg i(A). Then there are sequences {bn }, {cn } ⊂ F such that bn → b, cn → c, S(bn ) → 0 and, hence, bn cn → bc. Again from the submultiplicativity of i we deduce that S(bn cn ) ≤ γS(bn )S(cn ) ≤ γS(bn ) S(cn ), and therefore S(bc) = 0. It can be shown in the same way that S(cb) = 0. Thus, Ker S is a closed twosided ideal in alg i(A). Our next objective is to prove that Ker S = Qi (A). Since Ker S has been shown to be an ideal, we have Qi (A) ⊂ Ker S. To get the reverse inclusion, let b ∈ Ker S and choose a sequence {bn } ⊂ F such that bn → b and S(bn ) → 0. A little thought shows that bn − S(bn ) ∈ Qi (A) and passage to the limit n → ∞ yields that b ∈ Qi (A). Thus Ker S ⊂ Qi (A). ·
Putting the things together, we have alg i(A) = Ker S + Im S (because S is a bounded projection on alg i(A)) and at the same time Ker S = Qi (A) and Im S = i(A), which is the assertion.
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3 Symbol Analysis
3.43. Deﬁnitions. Let the hypothesis of Theorem 3.42 be fulﬁlled. We denote by the (necessarily continuous) projection of alg i(A) onto i(A) parallel to Qi (A); from the proof of the preceding theorem it is seen that Si ≤ γ in case i is γsubmultiplicative; i(−e) the linear homeomorphism given (correctly) by i(−e) : i(A) → A/Ker i, i(a) → a + Ker i; Smbi the continuous and linear mapping Smbi : alg i(A) → A/Ker i, B → i(−e) Si (B); Qi (A) and that it is clear Ker Smb i =
that 5 5 Smbi j k i(ajk ) = j k ajk + Ker i (the sum and the products ﬁnite); the linear homeomorphism given (correctly) by σi σi : alg i(A)/Qi (A) → A/Ker i, B + Qi (A) → Smbi (B).
Si
Thus, we have the following commutative diagram:
In the case where i is an embedding (Ker i = {0}) we regard Smbi as the mapping (3.34) Smbi : alg i(A) → A, A → i(−1) Si (A), where i(−1) is the inverse of i viewed as acting from A onto i(A). 3.44. Corollary. Suppose the hypotheses of Theorem 3.42 are satisﬁed. Then σi is a homeomorphic isomorphism of alg i(A)/Qi (A) onto A/Ker i. Proof. It remains to show that σi is multiplicative. Let A, B ∈ alg i(A). Due to Theorem 3.42 we have A = i(a) + K and B = i(b) + L with a, b ∈ A and K, L ∈ Qi (A). Hence, AB = i(a)i(b) + N = i(ab) + M with certain N, M ∈ Qi (A) and thus, Si (A) = i(a),
Si (B) = i(b),
Si (AB) = i(ab).
Consequently, σi ((A + Qi (A))(B + Qi (A))) = σi (AB + Qi (A)) = Smbi (AB) = i(−e) Si (AB) = i(−e) i(ab) = ab + Ker i = (a + Ker i)(b + Ker i) = i(−e) i(a) · i(−e) i(b) = i(−e) Si (A) · i(−e) Si (B) = Smbi (A)Smbi (B) = σi (A + Qi (A))σi (B + Qi (A)).
3.4 Harmonic Approximation: Algebraization
145
Before applying 3.42 and 3.44 to the concrete situation given by 3.40 we need two further results of technical nature. 3.45. Deﬁnitions. Let A be a Banach algebra and let S be a subset of A. We denote by algA S the smallest closed subalgebra of A containing S; closidA S the smallest closed twosided ideal of A containing S. It is clear that algA S = closA
! n m
# sjk : sjk ∈ S ,
j=1 k=1 ! n
# aj sj bj : sj ∈ S, aj ∈ A, bj ∈ A
closidA S = closA
j=1
For a ∈ A, we let a ⊗ Ijk denote the element in AN ×N whose jk entry is a and all other entries of which are zero. 3.46. Lemma. Let A be a Banach algebra and suppose an (admissible) Banach algebra norm is given in AN ×N . (a) If C is a closed twosided ideal in A, then CN ×N is a closed twosided ideal in AN ×N . (b) If S is a subset of A, then (algA S)N ×N = algAN ×N SN ×N . (c) If S is a subset of A, then (closidA S)N ×N = closidAN ×N SN ×N . Proof. (a) Obvious. (b) It is clear that the righthand side is contained in the lefthand side. In order to establish the reverse inclusion, we must show that s1 . . . sm ⊗ Ijk belongs to algAN ×N SN ×N for all s1 , . . . , sm ∈ S. This is obvious for m = 1 or for j = k. the general case follows from the identity s1 . . . sm ⊗ Ijk = (s1 ⊗ Ijk )(s2 . . . sm ⊗ Ikk ). (c) That the righthand side is a subset of the lefthand side is trivial. The reverse inclusion results from the identity asb ⊗ Ijk = (a ⊗ Ijk )(s ⊗ Ikk )(b ⊗ Ikk ).
3.47. Lemma. Let A and B be Banach algebras, let i : A → B be a linear mapping, suppose AN ×N and BN ×N are endowed with (admissible) Banach algebra norms, and deﬁne iN ×N : AN ×N → BN ×N ,
(ajk ) → (i(ajk )).
Then (algB i(A))N ×N = algBN ×N iN ×N (AN ×N ), (Qi (A))N ×N = QiN ×N (AN ×N ).
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3 Symbol Analysis
Proof. The ﬁrst equality follows immediately from Lemma 3.46(b). The second one will follow from Lemma 3.46(c) as soon as we have shown that (S1,1 )N ×N equals SN,N , where Sm,m is the collection of all quasicommutators im×m (ab) − im×m (a)im×m (b),
a, b ∈ Am×m .
The inclusion (S1,1 )N ×N ⊂ SN,N is a consequence of the identity i(ab) − i(a)i(b) ⊗ Ijk = iN ×N (a ⊗ Ijk )(b ⊗ Ikk ) − iN ×N (a ⊗ Ijk )iN ×N (b ⊗ Ikk ) and the reverse inclusion results from the observation that [iN ×N (ab) − iN ×N (a)iN ×N (b)]jk =i ajl blk − i(ajl )i(blk ) = [i(ajl blk ) − i(ajl )i(blk )]. l
l
l
3.48. Deﬁnitions. Let A be a closed subalgebra of L∞ containing the constants. Then AN ×N is a closed subalgebra of L∞ N ×N . Let {Kλ }λ∈Λ be an approximate identity and deﬁne the mapping K as in Proposition 3.40. Put alg K(AN ×N ) := algAN ×N K(AN ×N ), and let QK (AN ×N ) be the smallest closed twosided ideal of alg K(AN ×N ) containing all elements of the form {kλ (ab) − (kλ a)(kλ b)}λ∈Λ ,
a, b ∈ AN ×N
(these deﬁnitions are in accordance with 3.41 and 3.45). Lemma 3.47 tells us that alg K(AN ×N ) = (alg K(A))N ×N , QK (AN ×N ) = (QK (A))N ×N
(3.35) (3.36)
(note that the K in Proposition 3.40 is actually the KN ×N ). Proposition 3.40 and Theorem 3.42 give that ·
alg K(AN ×N ) = K(AN ×N ) + QK (AN ×N ),
(3.37)
and since K is 1submultiplicative, we have SK = 1 (recall 3.43). If {aλ } is in AN ×N , then there is an a ∈ L∞ N ×N such that M (aλ ) → M (a) strongly on L2N ; the (obviously linear and bounded) mapping AN ×N → L∞ N ×N which assigns that a to the sequence {aλ } will be denoted by LA for the meanwhile. 3.49. Proposition. (a) SK = K ◦ LA alg K(AN ×N ). (b) SmbK = LA alg K(AN ×N ). (c) Let {aλ } ∈ alg K(AN ×N ). Then {aλ } ∈ QK (AN ×N ) ⇐⇒ LA ({aλ }) = 0.
3.4 Harmonic Approximation: Algebraization
Proof. (a) If {aλ } =
j
kλ ajk
147
∈ alg K(AN ×N ),
k
the sum and the products are ﬁnite, then SK ({aλ }) = kλ = (K ◦ LA )({aλ }), ajk j
k
because M (aλ ) → M
j
ajk
k
strongly (see the proof of Proposition 3.40(a)). The continuity of SK and K ◦ LA give the assertion for general {aλ } ∈ alg K(AN ×N ). (b), (c) Immediate from (a).
3.50. Deﬁnition. Put NN,N := {aλ } ∈ A∞ → 0 as λ → ∞ N ×N : aλ L∞ N ×N and given a closed subalgebra A of L∞ let A NN,N = NN,N ∩ alg K(AN ×N ).
It is easy to see that NN,N is a closed twosided ideal of both A∞ N ×N and A is a closed twosided ideal of alg K(AN ×N ). It is clear AN ×N and that NN,N A = that NN,N = NN ×N , where N := N1,1 . This and (3.35) imply that NN,N A A A A NN ×N := (N )N ×N , where N := N1,1 . Therefore we shall henceforth write A , respectively. NN ×N and NNA×N in place of NN,N and NN,N 3.51. Proposition. (a) Let B be a closed subalgebra of QC containing the constants. Then QK (BN ×N ) = NNB×N . (b) Let B be a C ∗ subalgebra of QC. Then the mapping σK : alg K(BN ×N )/NNB×N → BN ×N ,
{aλ } + NNB×N → LA ({aλ })
is an isometric starisomorphism. (c) Let B be a C ∗ subalgebra of L∞ . If QK (BN ×N ) ⊂ NN ×N , then B is a subset of QC. Proof. (a) From Proposition 3.49(a) we deduce that NNB×N ⊂ Ker SK = QK (BN ×N ). On the other hand, the asymptotic multiplicativity of {Kλ } on the pair (QC, QC) (Theorem 3.23) implies that QK (BN ×N ) ⊂ NNB×N .
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3 Symbol Analysis
(b) Part (a), Corollary 3.44, and Proposition 3.49(b) give that σK is a homeomorphic starisomorphism. Finally, 1.26(e) shows that it is even an isometry. (c) By (3.36), it suﬃces to consider the case N = 1. If a ∈ B, then a ∈ B and so {kλ a}{kλ a} − {kλ (aa)} ∈ N , whence kλ a2 − kλ (a2 )∞ → 0 (λ → ∞) ∀ a ∈ B.
(3.38)
Let A(x) := a(eix ) (x ∈ R). Then, for x ∈ R, ( ∞ kλ,x (A − kλ,x A) = A(t) − (kλ A)(x)Kλ (x − t) dt −∞
( ≤ (
∞ −∞ ∞
= −∞
1/2 A(t) − (kλ A)(x)2 Kλ (x − t) dt 1/2 A(t)2 Kλ (x − t) dt − (kλ A)(x)2
1/2 = kλ,x (A2 ) − kλ,x A2 , hence, for t ∈ T, 1/2 kλ,t (a − kλ,t a) ≤ kλ,t (a2 ) − kλ,t a2 . Consequently, taking into account (3.38) we see that for each ε > 0 there is a λ0 ∈ Λ, λ0 > 1, such that aλ0 :=
sup
sup kλ,t (a − kλ,t a) < ε.
λ>λ0 ,λ∈Λ t∈T
The ﬁrst part of the proof of Theorem 3.21 shows that M2π/λ0 (a) ≤ caλ0 with some constant c independent of a and λ0 (recall 1.47). Therefore lim M2π/λ (a) = 0,
λ→∞
it follows that a ∈ V M O, and since a ∈ L∞ , we have a ∈ QC by Theorem 3.17. Remark. In case {Kλ }λ∈Λ is the approximate identity generated by the Poisson kernel, we shall write H, alg H(AN ×N ), etc. in place of K, alg K(AN ×N ), etc. Theorem 2.62(a) shows that the equality QK (BN ×N ) = NNB×N also holds for B = C + H ∞ .
3.5 Harmonic Approximation: Essentialization
149
3.5 Harmonic Approximation: Essentialization 3.52. Theorem. Suppose (a) A, B are Banach algebras and i : A → B is a γsubmultiplicative quasiembedding; (b) J is a closed twosided ideal of alg i(A); (c) Si (J) ⊂ J. Deﬁne the mapping iπ by iπ : A → alg i(A)/J,
a → i(a) + J.
Then (d) iπ is a γsubmultiplicative quasiembedding and Ker iπ = {a ∈ A : i(a) ∈ J}; (e) alg iπ (A) = alg i(A)/J; (f) σiπ is a homeomorphic isomorphism of the algebra alg iπ (A)/Qiπ (A) onto the algebra A/Ker iπ . Proof. (d) It is clear that iπ is linear and continuous. Let π denote the canonical projection of alg i(A) onto alg i(A)/J. Since i(A) + J = Ker π(I − Si ) is closed, πi(A) is normally solvable (see Gohberg, Krupnik [232, Chap. 4, Theorem 2.1]) and so iπ (A) is closed. We have a ∈ Ker iπ if and only if i(a) ∈ J. Hence, if a ∈ Ker iπ and b ∈ A, then i(ab) = Si (i(a)i(b)) ∈ J because of (c), and it follows that Ker iπ is a (necessarily closed) twosided ideal of A. It remains to show that iπ is γsubmultiplicative. Since " " " " "π " " " " "i " ajk " = "i ajk + J " " " j j k k " " " " " = "Si i(ajk ) + J " " j
k
" " " " " S = inf " i(a ) + G jk " i " G∈J
i
k " " " " " ≤ inf "Si i(ajk ) + Si (H)" " H∈J j k " " " " " ≤ Si inf " i(a ) + H jk " " H∈J j k " " " " " = Si " i(a ) + J jk " " j k " " " " π " = Si " i (a ) jk " " i
k
(due to (c))
(3.39)
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3 Symbol Analysis
and because Si ≤ γ, the γsubmultiplicativity follows. (e) The assertion is that the ﬁnite productsums iπ (ajk ) = i(ajk ) + J, ajk ∈ A, j
j
k
k
are dense in alg i(A)/J, which can be checked straightforwardly. (f) Immediate from Corollary 3.44. 3.53. Corollary. (a) If, in addition to the hypotheses 3.52(a)–(c), i is 1submultiplicative, then, for all a ∈ A, " " iπ (a) = "Siπ iπ (a) " = inf iπ (a) + G. G∈Qiπ (A)
(b) If, in addition to the hypotheses 3.52(a)–(c), i is 1submultiplicative and both A and alg i(A) are C ∗ algebras, then, for all a ∈ A, " " iπ (a) = "Smbiπ iπ (a) " = inf π a + g. g∈Ker i
Proof. (a) Since iπ (a) = Siπ (iπ (a) + G) for every G ∈ Qiπ (A), we have " " π "π " "Siπ i (a) + G " ≤ "i (a) + G" ≤ iπ (a). inf iπ (a) = inf G∈Qiπ (A)
G∈Qiπ (A)
(b) Theorem 3.52(f) and 1.26(e) imply that inf G∈Qiπ (A)
iπ (a) + G =
inf
g∈Ker iπ
a + g
and it remains to apply part (a). 3.54. Proposition. Let AπN ×N := AN ×N /NN ×N . Then π Kπ : L∞ N ×N → AN ×N ,
a → {kλ a}π := {kλ a} + NN ×N
is a 1submultiplicative isometry. Proof. Theorem 3.52(d) applied with A = L∞ N ×N , B = AN ×N , i = K, J = NN ×N (whose hypotheses are satisﬁed due to 3.40 and 3.49(a)) shows that Kπ is 1submultiplicative. If a ∈ L∞ N ×N and {cλ } ∈ NN ×N , then M (kλ a + cλ ) → M (a) strongly (see the proof of Proposition 3.40), so that ! # a ≤ inf lim inf kλ a + cλ : {cλ } ∈ NN ×N λ→∞ ! # ≤ inf sup kλ a + cλ : {cλ } ∈ NN ×N λ
= {kλ a}π ≤ {kλ a} = a, which implies that Kπ is an isometry.
3.5 Harmonic Approximation: Essentialization
151
Remark. Let A be a closed subalgebra of L∞ . In view of Theorem 3.52, alg Kπ (AN ×N ) equals alg K(AN ×N )/NNA×N . We also have ·
alg Kπ (AN ×N ) = Kπ (AN ×N ) + QKπ (AN ×N ), SKπ = 1, and alg Kπ (AN ×N )/QKπ (AN ×N ) is homeomorphically (isometrically in case A is a C ∗ algebra) isomorphic to AN ×N , because Ker Kπ = {0} by the preceding proposition. 3.55. Deﬁnition. Let {aλ }λ∈Λ ∈ A∞ N ×N . The (generalized) sequence {aλ } is said to be bounded away from zero (abbreviated as bafz ) if there exists a λ0 ∈ Λ such that (a) aλ ∈ GL∞ N ×N for all λ ∈ Λ, λ > λ0 ,
(b)
sup λ>λ0 ,λ∈Λ
∞ a−1 λ LN ×N < ∞.
Since aλ ∈ CN ×N , (a) is equivalent to the requirement that det aλ (t) = 0 for all t ∈ T, λ ∈ Λ, λ > λ0 . It is easily seen that the following equivalences are true: {aλ } bafz ⇐⇒ {det(aλ )} bafz ⇐⇒ {aλ }π ∈ G(A∞ N ×N /NN ×N ). Of particular importance is the case where aλ = kλ a for some a ∈ L∞ N ×N . For instance, the statement of Theorem 2.62(b) is that if a ∈ (C + H ∞ )N ×N , then a ∈ G(C + H ∞ )N ×N if and only if {kλ (det a)} is bounded away from zero, where {Kλ } is the approximate identity generated by the Poisson kernel. 3.56. Theorem. Let {aλ } ∈ alg K(AN ×N ), where A is a C ∗ subalgebra of L∞ containing the constants. Then {aλ } bafz ⇐⇒ {aλ }π ∈ G(alg Kπ (AN ×N )). Proof. We suppress the subscript N × N . The implication “⇐=” is obvious. So we are left with the implication “=⇒”. Thus, let {aλ }π = {aλ } + N ∈ G(A∞ /N ). Because alg K(A) is a C ∗ subalgebra of A∞ , it follows, that alg K(A) + N is a C ∗ subalgebra of A∞ (1.26(g)), hence (alg K(A) + N )/N is a C ∗ subalgebra of A∞ /N (1.26(f)). Therefore {aλ } + N belongs to G((alg K(A) + N )/N ) (1.26(d)). Taking into account 1.26(g) once more, we obtain that {aλ } + N A is in G(alg K(A)/N A ) = G(alg Kπ (A)). 3.57. Corollary. Let {aλ } ∈ alg K(L∞ N ×N ) be bounded away from zero. Then SmbK ({aλ }) ∈ GL∞ N ×N . Proof. The preceding theorem implies that {aλ }π is in G(alg Kπ (L∞ N ×N )). ) such that b a = I + c with some Hence, there is a {bλ } ∈ alg Kπ (L∞ λ λ λ N ×N {cλ } ∈ NN∞×N . From (3.37) we deduce that bλ = kλ b + dλ and aλ = kλ a + fλ , ∞ where {dλ }, {fλ } ∈ QK (L∞ N ×N ) and b, a ∈ LN ×N . By the deﬁnition of SmbK , we have a = SmbK ({aλ }) (see also Proposition 3.49). Thus
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3 Symbol Analysis
kλ b · kλ a + kλ b · fλ + dλ · kλ a + dλ · fλ = I + cλ . Now let SK act on both sides of this equality and take into account that SK QK (L∞ N ×N ) = 0 (Proposition 3.49). What results is that {kλ (ba)} = {I}. Now 3.14(c) yields ba = I, i.e., that a is leftinvertible in L∞ N ×N . It can be shown in the same way that a is rightinvertible.
3.6 Harmonic Approximation: Localization 3.58. Deﬁnitions. Suppose (a) A is a C ∗ subalgebra of L∞ containing the constants and F is a closed subset of M (A); let A = AN ×N ; (b) B is a C ∗ algebra with identity and i : A → B is a 1submultiplicative isometry. If a = (ajk )N j,k=1 ∈ A, then aF = 0 means that ajk F = 0 for all j, k. Deﬁne If = {a ∈ A : aF = 0},
JF = closidalg i(A) i(IF ),
πF : alg i(A) → alg i(A)/JF ,
G → G + JF ,
iF : A → alg i(A)/JF ,
a → i(a) + JF .
3.59. Lemma. Let A and B be Banach algebras, let i : A → B be a submultiplicative quasiembedding, and let I be a closed twosided ideal of A. Put J = closidalg i(A) i(I). Then (a) Si (J) ⊂ J,
(b)
i(a) ∈ J ⇐⇒ a ∈ I.
Proof. (a) J is the closure of the linear hull of all elements of the form i(a1 ) . . . i(am )i(c)i(b1 ) . . . i(bm ) where aj , bj ∈ A and c ∈ I. But Si i(a1 ) . . . i(am )i(c)i(b1 ) . . . i(bm ) = i(a1 . . . am cb1 . . . bm ), which is in i(I) and therefore in J. (b) If i(a) ∈ J, then i(a) = Si (i(a)) must be in the closure of the linear hull of all elements of the form i(a1 . . . am cb1 . . . bm ), aj , bj ∈ A, c ∈ I. Thus i(a) ∈ i(I), i.e., a ∈ I. 3.60. Deﬁnitions. Suppose 3.58(a), (b) are fulﬁlled. Theorem 3.52 and the preceding lemma show that iF is a 1submultiplicative quasiembedding and that Ker iF = IF . Thus, ·
alg iF (A) = alg i(A)/JF = iF (A) + QiF (A)
(3.40)
3.6 Harmonic Approximation: Localization
153
and σiF is an isometric starisomorphism of alg iF (A)/QiF (A) onto A/IF (Corollary 3.53(b)). The algebra alg iF (A) will be referred to as the local algebra (associated with F ⊂ M (A)) and if a ∈ A, then iF (a) will be called a local object. The local spectrum sp(iF (a)) is the spectrum of iF (a) as an element of alg iF (A). 3.61. Theorem. Let 3.58(a), (b) be satisﬁed. Then, for a ∈ A, iF (a) = aF , where aF := max a(x)L(CN ) and CN is endowed with the norm (1.9). x∈F
Proof. Due to Corollary 3.53(b) we have . iF (a) = inf a + gL∞ N ×N g∈IF
Let IF1 := {a ∈ A : aF = 0}. Then, by Lemma 3.46(b), inf a + gL∞ = N ×N
g∈IF
a + gL∞ . N ×N
inf
1) g∈(IF N ×N
(3.41)
We mentioned in 1.28 that the mapping ϕ : A/IF1 → AF,
a + IF1 → aF
is an isometric starisomorphism. Therefore, ϕN ×N : (A/IF1 )N ×N → (AF )N ×N ,
(ajk + IF1 ) → (ajk F )
is a starisomorphism. A C ∗ norm in (A/IF1 )N ×N is given by a + (IF1 )N ×N 1 :=
inf
1) h∈(IF N ×N
a + hL∞ N ×N
and a C ∗ norm in (AF )N ×N is aF 2 := max a(x)L(CN ) . x∈F
Thus, if a ∈ A = AN ×N , then by virtue of 1.26(e), " " aF 2 = "ϕN ×N a + (IF1 )N ×N "2 = a + (IF1 )N ×N 1 . Recall (3.41) to see that the proof is complete.
3.62. Corollary. Suppose 3.58(a), (b) are fulﬁlled. If a ∈ A is sectorial on F , then iF (a) is in G(alg iF (A)). Proof. Lemma 3.6(b) shows that there is a matrix d ∈ GCN ×N such that I − a(x)dL(CN ) < 1 for all x ∈ F . So Theorem 3.61 gives that iF (I) − iF (ad)alg iF (A) < 1. Since iF (I) is the identity in alg iF (A), it follows that iF (ad) = iF (a)d is in G(alg iF (A)), which implies the invertibility of iF (a) at once.
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3 Symbol Analysis
3.63. Corollary. Suppose 3.58(a), (b) hold. If iF (a) is left or right or twosided invertible in alg iF (A), then aF is invertible in the restriction algebra AF . Proof. There is a BF ∈ alg iF (A) such that BF iF (a) = iF (I). From (3.40) we deduce that BF = iF (b) + CF with certain b ∈ A and CF ∈ QiF (A). Hence, iF (b)iF (a) − iF (I) = −CF iF (a) ∈ QiF (A) and it follows that iF (ba − I) = SiF (iF (b)iF (a) − iF (I)) = 0. So Theorem 3.61 implies that (ba − I)F = 0, i.e., that det b(x) det a(x) = 1 for all x ∈ F . This gives the invertibility of aF in AF immediately. 3.64. Corollary. Let 3.58(a), (b) be satisﬁed and let N = 1. Then, if a ∈ A, a(F ) ⊂ sp (iF (a)) ⊂ conv a(F ). Proof. Immediate from the two preceding corollaries.
3.65. Lemma. Let A be a commutative C ∗ algebra with identity and let B be a C ∗ subalgebra of A containing the identity. Then, for ξ ∈ M (B), closidA c ∈ B : c(ξ) = 0 = a ∈ A : aMξ (A) = 0 . Proof. We know from 1.28 that there is a closed subset Fξ of M (A) such that (3.42) closidA c ∈ B : c(ξ) = 0 = IFξ . We must show that Fξ = Mξ (A). Fξ ⊂ Mξ (A): Let α ∈ Fξ . If c ∈ B, then c − c(ξ) ∈ B and (c − c(ξ))(ξ) = 0. Hence c − c(ξ) ∈ IFξ , and so (c − c(ξ))(α) = 0, i.e., c(α) = c(ξ). Consequently, α ∈ Mξ (A). Mξ (A) ⊂ Fξ : Let α ∈ Mξ (A). Because of (3.42), , n IF = closA ak ck : ak ∈ A, ck ∈ B, ck (ξ) = 0, n ∈ Z+ . k=1
Since
a k ck
k
(α) =
ak (α)ck (ξ) = 0
k
for all ﬁnite sums k ak ck ∈ IFξ , we have g(α) = 0 for all g ∈ IFξ . If α would not be in F , then there would exist a g ∈ IFξ such that g(α) = 0. This contradiction shows that α ∈ Fξ . 3.66. Deﬁnitions. Suppose (a) A is a commutative C ∗ algebra with identity I and B is a C ∗ subalgebra of A containing I; put A = AN ×N and let C = {cIN ×N ∈ A : c ∈ B}, where IN ×N is the N × N identity matrix in A;
3.6 Harmonic Approximation: Localization
155
(b) B is a C ∗ algebra with identity and i : A → B is a 1submultiplicative isometry. For ξ ∈ M (B), deﬁne Iξ = closidA {cIN ×N ∈ C : c(ξ) = 0},
Jξ = closidalg i(A) i(Iξ ),
πξ : alg i(A) → alg i(A)/Jξ ,
G → G + Jξ ,
iξ : A → alg i(A)/Jξ ,
a → i(a) + Jξ .
By Theorem 3.52 and Lemma 3.59, iξ is a 1submultiplicative quasiembedding whose kernel is Iξ and alg iξ (A) coincides with alg i(A)/Jξ . Lemma 3.65 and Lemma 3.46 show that Iξ = IMξ (A) , Jξ = JMξ (A) , πξ = πMξ (A) , iξ = iMξ (A)
(3.43) (3.44)
whenever A is a C ∗ subalgebra of L∞ (recall 3.58). 3.67. Theorem. Suppose that, in addition to 3.66(a), (b), the following holds: (c) i(caIN ×N ) = i(cIN ×N )i(aIN ×N ) ∀ c ∈ B,
∀ a ∈ A.
Then if Y ∈ alg i(A), Y ∈ G(alg i(A)) ⇐⇒ πξ Y ∈ G(alg iξ (A))
∀ ξ ∈ M (B).
Proof. The hypotheses imply that i(C) is a closed subalgebra of the center of alg i(A) which is isometrically isomorphic to B. Hence, to each N ∈ M (i(C)) there corresponds a ξ ∈ M (B) such that N = {i(cIN ×N ) : c ∈ B, c(ξ) = 0}. Thus JN := closidalg i(A) N = closidalg i(A) i(Iξ ) =: Jξ and the assertion follows from (the C ∗ version of) Theorem 1.35(a). 3.68. Deﬁnitions. Let a ∈ L∞ N ×N , let {Kλ }λ∈Λ be an approximate identity, and let F be a closed subset of M (L∞ ). The sequence {kλ a} is said to be F restricted bounded away from zero if there is a b ∈ L∞ N ×N such that aF = bF and {kλ b} is bounded away from zero. We are now in a position to establish one of the main results of the present chapter. 3.69. Corollary. Let B be a C ∗ algebra of QC containing the constants, let {Kλ }λ∈Λ be an approximate identity, and let a ∈ L∞ N ×N . (a) If {kλ a} is Mξ (L∞ )restricted bounded away from zero for each ξ in M (B), then {kλ a} is bounded away from zero. (b) If a is locally sectorial over B, then {kλ a} is bounded away from zero.
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3 Symbol Analysis
Proof. We apply Theorem 3.67 with A = L∞ , B = AπN ×N , i = Kπ . The hypothesis 3.66(a) is clearly satisﬁed and 3.66(b) is fulﬁlled due to Proposition 3.54. The hypothesis 3.67(c) requires that Kπ (ca) = Kπ (c)Kπ (a) ∀ c ∈ b,
∀ a ∈ L∞
(here we abbreviate cIN ×N and aIN ×N to c and a, respectively). But this is nothing else than the requirement that kλ (ca) − (kλ c)(kλ a)∞ → 0
(λ → ∞) ∀ c ∈ B,
∀ a ∈ L∞ ,
which is equivalent to the asymptotic multiplicativity of {Kλ } on the pair (B, L∞ ). Theorem 3.23 therefore shows that 3.67(c) is also satisﬁed. Thus, by virtue of Theorem 3.67 and Theorem 3.56, it suﬃces to show that Kξπ (a) (Kξπ := (Kπ )ξ ) is invertible in alg Kξπ (L∞ N ×N ) for each ξ ∈ M (B). Under the hypothesis (a), we deduce from (3.43), (3.44) that Kξπ (a) = π π ∞ Kξπ (b) with some b ∈ L∞ N ×N such that Kξ (b) ∈ G(alg K (LN ×N )) (Theoπ π rem 3.56), whence Kξ (a) ∈ G(alg Kξ (LN ×N )) (Theorem 3.67). Under the hypothesis (b), Corollary 3.62 applied with F = Mξ (L∞ ) in conjunction with (3.43), (3.44) gives the invertibility of Kξπ (a) in alg Kξπ (L∞ N ×N ). Remark. For (b) see also 4.31. 3.70. Theorem. Let B be a C ∗ algebra with identity and let i : L∞ N ×N → B be a 1submultiplicative isometry such that i(ϕa) = i(ϕ)i(a) for all ϕ ∈ QCN ×N and a ∈ L∞ N ×N . For τ ∈ M (C) = T and ξ ∈ M (QC), put i(cIN ×N ) : c ∈ C, c(τ ) = 0 , Jτ = closidalg i(L∞ N ×N ) Jξ = closidalg i(L∞ i(ϕIN ×N ) : ϕ ∈ QC, ϕ(ξ) = 0 , N ×N ) and deﬁne πτ , πξ , iτ , iξ as in 3.66. Then if Y ∈ alg i(L∞ N ×N ) and τ ∈ T, ∞ πτ Y ∈ G(alg iτ (L∞ N ×N )) ⇐⇒ πξ Y ∈ G(alg iξ (LN ×N )) ∀ ξ ∈ Mτ (QC).
Proof. Theorem 3.52 and Lemma 3.59 imply that ∞ iτ : L∞ N ×N → alg i(LN ×N )/Jτ
is a 1submultiplicative quasiembedding whose kernel is Iτ := a ∈ L∞ N ×N : a = cIN ×N , c ∈ C, c(τ ) = 0 . By virtue of Lemma 3.65 and 1.28 we have L∞ N ×N /Iτ = AN ×N , where A = L∞ Mτ (L∞ ). Therefore, the mapping ieτ deﬁned (correctly) by ieτ : AN ×N → alg i(L∞ N ×N )/Jτ ,
aMτ (L∞ ) → iτ (a)
3.7 Harmonic Approximation: Local Spectra
157
is a 1submultiplicative embedding (recall 3.38). From 1.26(e) we deduce that ieτ is even an isometry. Put B = QCMτ (L∞ ) and recall that by 2.81 the maximal ideal space of B is M (B) = Mτ (QC). For ξ ∈ Mτ (QC), let Jξτ := closidalg ieτ (AN ×N ) ieτ (bIN ×N ) : b ∈ B, b(ξ) = 0 . e ∞ It is clear that ieτ (AN ×N ) = iτ (L∞ N ×N ), alg iτ (AN ×N ) = alg iτ (LN ×N ), and Jξτ = closidalg iτ (L∞ iτ (ϕIN ×N ) : ϕ ∈ QC, ϕ(ξ) = 0 . (3.45) N ×N )
Now we apply Theorem 3.67 with i = ieτ and A, B as above. What results is that if Y ∈ alg iτ (L∞ N ×N ), then τ ∞ τ Y ∈ G(alg iτ (L∞ N ×N )) ⇐⇒ Y + Jξ ∈ G(alg iτ (LN ×N )/Jξ ) ∀ ξ ∈ Mτ (QC).
But Jτ is obviously a closed twosided ideal in Jξ and a little thought shows that Jξ /Jτ coincides with Jξτ (take into account (3.45)). Therefore τ ∞ alg iτ (L∞ N ×N )/Jξ = (alg i(LN ×N )/Jτ )/(Jξ /Jτ )
is naturally isomorphic to ∞ alg i(L∞ N ×N )/Jξ = alg iξ (LN ×N ),
which completes the proof.
3.71. Corollary. Let the hypotheses of Theorem 3.70 be satisﬁed. If Y is in alg i(L∞ N ×N ), then sp (Y ) =
/
/
sp (πτ Y ) =
τ ∈T
and, for τ ∈ T, sp (πτ Y ) =
sp (πξ Y )
(3.46)
ξ∈M (QC)
/
sp (πξ Y ).
(3.47)
ξ∈Mτ (QC)
Proof. Equalities (3.46) follow from Theorem 3.67 and equality (3.47) results from Theorem 3.70.
3.7 Harmonic Approximation: Local Spectra 3.72. Cluster sets. Let {Kλ }λ∈Λ be an approximate identity and let {aλ } be in alg K(L∞ ). Recall that Λ × T can be viewed as a subset of QC ∗ and that M (QC) = (closQC ∗ (Λ × T)) \ (Λ × T) (3.28 and 3.29). Let τ ∈ T and ξ ∈ M (QC). We deﬁne
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3 Symbol Analysis
ClK ({aλ }, Λ × T) as the set of all z ∈ C such that there is a sequence {(λn , tn )} ⊂ Λ × T with aλn (tn ) → z as n → ∞;
δ
ClK ({aλ }, T)
as the set of all z ∈ C such that there is a sequence {(λn , tn )} ⊂ Λ × T with λn → ∞ and aλn (tn ) → z as n → ∞;
ClK ({aλ }, τ )
as the set of all z ∈ C such that there is a sequence {(λn , tn )} ⊂ Λ × T with λn → ∞, tn → τ , and aλn (tn ) → z as n → ∞;
ClK ({aλ }, ξ)
as the set of all z ∈ C with the following property: for each ε > 0 and for each QC ∗ neighborhood U of ξ there is a (λ, t) ∈ (Λ × T) ∩ U such that z − aλ (t) < ε.
In these deﬁnitions “Cl ” is for “cluster.” Given {aλ } ∈ alg K(L∞ ) deﬁne by δ {aλ } : Λ × T → C, (µ, t) → aµ (t),
{aλ }
and for ν ∈ Λ, τ = eiθ0 ∈ T, ε > 0 put Λν := λ ∈ Λ : λ > ν , (τ − ε, τ + ε) := t = eiθ ∈ T : θ − θ0  < ε . It is easily seen that ClK ({aλ }, Λ × T) = clos δ {aλ } (Λ × T), $ ClK ({aλ }, T) = clos δ {aλ } (Λν × T), ν>0
ClK ({aλ }, τ ) =
$ $
clos δ {aλ } Λ × (τ − ε, τ + ε) .
(3.48) (3.49) (3.50)
ν>0 ε>0
It is also clear that ClK ({aλ }, ξ) =
$
clos δ {aλ } (Λ × T) ∩ Uε;q1 ,...,qn (ξ) ,
ε;q1 ,...,qn
the intersection over all ε > 0 and q1 , . . . , qn ∈ QC; here, of course, Uε;q1 ,...,qn (ξ) := η ∈ QC ∗ : η(qi ) − ξ(qi ) < ε ∀ i = 1, . . . , n . Each neighborhood of this form contains some neighborhood of the form U1;q (ξ), where q is a nonnegative function in QC: for instance take q=
1 q1 − ξ(q1 ) + . . . + qn − ξ(qn ) . ε
Thus, ClK ({a}, ξ) =
$ q∈QC
clos δ {aλ } (Λ × T) ∩ U1;q (ξ) .
3.7 Harmonic Approximation: Local Spectra
159
For a ∈ L∞ , the sets ClK ({kλ a}, . . .) will be simply denoted by ClK (a, . . .) and will be referred to as the cluster sets of a on Λ × T, on T, at τ ∈ T, or at ξ ∈ M (QC) (associated with the approximate identity {Kλ }λ∈Λ ). If {Kλ }λ∈(0,∞) is generated by the Poisson kernel, then ClK (a, . . .) will be written as ClH (a, . . .). In that case we have the familiar cluster sets of the harmonic extension a(ζ) (ζ ∈ D) of a: a(ζn ) → z as n → ∞ , ClH (a, T) = z ∈ C : ∃ {ζn } ⊂ D such that ζn  → 1, a(ζn ) → z as n → ∞ , ClH (a, τ ) = z ∈ C : ∃ {ζn } ⊂ D such that ζn → τ, ClH (a, ξ) = z ∈ C : ∀ ε > 0 ∀ QC ∗ neighborhood U of ξ ∃ ζ ∈ D ∩ U such that  a(ζ) − z < ε In this situation, Λ × T can be identiﬁed with a circular annulus, and if we let Λ = [0, ∞) (λ = 0 corresponds to ζ = 0), then Λ × T can be identiﬁed with D. For the harmonic extension it is obvious from (3.48)–(3.50) that ClH (a, D), ClH (a, T), and ClH (a, τ ) (τ ∈ T) are connected, compact, nonempty subsets of C. 3.73. Proposition. Let {aλ } ∈ alg K(L∞ ). (a) δ {aλ } : Λ × T → C is continuous on Λ × T equipped with the product topology of Λ ⊂ R and T. (b) If Λ×T is connected, then ClK ({aλ }, Λ×T), ClK ({aλ }, T), ClK ({aλ }, τ ) (τ ∈ T) are connected, compact, and nonempty. / ClK ({aλ }, τ ), (c) ClK ({aλ }, T) = τ ∈T
ClK ({aλ }, τ ) =
/
ClK ({aλ }, ξ)
(τ ∈ T).
ξ∈Mτ (QC)
Proof. (a) It suﬃces to prove that for each a ∈ L∞ the function δ {kλ a} is continuous on Λ × T. But this results from 3.14(a) and the fact that ( ∞ Kλ (x) − Kµ (x) dx → 0 as λ → µ, −∞
which is readily checked for continuous kernels K and extends to general kernels K by the density of the continuous functions with compact support in L1 (R). (b) If Λ × T is connected, then so are Λν × T and Λν × (τ − ε, τ + ε). By virtue of part (a), clos δ {aλ } (Λ × T),
clos δ {aλ } (Λν × T),
clos δ {aλ } (Λν × (τ − ε, τ + ε))
are therefore connected, compact, nonempty sets. So the assertion is a consequence of the following well known fact, which can be found in many textbooks: if K1 ⊃ K2 ⊃ K3 ⊃ . . . are connected, compact, and nonempty subsets of a Hausdorﬀ space, then K =
∞
n=1
Kn is connected, compact, and nonempty.
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3 Symbol Analysis
(c) The ﬁrst equality is obvious from the deﬁnition. Let us show the second equality. Suppose z ∈ ClK ({aλ }, τ ), (λn , tn ) ∈ Λ × T, λn → ∞, tn → τ , and aλn (tn ) → z. Since the sequence {(λn , tn )} is contained in a ball of QC ∗ centered at zero (3.14(b)), by the BanachAlaoglu theorem there is a ξ ∈ QC ∗ and a subsequence of {(λn , tn )}, which will again be denoted by {(λn , tn )}, such that (λn , tn ) → ξ in the weakstar topology. If ϕ, ψ ∈ QC, then by Theorem 3.23, ξ(ϕψ) − ξ(ϕ)ξ(ψ) equals lim (kλn ϕψ)(tn ) − (kλn ϕ)(tn ) · (kλn ψ)(tn ) = 0, n→∞
whence ξ ∈ M (QC), and if c ∈ C, then ξ(c) = lim (kλn c)(tn ) = c(τ ), n→∞
whence ξ ∈ Mτ (QC). It is clear that z ∈ ClK ({aλ }, ξ). Now let z ∈ ClK ({aλ }, ξ) for some ξ ∈ Mτ (QC). Then for each ε > 0 and each n ∈ N there is a (λn , tn ) ∈ (Λ×T)∩U1/n;χ1 (ξ) such that aλn (tn )−z < ε. Since (λn , tn ) ∈ U1/n;χ1 (ξ) and ξ(χ1 ) = τ , we have (kλn χ1 (tn )−τ  < 1/n, and 3.14(d) implies that λn → ∞ and tn → τ . Consequently, z ∈ ClK ({aλ }, τ ). 3.74. Open problems. Is ClK ({aλ }, ξ) (ξ ∈ M (QC)) connected whenever the set Λ × T is so? It would be suﬃciently interesting to know the answer for the case that K is the Poisson kernel and aλ = kλ a (a ∈ L∞ ). Under what conditions the conclusion that ClK ({aλ }, T) or ClK ({aλ }, τ ) (τ ∈ T) is connected remains valid when the hypothesis that Λ × T be connected is dropped? For instance, are ClK ({kλ a}, T) and ClK ({kλ a}, τ ) connected for every a ∈ L∞ in case {Kλ }λ∈Λ is generated by the Fej´er kernel and Λ = {1, 2, 3, . . .}? 3.75. Notation. Let {Kλ }λ∈Λ be an approximate identity. As in 3.66, for τ ∈ M (C) = T and ξ ∈ M (QC), let Jτ = closidalg K(L∞ ) {kλ c}λ∈Λ : c ∈ C, c(τ ) = 0 , Jξ = closidalg K(L∞ ) {kλ ϕ}λ∈Λ : ϕ ∈ QC, ϕ(ξ) = 0 . For {aλ } ∈ alg K(L∞ ), let {aλ }π , {aλ }πτ , {aλ }πξ denote the coset in ∞
alg Kπ (L∞ ) = alg K(L∞ )/N L , alg Kτπ (L∞ ) = alg Kπ (L∞ )/Jτ , alg Kξπ (L∞ ) = alg Kπ (L∞ )/Jξ , respectively, containing {aλ }.
3.7 Harmonic Approximation: Local Spectra
161
3.76. Theorem. Let {aλ }λ∈Λ ∈ alg K(L∞ ). Then (a) sp ({aλ }) = ClK ({aλ }, Λ × T); (b) sp ({aλ }π ) = ClK ({aλ }, T); (c) sp ({aλ }πτ ) = ClK ({aλ }, τ )
(τ ∈ T);
(d) sp ({aλ }πξ ) = ClK ({aλ }, ξ)
(ξ ∈ M (QC)).
Proof. (a) We have z∈ / sp ({aλ }) ⇐⇒ {aλ − z} ∈ G(alg K(L∞ )) ⇐⇒ {aλ − z} ∈ G(A∞ ) (3.39 and 1.26(d)) ⇐⇒ aλ (t) − z = 0 such that ⇐⇒
inf (λ,t)∈Λ×T
∀ (λ, t) ∈ Λ × T and
(aλ (t) − z)−1  ≤ M
∃M >0
∀ (λ, t) ∈ Λ × T
aλ (t) − z > 0
⇐⇒ z ∈ / ClK ({aλ }, Λ × T). (b) The assertion follows from part (c) combined with Corollary 3.71 and Proposition 3.73(c). However, there is a simple straightforward proof: z ∈ sp ({aλ }π ) ⇐⇒ {aλ − z} not bafz (Theorem 3.56) ⇐⇒ ∃ (λn , tn ) ∈ Λ × T : λn → ∞, aλn (tn ) → z as n → ∞ ⇐⇒ z ∈ ClK ({aλ }, T). (c) Taking into account Corollary 3.71 and Proposition 3.73(c) this is seen to be an immediate consequence of part (d). The proof we shall give for part (d) also works in the case at hand (where it is even a bit simpler); this provides a possibility of proving the assertion in a more direct way. (d) To establish the inclusion “⊃” it suﬃces to show that 0 ∈ / ClK ({aλ }, ξ) whenever {aλ }πξ is invertible. Thus, let {aλ }πξ be invertible. Then there are {bλ } ∈ alg K(L∞ ),
{cjλ } ∈ alg K(L∞ ),
ϕj ∈ QC
(j = 1, . . . , n)
such that ϕj (ξ) = 0 and " " 1 j " " {cλ }π {kλ ϕj }π " < . "{bλ }π {aλ }π − 1π − 8 j ∞
It follows that there is a {dλ } ∈ N L such that " 2 " j " " {cλ }{kλ ϕj } − {dλ }" < . "{bλ }{aλ } − 1 − 8 j
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3 Symbol Analysis
Since there is a λ0 ∈ Λ such that dλ < 1/8 for all λ > λ0 , we have " " j 3 " " ∀λ > λ0 , cλ · kλ ϕj " < "bλ aλ − 1 − 8 ∞ j hence
3 + {cjλ } · (kλ ϕj )(t) 8 j=1 for all λ > λ0 and t ∈ T. Put ε1 = 1/ 8n max {cjλ } . If n
bλ (t)aλ (t) − 1 <
j
(λ, t) ∈ (Λλ0 × T) ∩ Uε1 ;ϕ1 ,...,ϕn (ξ) = (Λλ0 × T) ∩ η ∈ QC ∗ : η(ϕj ) < ε1 then
∀ j = 1, . . . , n ,
1 3 4 + {cjλ }ε1 ≤ = 8 j=1 8 2 n
bλ (t)aλ (t) − 1 <
and therefore aλ (t) > 1/(2{bλ }) =: δ. Thus, there is no (λ, t) ∈ (Λλ0 × T) ∩ Uε1 ;ϕ1 ,...,ϕn (ξ) such that aλ (t) < δ. If ε2 > 0 is suﬃciently small, then, by 3.14(d), (Λ × T) ∩ Uε2 ;χ1 (ξ) ⊂ Λλ0 × T.
(3.51)
Consequently, if ε := min{ε1 , ε2 } and U := Uε;χ1 ,ϕ1 ,...,ϕn (ξ), then there is no (λ, t) ∈ (Λ × T) ∩ U such that aλ (t) < δ. But this implies that 0 is not in ClK ({aλ }, ξ), as desired. We now prove the inclusion “⊂”. To do this, assume 0 ∈ sp({aλ }πξ ) but 0 ∈ / ClK ({aλ }, ξ). So there are a QC ∗ neighborhood U1 of ξ and an m > 0 such that (3.52) aλ (t) ≥ m ∀ (λ, t) ∈ (Λ × T) ∩ U1 . Now choose ε1 > 0 so that ε1 < m/10. Since {aλ }πξ is not invertible in the (commutative) C ∗ algebra alg Kξπ (L∞ ), we deduce from 1.20(c) that there exists a {uλ } ∈ alg K(L∞ ) such that {uλ }πξ = 1 and {uλ }πξ {aλ }πξ < ε1 .
(3.53)
Hence there are {cjλ } ∈ alg K(L∞ ) and ϕj ∈ QC (j = 1, . . . , n) such that ϕj (ξ) = 0 and " " j " " {cλ }π {kλ ϕj }π " < 2ε1 , "{uλ }π {aλ }π − j ∞
and there is a {dλ } ∈ N L
such that
3.7 Harmonic Approximation: Local Spectra
" " j " " sup "uλ aλ − cλ kλ ϕj − dλ "
λ∈Λ
j
∞
163
< 3ε1 .
There exists a λ0 ∈ Λ such that dλ ∞ < ε1 for all λ > λ0 . If ε2 > 0 is suﬃciently small, then (3.51) holds, and so dλ (t) < ε1 for all (λ, t) in (Λ × T) ∩ U2 , where U2 := Uε2 ;χ1 (ξ). Thus, + + j + + cλ (t)(kλ ϕj )(t)+ < 4ε1 ∀ (λ, t) ∈ (Λ × T) ∩ U2 . +uλ (t)aλ (t) − j
Further, there is a QC ∗ neighborhood U3 of ξ such that {cjλ } (kλ ϕj )(t) < ε1 ∀ (λ, t) ∈ (Λ × T) ∩ U3 , j
(recall that ϕj (ξ) = 0), whence uλ (t)aλ (t) < 5ε1
∀ (λ, t) ∈ (Λ × T) ∩ U2 ∩ U3 ,
and ﬁnally, by (3.52), uλ (t) <
1 5ε1 < m 2
∀ (λ, t) ∈ (Λ × T) ∩ U1 ∩ U2 ∩ U3 .
Now choose a q ∈ QC such that 0 ≤ q ≤ 1, q(ξ) = 0, and Uδ0 ;q (ξ) ⊂ U1 ∩ U2 ∩ U3 for some δ0 > 0 (recall 3.72 to see that this is possible). Thus, we have proved that uλ (t) <
1 2
∀ (λ, t) ∈ Λ × T
for which 0 ≤ (kλ q)(t) < δ0 .
(3.54)
The equality {uλ }πξ = 1 in (3.53) implies that l " " " " {fλi }{kλ ψi } + {eλ }" ≥ 1 "{uλ } +
(3.55)
i=1 ∞
for all {fλi } ∈ alg K(L∞ ), {eλ } ∈ N L , ψi ∈ QC, ψi (ξ) = 0. The function q is in QC and q(ξ) equals zero. Therefore, by (3.55), {uλ }(1 − {kλ q})n ≥ 1 ∀ n ∈ Z+ , or, equivalently, sup sup uλ (t) 1 − (kλ q)(t)n ≥ 1 ∀ n ∈ Z+ . λ∈Λ t∈T
If 0 ≤ (kλ q)(t) < δ0 , then 1 − δ0 < 1 − (kλ q)(t) ≤ 1, and so, by (3.54), uλ (t) 1 − (kλ q)(t)n ≤ uλ (t) <
1 . 2
(3.56)
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3 Symbol Analysis
On the other hand, if δ0 /2 < (kλ q)(t) ≤ 1, then 0 ≤ 1 − (kλ q)(t) < 1 − δ0 /2, whence δ0 n0 1 < uλ (t) 1 − (kλ q)(t)n0 ≤ {uλ } 1 − 2 2 if only n0 is large enough. Because Λ × T = (λ, t) : 0 ≤ (kλ q)(t) < δ0 ∪ (λ, t) : δ0 /2 < (kλ q)(t) ≤ 1 , we arrive at the inequality sup sup uλ (t) 1 − (kλ q)(t)n0 < λ∈Λ t∈T
1 2
which contradicts (3.56). The proof is complete. 3.77. Corollary. If a ∈ L∞ , τ ∈ T, ξ ∈ M (QC), then (a) R(a) ⊂ sp ({kλ a}π ) = ClK (a, T) ⊂ conv R(a), (b) a(Xτ ) ⊂ sp ({kλ a}πτ ) = ClK (a, τ ) ⊂ conv a(Xτ ), (c) a(Xξ ) ⊂ sp ({kλ a}πξ ) = ClK (a, ξ) ⊂ conv a(Xξ ). Proof. The inclusions for the spectra follow from Corollary 3.64 and those for the cluster sets then result from Theorem 3.76. Remark. In particular, if a ∈ L∞ is continuous at τ ∈ T, then ClK (a, τ ) = ClK (a, ξ) = {a(τ )} (ξ ∈ Mτ (QC)). Furthermore, if a ∈ C, then ClK (a, T) = R(a) = a(T). If a ∈ QC, then, by Proposition 3.73(c), ClK (a, T) = R(a),
ClK (a, τ ) = a(Xτ ),
ClK (a, ξ) = {a(ξ)}
(τ ∈ T, ξ ∈ M (QC)). A rather puzzling consequence of this and Proposition 3.73(b) is that for a ∈ QC the sets a(Xτ ) and R(a) are always connected. This in turn implies that M (QC) as well as each ﬁber Mτ (QC) is connected. 3.78. Corollary. Assume Λ is connected and let a ∈ L∞ . (a) If τ ∈ T and a(Xτ ) is contained in some line segment, then sp ({kλ a}πτ ) = ClK (a, τ ) = conv a(Xτ ). (b) If a ∈ P2 C and if a1τ , a2τ (it may be that a1τ = a2τ ) denote the two values taken by a on Xτ (τ ∈ T), then / [a1τ , a2τ ], sp ({kλ a}π ) = ClK (a, T) = τ ∈T
sp ({kλ a}πτ ) = ClK (a, τ ) = [a1τ , a2τ ] ∀ τ ∈ T.
3.7 Harmonic Approximation: Local Spectra
165
Proof. (a) Combine Proposition 3.73(b) and Corollary 3.77(b). (b) Immediate from (a) and Proposition 3.73(c) or (3.46). Apart from some special cases, it is not easy to describe ClK (a, ξ) for ξ ∈ M (QC). The following theorem provides a situation where this is possible. Notice however that its proof heavily relies on the entire analysis of M (QC) and M (P QC) originated by Sarason (see 3.24–3.36). 3.79. Theorem. Let {Kλ }λ∈(1,∞) be an approximate identity. If a ∈ P QC and ξ ∈ M (QC), then sp ({kλ a}πξ ) = ClK (a, ξ) = conv a(Xξ ) = conv a(Mξ (P QC)). Proof. The ﬁrst and the third equalities follow from Theorem 3.76(c) and Theorem 3.36(a), respectively. By Theorem 3.36(b), aXξ is in P CXξ , and so we may suppose that a ∈ P C. Without loss of generality assume ξ ∈ Mτ (QC) and τ = 1. Then a can be written as a = cχ + g, where c ∈ C, χ is the characteristic function of the upper halfcircle, and g ∈ P C is continuous at τ = 1. Therefore it suﬃces to prove the assertion for χ. Because χ(Xξ ) = χ(Mξ (P QC)) (Theorem 3.36(a)), χ(Xξ ) is a singleton for ξ ∈ / M10 (QC) (Theorem 3.36(c)) and the assertion is trivial in that case. Thus, suppose ξ ∈ M10 (QC). Then χ(Xξ ) = {0, 1} (Theorem 3.36(d)), so {0, 1} ⊂ ClK (χ, ξ) ⊂ [0, 1] by Corollary 3.77(c), and we must show that each µ ∈ (0, 1) belongs to ClK (χ, ξ). Let ε > 0 be given arbitrarily and let Uδ;ϕ (ξ) = η ∈ QC ∗ : η(ϕ) − ϕ(ξ) < δ be any QC ∗ neighborhood of ξ. Lemma 3.26 shows that there exist ν ∈ R and λ0 ∈ (1, ∞) such that (kλ H)(ν/λ) − µ < ε for all λ > λ0 . In the language of 3.30, this means that δ(λ,ν/λ) H − µ < ε for all (λ, ν/λ) ∈ Kν with λ > λ0 . If δ1 < δ, then the neighborhood Uδ1 ;ϕ,χ1 (ξ) = η ∈ QC ∗ : η(ϕ) − ϕ(ξ) < δ1 , η(χ1 ) − 1 < δ1 is contained in Uδ;ϕ (ξ). By Lemma 3.31, there is a (λ, ν/λ) ∈ Kν ∩ Uδ1 ;ϕ,χ1 (ξ), and by choosing δ1 suﬃciently small we can guarantee that λ > λ0 (see 3.14(d)). Thus, there exists a (λ, t) ∈ (Λ × T) ∩ Uδ;ϕ (ξ) with (kλ χ)(t) − µ < ε, which completes the proof.
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3 Symbol Analysis
3.8 Local Sectoriality Continued 3.80. Theorem. Let {Kλ }λ∈Λ be any approximate identity and suppose Λ is connected. (a) If a ∈ L∞ N ×N , τ ∈ T, and conv a(Xτ ) is a line segment, then a is sectorial on Xτ ⇐⇒ {kλ a}πτ ∈ G(alg Kτπ (L∞ N ×N )) ⇐⇒ 0 ∈ / ClK {det(kλ a)}, τ . (b) If a ∈ (P QC)N ×N and ξ ∈ M (QC), then a is sectorial on Xξ ⇐⇒ {kλ a}πξ ∈ G(alg Kξπ (L∞ N ×N )) ⇐⇒ 0 ∈ / ClK {det(kλ a)}, ξ . Proof. In both cases the second equivalence “⇐⇒” and the ﬁrst implication “=⇒” follow from Theorem 3.76 and Corollary 3.62, respectively. So we are left with the ﬁrst implication “⇐=”. Let β be τ (∈ T) or ξ(∈ M (QC)) and suppose conv a(Xβ ) = [E, F ]. By Corollary 3.63, E and F are invertible matrices. Due to Theorem 3.4, the sectoriality of a on Xβ will follow as soon as we have shown that det(µE + (1 − µ)F ) = 0 for all µ ∈ [0, 1]. There is no loss of generality in assuming that E is the identity matrix I and that F is the Jordan canonical form, F = J (see the proof of Lemma 3.3(b)). Let ajj ∈ L∞ (j = 1, . . . , N ) denote the diagonal entries of a. Since a(x) is an uppertriangular matrix for each x ∈ Xβ , we have {det(kλ a)}πβ =
N
{kλ ajj }πβ .
j=1
Hence, if {kλ a}πβ is invertible, so also is {kλ ajj }πβ for each j. It is clear that conv ajj (Xβ ) = [1, θj ], where θj is an eigenvalue of J. Thus, by Corollary 3.78(a) for β = τ and by Theorem 3.79 for β = ξ, the line segments [1, θj ] do not contain the origin. This gives det(µI + (1 − µ)J) =
N
(µ + (1 − µ)θj ) = 0
∀ µ ∈ [0, 1].
j=1
3.81. Open problems. Does the ﬁrst implication “⇐=” of part (b) in Theorem 3.80 hold under the hypothesis that a ∈ L∞ N ×N and conv a(Xξ ) is a line segment? Does that implication hold for every a ∈ (P2 QC)N ×N ? Equivalently: is Theorem 3.79 true with P QC replaced by P2 QC? Suﬃciently interesting special case: is Theorem 3.79 valid for a = χE , the characteristic function of a measurable subset E of T (well, say of the “simple” kind as in 3.37)? Note that, by Corollary 3.62 and Theorem 3.76, the ﬁrst implications “=⇒” as well as the second equivalences “⇐⇒” of Theorem 3.80 hold for every a ∈ L∞ N ×N .
3.8 Local Sectoriality Continued
167
3.82. Corollary. Let {Kλ }λ∈Λ be any approximate identity and suppose Λ is connected. (a) If a ∈ L∞ N ×N and conv a(Xτ ) is a line segment for each τ ∈ T, then a is locally sectorial over C ⇐⇒ {kλ a} bafz ⇐⇒ 0 ∈ / ClK {det(kλ a)}, T . (b) If a ∈ (P QC)N ×N , then a is locally sectorial over QC ⇐⇒ {kλ a} bafz ⇐⇒ 0 ∈ / ClK {det(kλ a)}, T . Proof. Theorem 3.80, Theorem 3.67 (or Corollary 3.71), and Proposition 3.73(c). Remark. By Corollary 3.69(b) and Theorem 3.76 the ﬁrst implications “=⇒” and the second equivalences “⇐⇒” hold for every a ∈ L∞ N ×N . 3.83. Deﬁnition. Let OCN ×N denote the (closed) set of noninvertible matrices, OCN ×N = CN ×N \ GCN ×N . Note that OC1×1 is nothing else but the origin in C. For a ∈ (P QC)N ×N and τ = eiθ ∈ T, let βτ (a, δ) (δ > 0) denote the distance between OCN ×N and the line segment ( ( 1 δ 1 θ+δ ix ix a(e ) dx, a(e ) dx δ θ−δ δ θ and put βτ (a) := lim inf βτ (a, δ). δ→0
3.84. Proposition. Let a ∈ (P QC)N ×N , let τ ∈ T, and let {Kλ }λ∈(1,∞) be an approximate identity. (a) βτ (a) > 0 ⇐⇒ a is sectorial on Xξ for all ξ ∈ Mτ0 (QC). (b) If a ∈ GL∞ N ×N , then βτ (a) > 0 ⇐⇒ a is sectorial on Xξ for all ξ ∈ Mτ (QC) ⇐⇒ 0 ∈ / ClK {det(kλ a)}, τ . (c) If N = 1, then βτ (a) > 0 ⇐⇒ a is sectorial on Xξ for all ξ ∈ Mτ (QC) ⇐⇒ 0 ∈ / ClK (a, τ ). Proof. The second equivalences in (b) and (c) follow from Theorem 3.80 and Proposition 3.73(c). Without loss of generality assume τ = 1.
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3 Symbol Analysis
(a) Let β1 (a) = 0. Then, by a compactness argument, there are a sequence {δn } of positive numbers tending to zero and a matrix b ∈ OCN ×N such that the limits ( 0 ( δn 1 1 ix a0 = lim a(e ) dx, a1 = lim a(eix ) dx (3.57) n→∞ δn −δ n→∞ δn 0 n exist and b ∈ [a0 , a1 ]. Put λn = π/δn . Due to the compactness of the unit ball in QC ∗ and Proposition 3.29(b) there are a subsequence {λnk } of {λn } and a ξ ∈ M10 (QC) such that (λnk , 1) → ξ (recall 3.30). From Theorem 3.36(d) we deduce that a0 = a(ξ, 0) and a1 = a(ξ, 1). Thus [a(ξ, 0), a(ξ, 1)] contains b ∈ OCN ×N , and so Proposition 3.2 in conjunction with Theorem 3.36(a), (d) implies that a is not sectorial on Xξ . Now suppose there is a ξ ∈ M10 (QC) such that a is not sectorial on Xξ . Then, again by Proposition 3.2 and Theorem 3.36(a), (d), there is a b in OCN ×N belonging to [a(ξ, 0), a(ξ, 1)]. If {λn } ⊂ (1, ∞) is any sequence such that (λn , 1) → ξ and if we set δn = π/λn , then the limits (3.57) exist and are equal to a(ξ, 0) and a(ξ, 1), respectively (Theorem 3.36(d)). Thus, β1 (a) = 0. (b) The (ﬁrst) implication “⇐=” is immediate from (a). To get the reverse implication notice that a(Xξ ) is a singleton for ξ ∈ M1 (QC)\M10 (QC) (Theorem 3.36(a), (c)) and that therefore the invertibility of a yields the sectoriality of a on Xξ . (c) The (ﬁrst) implication “⇐=” is again a consequence of (a). So suppose β1 (a) > 0. Then a is sectorial on Xξ for each ξ ∈ M10 (QC) by virtue of (a). Assume there is a ξ0 ∈ M1 (QC) \ M10 (QC) such that a is not sectorial on Xξ0 . Since a(Xξ0 ) is a singleton (Theorem 3.36(a), (c)), we have a(x) = 0 for all x ∈ Xξ0 . Proposition 3.29(a) for A = L∞ shows that there are (λn , tn ) in (1, ∞) × T such that (mλn a)(tn ) = (mλn a)(tn ) − a(x0 ) <
1 n
(x0 ∈ Xξ0 )
(3.58)
and in view of 3.14(d) we may assume that λn → ∞ and tn → τ = 1 (recall (3.17)). By the BanachAlaoglu theorem and Proposition 3.29(a) for A = QC, there are a ξ ∈ M10 (QC) and a subsequence {λnk } of {λn } such that (λnk , tnk ) → ξ. Now (3.58) implies that 0 is in the cluster set of a at ξ associated with {mλ } (the moving average). Hence, by Theorem 3.79, a cannot be sectorial on Xξ . We arrive at a contradiction, because a is sectorial on Xξ for all ξ ∈ M10 (QC). This proves the ﬁrst implication “=⇒”. 3.85. Corollary. (a) If a ∈ (P QC)N ×N , then a is locally sectorial over QC ⇐⇒ a ∈ GL∞ N ×N and βτ (a) > 0 ∀ τ ∈ T. (b) If a ∈ P QC, then a is locally sectorial over QC ⇐⇒ βτ (a) > 0
∀ τ ∈ T.
3.9 Notes and Comments
169
Proof. (a) Notice that locally sectorial matrix functions are necessarily in GL∞ N ×N . So the assertion is a straightforward consequence of Proposition 3.84(b). (b) Immediate from Proposition 3.84(c). Remark. We must confess that we have not been able to remove the assumption “a ∈ GL∞ N ×N ” in (a).
3.9 Notes and Comments 3.1. Matrix functions which are analytically sectorial over C were ﬁrst studied by Simonenko [492]. Matrix functions which are geometrically sectorial over C were introduced by Douglas and Widom [166], who also raised the question of whether geometric sectoriality implies analytic sectoriality. Azoﬀ and Clancey [16] then showed that the answer is no in general. 3.3–3.4. Clancey [134]. 3.6–3.10. Theorem 3.9 is known. Theorem 3.8 is due to the authors but its proof makes essential use of an argument by Roch [421]. 3.11. Toeplitz operators with P2 C symbols were ﬁrst considered by Clancey [134] and Clancey, Morrel [140]. Douglas [160], [161] studied operators whose symbol a has the property that a(Xξ ) is contained in some straight line segment for each ξ ∈ M (QC). This class of symbols contains P2 QC. See also Silbermann [483]. 3.12–3.14. All the facts stated here can be found in Ahiezer [3], for example. 3.15. The asymptotic multiplicativity of the Poisson kernel on the pair (C + H ∞ , C + H ∞ ) was discovered by Douglas [159] and its asymptotic multiplicativity on the pair (QC, L∞ ) by Sarason [456]. 3.17. Sarason [454]. 3.21–3.22. These results are certainly known to specialists. The proof of Theorem 3.21 is patterned after Garnett [211, Theorem VI.1.2]. The proof of this theorem motivates why it is more convenient to deﬁne kλ ϕ as a convolution over the real line rather than over the circle. 3.23. B¨ottcher, Silbermann [112]. 3.24–3.25. Wellknown. 3.26–3.36. All these results as well as the basic ideas for their proofs are due to Sarason [456]. Our minor contribution is the extension of Sarason’s results to the case of arbitrary approximate identities and the simpliﬁcation of some of his arguments.
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3 Symbol Analysis
3.38–3.71. This approach was developed by Silbermann [481], [483]. Theorem 3.42 and Corollary 3.44 had been earlier established in B¨ ottcher, Silbermann [106], and similar results and ideas are also in Clancey [136]. In connection with Deﬁnition 3.60 we remark that local Toeplitz operators were introduced by Douglas [161]. The results of 3.52, 3.61, 3.62–3.64, 3.67 can be found in Silbermann [481], [483]. There Corollary 3.69 is proved for the Poisson kernel; its extension to arbitrary approximate identities was obtained in B¨ ottcher, Silbermann [112]. 3.72–3.79. B¨ottcher, Silbermann [112]. 3.80–3.84. These results were established in Silbermann [481], [482], [483] and B¨ ottcher, Silbermann [112]. The parts (b) of 3.80 and 3.82 as well as 3.84 were proved by Sarason [456] for N = 1 and the Poisson kernel.
4 Toeplitz Operators on H 2
4.1 Fredholmness We begin by applying Theorem 3.42 to the Fredholm theory of Toeplitz operators. Although this chapter is concerned with Toeplitz operators on H 2 ∼ = 2 , p p we ﬁrst state some results for Toeplitz operators on H and , because there do not arise any substantial diﬃculties when passing from the case p = 2 to the case p = 2. 4.1. Proposition. (a) If 1 < p < ∞, then p T : L∞ N ×N → L(HN ),
a → T (a)
is a submultiplicative embedding. (b) If 1 ≤ p < ∞ and if the norm in the space pN (Z) is chosen so that P L(pN (Z)) = 1, then p p T : MN ×N → L(N ),
a → T (a)
is a 1submultiplicative isometry. Proof. (a) Let ajk be a ﬁnite collection of functions in L∞ N ×N . Then " " " " " T (ajk )" " " j
p L(HN )
k
" !" # " " p " " = sup " P M (ajk )P f " : f ∈ HN , f HNp ≤ 1 Lp j k N " !" # " " p " " = sup " P M (ajk )P g " : g ∈ LN , P gLpN ≤ 1 j
k
Lp N
" " " 1 " " " ≥ P M (a )P jk " " p , cp L(L ) j k
N
(4.1)
172
4 Toeplitz Operators on H 2
where cp = P L(LpN ) . Let U ±n = M (χ±n I) denote the bilateral shifts on LpN . Then " " " " P M (ajk )P " " j
k
" " " " = "U −n P M (ajk )P U n " j
k
" " " " =" (U −n P U n )(U −n M (ajk )U n )(U −n P U n )" j
k
j
k
" " " " =" (U −n P U n )M (ajk )(U −n P U n )" " " " " ≥" M (ajk )" j
(U −n P U n → I strongly + 1.1(e))
k
" " " " ajk ". = "M j
(4.2)
k
N For 1 ≤ n ≤ N , deﬁne En ∈ L(LpN ) by En : (fk )N k=1 → (δnk fk )k=1 . Given ∞ ∈ L , we have f = (fmn )N m,n=1 N ×N " " " " 2 ) = " = M (f ) E M (f )E f L∞ " 2 m n L(L N ×N N
≤ const
L(LN )
m,n
M (fmn )L(L2 )
m,n
= const
M (fmn )L(Lp )
(Proposition 2.2)
m,n
≤ const
Em M (f )En L(LpN )
m,n
≤ const M (f )L(LpN ) . Hence, (4.1), (4.2), (4.3) give " " " " T (ajk )" " j
k
p L(HN )
(4.3)
" " " " ≥ const " ajk " j
k
L∞ N ×N
.
(4.4)
In particular, T (a)L(HNp ) ≥ const aL∞ N ×N
∀ a ∈ L∞ N ×N .
(4.5)
∀ a ∈ L∞ N ×N .
(4.6)
On the other hand, it is readily seen that T (a)L(HNp ) ≤ const aL∞ N ×N
The mapping T is clearly linear, (4.5) shows that T is onetoone, (4.6) implies that T is continuous, and (4.5), (4.6) together imply that T has a closed image.
4.1 Fredholmness
173
Thus, T is an embedding. Finally, by combining (4.4) and (4.6) we see that T is submultiplicative. (b) The proof is essentially the same as that of part (a). We have " " " " " " " " T (ajk )" p ≥ " P M (ajk )P " p " j
L(N )
k
" " " " ≥ "M ajk " j
k
j
L(p N (Z))
L(N (Z))
k
" " " " ≥ "T ajk " j
k
L(p N)
,
which gives the 1submultiplicativity of T . Since, in particular, T (a)L(pN ) ≥ M (a)L(pN (Z)) ≥ T (a)L(pN ) and, by deﬁnition, M (a)L(pN (Z)) = aMNp ×N , we ﬁnally conclude that T is an isometry. 2 4.2. Corollary. If a ∈ L∞ N ×N is sectorial then T (a) is invertible on HN .
Proof. This follows from the preceding proposition and Corollary 3.62, applied 2 ), i = T . with A = L∞ , F = X, B = L(HN Remark. Recall that “sectorial” means “analytically sectorial.” Azoﬀ and Clancey [16] showed that there exist geometrically sectorial matrix functions 2 a ∈ L∞ 2×2 such that T (a) is not even semiFredholm on H2 . 4.3. Corollary. (a) If 1 < p < ∞ and if A is a closed subalgebra of L∞ N ×N , then · algL(HNp ) T (A) = T (A) + QT (A) and algL(HNp ) T (A)/QT (A) is homeomorphically isomorphic to A. p (b) If 1 ≤ p < ∞ and if A is a closed subalgebra of MN ×N , then ·
algL(pN ) T (A) = T (A) + QT (A) and algL(pN ) T (A)/QT (A) is homeomorphically isomorphic to A. Proof. Proposition 4.1, Theorem 3.42, Corollary 3.44. The next proposition provides a description of the quasicommutator ideal QT (L∞ N ×N ). In accordance with 3.43, we let ST denote the projection of ∞ ∞ alg T (L∞ N ×N ) onto T (LN ×N ) parallel to QT (LN ×N ). Analogously ST is unp derstood on alg T (MN ×N ). p and pN as 4.4. Proposition. For n ≥ 0, deﬁne V n and V (−n) on HN
V n = T (χn I),
V (−n) = T (χ−n I).
4 Toeplitz Operators on H 2
174
(a) If 1 < p < ∞ and A ∈ algL(HNp ) T (L∞ N ×N ), then the strong limit s lim V (−n) AV n n→∞
exists and equals ST (A). p (b) If 1 ≤ p < ∞ and A ∈ algL(pN ) T (MN ×N ), then the strong limit
s lim V (−n) AV n n→∞
exists and equals ST (A). (c) Under the hypotheses of (a) or (b), p (−n) A ∈ QT (L∞ AV n = 0. N ×N ) (resp. QT (MN ×N )) ⇐⇒ s lim V n→∞
(d) The only compact Toeplitz operator on p ∞ Moreover, if a ∈ MN ×N and b ∈ LN ×N , then T (a)L(pN ) = T (a)Φ(pN ) ,
p HN
or pN is the zero operator.
T (b)L(HNp ) ≤ cp T (b)Φ(HNp ) ,
where cp := P L(Lp ) . p , then Proof. (a) If f ∈ HN T (ajk ) V n f = P U −n P M (ajk )P U n P f, V (−n) j
j
k
k
and the arguments used in the proof of Proposition 4.1(a) show that this p to converges in the norm of HN ajk P f = T ajk f = ST T (ajk ) f. PM j
k
j
k
j
k
This proves the assertion for the case where A is a ﬁnite productsum of Toeplitz operators. The general case now follows from 1.1(d). (b) The proof is that of part (a). (c) Immediate from (a) resp. (b). (d) If K is a compact operator, then V (−n) KV n → 0 uniformly, because p → 0 strongly. It is clear that V (−n) T (c)V n = T (c) for every c ∈ MN V ×N ∞ resp. c ∈ LN ×N . Thus, by 1.1(e), " " T (c) ≤ lim inf "V (−n) (T (c) + K)V n " n→∞ ≤ sup P V (−n) T (c) + K V n = P T (c) + K (−n)
n
for every compact operator K.
4.1 Fredholmness
175
4.5. Proposition. (a) Let 1 < p < ∞ and let B be a closed subalgebra of C + H ∞ containing C. Then p QT (BN ×N ) = C∞ (HN ).
(b) Let 1 < p < ∞ and let B be a closed subalgebra of Cp + Hp∞ containing Cp . Then QT (BN ×N ) = C∞ (pN ). Proof. (a) Formula (2.18) and Theorem 2.42(a) imply that QT (BN ×N ) ⊂ p p ). So it remains to show that C∞ (HN ) ⊂ QT (CN ×N ). By virtue of C∞ (HN Lemma 3.47 we may assume that N = 1. Let K ∈ C∞ (H p ) and notice ﬁrst that (I − T (χn )T (χ−n ))K converges uniformly to K as n → ∞, because T (χn )T (χ−n ) → 0 strongly. Since I −T (χn )T (χ−n ) has ﬁnite rank, it remains to show that C0 (H p ) ⊂ QT (C). This on its hand will follow as soon as we have shown that every operator L ∈ L(H p*) of the form Lf = (f, χk )χn (k, n ≥ 0) is in QT (C), where (f, χk ) := 1/(2π) T f χ−k dm. But this is immediate from the identity L = T (χn ) T (χ1 χ−1 ) − T (χ1 )T (χ−1 ) T (χ−k ). (b) The proof is the same. Remark. Let Lf := (f, g)e0 , where g = (1, 1, . . .) ∈ ∞ . Then L ∈ C0 (1 ), but it can be shown that L is not in algL(1 ) T (W ). Using Proposition 4.4(c) one can easily prove that QT (W ) = C∞ (1 ) ∩ algL(1 ) T (W ). We are now in a position to prove the following important spectral inclusion theorem (recall Theorems 2.30 and 2.93). 2 4.6. Corollary. If A ∈ algL(HN2 ) T (L∞ N ×N ) is Fredholm on HN , then
SmbT (A) ∈ GL∞ N ×N . 2 ∞ In particular, if a ∈ L∞ N ×N and T (a) ∈ Φ(HN ), then a ∈ GLN ×N .
Proof. We suppress the subscript N . If A ∈ Φ(H 2 ), then A + C∞ (H 2 ) belongs to G(L(H 2 )/C∞ (H 2 )). Proposition 4.5 implies that C∞ (H 2 ) ⊂ alg T (H 2 ) and so 1.26(d) shows that A + C∞ (H 2 ) must be invertible in alg T (L∞ )/C∞ (H 2 ). Moreover, Proposition 4.5 even states that C∞ (H 2 ) ⊂ QT (L∞ ), therefore A + QT (L∞ ) must belong to G(alg T (L∞ )/QT (L∞ )). Corollary 4.3 applied with A = L∞ N ×N completes the proof. The next two corollaries settle the Fredholm theory for operators belonging to the closed algebra generated by Toeplitz operators with C + H ∞ symbols. 4.7. Corollary. (a) Let 1 < p < ∞ and let B be a closed subalgebra of C +H ∞ containing C. Then
176
4 Toeplitz Operators on H 2 ·
p algL(HNp ) T (BN ×N ) = T (BN ×N ) + C∞ (HN ) p and algL(HNp ) T (BN ×N )/C∞ (HN ) is homeomorphically isomorphic to BN ×N .
(b) Let 1 < p < ∞ and let B be a closed subalgebra of Cp + Hp∞ containing Cp . Then ·
algL(pN ) T (BN ×N ) = T (BN ×N ) + C∞ (pN ) and algL(pN ) T (BN ×N )/C∞ (pN ) is isometrically isomorphic to BN ×N (as a p subalgebra of MN ×N ). Proof. Corollary 4.3 and Proposition 4.5 show that the corresponding algebras are homeomorphically isomorphic. On combining Proposition 4.1(b) and Proposition 4.4(d) we see that in the case (b) the isomorphism is even isometric. 4.8. Corollary. (a) Let 1 < p < ∞ and A ∈ algL(HNp ) T ((C + H ∞ )N ×N ). Then p ) ⇐⇒ det SmbT (A) ∈ G(C + H ∞ ). A ∈ Φ(HN (b) Let 1 < p < ∞ and A ∈ algL(pN ) T ((Cp + Hp∞ )N ×N ). Then A ∈ Φ(pN ) ⇐⇒ det SmbT (A) ∈ G(Cp + Hp∞ ). (c) Under the hypotheses (a) or (b), if A is Fredholm then Ind A = Ind T (det SmbT (A)). Proof. (a), (b) The implications “⇐=” follow from Corollary 4.7. To prove the reverse implications, note ﬁrst that, again by Corollary 4.7, A = T (SmbT (A)) + K with some compact operator K. Thus, if A is Fredholm, then so is T (SmbT (A)) and Theorem 2.94 completes the proof. (c) Note that
Ind A = Ind T (SmbT (A)) + K = Ind T SmbT (A)
and apply Theorem 2.94. 4.9. Remark. If A ∈ algL(1N ) T (WN ×N ), then A ∈ Φ(1N ) ⇐⇒ det SmbT (A) ∈ GWN ×N . The implication “⇐=” results from Corollary 4.3(b) and the remark in 4.5. That Ind A = Ind T (SmbT (A)) = −ind det SmbT (A) can be shown as above, and so an index perturbation argument yields the implication “=⇒”. We now show how the machinery developed in Chapter 3 to study harmonic approximation can be applied to the Fredholm theory of (block) 2 . Toeplitz operators on HN
4.1 Fredholmness
177
4.10. Essentialization. We know from Proposition 4.1 that the mapping 2 T : L∞ N ×N → L(HN ),
a → T (a)
2 is a 1submultiplicative isometry. Proposition 4.5 tells us that C∞ (HN ) is ∞ contained in alg T (LN ×N ) and is therefore a closed twosided ideal of that algebra. Finally, Proposition 4.4 shows that 2 ST (K) = s lim V (−n) KV n = 0 ∀ K ∈ C∞ (HN ), n→∞
because V n → 0 weakly (recall 1.1(f)). So Theorem 3.52 can be used to see that the mapping π ∞ T π : L∞ N ×N → alg T (LN ×N ),
2 a → T π (a) := T (a) + C∞ (HN )
is a 1submultiplicative quasiembedding. From Proposition 4.1(b) and Proposition 4.4(d) we deduce that T π is even an isometry. Since alg T π (L∞ N ×N ) is 2 2 )/C∞ (HN ), we conclude from a C ∗ subalgebra of the Calkin algebra L(HN 1.26(d) that, for a ∈ L∞ N ×N , 2 ) ⇐⇒ T π (a) ∈ G(alg T π (L∞ T (a) ∈ Φ(HN N ×N )).
(4.7)
4.11. Localization. Let F be a closed subset of X = M (L∞ ). In accordance with 3.58 we deﬁne IF = a ∈ L∞ JFπ = closidalg T π (L∞ T π (IF ). N ×N : aF = 0 , N ×N ) Lemma 3.59(a) implies that ST π (JFπ ) ⊂ JFπ . Consequently, again by Theorem 3.52, the mapping π ∞ TFπ : L∞ N ×N → alg TF (LN ×N ),
a → TFπ (a) := T π (a) + JFπ
is a 1submultiplicative quasiembedding with Ker TFπ = IF (Lemma 3.59(b)). Keeping in mind 3.60, we call TFπ (a) (a ∈ L∞ N ×N ) a local Toeplitz operator. Theorem 3.61 specializes to give that TFπ (a) = aF ∀ a ∈ L∞ N ×N .
(4.8)
In case F is a ﬁber Xξ , where ξ ∈ M (B) and B is a C ∗ subalgebra of L∞ containing the constants, we also have from (3.43) IXξ = closidL∞ cIN ×N : c ∈ B, c(ξ) = 0 , N ×N where IN ×N denotes the N × N identity matrix. In that case we abbreviate π π , TX (a) to Iξ , Jξπ , Tξπ (a), respectively. IXξ , JX ξ ξ Local Toeplitz operators will be studied in some more detail later. In the meanwhile we conﬁne ourselves to stating the following.
178
4 Toeplitz Operators on H 2
4.12. Theorem. Let B be a C ∗ subalgebra of QC containing the constants. Then if a ∈ L∞ N ×N , 2 ) ⇐⇒ Tξπ (a) ∈ G(alg Tξπ (L∞ T (a) ∈ Φ(HN N ×N )) ∀ ξ ∈ M (B).
Proof. This is a consequence of (4.7) and Theorem 3.67, applied with A = L∞ , 2 2 )/C∞ (HN ), i = T π . Note that T π (ca) = T π (c)T π (a) for all c in B = L(HN ∞ B ⊂ C + H and a ∈ L∞ by virtue of formula (2.18) and Theorem 2.42(a). 4.13. Corollary. Let B be a C ∗ subalgebra of QC containing the constants and let a ∈ L∞ N ×N . (a) If for each ﬁber Xξ , ξ ∈ M (B), there exists a bξ ∈ L∞ N ×N such that 2 2 ), then T (a) ∈ Φ(HN ). aXξ = bξ Xξ and T (bξ ) ∈ Φ(HN 2 (b) If a is locally sectorial over B, then T (a) ∈ Φ(HN ).
Proof. (a) If T (bξ ) is Fredholm, then Tξπ (bξ ) is invertible in alg Tξπ (L∞ N ×N ) by the preceding theorem. From (4.8) we deduce that Tξπ (a) = Tξπ (bξ ), and again applying the preceding theorem we get the Fredholmness of T (a). (b) We have Tξπ (a) ∈ G(alg Tξπ (L∞ N ×N )) for all ξ ∈ M (B) due to Corollary 3.62. Note that part (a) of this corollary is identical with Theorem 2.96. For (b) see also 4.31.
4.2 Stable Convergence 4.14. Deﬁnition. Let Λ be either of the index sets Λ = {l0 , l0 + 1, l0 + 2, . . .} (l0 ∈ N) or Λ = (r0 , ∞) (r0 ∈ R+ ). Let X be a Banach space, let A ∈ L(X), and let {Aλ }λ∈Λ be a (generalized) sequence of operators Aλ ∈ L(X). The sequence {Aλ }λ∈Λ is said to converge stably to A if (a) Aλ → A strongly on X as λ → ∞, (b) there is a λ0 ∈ Λ such that Aλ ∈ GL(X) for all λ > λ0 , (c) sup A−1 λ L(X) < ∞. λ>λ0
For instance, if {Kλ }λ∈Λ is an approximate identity and a ∈ L∞ N ×N , then M (kλ a) converges stably to M (a) on L2N if and only if {kλ a} is bounded away from zero (see 3.55 for (b), (c) and Proposition 3.40(a) for (a)). Here our concern is the study of questions connected with the stable convergence 2 . On this basis we shall then propose an approach to of T (kλ a) to T (a) on HN establishing index formulas for Toeplitz operators with locally sectorial matrix symbol.
4.2 Stable Convergence
179
∞ 4.15. Algebraization. Let BN,N denote the collection of all (generalized) 2 sequences {Aλ }λ∈Λ of operators Aλ ∈ L(HN ) such that sup Aλ L(HN2 ) < ∞. λ∈Λ
On deﬁning α{An } := {αAn }, {Aλ } + {Bλ } := {Aλ + Bλ }, {Aλ }{Bλ } := {Aλ Bλ }, {An }∗ := {A∗n }, and {Aλ } := sup Aλ L(HN2 )
(4.9)
λ∈Λ
∞ we make BN,N become a C ∗ algebra. Let
∞ 2 : there exists an A ∈ L(HN ) such that BN,N = {Aλ } ∈ BN,N ∗ ∗ 2 Aλ → A and Aλ → A strongly on HN as λ → ∞ . ∞ It is not diﬃcult to see that BN,N is a C ∗ subalgebra of BN,N . If {Aλ } ∈ BN,N converges stably to its strong limit A, then A is in2 vertible. Indeed, we have f ≤ A−1 λ Aλ f for all f ∈ HN , hence −1 Aλ f ≥ (1/M )f with M := sup Aλ , and passage to the limit λ → ∞ λ∈Λ
2 ; the stable convergence of Aλ to A gives Af ≥ (1/M )f for all f ∈ HN ∗ implies the stable convergence of Aλ to A∗ , and so the same argument applied 2 ; the conclusion to A∗λ yields the inequality A∗ f ≥ (1/M )f for all f ∈ HN is that A must be invertible.
4.16. Essentialization. Now put MN,N = {Aλ } ∈ BN,N : Aλ L(HN2 ) → 0 as λ → ∞ , 2 JN,N = {Aλ } ∈ BN,N : Aλ = K + Cλ , K ∈ C∞ (HN ), {Cλ } ∈ MN,N . It can be checked straightforwardly that MN,N is a closed twosided ideal of ∞ and BN,N and that JN,N is a closed twosided ideal of BN,N . both BN,N ∞ ∞ ∞ := B1,1 ), we Since (4.9) deﬁnes an admissible norm on BN ×N (with B ∞ ∞ have BN,N = BN ×N , BN,N = BN ×N , MN,N = MN ×N , JN,N = JN ×N , where B := B1,1 , M := M1,1 , J := J1,1 . For {Aλ } ∈ BN ×N , let {Aλ }πM and {Aλ }πJ denote the cosets {Aλ } + MN ×N and {Aλ } + JN ×N , respectively. A little thought shows that, for {Aλ } ∈ BN ×N , ∞ Aλ converges strongly to its limit ⇐⇒ {Aλ }πM ∈ G(BN ×N /MN ×N ).
4.17. Proposition. Let {Aλ } ∈ BN ×N and let A denote the strong limit of Aλ as λ → ∞. Then the following are equivalent: (i) Aλ converges stably to A. (ii) {Aλ }πM ∈ G(BN ×N /MN ×N ). 2 (iii) A ∈ GL(HN ) and {Aλ }πJ ∈ G(BN ×N /JN ×N ).
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4 Toeplitz Operators on H 2
Proof. We drop the subscript N × N . (i) ⇐⇒ (ii): Since B/M is a C ∗ subalgebra of B∞ /M, {Aλ }πM is invertible in B/M if it is invertible in B∞ /M (1.26(d)). (i)+(ii) =⇒ (iii): The invertibility of A was shown in 4.15 and the invertibility of {Aλ }πJ is a consequence of (ii). (iii) =⇒ (ii): There is a {Bλ } ∈ B such that K ∈ C∞ (H 2 ),
Bλ Aλ = I + K + Cλ ,
{Cλ } ∈ M.
Passage to the strong limit λ → ∞ gives BA = I + K. Hence (Bλ − KA−1 )Aλ = I + K + Cλ − KA−1 Aλ = I + K + Cλ − KA−1 A + Cλ = I + Cλ + Cλ with some {Cλ } ∈ M, because KA−1 ∈ C∞ (H 2 ) and A∗λ → A∗ strongly (recall 1.3(d)). It follows that {Aλ }πM is leftinvertible in B/M. The rightinvertibility can be shown analogously. 4.18. alg T K(AN ×N ). Let A be a closed subalgebra of L∞ containing the constants, put A = AN ×N , and let {Kλ } be an approximate identity. (a) If a ∈ L∞ N ×N , then {T (kλ a)} ∈ BN ×N . This follows from the fact that {kλ a} ∈ AN ×N (Proposition 3.40(a)). (b) The mapping deﬁned by T K : L∞ N ×N → BN ×N ,
a → {T (kλ a)}
is a 1submultiplicative isometry. That T K is an isometry follows from the equalities = sup kλ aL∞ = sup T (kλ a)L(HN2 ) , aL∞ N ×N N ×N λ
(4.10)
λ
the ﬁrst of which holds by virtue of Proposition 3.40(b), while the second is true because of Proposition 4.1(b) with p = 2. Since " " " " " " " " aij " = "T aij " (by (4.10)) " T kλ i
j
i j " " " " T (aij )" ≤ " i
(Proposition 4.1(b))
j
" " " " ≤ lim inf " T (kλ aij )" λ→∞
i
j
" " " " ≤ sup " T (kλ aij )" λ
" i j " " " T (kλ aij ) ", =: " i
j
it follows that T K is 1submultiplicative.
(by 1.1(e))
4.2 Stable Convergence
181
(c) So Theorem 3.42 gives that ·
alg T K(A) = T K(A) + QT K (A).
(4.11)
Because T K is 1submultiplicative, the projection ST K has norm 1. Let LB denote the (linear and continuous) mapping which is deﬁned by 2 ), LB : BN ×N → L(HN
{Bλ } → s−lim Bλ . λ→∞
The mappings ST K and SmbT K , which are given at ﬁnite productsums (correctly) by {T (kλ aij )} → T kλ aij , ST K : i
j
SmbT K :
i
i
{T (kλ aij )} →
j
i
j
aij ,
j
can then be represented in the form ST K = T K ◦ SmbT ◦ LB alg T K(A), SmbT K = SmbT ◦ LB alg T K(A). (d) The restriction of LB to alg T K(A) will be denoted by Φ. It is clear that Φ is a continuous algebraic homomorphism whose range is dense in alg T (A). Thus, if A is a C ∗ algebra, then Φ is a starhomomorphism of alg T K(A) onto alg T (A) (see 1.26(e)). Notice that Φ(QT K (A)) is contained in QT (A). (e) Deﬁne the mapping Ψ at ﬁnite productsums by {T (kλ aij )} → {kλ aij }. Ψ: i
j
i
j
Because " " " " " " " " {kλ aij }" := sup " kλ aij " " i
j
λ
i
j
" " " " kλ aij " (Proposition 4.1(b)) = sup "T λ
i
j
" " " " T (kλ aij )" ≤ sup " λ
i
(Proposition 4.1(b))
j
" " " " {T (kλ aij )}", =: " i
j
it follows that the mapping Ψ extends to a continuous algebraic homomorphism of alg T K(A) into alg K(A). Moreover, if A is a C ∗ algebra, then
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4 Toeplitz Operators on H 2
Ψ is a starhomomorphism of alg T K(A) onto alg K(A). We always have Ψ (QT K (A)) ⊂ QK (A). Finally, given µ ∈ Λ deﬁne the continuous algebraic homomorphism Fixµ by Fixµ : alg T K(A) → alg T (A), {Aλ } → Aµ . It is easy to see that we can now write (Ψ {Aλ })µ = (SmbT ◦ Fixµ ){Aλ }. (f) Thus, we have the following picture:
π 4.19. alg T Kπ M (AN ×N ) and alg T KJ (AN ×N ). (a) For a closed subalge∞ bra of L containing C, put A = AN ×N and
MA N ×N = MN ×N ∩ alg T K(A),
JNA×N = JN ×N ∩ alg T K(A)
A (also recall Lemma 3.47). Thus, both MA N ×N and JN ×N are closed twosided ideals of alg T K(A).
(b) We have ST K (JNA×N ) = {0}. Indeed, if {Bλ } ∈ JNA×N , then Bλ = 2 ) ⊂ QT (A) (Proposition 4.5) and {Cλ } ∈ MN ×N , K + Cλ with K ∈ C∞ (HN whence ST K {Bλ } = (T K ◦ SmbT ◦ LB ){Bλ } = (T K ◦ SmbT )(K) = T K(0) = 0. (c) So Theorem 3.52 gives the following: the mappings π T KM : A → alg T K(A)/MA N ×N ,
a → {T (kλ a)} + MA N ×N ,
π T KJ : A → alg T K(A)/JNA×N ,
a → {T (kλ a)} + JNA×N ,
are 1submultiplicative quasiembeddings and π alg T KM (A) = alg T K(A)/MA N ×N ,
π alg T KJ (A) = alg T K(A)/JNA×N .
4.2 Stable Convergence
183
π π The mappings T KM and T KJ are actually isometries: if {T (kλ a)} ∈ JNA×N , 2 ) and this can only happen if a = 0 then T (kλ a) converges to T (a) ∈ C∞ (HN (Proposition 4.4(d)). For {Bλ } ∈ alg T K(A), we denote the cosets {Bλ } + A π π MA N ×N and {Bλ } + JN ×N by {Bλ }M and {Bλ }J , respectively.
(d) Let θ denote the (continuous) algebraic homomorphism deﬁned by π π θ : alg TM K(A) → alg T KJ (A),
{Bλ }πM → {Bλ }πJ .
(e) The mapping Φ deﬁned in 4.18(d) maps the ideal JNA×N into the ideal 2 ). We can therefore deﬁne the quotient mapping C∞ (HN Φπ : alg TJπ K(A) → alg T π (A), which is the extension to the whole algebra of the mapping given at ﬁnite productsums (correctly) by Φπ :
i
j
{T (kλ aij )}πJ →
i
T π (aij ).
j
Clearly, Φπ (QT KπJ (A)) ⊂ QT π (A). Finally, Φπ is a surjective algebraic starhomomorphism whenever A is a C ∗ algebra. (f) The mapping Ψ introduced in 4.18(c) has the property that Ψ (JNA×N ) ⊂ Indeed, if {K + Cλ } ∈ JNA×N , then
NNA×N .
(Ψ {K + Cλ })µ = (SmbT ◦ Fixµ ){K + Cλ } = SmbT (K + Cµ ) = SmbT (Cµ ) and {SmbT (Cλ )}λ∈Λ is obviously in NNA×N whenever {Cλ }λ∈Λ ∈ MA N ×N . Thus, the quotient mapping π Ψ π : alg T KJ (A) → alg Kπ (A)
can be deﬁned. It is the extension to the whole algebra of the mapping given on ﬁnite productsums (correctly) by Ψπ :
{T (kλ aij )}πJ → {kλ aij }π . i
j
i
j
Again we have Ψ π (QT KπJ (A)) ⊂ QKπ (A), and if A is a C ∗ algebra, then Ψ π is a surjective algebraic starhomomorphism. (g) Thus, the “quotient picture” or the “essentialization” of 4.18(f) looks as in the following diagram:
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4 Toeplitz Operators on H 2
4.20. Theorem. Let A be a C ∗ subalgebra of L∞ containing C, let {Bλ } be in alg T K(AN ×N ), and let B ∈ alg T (AN ×N ) denote the strong limit of Bλ as λ → ∞. Then the following are equivalent: (i) Bλ converges stably to B. π (AN ×N )). (ii) {Bλ }πM ∈ G(alg T KM 2 π ) and {Bλ }πJ ∈ G(alg T KJ (AN ×N )). (iii) B ∈ GL(HN
Proof. The proof is a combination of that of Proposition 4.17 and some standard C ∗ arguments as they have already been used in the proof of Theorem 3.56. The following proposition provides two important situations in which the quasicommutator ideal of alg T K(AN ×N ) can be identiﬁed. 4.21. Proposition. (a) Let B be a closed subalgebra of QC containing C and let {Kλ }λ∈Λ be any approximate identity. Then QT K (BN ×N ) = JNB×N . (b) Let B be a closed subalgebra of C + H ∞ containing C and let {Kλ }λ∈Λ be the approximate identity generated by the Poisson kernel. Then QT H (BN ×N ) = JNB×N . Proof. By virtue of Lemma 3.47 it suﬃces to consider the case N = 1. If {Aλ } ∈ J B , then ST K {Aλ } = 0 by 4.19(b). Consequently, J B ⊂ QT K (B). Now let ϕ, ψ ∈ B. Then, by formula (2.18), T (kλ ϕ)T (kλ ψ) − T (kλ (ϕψ)) ) = T [(kλ ϕ)(kλ ψ) − kλ (ϕψ)] − H(kλ ϕ)H(kλ ψ)
(4.12)
4.2 Stable Convergence
185
) where f)(t) := f (1/t) for t ∈ T). The ﬁrst term in (note that (kλ ψ)) = kλ ψ, (4.12) converges uniformly to zero as λ → ∞, since {Kλ } is asymptotically multiplicative on the pair (B, B) (Theorem 3.23 resp. Theorem 2.62(a)). Further, we have ψ) = c + h with c ∈ C and h ∈ H ∞ . For ﬁxed λ, the operator a → kλ a is bounded on L2 , and because it maps PA into PA (by (3.5)), it maps H 2 into H 2 . Hence, kλ h ∈ H ∞ and therefore ) = H(kλ ϕ)H(kλ c) H(kλ ϕ)H(kλ ϕ) converges uniformly to H(ϕ)H(c) ∈ C∞ (H 2 ), since H(kλ ϕ) converges strongly to H(ϕ) and H(kλ c) converges uniformly to H(c). Thus, we have shown that every quasicommutator in alg T K(B) belongs to J , which implies that QT K (B) ⊂ J B . 4.22. Corollary. (a) Let {Aλ } ∈ alg T K(QCN ×N ), where {Kλ }λ∈Λ is any approximate identity. Then Aλ converges stably to its strong limit A as λ → ∞ 2 ). if and only if A ∈ GL(HN (b) Let {Aλ } ∈ alg T H((C + H ∞ )N ×N ). Then Aλ converges stably to its 2 ). strong limit A as λ → ∞ if and only if A ∈ GL(HN Proof. (a) We showed in 4.15 that A is invertible if Aλ converges stably to A. To get the reverse implication it suﬃces by virtue of Theorem 4.20 to show that {Aλ }πJ ∈ G(alg T K(QCN ×N )/JNQC ×N ), which, in view of Proposition 4.21, Corollary 3.44, and the fact that Ker T K = {0}, is equivalent to the requirement that SmbT K {Aλ } be in GQCN ×N . But we know from 4.18(c) that SmbT K {Aλ } = SmbT A, and the invertibility of SmbT A results from Corollary 4.6 in conjunction with 1.26(d). 2 ), then SmbT A is invertible in (C + H ∞ )N ×N by Corol(b) If A ∈ GL(HN lary 4.8. As in the proof of part (a), this implies that {Aλ }πJ is invertible in π ((C + H ∞ )N ×N ), which, in turn, gives the invertibility of {Aλ }πJ in alg T HJ π alg T HJ (L∞ N ×N ). So Theorem 4.20 applies again.
4.23. Localization. Let F be a closed subset of X = M (L∞ ), let A be a closed subalgebra of L∞ containing C, and put A = AN ×N . (a) In accordance with 3.58 we deﬁne π (IF ). IF = a ∈ A : aF = 0 , JFπ = closidalg T KπJ (A) T KJ From Lemma 3.59 and Theorem 3.52 we deduce that T KFπ : A → alg T KFπ (A) = alg T Kπ (A)/JFπ , a → {T (kλ a)}πF := {T (kλ a)}πJ + JFπ is a 1submultiplicative quasiembedding whose kernel is IF . If F is a ﬁber Xξ , where ξ ∈ M (B) and B is a C ∗ algebra of L∞ containing the constants, π and {Aλ }πXξ to T Kξπ and {Aλ }πξ , respectively. we abbreviate T KX ξ
186
4 Toeplitz Operators on H 2
(b) It can be checked straightforwardly that Φπ and Ψ π (deﬁned in 4.19(e) and 4.19(f)) map JFπ into the corresponding ideal JFπ of alg T π (A) and alg Kπ (A). Therefore we can deﬁne the quotient mappings ΦπF and ΨFπ in a natural way. These mappings are continuous algebraic homomorphisms, which are surjective whenever A is a C ∗ algebra. (c) So we arrive at the following “localization” of the diagram 4.18(f):
4.24. Theorem. Let {Kλ }λ∈Λ be any approximate identity, let F be a closed subset of X, and let a ∈ L∞ N ×N . Then the following spectral inclusions hold: sp TFπ (a) ⊂ sp {T (kλ a)}πF ⊃ sp {kλ a}πF ∪ ∪ ∪ sp (aF ) = sp (aF ) = sp (aF ). In the case N = 1 we have sp (aF ) = a(F ) and, in addition, sp {T (kλ a)}πF ⊂ conv a(F ). Proof. The inclusions in the ﬁrst row follow from the fact that ΦπF and ΨFπ are algebraic homomorphisms. The vertical inclusions result from Corollary 3.63. Finally, the last inclusion is a consequence of Corollary 3.62 (or Corollary 3.64). 4.25. Lemma. Suppose ϕ ∈ QC and let {Kλ }λ∈Λ be any approximate identity. Then T (kλ (ϕa)) − T (kλ ϕ)T (kλ a) ∈ J . Proof. Due to Theorem 3.23, T (kλ (ϕa)) − T (kλ ϕ · kλ a)L(H 2 ) → 0. By formula (2.18), T (kλ ϕ · kλ a) − T (kλ ϕ)T (kλ a) = H(kλ ϕ)H(kλ ) a). We have ϕ = c + h with c ∈ C and h ∈ H ∞ , and since kλ h ∈ H ∞ (see the proof of Proposition 4.21), we conclude that a) = H(kλ c − c)H(kλ ) a) + H(c)H(kλ ) a). H(kλ ϕ)H(kλ )
(4.13)
4.3 Index Computation
187
Because H ∗ (kλ ) a) converges strongly to H ∗ () a) (see the proof of Proposition 3.40(a)) and kλ c converges uniformly to c, it follows that the ﬁrst term in (4.13) converges uniformly to zero while the second converges uniformly to H(c)H() a) ∈ C∞ (H 2 ). 4.26. Theorem. Let {Kλ }λ∈Λ be any approximate identity and let a be in ∗ L∞ N ×N . Suppose B is a C subalgebra of QC containing the constants. (a) If for each ﬁber Xξ , ξ ∈ M (B), there is a bξ ∈ L∞ N ×N such that 2 ), aXξ = bξ Xξ and T (kλ bξ ) converges stably to T (bξ ) and if T (a) ∈ GL(HN then T (kλ a) converges stably to T (a). (b) If a is locally sectorial over B, then for T (kλ a) to converge stably to 2 . T (a) it is necessary and suﬃcient that T (a) be invertible on HN Proof. Without loss of generality suppose C ⊂ B. (a) Theorem 4.20(a) (with A = L∞ ) and Theorem 3.67 (with A = L∞ , π π (L∞ B = alg T KJ N ×N ), i = T KJ ), whose hypothesis (c) is fulﬁlled by the preceding lemma, imply that {T (kλ bξ )}πξ is invertible in alg T Kξπ (L∞ N ×N ) for each ξ ∈ M (B). Theorem 3.61 shows that {T (kλ a)}πξ = {T (kλ bξ )}πξ and so, π (L∞ again by Theorem 3.67, {T (kλ a)}πJ ∈ G(alg T KJ N ×N )). Now the assertion follows from Theorem 4.20. (b) We have {T (kλ a)}πξ ∈ G(alg T Kξπ (L∞ N ×N )) as a consequence of Corollary 3.62. So it remains to apply Theorem 3.67 and Theorem 4.20.
4.3 Index Computation We now establish an index formula for operators A belonging to a relatively large subclass of alg T (L∞ N ×N ). However, note that unless A is a scalar Toeplitz operator, A = T (a) with a ∈ L∞ , the knowledge of an index formula does, in general, not solve the invertibility problem. In particular, although it is an easy matter to derive the index formulas for A ∈ alg T (C) or A ∈ T (CN ×N ) (recall Corollary 4.8), it is a very delicate question to decide whether such an operator is invertible. 4.27. Deﬁnition. Let {Kλ }λ∈Λ be an approximate identity whose index set is Λ = {l0 , l0 + 1, l0 + 2, . . .} (l0 ∈ N) or Λ = (r0 , ∞) (r0 ∈ R+ ). In either case, conv Λ, the convex hull of Λ, is a connected subset of R+ . Now let {aλ }λ∈Λ ∈ alg K(L∞ N ×N ) and assume aλ is well deﬁned for all λ ∈ conv Λ. For example, if aλ = kλ b, where b ∈ L∞ N ×N , then aλ is deﬁned for all λ ∈ R+ in a natural manner. If Λ is connected, then aλ is of course also well deﬁned for all λ ∈ conv Λ, since Λ = conv Λ. We shall say that {aλ }λ∈Λ is bounded away from zero on conv Λ if there is a λ0 ∈ conv Λ such that {aλ }λ∈(λ0 ,∞) is bounded away from zero. In that case det aλ (t) = 0 for all (λ, t) ∈ (λ0 , ∞) × T. By Proposition 3.73(a) and 2.41(b), ind det aλ depends
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4 Toeplitz Operators on H 2
on λ ∈ (λ0 , ∞) continuously, and since (λ0 , ∞) is connected, it follows that lim ind det aλ is a ind det aλ is constant for λ > λ0 . This implies that λ→∞,λ∈Λ
welldeﬁned integer. This integer will simply be denoted by ind {det aλ }. Note that Corollary 3.69(b) actually states that {kλ a} is bounded away ∗ from zero on conv Λ whenever a ∈ L∞ N ×N is locally sectorial over a C subalgebra of QC. A similar remark can be made for Theorem 4.26(b). 4.28. Theorem. Let {Aλ } ∈ alg T K(L∞ N ×N ), put A := Φ{Aλ } := s−lim Aλ ∈ alg T (L∞ N ×N ) , λ→∞ {aλ } := Ψ {Aλ } ∈ alg K(L∞ N ×N ) and suppose {Aλ }πJ ∈ G(alg T KJ (L∞ N ×N )) and Λ is connected. Then 2 (a) A ∈ Φ(HN ),
(b) {aλ } is bounded away from zero, (c) Ind A = −ind {det aλ }. Proof. (a), (b) Since Φπ and Ψ π are algebraic homomorphisms, the invertibility of {Aλ }πJ implies that of both Aπ = Φπ {Aλ }πJ and {aλ }π = Ψ π {Aλ }πJ . This in turn gives (a) and (b) at once. (c) Let Ind A = κ. Deﬁne χ(t) = diag (tκ , 1, . . . , 1), t ∈ T. Then T (χ) 2 2 ) and Ind T (χ) = −κ. Consequently, AT (χ) ∈ Φ(HN ) and belongs to Φ(HN 2 Ind AT (χ) = 0. It follows that there is an R0 ∈ C∞ (HN ) such that AT (χ)+R0 2 is invertible. Because C∞ (HN ) equals QT (CN ×N ) (Proposition 4.5), there is even a ﬁnite productsum 2 T (ϕij ) ∈ C∞ (HN ), ϕij ∈ CN ×N , R= i
j
such that 2 ). (4.14) AT (χ) + R ∈ GL(HN Put {Rλ } := i j T (kλ ϕij ) . Since ϕij ∈ CN ×N , it follows that Rλ converges uniformly to the compact operator R as λ → ∞ and hence
5
∞
{Rλ } ∈ JNC×N ⊂ JNL×N .
(4.15)
It is clear that Aλ T (kλ χ) + Rλ → AT (χ) + R
strongly.
Because of (4.15), {Aλ T (kλ χ) + Rλ }πJ = {Aλ T (kλ χ)}πJ = {Aλ }πJ {T (kλ χ)}πJ
(4.16)
4.3 Index Computation
189
and since {Aλ }πJ is invertible by our hypothesis and {T (kλ χ)}πJ has the inverse {T (kλ χ−1 )}πJ (Lemma 4.25 for ϕ ∈ C and a ∈ C), we conclude that π {Aλ T (kλ χ) + Rλ }πJ ∈ G(alg T KJ (L∞ N ×N )).
(4.17)
Now (4.14), (4.16), (4.17), and Theorem 4.20 give that π {Aλ T (kλ χ) + R}πM = {Aλ T (kλ χ) + Rλ }πM ∈ G(alg T KM (L∞ N ×N )).
In particular, Aλ T (kλ χ) + R must be invertible for λ > λ0 , whence 0 = Ind (Aλ T (kλ χ) + R) = Ind Aλ − κ
(4.18)
for λ > λ0 . Since {aλ } = Ψ {Aλ }, we have, for λ > λ0 , aλ = (SmbT ◦ Fixλ ){Aµ } = SmbT Aλ , which implies that ST (Aλ ) = T (Aλ ). Thus, Aλ = T (aλ ) + Kλ with 2 ), so Ind Aλ = Ind T (aλ ) = −ind det aλ some Kλ ∈ QT (CN ×N ) = C∞ (HN (Theorem 2.94 for the case of a continuous symbol), and therefore (4.18) ﬁnally gives ind det aλ = −κ for λ > λ0 . This ends the proof. 4.29. Remark. Let {Kλ }λ∈(1,∞) be the approximate identity generated by the Poisson kernel. Taking into account Theorem 2.62(a) it is easily seen that if a ∈ G(C + H ∞ ), then {T (kλ a−1 )}πJ is the inverse of {T (kλ a)}πJ in π (L∞ ). Thus, the preceding theorem immediately implies that alg T KJ Ind T (a) = −ind {kλ a} ∀ a ∈ G(C + H ∞ ),
(4.19)
a fact which had already been established in the proof of Theorems 2.64 and 2.65. Further, once (4.19) has been proved, Corollary 4.8(c) is all what then is needed to deduce that Ind A = −ind {kλ det SmbT A} whenever A ∈ algL(HN2 ) T (C + H ∞ )N ×N is Fredholm. 4.30. Corollary. Let B be a C ∗ subalgebra of QC containing the constants and let {Kλ }λ∈Λ be any approximate identity. If a ∈ L∞ N ×N is locally sectorial over B, then 2 ), (a) T (a) ∈ Φ(HN
(b) {kλ a} is bounded away from zero on conv Λ, (c) Ind T (a) = −ind {det(kλ a)}. Proof. (a) and (b) follow from Corollary 4.13(b) and Corollary 3.69(b), respectively, (c) will follow from Theorem 4.28 as soon as we have proved π (L∞ that {T (kλ a)}πJ is invertible in alg T KJ N ×N ). But this follows from Corolπ lary 3.62 (which shows that {T (kλ a)}ξ is in G(alg T Kξπ (L∞ N ×N )) for each ξ ∈ M (B)) in conjunction with Theorem 3.67 and Lemma 4.25.
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4.31. Remark. Theorem 3.8 oﬀers another possibility of proving the previous corollary. We have a = sb where s ∈ GL∞ N ×N is sectorial and b ∈ GBN ×N . There are c, d ∈ GCN ×N and an ε > 0 such that, for z ∈ CN and t ∈ T, Re c(kλ s)(t)dz, z = kλ [Re (csdz, z)] (t) ≥ (kλ [εz2 ])(t) = εz2 , which implies that kλ s ∈ GCN ×N is sectorial for all λ ∈ Λ and that {kλ s} is bafz on conv Λ. From Theorem 3.23 we deduce that (kλ b)(kλ b−1 ) − IL∞ → 0 as N ×N
λ → ∞,
and this shows that {kλ b} is bafz on conv Λ. Since, again by Theorem 3.23, kλ (sb) − (kλ s)(kλ b)L∞ → 0 as N ×N
λ → 0,
(4.20)
we see that {kλ a} is bafz on conv Λ. Because T (a) = T (s)T (b)+ compact operator, since T (s) is invertible (Corollary 4.2) and T (b) is Fredholm (a regularizer is T (b−1 )), it follows that 2 ) and that Ind T (a) = Ind T (b). Thus, it remains to prove that T (a) ∈ Φ(HN Ind T (b) = −ind {det(kλ a)}. Taking into account (4.20) it is easily seen that ind {det(kλ a)} = ind {det(kλ s)(kλ b)} = ind {det(kλ s)} + ind {det(kλ b)} = ind {det(kλ b)} (recall that kλ s is sectorial). Because, by Theorem 3.23, ind {det(kλ b)} = ind {kλ (det b)}, we are left with the equality Ind T (b) = −ind {kλ (det b)}. Theorem 1.14(c) shows that T (det b) ∈ Φ(H 2 ). Choose matrix functions → 0 as n → ∞. Then, obviously, bn ∈ (PA +H ∞ )N ×N such that b−bn L∞ N ×N det b − det bn L∞ → 0 as n → ∞, and we have Ind T (b) = Ind T (bn ),
Ind T (det b) = Ind T (det bn )
(4.21)
for all suﬃciently large n. Since the operator entries of T (bn ) commute modulo ﬁniterank operators (Proposition 2.14), we deduce from Theorem 1.15(a) that Ind T (bn ) = Ind T (det bn ). So (4.21) implies that Ind T (b) = Ind T (det b). Hence, it remains to show that Ind T (ϕ) = −ind {kλ ϕ} for every ϕ ∈ GB. If B = C, then T (kλ ϕ) converges uniformly to T (ϕ), so that the desired index equality is an immediate consequence of Theorem 2.42. Thus let B = QC. Let Ind T (ϕ) = m, then T (ϕχm ) is invertible (Corollary 2.40). Put ψ = ϕχm and notice that ψ ∈ GQC. Since T (ψ −1 ) is a regularizer of T (ψ), it follows that T (ψ −1 ) also has index zero and is therefore invertible. Consequently, T (ψ)T (ψ −1 ) = I − H(ψ)H(ψ)−1 )
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is also invertible. We have T (kλ ψ)T (kλ ψ −1 ) = I − H(kλ ψ)H(kλ ψ)−1 ), and since H(kλ ψ) and H(kλ ψ)−1 ) converge uniformly to H(ψ) and H(ψ)−1 ), respectively (note that ψ and ψ)−1 are in C +H ∞ ), we see that T (kλ ψ)T (kλ ψ −1 ) must be invertible for all λ large enough. Hence ind kλ ψ = −Ind T (kλ ψ) = 0 for all suﬃciently large λ (Theorem 2.42) and since, by Theorem 3.23, ind {kλ ψ} = ind {kλ ϕ} + ind {kλ χm } = ind {kλ ϕ} + m, we ﬁnally obtain that ind {kλ ϕ} = −m, as desired.
4.4 Transﬁnite Localization We now present the transﬁnite induction approach to maximal antisymmetric sets for C + H ∞ and shall give two applications of this approach: the ﬁrst consists in proving Axler’s theorem 2.83 for the matrix case and the second is the determination of the norm of a “local Hankel operator.” We begin by extending Theorems 2.11 and 2.54 to block Hankel operators on Hilbert spaces. The bulk of the work that is necessary to prove Nehari’s theorem for the matrix case is done by the following theorem. 4.32. Theorem (Parrott). Let H and K be Hilbert spaces with the orthogonal decompositions H = H1 ⊕ H2 and K = K1 ⊕ K2 , and let MX ∈ L(H, K) have the operator matrix K1 XC H1 → : H2 K2 AB with respect to these decompositions. Then "# " " !" " 0 0 " " 0C " " . " " " , inf MX = max " AB " " 0B " X∈L(H1 ,K1 ) Proof. For a proof we refer to Parrott [376] or Power [399]. 4.33. Theorem. If a ∈ L∞ N ×N , then ∞ (a, HN H(a)L(HN2 ) = distL∞ ×N ). N ×N
Proof. We ﬁrst show that there is a sequence a0 , a−1 a−2 , . . . of N × N matrices such that H(a) = Sn , where Sn ∈ L(2N (Z)) is deﬁned as the operator whose (block) matrix representation with respect to the decomposition 2N (Z) = 2N (Z− ) ⊕ 2N (Z+ ) is given by
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⎞ 0 0 0 ⎜ . . . a−n+1 a−n . . . a−2n+1 ⎟ ⎟ ⎜ ⎟ ⎜ .. .. .. 0 ⎟ ⎜ . . . ⎟ ⎜ ⎠ ⎝ . . . a0 a−1 . . . a−n a0 . . . a−n+1 0 H(a)J ⎛
(recall 2.13). This, on its hand, will follow once we have shown that the following inductive process is valid: deﬁne Rn ∈ L(2N (Z)) for n = 0, 1, 2, . . . by ⎞ ⎛ a−n+1 a−n+2 a−n+3 . . . ⎟ ⎜a ⎟ ⎜ −n+2 a−n+3 . . . Rn = ⎜ ⎟; ⎠ ⎝ a−n+3 . . . ... then, given Rn there is an a−n ∈ CN ×N such that Rn+1 = Rn . But this is an immediate consequence of Parrott’s theorem 4.32, because in the case at hand " " " " " 0 0 " " 0C " " "=" " " A B " " 0 B " = Rn . Thus, the existence of the sequence a0 , a−1 , a−2 , . . . with desired property is proved. It is easily seen that {Sn ϕ} is convergent for every ϕ ∈ 2N (Z) with ﬁnite support, and an ε/3 argument then gives the convergence of {Sn ϕ} for every ϕ ∈ 2N (Z). So 1.1(e) implies that the operator S deﬁned by Sϕ = lim Sn ϕ n→∞
is bounded on 2N (Z) and that S ≤ H(a). Now Proposition 2.2 can be applied to deduce that there is a b ∈ L∞ N ×N such that the nth matrix Fourier coeﬃcient of b equals an (n ∈ Z) and that S = M (b). Therefore, bL∞ := M (b) = S ≤ H(a) N ×N ∞ ∞ and since b − a ∈ HN ×N , it follows that dist(a, HN ×N ) ≤ H(a). As the reverse inequality is obvious, we are done.
4.34. Theorem. If a ∈ L∞ N ×N , then ∞ (a, CN ×N + HN H(a)Φ(HN2 ) = distL∞ ×N ). N ×N
Proof. Once the matrix analogue of Nehari’s theorem has been established, the proof is almost literally the one given for Theorem 2.54 (with p = 2 and cp = 1). Our next objective is to state some results on Toeplitz operators with unitaryvalued symbols, which almost immediately follow from the preceding two theorems.
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4.35. Lemma. Let A be a C ∗ algebra with identity element e. Suppose u ∈ GA satisﬁes u−1 = u∗ and p, q ∈ A satisfy p2 = p = p∗ , q 2 = q = q ∗ , p + q = e. Then pup + q is leftinvertible in A ⇐⇒ qup < 1; pup + q is rightinvertible in A ⇐⇒ puq < 1. Proof. By the GelfandNaimark theorem 1.26(b), we may assume that A is a C ∗ subalgebra of L(H), where H is some Hilbert space, that u is a unitary operator and p, q are orthogonal and complementary projections on H. From 1.26(d) we know that an operator belonging to a C ∗ subalgebra of L(H) is left (resp. right) invertible in that C ∗ algebra if and only if it is left (resp. right) invertible in L(H). The image Im p of p is equal to Ker q and is therefore a closed subspace of H. It is easy to see that pup + q is left invertible if and only if pupIm p is leftinvertible on Im p. For f ∈ Im p, we have f 2 = uf 2 = p(uf )2 + q(uf )2 = (pup)f 2 + (qup)f 2 .
(4.22)
But pupIm p is leftinvertible on Im p if and only if there is an ε > 0 such that (pup)f 2 ≥ εf 2 for all f ∈ Im p, and due to (4.22) this is valid if and only if qup < 1. The assertion on the rightinvertibility follows by taking adjoints. 4.36. Corollary. Let u ∈ GL∞ N ×N be unitaryvalued. Then the following are equivalent: 2 (i) T (u) is left (resp. right) invertible on HN ; ∞ ∞ (ii) dist(u, HN ×N ) < 1 (resp. dist(u, HN ×N ) < 1); ∞ (iii) u = sh (resp. u = hs), where s ∈ GL∞ N ×N is sectorial and h ∈ HN ×N ∞ (resp. h ∈ HN ×N ).
Moreover, one has 2 ∞ T (u) ∈ GL(HN ) ⇐⇒ dist(u, GHN ×N ) < 1 ∞ ∞ ⇐⇒ u = sh, where s ∈ GLN ×N is sectorial and h ∈ GHN ×N .
Proof. If T (u) is leftinvertible, then so is P M (u)P + Q, the preceding lemma ∞ gives QM (u)P L(L2N ) = H() u)L(HN2 ) < 1, and hence dist(u, HN ×N ) < 1 by Theorem 4.33. The implication (ii) =⇒ (iii) results from Lemma 3.6(d). Finally, if (iii) holds then h−1 ∈ L∞ N ×N and so formula (2.20) and Corollary 4.2 show that T (h−1 )T −1 (s) (resp. T −1 (s)T (h−1 )) is a left (resp. right) inverse of T (u). Now suppose T (u) is invertible. By what has just been proved, there is an ∞ h ∈ HN ×N such that u − h < 1. So, due to Proposition 4.1(b), I − T (u∗ h) = I − u∗ h = u − h < 1,
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which implies the invertibility of T (u∗ h) = T ∗ (u)T (h). Hence T (h) is in 2 ), consequently the equation T (h)f = hf = IN ×N has a solution GL(HN 2 ∞ f ∈ HN ×N and from Theorem 2.93 we deduce that f ∈ LN ×N . Thus, ∞ ∞ f ∈ HN ×N , i.e., h ∈ GHN ×N . ∞ ∞ If dist(u, GHN ×N ) < 1, then u = sh with s ∈ GLN ×N sectorial and ∞ h ∈ GHN ×N by virtue of Lemma 3.6(d). Finally, if u has such a representation, then T (u) is obviously invertible (formula (2.20) and Corollary 4.2). 4.37. Corollary. Let u ∈ GL∞ N ×N be unitaryvalued. Then 2 ∞ (a) T (u) ∈ Φ+ (HN ) ⇐⇒ dist(u, CN ×N + HN ×N ) < 1 ⇐⇒ u = sb with ∞ ∞ s ∈ GLN ×N sectorial and b ∈ CN ×N + HN ×N ; 2 ∞ (b) T (u) ∈ Φ− (HN ) ⇐⇒ dist(u, CN ×N + HN ×N ) < 1 ⇐⇒ u = bs with ∞ ∞ s ∈ GLN ×N sectorial and b ∈ CN ×N + HN ×N ; 2 ∞ (c) T (u) ∈ Φ(HN ) ⇐⇒ dist(u, G(CN ×N + HN ×N )) < 1 ⇐⇒ u = sb with ∞ ∞ s ∈ GLN ×N sectorial and b ∈ G(CN ×N + HN ×N ). 2 ). Then P M (u)P +Q+C∞ (L2N ) is leftinvertible Proof. (a) Let T (u) ∈ Φ+ (HN 2 2 in L(LN )/C∞ (LN ) (recall Remark 2.70), Lemma 4.35 gives the inequality
u)Φ(HN2 ) < 1, QuP + C∞ (L2N )Φ(L2N ) = H() ∞ and Theorem 4.34 implies that dist(u, CN ×N + HN ×N ) < 1. If this distance estimate holds, then, by virtue of Lemma 3.6(d), u = sb with s, b as desired. Finally, if u has such a representation, then T (b−1 )T −1 (s) is a left regularizer of T (u).
(b) Analogous (or take adjoints). 2 (c) Suppose T (u) ∈ Φ(HN ). We know from (a) that there is a matrix ∞ function b ∈ CN ×N + HN ×N with u − b < 1. In view of 4.10,
= u − bL∞ < 1, I − T (u∗ b)Φ(HN2 ) = I − u∗ bL∞ N ×N N ×N 2 so T ∗ (u)T (b) = T (u∗ b) + compact operator is in Φ(HN ), and hence T (b) is in 2 ∞ Φ(HN ). Theorem 2.94 now gives the invertibility of b in CN ×N + HN ×N , and the desired distance estimate follows. That the distance estimate yields the required factorization is a consequence of Lemma 3.6(d), and if u possesses that factorization u = sb, then T (b−1 )T −1 (s) is a regularizer of T (u).
4.38. Remark. Pousson [398] and Rabindranathan [409] showed that ev∞ ery a ∈ GL∞ N ×N can be factored in the form a = uh with h ∈ GHN ×N and a unitaryvalued function u ∈ GL∞ . Using this result one can see N ×N 2 if and only that a Toeplitz operator is Fredholm (resp. invertible) on HN if its symbol is of the form sb where s ∈ GL∞ N ×N is sectorial and b is in ∞ ∞ ) (resp. in GH ). Indeed, if a = sb with s sectorial and b G(CN ×N + HN ×N N ×N ∞ ), then T (a) = T (s)T (b)+ compact operator, which gives in G(CN ×N + HN ×N
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the Fredholmness of T (a). On the other hand, if T (a) = T (uh) = T (u)T (h) is Fredholm, then T (u) must also be Fredholm, and from Corollary 4.37(c) ∞ we deduce that u = sb where s is sectorial and b ∈ G(CN ×N + HN ×N ). It is ∞ clear that bh is also in G(CN ×N + HN ×N ), so that a = s(bh) is the desired factorization. The argument is analogous for the case of invertibility. 4.39. Corollary. Let u ∈ GL∞ N ×N be unitaryvalued. (a) If for each maximal antisymmetric set S for C + H ∞ there exists a 2 unitaryvalued uS ∈ GL∞ N ×N such that uS = uS S and T (uS ) ∈ Φ+ (HN ) 2 2 2 (resp. Φ− (HN )), then T (u) ∈ Φ+ (HN ) (resp. Φ− (HN )). 2 (b) If u is locally sectorial over C + H ∞ , then T (u) ∈ Φ(HN ). 2 Proof. (a) Let T (uS ) ∈ Φ+ (HN ) for each S in question. Then Corollary 4.38(a) gives the estimate ∞ (4.23) distS (u, CN ×N + HN ×N ) < 1, 2 ). and Theorem 3.5 with Corollary 4.37(a) implies that T (u) ∈ Φ+ (HN
(b) In this case (4.23) results from Lemma 3.6(b), and since the maximal antisymmetric sets for C +H ∞ are the same as those for C +H ∞ , the assertion follows from Theorem 3.5 and Corollary 4.37(a), (b). We now turn to Axler’s method of transﬁnite localization. In particular, using this method we shall remove the twice occurring “unitaryvalued” in the previous corollary. 4.40. Transﬁnite induction. Let W be a set and let < be a relation on W . The set W is said to be ordered by the relation < if this relation is irreﬂexive (i.e., there is no u ∈ W such that u < u), connected (i.e., for any u, v ∈ W either u = v, or u < v, or v < u holds), and transitive (i.e., u, v, w ∈ W , u < v, v < w always implies that u < w). A nonempty subset U of W is said to have a ﬁrst element if there is a u ∈ U such that u < v for all v ∈ U with u = v. A set W is said to be well ordered by a relation < if W is ordered by the relation < and each nonempty subset of W has a ﬁrst element. It follows from the axiom of choice that every set can be well ordered. Let W be a wellordered set. As usual, the ﬁrst element of W will be denoted by 1 and w + 1 will denote the ﬁrst element of the set {v ∈ W : w < v}. We say that w ∈ W has a predecessor if there is a v ∈ W such that w = v + 1. The principle of transﬁnite induction consists in the following: to show that a statement (*) holds for all elements of a wellordered set W it suﬃces to show that (i) (*) holds for w = 1, (ii) if w ∈ W and (*) holds for all v < w, then (*) holds for w.
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We shall usually break (ii) into two cases: (a) w has a predecessor and (b) w does not have a predecessor. 4.41. Transﬁnite decomposition of M (L∞ ). Let W be a set whose carM (L∞ ) and let < be a relation on W which makes W into a dinality is 22 wellordered set. We use transﬁnite induction to deﬁne for each w ∈ W a partition ∆w of X = M (L∞ ). Thus if w ∈ W , then ∆w will be a collection of disjoint subsets of X whose union is X. The partitions ∆w are deﬁned as follows. (i) ∆1 is the partition of X whose only element is X. (ii) Suppose that w has a predecessor v and that ∆v has been deﬁned. Deﬁne an equivalence relation on X by saying that x is equivalent to y if (a) there exists an S ∈ ∆v such that x ∈ S and y ∈ S and (b) f (x) = f (y) for every f ∈ C + H ∞ such that f S is realvalued. The elements of ∆w are now deﬁned to be the equivalence classes of X under this equivalence relation. (iii) Suppose that w has no predecessor and that ∆v is deﬁned for all v < w. Deﬁne an equivalence relation on X by saying that x is equivalent to y if for each v < w there exists an Sv ∈ ∆v such that x ∈ Sv and y ∈ Sv , and then deﬁne the elements of ∆w as the equivalence classes of X under this equivalence relation. It is clear that if v < w then ∆w is a reﬁnement of ∆v in the sense that for each Sw ∈ ∆w there exists an Sv ∈ ∆v such that Sw ⊂ Sv . Also note that each S ∈ ∆w is a closed subset of X. The following proposition identiﬁes ∆2 := ∆1+1 and shows that the above construction terminates with the partition of X into maximal antisymmetric sets for C + H ∞ . 4.42. Proposition. (a) ∆2 is the partition of X into ﬁbers over M (QC). (b) There exists a w ∈ W such that each S ∈ ∆w is a maximal antisymmetric set for C + H ∞ . Proof. (a) x is equivalent to y ⇐⇒ f (x) = f (y) ∀ f ∈ C + H ∞ realvalued ⇐⇒ f (x) = f (y) ∀ f ∈ QC realvalued ⇐⇒ f (x) = f (y) ∀ f ∈ QC. (b) For x ∈ X, let Sx be the maximal antisymmetric set for C +H ∞ which contains x. Using transﬁnite induction we ﬁrst show that for each w ∈ W there is an S ∈ ∆w such that Sx ⊂ S. (i) This is obvious for w = 1. (ii) Let w = v + 1 and suppose Sx ⊂ Sv ∈ ∆v . If y is in Sx , then y is equivalent to x: indeed, if f ∈ C + H ∞ and f Sv is realvalued, then f Sx is realvalued, so f Sx must be constant, whence f (y) = f (x). This implies that there is an S ∈ ∆w which contains all y ∈ Sx .
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(iii) Suppose w has no predecessor and that for each v < w there is an Sv ∈ ∆v such that Sx ⊂ Sv . If y ∈ Sx , then y ∈ Sv for all v < w, so y is equivalent to x, and hence there exists an S ∈ ∆w such that y ∈ S and x ∈ S, which gives the inclusion Sx ⊂ S. Thus if w ∈ W and S ∈ ∆w and S is an antisymmetric set for C + H ∞ , then S is a maximal antisymmetric set for C + H ∞ . If w ∈ W is such that ∆w = ∆w+1 , then the deﬁnition of ∆w+1 implies that each S ∈ ∆w is an antisymmetric set for C + H ∞ . Consequently, to prove the proposition it suﬃces to show that there is a w ∈ W such that ∆w = ∆w+1 . Assume this is false. Then for each w ∈ W there is a set Sw ∈ ∆w+1 \ ∆w . ∞ Clearly, if w = v, then Sw = Sv . Hence the mapping W → 2M (L ) , w → Sw is onetoone. However, the cardinality of W is too large for there to exist ∞ any injective mappings from W into 2M (L ) . This contradiction completes the proof. We now state some lemmas in order to prepare the proof of Theorem 4.48. Recall the terminology introduced in 4.10 and 4.11. For A ∈ alg T (L∞ N ×N ) and S a closed subset of X, let Aπ and AπS denote the cosets in alg T π (L∞ N ×N ) ), respectively, which contain A. and alg TSπ (L∞ N ×N 4.43. Lemma. Suppose w ∈ W has no predecessor and let {Sv }v<w be a Sv . collection of sets Sv ∈ ∆v such that Su ⊂ Sv for v < u. Put Sw := v<w π π Then Sw ∈ ∆w and JSw = clos JSv . v<w
Proof.That Sw belongs to ∆w is easily veriﬁed. It is clear that both JSπw and JSπv are closed twosided ideals of alg T π (L∞ ). It is also clear that clos v<w π JSπw ⊂ clos JSv . To see the opposite inclusion let N = 1 (which in view of v<w
Lemma 3.46 is no loss of generality) and let a ∈ L∞ be such that aSw = 0. Let ε > 0 and deﬁne U = {x ∈ X : a(x) < ε}. Since U is an open set and Sw ⊂ U is the intersection of the compact sets Sv (v < w), which satisfy Su ⊂ Sv for v < u, there exists a v < w such that Sv ⊂ U . Choose b ∈ L∞ so that bSv = 0, π π b(X \ U ) = 1, 0 ≤ b ≤ 1. Then a(1 − b) < ε and thusT (a) − T (ab) < ε. But T π (ab) ∈ JSπv , and so dist T π (a), clos JSπv < ε. It follows that v<w π π JSv and thus JSπw ⊂ clos JSv . T π (a) ∈ clos v<w
v<w
4.44. Lemma. If w ∈ W and S ∈ ∆w , then S is a weak peak set for C + H ∞ . Proof. We prove this by transﬁnite induction. (i) The case w = 1 is trivial. (ii) Suppose that w ∈ W does not have a predecessor and that the lemma holds for all v < w. Let S ∈ ∆w . Then for each v < w there exists an Sv ∈ ∆v
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such that S ⊂ Sv . The deﬁnition of ∆w shows that S =
Sv . Since each v<w
Sv is a weak peak set, S is also a weak peak set. (iii) Now suppose that w has a predecessor v and that the lemma holds for v. Let S ∈ ∆w and let Sv ∈ ∆v be such that S ⊂ Sv . Thus, Sv is a weak peak set for C + H ∞ and therefore (C + H ∞ )Sv is a closed algebra. By the deﬁnition of ∆w , there is a λ ∈ R such that $ x ∈ Sv : f (x) = λ , S= f
the intersection over all f ∈ C + H ∞ whose restriction to Sv is realvalued. But x ∈ Sv : f (x) = λ = x ∈ Sv : 1 − ε(f (x) − λ)2 = 1 (ε > 0 suﬃciently small) is a peak set for (C + H ∞ )Sv , and so S must be a weak peak set for (C + H ∞ )Sv . A result from the theory of function algebras (see, e.g., Gamelin [209, Chap. II, Corollary 12.9]) now implies that S is a weak peak set for C + H ∞ . 4.45. Deﬁnition. Let v ∈ W and Sv ∈ ∆v . Then, by 1.28 and the preceding lemma, both (C + H ∞ )Sv and (C + H ∞ )Sv are closed subalgebras of L∞ Sv . Let QSv denote the C ∗ subalgebra of L∞ Sv deﬁned by QSv := (C + H ∞ )Sv ∩ (C + H ∞ )Sv . Note that if v > 1, then QSv is not equal to (C + H ∞ ) ∩ (C + H ∞ ) Sv = QCSv ∼ = C. Also notice that S1 = X and thus QS1 = (C + H ∞ )X ∩ (C + H ∞ )X is nothing else than QC. 4.46. Lemma. Let v ∈ W and Sv ∈ ∆v . Then DSv := TSπv (ϕ) : ϕ ∈ L∞ , ϕSv ∈ QSv is a C ∗ subalgebra of the center of alg TSπv (L∞ ). Proof. Since QSv is a closed subalgebra of L∞ Sv , it follows from Theorem 3.61 (with A = L∞ , B = L(H 2 )/C∞ (H 2 ), i = T π , F = Sv , N = 1) that DSv is a closed subspace of alg TSπv (L∞ ). To see that DSv is contained in the center of alg TSπv (L∞ ), let ϕ ∈ L∞ be such that ϕSv ∈ QSv and let a ∈ L∞ . Let ϕSv = h1 Sv = h2 Sv , where h1 and h2 are functions in C + H ∞ . Then TSπv (ϕ)TSπv (a) − TSπv (a)TSπv (ϕ) = TSπv (h2 )TSπv (a) − TSπv (a)TSπv (h1 ) = TSπv (h2 a) − TSπv (ah1 ) = TSπv (ϕa) − TSπv (aϕ) = 0, and it follows that DSv ⊂ Cen (alg TSπv (L∞ )). If ϕSv ∈ QSv and ψSv ∈ QSv , then TSπv (ϕ)TSπv (ψ) = TSπv (ϕψ) and thus DSv is an algebra.
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4.47. Lemma. Suppose w ∈ W has a predecessor v and let Sv ∈ ∆v . (a) The maximal ideal space M (QSv ) of QSv can be identiﬁed with the set {S ∈ ∆w : S ⊂ Sv } and the Gelfand map is given by Γ : QSv → C({S ∈ ∆w : S ⊂ Sv }),
f → f S.
(b) For S ∈ ∆w and S ⊂ Sv , deﬁne ES := closidalg TSπ (L∞ ) TSπv (ϕ) : ϕ ∈ L∞ , ϕSv ∈ QSv , ϕS = 0 , v ES := closidalg TSπ (L∞ ) TSπv (a) : a ∈ L∞ , aS = 0 . v
Then ES = ES . (c) If Sw ∈ ∆w and Sw ⊂ Sv , then ESw is isometrically isomorphic to JSπw /JSπv and alg TSπv (L∞ )/ESw is isometrically isomorphic to alg TSπw (L∞ ). Remark. In the proof of Theorem 4.48, when we shall be applying the local principle 1.35, the ideal ES will appear and there will be a point at which we must show that alg TSπv (L∞ )/ESw ∼ = alg TSπw (L∞ ). However, the latter isomorphism is not obvious. What is “obvious” is the isomorphism alg TSπv (L∞ )/ES w ∼ = alg TSπw (L∞ ) (see the proof of part (c)). This is the justiﬁcation of part (b) of the present lemma. Proof. (a) This follows from 1.27(b). (b) What we must prove is that ES ⊂ ES . Let a ∈ L∞ and suppose aS = 0. Let ε > 0 be given arbitrarily. For σ ∈ M (QSv ), let Mσ (L∞ Sv ) denote the ﬁber of L∞ Sv over σ and put U = σ ∈ M (QSv ) : a(x) < ε ∀ x ∈ Mσ (L∞ Sv ) . Assume that U is not an open subset of M (QSv ). Then there is a σ ∈ U and a net σi in M (QSv ) such that σi → σ and such that for each i there exists an xi ∈ Mσi (L∞ Sv ) with a(xi ) ≥ ε. Taking a subnet, we can assume that there is an x ∈ M (L∞ Sv ) such that xi → x and it follows that x is even in Mσ (L∞ Sv ) (recall that the mapping τ : M (L∞ Sv ) → M (QSv ) which sends a functional to its restriction functional is continuous). However, a(x) ≥ ε, which contradicts our assumption that σ ∈ U . The conclusion is that U is an open subset of M (QSv ). Since S ∈ ∆w and S ⊂ Sv , there is exactly one σ ∈ M (QSv ) such that S ⊂ Mσ (L∞ Sv ) (recall part (a)). Because aS = 0, it is clear that σ ∈ U . Thus, there is an f ∈ QSv such that f (σ) = 0, f (M (QSv )\U ) = 1, 0 ≤ f ≤ 1. Let h be a function in C + H ∞ for which hSv = f . Note that the choice of f implies that a(1 − h)Sv < 1. Thus, ε > TSπv (a) − TSπv (ah) = TSπv (a) − TSπv (a)TSπv (h).
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4 Toeplitz Operators on H 2
But TSπv (h) ∈ ES , so TSπv (a)TSπv (h) ∈ ES , whence dist(TSπv (a), ES ) ≤ TSπv (a) − TSπv (a)TSπv (h) < ε. Letting ε go to zero, we conclude that TSπv (a) ∈ ES . (c) We have JSπv = closidalg T π (L∞ ) T π (a) : a ∈ L∞ , aSv = 0 , JSπw = closidalg T π (L∞ ) T π (a) : a ∈ L∞ , aSw = 0 , ES v = closidalg TSπ (L∞ ) TSπv (a) : a ∈ L∞ , aSw = 0 . v
A little thought therefore shows that JSπw /JSπv ∼ = ES w and so part (b) gives π π ∼ that JSw /JSv = ESw (note that all algebras occurring are C ∗ algebras and take into account 1.26(e)). The second assertion now results as follows: alg TSπv (L∞ )/ESw ∼ = (alg T π (L∞ )/JSπv )/(JSπw /JSπv ) ∼ alg T π (L∞ )/J π = alg T π (L∞ ). = Sw Sw π 4.48. Theorem (Axler). Let A ∈ alg T (L∞ N ×N ) and suppose A is not left π ∞ (right, resp. twosided) invertible in alg T (LN ×N ). Then there exists a collection {Sw }w∈W of subsets of X such that
(a) Sw ∈ ∆w for each w ∈ W ; (b) Sw ⊂ Sv if v < w; (c) if w ∈ W , then AπSw is not left (right, resp. twosided) invertible in alg TSπw (L∞ N ×N ). Proof. For the sake of deﬁniteness, let us prove the theorem for the case of leftinvertibility. The collection {Sw }w∈W of subsets of X that satisﬁes conditions (a)–(c) will be deﬁned by transﬁnite induction. (i) For w = 1, let S1 = X. Then JSπ1 = {0} and so conditions (a)–(c) are obviously satisﬁed. (ii) Suppose that w has no predecessor, that Sv has been deﬁned for v < w, Sv . It is and that conditions (a)–(c) are satisﬁed for v < w. Put Sw := v<w
obvious that (b) holds for w and (we mentioned this in Lemma 4.43) it can be easily veriﬁed that (a) is also true for w. Now assume (c) does not hold. Thus, AπSw is leftinvertible in the algebra ∞ π π π π alg TSπw (L∞ N ×N ). So there is a B ∈ alg T (LN ×N ) for which B A − I ∈ JSw . ∞ Lemma 4.43 shows that, for D ∈ alg T (LN ×N ),
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/ DSπw := dist(Dπ , JSπw ) = dist Dπ , clos JSπv , v<w
such that BSπv AπSv − alg TSπv (L∞ N ×N ), which
ISπv < 1. Conseand hence there is a v < w π π implies that AπSv is quently, BSv ASv is invertible in leftinvertible in that algebra. This, however, contradicts the induction hypothesis. Thus, we have proved that (c) holds for w. (iii) Now suppose w ∈ W has a predecessor v, that Sv has been deﬁned and satisﬁes (a)–(c). Thus, AπSv is not leftinvertible in alg TSπv (L∞ N ×N ). We apply Theorem 1.35(a) (in the setting A = alg TSπv (L∞ N ×N ), B = DSv := {TSπv (ϕIN ×N ) : ϕ ∈ L∞ , ϕSv ∈ QSv }, a = AπSv ). Lemma 4.46 tells us that DSv is a C ∗ subalgebra of Cen (alg TSπv (L∞ N ×N )). By Theorem 3.61 (in the setting A = L∞ , B = L(H 2 )/C∞ (H 2 ), i = T π , F = Sv , N = 1), the mapping QSv → DSv , ϕSv → TSπv (ϕIN ×N ) is an isometric isomorphism. It follows that M (DSv ) can be identiﬁed with M (QSv ) = {S ∈ ∆w : S ⊂ Sv } (Lemma 4.47(a)). So the ideal of the algebra alg TSπv (L∞ N ×N ) generated by Sw ∈ M (QSv ) (i.e., the JSw in the terminology of 1.33) coincides with closid TSπv (ϕIN ×N ) ∈ DSv : ϕSw = 0 = closid TSπv (ϕIN ×N ) : ϕ ∈ L∞ , ϕSv ∈ QSv , ϕSw = 0 , that is, with (ESw )N ×N , where ESw is deﬁned as in Lemma 4.47(b). Thus, what results is that there exists an Sw ∈ ∆w (property (a)) such that Sw ⊂ Sv (property (b)) and AπSv + (ESw )N ×N is not leftinvertible in alg TSπv (L∞ N ×N )/(ESw )N ×N . Lemma 4.47(c) in conjunction with Lemmas 3.46 and 3.47 ﬁnally implies that AπSw is not leftinvertible in alg TSπw (L∞ N ×N ) (property (c)). 4.49. Corollary. (a) Let A ∈ alg T (L∞ N ×N ) and w ∈ W . Then A is left (right, 2 if and only if AπS is left (right, resp. tworesp. twosided) Fredholm on HN sided) invertible in alg TSπ (L∞ N ×N ) for all S ∈ ∆w . ∞ (b) Let a ∈ L∞ N ×N . If for each maximal antisymmetric set S for C + H ∞ 2 there exists an aS ∈ LN ×N such that aS = aS S and T (aS ) ∈ Φ+ (HN ) (resp. 2 2 2 )), then T (a) ∈ Φ+ (HN ) (resp. Φ− (HN )). Φ− (HN ∞ 2 then T (a) ∈ Φ(HN ). (c) If a ∈ L∞ N ×N is locally sectorial over C + H
Proof. (a) The “if” part is immediate from the preceding theorem, the “only if” portion results from that theorem in conjunction with 1.26(d). (b) Note that TSπ (a) = TSπ (aS ), take into account Proposition 4.42(b), and apply part (a). (c) Corollary 3.62 gives the invertibility of TSπ (a) in alg TSπ (L∞ N ×N ) for each maximal antisymmetric set S for C + H ∞ , so that Proposition 4.42(b) and part (a) apply once more.
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4 Toeplitz Operators on H 2
Remark. Combining 4.49(c) and Remark 4.38 we see that every matrix function which is locally sectorial over C +H ∞ can be written in the form sb where ∞ s ∈ GL∞ N ×N is sectorial and b is in G(CN ×N + HN ×N ). Of course, it would be desirable to have a more direct proof of this result (see the proof of 2.86(a)). 4.50. Compactness of quasicommutators. (a) Let a, b ∈ L∞ and let B be a C ∗ subalgebra of QC containing the constants. Suppose for each ξ ∈ M (B) either aXξ ∈ H ∞ Xξ or bXξ ∈ H ∞ Xξ . Then T (ab) − T (a)T (b) = H(a)H()b) ∈ C∞ (H 2 ).
(4.24)
Indeed, if aXξ = hXξ where h ∈ H ∞ , then Tξπ (ab) − Tξπ (a)Tξπ (b) = Tξπ (hb) − Tξπ (h)Tξπ (b) = 0, the situation is analogous for bXξ ∈ H ∞ Xξ , so $ T π (ab) − T π (a)T π (b) ∈
Jξπ ,
ξ∈M (B)
and Theorem 1.35(c) gives the assertion (note that alg T π (L∞ ) as a C ∗ algebra is semisimple). (b) Axler [12] also established the following theorem. Let A ∈ alg T (L∞ ). Then there exists a collection {Sw }w∈W of subsets of X such that (a) Sw ∈ ∆w for each w ∈ W , (b) Sw ⊂ Sv if v < w, (c) Aπ = AπSw for each w ∈ W . The proof of this theorem is similar in spirit to the proof of Theorem 4.48. We therefore only indicate how the collection {Sw }w∈W is deﬁned by transﬁnite induction. For w = 1, let S1 = X. If w has no predecessor and Sv has Sv . If w has a predecessor v and Sv been deﬁned for v < w, then Sw := v<w
has been deﬁned, then, by Theorem 1.35(d), AπSv = max dist(AπSv , ES ) : S ∈ ∆w , S ⊂ Sv and Sw is deﬁned by AπSv = dist(AπSv , ESw ). An immediate consequence of the above theorem is that Aπ = max AπS ∀ w ∈ W S∈∆w
∀ A ∈ alg T (L∞ ).
(4.25)
(c) The result in (a) can now be reﬁned as follows. Let a, b ∈ L∞ and suppose for each maximal antisymmetric set S for C + H ∞ either aS ∈ H ∞ S or bS ∈ H ∞ S. Then (4.24) holds. To see this, apply the argument of part (a) to show that TSπ (ab) − TSπ (a)TSπ (b) = 0 for each S in question and then apply Proposition 4.42(b) and (4.25).
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(d) (The AxlerChangSarasonVolberg theorem). Let a, b ∈ L∞ . Then (4.24) holds if and only if alg (H ∞ , a) ∩ alg (H ∞ , b) ⊂ C + H ∞ . This is the ﬁnal solution of the compactness problem for the quasicommutators T (ab) − T (a)T (b). Here alg (H ∞ , f ) denotes the smallest closed subalgebra of L∞ containing H ∞ and f . In connection with this criterion notice the following well known fact (see, e.g., Douglas [162]): If A is a closed subalgebra of L∞ containing H ∞ , then either A = H ∞ or C + H ∞ ⊂ A. (e) Open problems. Establish an AxlerChangSarasonVolberg theorem for harmonic approximation or stable convergence, i.e., ﬁnd necessary and suﬃcient conditions for
or
{kλ (ab) − (kλ a)(kλ b)}π ∈ N
(4.26)
π T (kλ (ab)) − T (kλ a)T (kλ b) ∈ J
(4.27)
to hold. In the original edition of this book we wrote that a reasonable conjecture would be that (4.26) is true (say, for the approximate identity generated by the Poisson kernel) if and only if alg (H ∞ , a) ∩ alg (H ∞ , b) ⊂ C + H ∞ and
alg (H ∞ , a) ∩ alg (H ∞ , b) ⊂ C + H ∞ .
In 1998, Gorkin and Zheng [240] proved that the “if” part of this conjecture is true for the Poisson kernel and gave counterexamples showing that the “only if” portion is not true for the Poisson kernel. One of their counterexamples is a = (B + B)/2, b = (B − B)/2, where B is an inﬁnite Blaschke product. Theorem 3.23 and Lemma 4.25 in conjunction with Theorem 1.35(c) imply that (4.26) and (4.27) are valid if either aXξ ∈ CXξ or bXξ ∈ CXξ for each ξ ∈ M (QC). Trying to reﬁne this result to maximal antisymmetric sets for C + H ∞ immediately leads to the following problem. Can transﬁnite localization be applied to harmonic approximation or staπ π (T HJ ) in place of T π ? We suspect ble convergence, i.e., to Kπ (Hπ ) or T KJ that in these cases localization with respect to ﬁbers over QC (Corollary 3.69 and Theorem 4.26) is the ﬁnal stage. In order to support this, we remark that we do not know any good analogue of formula (2.20) which forms the basis for the “deleting of H ∞ symbols.” See also the end of the Section 4.77. Find a criterion for H(a)H()b) to be a trace class operator on H 2 (or, more generally, to be in Cp (H 2 )). This problem is of interest in connection with the theory of Toeplitz determinants (see 10.12, 10.27, 10.61, 10.63). We now use the transﬁnite induction approach to determine the norm of local Hankel operators, since the knowledge of this norm will enable us to
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4 Toeplitz Operators on H 2
employ Lemma 4.35 to establish criteria for the invertibility of local Toeplitz operators. However, it is necessary to modify some of the above arguments. The reason is that not every bounded Hankel operator belongs to alg T (L∞ ). To show this is the content of the following proposition. 4.51. Proposition. Let f ∈ C be any function such that f (−1) = f (1) = 0. If a ∈ L∞ and H(a) ∈ alg T (L∞ ), then af ∈ C + H ∞ . In particular, if a ∈ L∞ has a jump discontinuity at some point in T \ {±1}, then H(a) ∈ / alg T (L∞ ). Proof. Suppose H(a) ∈ alg T (L∞ ). If ϕ ∈ C, then T (ϕ) commutes modulo compact operators with every operator in alg T (L∞ ) and hence T π (ϕ) belongs to the center of alg T π (L∞ ). Thus, T (ϕ)H(a) − H(a)T (ϕ) ∈ C∞ (H 2 )
∀ ϕ ∈ C.
But T (ϕ)H(a) − H(a)T (ϕ) equals H(ϕa) − H(ϕ)T () a) − H(aϕ) ) + T (a)H(ϕ) ) (this can be veriﬁed by using the P ’s and Q’s as in the proof of Proposition 2.14), and since H(ϕ) and H(ϕ) ) are compact for ϕ ∈ C, it follows that H(a(ϕ − ϕ)) ) must be compact for every ϕ ∈ C. So Theorem 2.54 shows that a(ϕ − ϕ) ) ∈ C + H ∞ and thus af (ϕ − ϕ) ) ∈ C + H ∞ for every ϕ ∈ C. Let S be any maximal antisymmetric set for C + H ∞ . Because C ⊂ C + H ∞ , there is a τ ∈ T such that S ⊂ Xτ . If τ = ±1, then there exists a ϕ ∈ C such that ϕ(τ ) = ϕ(τ ) ) and we conclude that af S ∈ C + H ∞ S. If τ = ±1, then obviously af S = 0S ∈ C + H ∞ S. Thus, by Corollary 1.23, af ∈ C + H ∞ . It remains to notice that functions in H ∞ and thus in C + H ∞ cannot have jumps. 4.52. The algebra generated by singular integral operators. (a) Let A ∞ denote the direct product L∞ N ×N × LN ×N . On deﬁning α (a, b) := (α a, α b), (a, b) + (c, d) := (a + c, b + d), (a, b)(c, d) = (ac, bd), (a, b)∗ := (a∗ , b∗ ), and , (a, b)A := max aL∞ , bL∞ N ×N N ×N we make A become a C ∗ algebra. (b) It will be convenient to denote the multiplication operator on L2N generated by f ∈ L∞ N ×N simply by f . The mapping σ given by σ : A → L(L2N ),
(a, b) → aP + bQ
(recall that P := diag (P, . . . , P ), Q := diag (Q, . . . , Q)) is a submultiplicative embedding. To see this note ﬁrst that " " " " " " " " (ajk P + bjk Q)" = "U −n (ajk P + bjk Q) U n " " j
j
k
k
" " " " " " " " =" (ajk U −n P U n + bjk U −n QU n )" ≥ " ajk ", j
k
j
k
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205
because U −n P U n → I and U −n QU n → 0 strongly. It can be shown analogously that " " " " " " " " (ajk P + bjk Q)" ≥ " bjk ". " j
j
k
k
Since, in particular, aP + bQ ≥ max{a, b}, it follows that Ker σ = {0} and that Im σ is closed. Finally, we have " " " " " " " " (ajk , bjk ) " = "σ ajk , bjk " "σ j
j
k
j
k
k
" " " " " " " " " " " " =" ajk P + bjk Q" ≤ " ajk " + " bjk " j
k
j
k
j
j
k
k
" " " " " " " " ≤ 2" (ajk P + bjk Q)" = 2" σ(ajk , bjk )", j
j
k
k
which shows that σ is 2submultiplicative. (c) The algebra alg σ(A) contains QaP (= (0 · P + 1 · Q)(a · P + 0 · Q)) and P aQ for every a ∈ L∞ N ×N . Note that QaP and P aQ can be identiﬁed with H() a) and H(a), respectively (see 2.10, 2.15, and also 4.36). (d) The collection C∞ (L2N ) of all compact operators on L2N is a subset of the quasicommutator ideal Qσ (A). Indeed, by Lemma 3.47 it suﬃces to consider the case N = 1, the operators (0 · P + 1 · Q) − (0 · P + χ−n Q)(0 · P + χn Q) = χ−n P χn Q, (1 · P + 0 · Q) − (χn P + 0 · Q)(χ−n P + 0 · Q) = χn Qχ−n P
−1
n−1 2 take k∈Z fk χk ∈ L into k=−n fk χk and k=0 fk χk , respectively, and therefore the operator f → (f, χk )χm belongs to Qσ (A) for all k, m ∈ Z. The rest is as in the proof of Proposition 4.5. (e) By (d) and Theorem 3.52, the mapping σ π : A → alg σ π (A) := alg σ(A)/C∞ (L2N ), (a, b) → (aP + bQ)π := aP + bQ + C∞ (L2N ) is a submultiplicative quasiembedding whose kernel is {(a, b) ∈ A : aP + bQ ∈ C∞ (L2N )}. But if aP + bQ ∈ C∞ (L2N ), then P aP = P (aP + bQ)P and QbQ = Q(aP + bQ)Q are in C∞ (L2N ), which implies that a = b = 0 (Lemma 4.4(d)). Thus, σ π is actually an embedding. (f) For F a closed subset of X = M (L∞ ), deﬁne RπF = closidalg σπ (A) (aP + bQ)π : a, b ∈ L∞ N ×N , aF = bF = 0 .
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Theorem 3.52 and Lemma 3.59 give that the mapping σFπ : A → alg σFπ (A) := alg σ π (A)/RπF , (a, b) → (aP + bQ)πF := (aP + bQ)π + RπF is a submultiplicative quasiembedding. Put JFπ = closidalg σπ (A) (P aP )π : a ∈ L∞ N ×N , aF = 0 (note that P aP ∈ alg σ(A)). (g) The equality P π RπF P π = JFπ holds. The inclusion “⊃” is obvious. We prove the reverse inclusion. By Lemma 3.46, it is enough to consider the case N = 1. Let n n A= (ak P + bk Q), B = (ck P + dk Q), k=1
k=1
∞
∞
where ak , bk , ck , dk ∈ L , and let f, g ∈ L be such that f F = gF = 0. We must show that (P A(f P + gQ)BP )π is in JFπ . Put A1 =
n−1
(ak P + bk Q),
k=1
B1 =
n
(ck P + dk Q).
k=2
We have (P A(f P + gQ)BP )π = (P A(P f P + Qf P + P gQ + QgQ)BP )π ,
(4.28)
and (P AP f P BP )π is clearly in JFπ . Let us write C π ≡ Dπ in case C π − Dπ is in JFπ . Then (P AQf P BP )π = = = π (P AP gQBP ) ≡ ≡
(P A1 (an P + bn Q)Qf P BP )π (P A1 bn Qf P BP )π ≡ (P A1 bn f P BP )π (P A1 Qbn f P BP )π , (P AP gBP )π = (P AP (gc1 P + gd1 Q)B1 P )π (P AP gd1 QB1 )π ,
and it follows by induction with respect to n that the second and third terms in (4.28) always belong to JFπ . Finally, using this we get (P AQgQBP )π = (P AQgBP )π − (P AQgP BP )π ≡ (P AQgBP )π = (P AQgc1 P B1 P )π + (P AQgd1 QB1 P )π ≡ (P AQgd1 QB1 P )π , and so again by induction with respect to n we conclude that the fourth term in (4.28) also belongs to JFπ .
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π ∞ (h) Deﬁne alg τ π (L∞ N ×N ) := algalg σ π (A) {(P aP ) : a ∈ L }. Then the π π π π ∞ equality P alg σ (A)P = alg τ (LN ×N ) holds. The inclusion “⊃” is again trivial. To show the opposite inclusion, assume N = 1 (Lemma 3.47) and let B and B1 be as in (g). Because P BP = P c1 P B1 P + P d1 QB1 P , it suﬃces to show that (P f P BP )π and (P f QBP )π belong to alg τ π (L∞ N ×N ) for every . This is readily veriﬁed for n = 1 and since f ∈ L∞ N ×N
(P f P BP )π = (P f P c1 P B1 P )π + (P f P d1 QB1 P )π , (P f QBP )π = (P f Qc1 P B1 P )π + (P f Qd1 QB1 P )π = (P f c1 P B1 P )π − (P f P c1 P B1 P )π +(P f d1 QB1 P )π − (P f P d1 QB1 P )π , the assertion for general n follows by induction with respect to n. π (k) Let F be a closed subset of X and let a ∈ L∞ N ×N . Then TF (a) is left π ∞ (right, resp. twosided) invertible in alg TF (LN ×N ) if and only if (P aP + Q)πF is so in alg σFπ (A). To see this suppose, e.g., that TFπ (a) is leftinvertible. Thus, there exists ∞ an operator B ∈ alg τ (L∞ N ×N ) := algalg σ(A) {P aP : a ∈ LN ×N } such that π π π π B (P aP ) − P ∈ JF . Taking into account that B = P BP , it is easy to deduce from (g) that
(P BP + Q)π (P aP + Q)π − I π = B π (P aP )π − P π belongs to JFπ ⊂ P π RπF P π ⊂ RπF , i.e., that (P aP + Q)π is left invertible in alg σFπ (A). On the other hand, if there is a B π in the algebra alg σ π (A) such that π B (P aP + Q)π − I π ∈ RπF , then (P BP )π (P aP )π − P π ∈ P π RπF P π , and so (g) and (h) imply that TFπ (a) is leftinvertible in alg TFπ (L∞ N ×N ). (l) To every operator A ∈ alg T (L∞ N ×N ) there corresponds in a natural way an element (P AP )π ∈ alg τ π (L∞ N ×N ). If F is a closed subset of X and ), then A ∈ alg T (L∞ N ×N (P AP )π + RπF alg σπ (A) = Aπ + JFπ alg T π (L∞ . N ×N ) Indeed, Aπ + JFπ alg T π (L∞ = (P AP )π + JFπ alg σπ (A) N ×N ) = (P AP )π + P π RπF P π alg σπ (A)
(by (g))
= P π (P AP )π P π + P π RπF P π alg σπ (A) ≤ (P AP )π + RπF alg σπ (A)
(since P π = P = 1)
= (P AP )π + JFπ alg σπ (A)
(because JFπ ⊂ RπF )
= Aπ + JFπ alg T π (L∞ . N ×N )
4 Toeplitz Operators on H 2
208
∞ Recall that, for a closed subset F of X, for a ∈ L∞ N ×N , and B ⊂ LN ×N , we deﬁned distF (a, B) := inf max a(x) − b(x)L(CN ) . b∈B x∈F
4.53. Theorem. Let a ∈ L∞ N ×N and deﬁne the partitions ∆w (w ∈ W ) of X as in 4.41. If w ∈ W and S ∈ ∆w , then ∞ (QaP )πS alg σSπ (A) = distS (a, CN ×N + HN ×N ).
(4.29)
∞ Proof. Choose h ∈ CN ×N + HN ×N so that ∞ (a − h)S ≤ distS (a, CN ×N + HN ×N ) + ε.
Let IS := {g ∈ L∞ N ×N : gS = 0}. Because (QaP )πS = (Q(a − h)P )πS (since (QhP )πS = 0) ≤ inf (Q(a − h − g)P )π ≤ inf a − h − gL∞ N ×N g∈IS
g∈IS
=: a − hL∞ = (a − h)S N ×N /IS
(proof of Theorem 3.61)
and ε > 0 can be chosen arbitrarily, it follows that for each w ∈ W and each S ∈ ∆w in (4.29) the inequality “≤” holds. That actually equality holds will be proved by transﬁnite induction. (i) For w = 1 this is Theorem 4.34. (ii) Suppose that w has no predecessor and that (4.29) holds for all S ∈ ∆v with v < w. Let S ∈ ∆w . The deﬁnition of ∆w shows that there is a family Sv . Given {Sv }v<w of sets Sv ∈ ∆v such that Su ⊂ Sv for v < u and S = v<w
any ε > 0 there is a v0 < w such that / dist (QaP )π , clos RπSv ≥ dist (QaP )π , RπSv0 − ε. v<w
Thus, ∞ ∞ distS (a, CN ×N + HN ×N ) ≤ distSv0 (a, CN ×N + HN ×N )
= (QaP )πSv0 (induction hypothesis) / = dist QaP )π , RπSv0 ≤ dist (QaP )π , clos RπSv + ε v<w
≤ dist (QaP )π , RπS + ε = (QaP )πS + ε, π where the last “≤” results from the inclusion clos RSv ⊂ RπS , which can v<w π be proved in the same way as Lemma 4.43 (of course, actually clos RSv v<w
is equal to RπS ). Letting ε go to zero, we arrive at the desired inequality.
4.4 Transﬁnite Localization
209
(iii) Now suppose w has a predecessor v and (4.29) holds for all S ∈ ∆v . Let S0 ∈ ∆w . There is an Sv ∈ ∆v such that S0 ⊂ Sv . Let A denote the operator T (a∗ a) − T (a∗ )T (a) ∈ alg T (L∞ N ×N ). Recall that, by Lemma 4.47(a), M (QSv ) can be identiﬁed with {S ∈ ∆w : S ⊂ Sv }. Now let ε > 0 and put U (S0 ) := S ∈ ∆w : S ⊂ Sv , AπS < AπS0 + ε . From Theorem 1.35(b) applied in the same setting as in the proof of Theorem 4.48 we deduce that the mapping M (QSv ) → R+ , S → AπS is upper semicontinuous. Therefore U (S0 ) is an open subset of M (QSv ). Thus, there is a ϕ ∈ QSv such that ϕS0 = 1, ϕ(M (QSv )\U (S0 )) = 0, 0 ≤ ϕ ≤ 1. Choose a function f ∈ C + H ∞ so that f Sv = ϕ. If S ∈ U (S0 ), then f S AπS 1/2 < (AπS0 + ε)1/2
(4.30)
and if S ∈ M (QSv ) \ U (S0 ), then f S AπS 1/2 = 0.
(4.31)
Now deﬁne g ∈ L∞ N ×N as g = f IN ×N . Then ∞ ∞ distS0 (a, CN ×N + HN ×N ) = distS0 (ag, CN ×N + HN ×N ) ∞ ≤ distSv (ag, CN ×N + HN ×N ) (because S0 ⊂ Sv )
= (QagP )πSv (induction hypothesis) " π "1/2 (C ∗ norm property) = " (QagP )∗ (QagP ) Sv " = (P ga∗ QagP )πSv 1/2
(P gQ)π = (QgP )π = 0 = (P gP a∗ QaP gP )πSv 1/2 " "1/2 π = " P gP (P a∗ QaP )P gP + RπSv " " "1/2 π = " P gP (P a∗ aP − P a∗ P aP )P gP + RπSv " = T π (g)Aπ T π (g) + JSπv 1/2 = TSπv (g)AπSv TSπv (g)1/2 .
(by 4.52(l)) (4.32)
Now Theorem 1.35(d) applied in the same context as in the proof of Theorem 4.48 gives that (4.32) equals max TSπ (g)AπS TSπ (g)1/2 : S ∈ ∆w , S ⊂ Sv (4.33) ≤ max f S AπS 1/2 : S ∈ ∆w , S ⊂ Sv . Taking into account (4.30) and (4.31) we see that (4.33) is not greater than (AπS0 + ε)1/2 and since ε > 0 can be chosen arbitrarily, we get
210
4 Toeplitz Operators on H 2 ∞ π 1/2 distS0 (a, CN ×N + HN . ×N ) ≤ AS0
But
" "1/2 π AπS0 1/2 = " T (a∗ a) − T (a∗ )T (a) + JSπ0 " = (P a∗ aP − P a∗ P aP )π + RπS0 1/2 (by 4.52(l)) " π "1/2 = (P a∗ QaP )πS0 1/2 = " (QaP )∗ (QaP ) S0 " = (QaP )πS0 .
Remark. If w > 1 and S ∈ ∆w , then S is contained in some ﬁber Xξ over ξ ∈ M (QC) (Proposition 4.42(a)). Thus, in that case CS ∼ = C and hence, for , a ∈ L∞ N ×N ∞ (QaP )πS alg σSπ (A) = distS (a, HN ×N ). 4.54. Corollary. Let u ∈ GL∞ N ×N be unitaryvalued, let B be C, QC, or C + H ∞ , and let F be a maximal antisymmetric set for B. Then TFπ (u) is ∞ left resp. right invertible in alg TFπ (L∞ N ×N ) if and only if distF (u, HN ×N ) < 1 ∞ resp. distF (u, HN ×N ) < 1. Proof. If B = QC or B = C +H ∞ , then there is a w ∈ W such that F belongs to ∆w . So it remains to apply 4.52(k), Lemma 4.35, and the remark in 4.53. The case B = C can be reduced to the case B = QC by using Theorem 3.70 (whose proof shows that the statement of the theorem is also true for onesided invertibility) along with Theorem 3.5. Instead of Theorem 3.5 one can also apply Theorem 1.35(d) in the spirit of the proof of Theorem 3.70. ∞ We ﬁnally show that distF (u, HN ×N ) is equal to some other quantity, a fact that will be needed later.
4.55. The algebras HF∞ . Let F be a weak peak set for H ∞ . Then, by 1.28, H ∞ F is a closed subalgebra of L∞ F and hence, HF∞ := f ∈ L∞ : f F ∈ H ∞ F is a closed subalgebra of L∞ . Clearly, HF∞ = H ∞ + IF
where
IF := g ∈ L∞ : gF = 0 .
If F is a maximal antisymmetric set for C + H ∞ , then, we mentioned this in 1.28, F is a weak peak set for C + H ∞ and thus for H ∞ . So in that case HF∞ makes a sense. If F = Xτ (τ ∈ T) or F = Xξ (ξ ∈ M (QC)), then F is a peak set for C + H ∞ (recall 2.81) and thus for H ∞ . So we now consider the ∞ ∞ and HX . algebras HX τ ξ 4.56. Proposition. Let F be a weak peak set for H ∞ and let a ∈ L∞ N ×N . Then ∞ inf max a(x) − h(x)L(CN ) distF (a, HN ×N ) := ∞ h∈HN ×N x∈F
is equal to
dist(a, (HF∞ )N ×N ) :=
inf
∞) f ∈(HF N ×N
a − f L∞ . N ×N
4.5 Local Toeplitz Operators
211
∞ ∞ Proof. Let m := distF (a, HN ×N ). For each ε > 0, there is an h ∈ HN ×N such that (a − h)F < m + ε and there is an open set U ⊃ F such that sup a(x) − h(x) < m + ε. Since F is a weak peak set for H ∞ , there exists a x∈U
peak set P for H ∞ such that F ⊂ P ⊂ U . Thus, (a − h)P < m + ε. Choose ∞ ϕ ∈ H ∞ so that ϕP = 1 and ϕ(x) < 1 for x ∈ X \ P , and deﬁne g ∈ HN ×N n as g = ϕIN ×N . There exists an n ∈ Z+ such that g (x)(a(x) − h(x)) < ε = 1, it follows that for all x in the (compact) set X \ U , and since gL∞ N ×N n g n (a − h)L∞ < m + ε. But g (a − h) = a − h + f with some f ∈ L∞ N ×N N ×N satisfying f F = 0, whence dist(a, (HF∞ )N ×N ) ≤ a − h + f L∞ < m + ε. N ×N On the other hand, if we let m := dist(a, (HF∞ )N ×N ), then for each ε > 0 there < m + ε, and since h = f + g with is an h ∈ (HF∞ )N ×N such that a − hL∞ N ×N ∞ f ∈ HN and gF = 0, we arrive at the inequality (a − f )F < m + ε. ×N
4.5 Local Toeplitz Operators Let B be C, QC, or C + H ∞ and let S denote the collection of the maximal antisymmetric sets for B. Let a ∈ L∞ . We know from Theorem 2.83 that T (a) is Fredholm if and only if T (a) is Srestricted invertible for all S ∈ S. On the other hand, Theorem 4.12 (for B = C or QC) and Corollary 4.49(a) (for B = QC or C + H ∞ ) tell us that T (a) is Fredholm if and only if the local operators TSπ (a) are invertible for all S ∈ S. The conclusion is that T (a) is Srestricted invertible for all S ∈ S if and only if TSπ (a) is invertible for all S ∈ S. The question we are interested in here is as follows: given an individual F ∈ S, is it true that T (a) is F restricted invertible if and only if TFπ (a) is invertible? We shall show that the answer is yes. Note that the “only if” part is trivial. 4.57. Lemma. Let F be a closed subset of X = M (L∞ ), let a ∈ L∞ , and suppose a(x) = 0 for x ∈ F . Then there exists a b ∈ GL∞ such that bF = aF . Proof. Let U be a clopen (:= simultaneously closed and open) neighborhood of F such that a(x) = 0 for x ∈ U (recall that X is totally disconnected). Then the characteristic function χU of U is (the Gelfand transform of a function) in L∞ . Put b = aχU + 1 − χU . Thus bF = aF and the function c ∈ L∞ given by c(x) = 1/a(x) for x ∈ U and c(x) = 1 for x ∈ X \ U is the inverse of b. 4.58. Proposition. Let F be a closed subset of X and let a ∈ L∞ . Then T (a) is F restricted left, right, or twosided invertible (resp. TFπ (a) is left, right, twosided invertible) if and only if there is a b ∈ GL∞ such that bF = aF and T (b/b) (resp. TFπ (b/b)) has the corresponding property.
4 Toeplitz Operators on H 2
212
Proof. If T (a) is F restricted left, right, or twosided invertible, then a(x) = 0 for x ∈ F due to Theorem 2.30. The same conclusion can be drawn from the left, right, or twosided invertibility of TFπ (a) by using Corollary 3.63. Hence, by the preceding lemma, there is a b ∈ GL∞ such that bF = aF . We saw in the proof of Proposition 2.19 that b/b factors as hbh with h ∈ GH ∞ . Since T (b/b) = T (h)T (b)T (h), all assertions of the proposition follow at once. 4.59. The ChangMarshall theorem. A closed subalgebra of L∞ is called a Douglas algebra if it contains H ∞ . If B is an arbitrary subset of L∞ , then algL∞ (B, H ∞ ) is clearly a Douglas algebra and vice versa, every Douglas algebra is of this form. The following remarkable fact was conjectured by Douglas and proved by Chang and Marshall: Every Douglas algebra A is of the form A = algL∞ (B, H ∞ ), where B is some collection of inner functions. Given a Douglas algebra A deﬁne ΣA := {ϕ ∈ H ∞ : ϕ inner , ϕ ∈ A}. Then, by the ChangMarshall theorem, A = algL∞ (ΣA , H ∞ ). The algebras HF∞ deﬁned in 4.55 are obviously Douglas algebras. So the ChangMarshall theorem implies that HF∞ = algL∞ (ΣF , H ∞ ) where ΣF := ϕ ∈ H ∞ : ϕ inner , ϕ ∈ HF∞ . Thus, HF∞ = clos
n
hi ϕi : hi ∈ H ∞ , ϕi ∈ ΣF ,
i=1
and because h1 ϕ1 + h2 ϕ2 = (h1 ϕ2 + h2 ϕ1 )ϕ1 ϕ2 , we have HF∞ = closL∞ hϕ : h ∈ H ∞ , ϕ ∈ H ∞ , ϕ inner, ϕ ∈ HF∞ .
(4.34)
Actually the full strength of the ChangMarshall theorem is not required for our purposes, since all what we need is equality (4.34), i.e., the ChangMarshall theorem for the special case that A = HF∞ . For this case the theorem was proved by Axler [12] by employing techniques that are simpler than those required for the general case. 4.60. ClanceyGosselin sets. A closed subset F of X = M (L∞ ) will be called a ClanceyGosselin set (brieﬂy a CGset) if it has the following properties: (a) F is a weak peak set for H ∞ . (b) ϕ ∈ H ∞ , ϕ inner, ϕ ∈ HF∞ , implies that ϕ is constant on F . In other words, the CGsets are those weak peak sets for H ∞ for which ΣF F ∼ = C. Clancey and Gosselin [139] showed that the ﬁbers Xτ (τ ∈ T), the ﬁbers Xξ (ξ ∈ M (QC)), and the maximal antisymmetric sets for C +H ∞ have property
4.5 Local Toeplitz Operators
213
(b). The simplest case is that in which F is a maximal antisymmetric set for C + H ∞ : if ϕ ∈ H ∞ is inner and ϕF ∈ H ∞ F , then (ϕ + ϕ)F ∈ H ∞ F and (1/i)(ϕ − ϕ)F ∈ H ∞ F are realvalued, so the antisymmetry property of F implies that (ϕ + ϕ)F and (ϕ − ϕ)F are constant, and so ϕF must also be constant. The veriﬁcation of (b) for the ﬁbers Xτ and the ﬁbers Xξ is not trivial. It requires a series of ingredients from the theory of function algebras, so its proof must be omitted here. The extension result stated in Proposition 4.62 below may serve as a motivation for the introduction of CGsets. 4.61. Lemma. Let F be a CGset and let a ∈ L∞ . Then distF (a, GH ∞ ) = dist(a, GHF∞ ).
(4.35)
Proof. Choose an f ∈ GHF∞ so that a − f < dist(a, GHF∞ ) + ε. By virtue of (4.34) we may assume that f = gϕ with g ∈ H ∞ , ϕ ∈ HF∞ , ϕ inner. Since ϕ ∈ GHF∞ (ϕ−1 = ϕ ∈ H ∞ ⊂ HF∞ ), it follows that g ∈ GHF∞ . Write g = ψh with ψ inner and h ∈ GH ∞ (see 1.41). Then ψ = gh−1 ∈ GHF∞ , hence ψ = ψ −1 ∈ HF∞ . Because F is a CGset, we conclude that ψF and ϕF are constant. Without loss of generality assume ψF = ϕF = 1 (otherwise write f = (cg)(c ϕ), g = (dψ)(dh)). Thus, (a − h)F = (a − ψhϕ)F ≤ a − ψhϕ∞ = a − f ∞ and letting ε go to zero we obtain “≤” in (4.35). We now prove the opposite inequality. Let ε > 0 and let h0 ∈ GH ∞ satisfy (a − h0 )F < distF (a, GH ∞ ) + ε/2. Then let V0 be a clopen neighborhood of F such that (4.36) (a − h0 )V0 < distF (a, GH ∞ ) + ε. n Vi , where each Vi is clopen and Since X \ V0 is compact, we have X \ V0 = i=1
there is an xi ∈ Vi such that max a(xi ) − a(y) < y∈Vi
ε 2
∀ i = 1, . . . , n.
(4.37)
Put hi = a(xi ) if a(xi ) if a(xi ) = 0 and let hi = ε/2 if a(xi ) = 0. Let χVi n denote the characteristic function of Vi and put h = i=0 hi χVi . Because because hF = h F , h is even in HF∞ . The each Vi is clopen, h is in L∞ , and 0
n −1 −1 −1 −1 inverse of h is clearly h = i=0 hi χVi , and since h F = h−1 0 F , h belongs to HF∞ . Thus, h ∈ GHF∞ . We have dist(a, GHF∞ ) ≤ a − h∞ = max (a − h)Vi , i=0,...,n
and because of (4.36) and the inequalities (a − b)Vi < ε for i = 1, . . . , n (resulting from (4.37)), it follows that dist(a, GHF∞ ) < distF (a, GH ∞ ) + ε.
214
4 Toeplitz Operators on H 2
4.62. Proposition. Let F be a CGset and suppose a function u ∈ L∞ is unimodular on F (i.e., u(x) = 1 for x ∈ F ). If distF (u, H ∞ ) < 1 (resp. distF (u, GH ∞ ) < 1), then there exists a unimodular function v ∈ GL∞ (i.e., v(x) = 1 for all x ∈ X) such that vF = uF and dist(v, H ∞ ) < 1 (resp. dist(v, GH ∞ ) < 1). Proof. Due to Lemma 4.57 there is a w ∈ GL∞ such that wF = uF , and since w/w also coincides with u on F , it can be a priori assumed that u(x) = 1 for all x ∈ X. Suppose distF (u, H ∞ ) < 1. Then, by Proposition 4.56, dist(u, HF∞ ) < 1 and so (4.34) implies that there are a function g ∈ H ∞ and an inner function ϕ ∈ H ∞ such that ϕ ∈ HF∞ and u − gϕ∞ < 1. Since F is a CGset, ϕF is constant, say ϕF = 1. Put v = ϕu. Then vF = uF , v(x) = ϕ(x)u(x) = 1 for x ∈ X, and because v − g∞ = ϕu − g∞ = u − ϕg∞ < 1, it follows that dist(v, H ∞ ) < 1. Now suppose distF (u, GH ∞ ) < 1. Lemma 4.61 shows that u−f ∞ < 1 for some f ∈ GHF∞ . By (4.34) and an argument used in the proof of Lemma 4.61, we may assume that f = ψhϕ with h ∈ GH ∞ , ϕ and ψ inner, ϕ and ψ in HF∞ . Since F is a CGset, it may be assumed that ϕF = ψF = 1. Thus, if we let v = ϕψu, then vF = uF , v(x) = 1 for x ∈ X, and since v − h∞ = ϕψu − h∞ = u − ψhϕ∞ = u − f ∞ < 1, we ﬁnally see that dist(v, GH ∞ ) < 1. 4.63. Theorem (Clancey/Gosselin). Let B be C, QC, or C + H ∞ and let F be a maximal antisymmetric set for B. Let a ∈ L∞ . Then the following are equivalent: (i) T (a) is F restricted left (resp. right) invertible; (ii) TFπ (a) is left (resp. right) invertible; (iii) we have a(x) = 0 for all x ∈ F and distF (a/a, H ∞ ) < 1 (resp. distF (a/a, H ∞ ) < 1). Proof. We only consider the case of leftinvertibility. (i) =⇒ (ii). Obvious. (ii) =⇒ (iii). By Proposition 4.58, there is a b ∈ GL∞ such that bF = aF and TFπ (b/b) is leftinvertible. From Corollary 4.54 we deduce that distF (a/a, H ∞ ) = distF (b/b, H ∞ ) < 1. (iii) =⇒ (i). Lemma 4.57 shows that there is a b ∈ GL∞ such that bF = aF . Clearly, distF (b/b, H ∞ ) < 1. Now Proposition 4.62 yields the
4.5 Local Toeplitz Operators
215
existence of a unimodular function v ∈ GL∞ such that vF = (b/b)F and dist(v, H ∞ ) < 1. By Theorem 2.20(a), T (v) is leftinvertible, and hence T (b/b) is F restricted leftinvertible. It remains to apply Proposition 4.58. 4.64. Theorem (Clancey/Gosselin). Assume B is C, QC, or C +H ∞ and F is a maximal antisymmetric set for B. Let a ∈ L∞ . Then the following are equivalent: (i) T (a) is F restricted invertible; (ii) T (a) is F restricted leftinvertible and F restricted rightinvertible; (iii) TFπ (a) is invertible; (iv) a(x) = 0 for x ∈ F , distF (a/a, H ∞ ) < 1, and distF (a/a, H ∞ ) < 1; (v) a(x) = 0 for x ∈ F and distF (a/a, GH ∞ ) < 1. Proof. (i) =⇒ (ii) =⇒ (iii). Obvious. (ii) ⇐= (iii) ⇐⇒ (iv). Theorem 4.63. (v) =⇒ (i). First choose a b ∈ GL∞ satisfying bF = aF (Lemma 4.57). Then apply Proposition 4.62 to deduce that there is a unimodular v ∈ GL∞ such that vF = (b/b)F = (a/a)F and dist(v, GH ∞ ) < 1. Thus, by Theorem 2.20(c), T (v) ∈ GL(H 2 ), which implies the F restricted invertibility of T (b/b), and Proposition 4.58 then gives that of T (a). (ii) =⇒ (v). Again choose b ∈ GL∞ so that bF = aF , put u = b/b, and deduce from Proposition 4.58 that T (u) is both F restricted leftinvertible and F restricted rightinvertible. So, by Theorem 4.63 and Proposition 4.56, dist(u, HF∞ ) < 1,
dist(u, HF∞ ) < 1.
(4.38)
Now (4.34) implies that there are an h ∈ H ∞ and an inner function ϕ such that ϕ ∈ HF∞ and u − hϕ∞ < 1. We show that hϕ ∈ GHF∞ , which, by Lemma 4.61, will complete the proof. Since F is a CGset, it can be assumed that ϕF = 1. Write h = ψg with an inner function ψ and an outer function g ∈ H ∞ (see 1.41(a)). Because 1 − u ϕψg∞ = u − hϕ∞ < 1,
(4.39)
it follows that g ∈ GL∞ and hence g ∈ GH ∞ (1.41(g)). Also because of (4.39), TFπ (u ϕψg) = TFπ (u)IFπ TFπ (g)TFπ (ψ) (recall that ϕF = 1) is invertible. From (4.38) and Theorem 4.63 we get the invertibility of TFπ (u), and since g ∈ GH ∞ , TFπ (g) is also invertible. The conclusion is that TFπ (ψ) must also be invertible. So Theorem 4.63 in conjunction with Proposition 4.56 shows that there is an f ∈ HF∞ such
216
4 Toeplitz Operators on H 2
that ψ − f ∞ < 1. Hence 1 − ψf ∞ < 1 and since ψf ∈ HF∞ , we obtain that actually ψf ∈ GHF∞ . Thus, ψ = f (ψf )−1 ∈ HF∞ and since ψ ∈ H ∞ ⊂ HF∞ , it follows that ψ ∈ GHF∞ . Finally, taking into account that ϕ ∈ GHF∞ (ϕ−1 = ϕ ∈ H ∞ ⊂ HF∞ ) and that g ∈ GH ∞ we deduce that hϕ = ψgϕ ∈ GHF∞ , as desired. 4.65. Corollary. Let B be C, QC, or C + H ∞ and let S denote the family of the maximal antisymmetric sets for B. Then if S ∈ S and a ∈ L∞ , $ $ sp T π (f ) = sp T (f ), (4.40) sp TSπ (a) = f ∈a+IS
f ∈a+IS
where IS = {g ∈ L∞ : gS = 0}. Proof. Clearly, if f ∈ a + IS , then sp T (f ) ⊃ sp T π (f ) ⊃ sp TSπ (f ) = sp TSπ (a) and thus (4.40) holds with the two “=” replaced by “⊂”. It remains to show that sp T (f ) is contained in sp TSπ (a), i.e., that the invertibilf ∈a+IS
ity of TSπ (a − λ) (λ ∈ C) implies the existence of an f ∈ a + IS such that T (f − λ) is invertible. But this is equivalent to saying that the invertibility of TSπ (a − λ) implies the Srestricted invertibility of T (a − λ), and this was proved in Theorem 4.64. Remark. As an immediate consequence of Theorem 4.12 (B = C or QC) and Corollary 4.49(a) (B = QC or B = C + H ∞ ) we have that / sp TSπ (a), sp T π (a) = S∈S
i.e., the essential spectrum of a Toeplitz operator is the union of all its “local spectra.” The above theorem lies deeper and involves the following identiﬁcation of the local spectrum: the spectrum of a local Toeplitz operator TSπ (a) is precisely the common part of the essential spectra of all Toeplitz operators whose symbols coincide on S with aS. 4.66. Open problems. (a) What can be said for matrix symbols about the connection between the invertibility of local Toeplitz operators and restricted invertibility? (b) Find an analogue of Theorem 4.64 (well, say with B = C or B = QC) for harmonic approximation or stable convergence, i.e., for the case that T π π is replaced by K or T KJ . Note that it is again the lack of an analogue of formula (2.20) and the observation that will be made in 4.77 which complicate the things and require other techniques (the “again” is because of 4.50(e)).
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217
4.6 Symbols with Speciﬁc Local Range 4.67. Theorem. Let ξ ∈ M (B), where B is C or QC. Let a ∈ L∞ and assume the set a(Xξ ) is contained in some straight line segment. Then sp Tξπ (a) = conv a(Xξ ). Proof. If a(Xξ ) is a singleton, Corollary 3.64 applies. Let conv a(Xξ ) = [z1 , z2 ], where z1 , z2 ∈ C and z1 = z2 . Put b = 2(z2 − z1 )−1 [a − (z1 + z2 )/2]. Then {−1, 1} ⊂ b(Xξ ) ⊂ [−1, 1] and it is clear that sp Tξπ (a) = [z1 , z2 ] if and only if sp Tξπ (b) = [−1, 1]. From Corollary 3.64 (or Theorem 4.24) we deduce that {−1, 1} ⊂ sp Tξπ (b) ⊂ [−1, 1]. Let µ ∈ (−1, 1) and assume µ ∈ / sp Tξπ (b). By Proposition 4.58, there is a ∞ c ∈ GL such that cXξ = (b − µ)Xξ and Tξπ (c/c) ∈ GL(H 2 ). Hence, by Corollary 4.54, (4.41) distXξ (c/c, H ∞ ) < 1. But the range of c/c on Xξ is the doubleton {−1, 1}. So Lemma 2.90 shows that (4.41) is impossible and this contradiction completes the proof. 4.68. Open problems. Let F be a maximal antisymmetric set for C, QC, or C + H ∞ and let {Kλ }λ∈Λ be an approximate identity whose index set Λ is connected. Is it true that the local spectra sp TFπ (a),
sp {kλ a}πF ,
sp {T (kλ a)}πF
(4.42)
are connected for every a ∈ L∞ ? There is only one case in which we know that the answer is yes: if τ ∈ T = M (C), then sp {kλ a}πτ is connected for every a ∈ L∞ (Proposition 3.73(b) and Theorem 3.76(c)). We do not know any symbol a ∈ L∞ for which any of the local spectra (4.42) is disconnected. However, there are certain classes of symbols for which the connectedness of some of the local spectra (4.42) is known. For instance, if ξ ∈ M (C) or ξ ∈ M (QC) and if a(Xξ ) is contained in some straight line segment, then, by Theorems 4.67 and 4.24, π sp TX (a) = sp {T (kλ a)}πXξ = conv a(Xξ ) ξ
are connected; from Theorem 4.24 we also know that sp {kλ a}πXξ ⊂ conv a(Xξ ) and we conjecture that “⊂” can be replaced by equality. This conjecture is supported by the fact that equality holds for B = C (Corollary 3.78(a)) or for B = QC and a ∈ P QC (Theorem 3.79).
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4.69. Lemma. Suppose A is a C ∗ algebra with identity element. Suppose a = diag (a1 , . . . , aM ) ∈ AN ×N is a blockdiagonal matrix each block ai (i = 1, . . . , M ) of which is uppertriangular with equal entries bi on the diagonal. Then a ∈ GAN ×N if and only if bi ∈ GA for i = 1, . . . , M . Proof. Due to the GelfandNaimark theorem 1.26(b) it can be assumed that A is a C ∗ subalgebra of L(H), where H is some Hilbert space. Then a may be thought of as an operator in L(HN ). It is easily seen that the invertibility of a on HN implies that each diagonal block ai is an invertible operator on the direct sum of the corresponding number of copies of H. Consideration of the southeast entry of ai shows that bi must be onto, while consideration of the northwest entry of ai yields that bi is onetoone. 4.70. Theorem. Let B = C or B = QC, let a ∈ L∞ N ×N , and suppose for each ξ ∈ M (B) the set conv a(Xξ ) is a (possibly degenerate) straight line segment. Then 2 ) ⇐⇒ a is locally sectorial over B. T (a) ∈ Φ(HN In that case {kλ a} is bounded away from zero on conv Λ for every approximate identity {Kλ }λ∈Λ and Ind T (a) = −ind {det kλ a}. Proof. In the scalar case (N = 1) the Fredholm criterion is immediate from Theorem 4.12 in conjunction with Theorem 4.67. In the general case we are by virtue of Corollary 4.13(b) and Corollary 4.30 left with the proof of the implication “=⇒”. 2 ). Then, by Theorem 4.12, Tξπ (a) is invertible for So assume T (a) ∈ Φ(HN each ξ ∈ M (B). Fix ξ ∈ M (B) and let conv a(Xξ ) = µE + (1 − µ)F : µ ∈ [0, 1] . In particular, there are x1 , x2 ∈ Xξ such that a(x1 ) = E and a(x2 ) = F . Due to Theorem 2.93, both E and F are invertible matrices. There exists an invertible matrix D such that J := D−1 E −1 F D is in Jordan canonical −1 −1 E aD. Then b(x) is an uppertriangular from. Deﬁne b ∈ L∞ N ×N as b := D matrix for each x ∈ Xξ , and if we let bii (i = 1, . . . , N ) denote the diagonal entries of b, then conv bii (Xξ ) = [1, λi ], where λi is an eigenvalue of J (note that bii (x1 ) = 1 and bii (x2 ) = λi ). The invertibility of Tξπ (a) implies that Tξπ (b) is invertible, and hence, by Lemma 4.69, Tξπ (bii ) is invertible for each i. Since sp Tξπ (bii ) = [1, λi ] (Theorem 4.67), it follows that the origin does not belong to any of the line segments [1, λi ]. Thus, det(µE + (1 − µ)F )(det E)−1 = det(µI + (1 − µ)J) =
n
(µ + (1 − µ)λi ) = 0
i=1
for all µ ∈ [0, 1]. Now Theorem 3.4 completes the proof.
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219
Remark. The preceding theorem reduces the Fredholm theory of Toeplitz operators with (P2 C)N,N or (P2 QC)N,N symbols (or, more generally, with symbols whose range on each ﬁber over C or QC is contained in some line segment) to the question of ﬁnding conditions for the local sectoriality of such symbols. This is one reason for the due attention we paid to local sectoriality in Chapter 3. In particular, recall Theorem 3.4 (along with Proposition 3.2), Proposition 3.11 (together with Theorem 3.9), Corollary 3.82, and Corollary 3.85. 4.71. Pn C symbols. Things are more complicated for symbols taking n ≥ 3 values on some ﬁber Xτ (τ ∈ T). The knowledge we have about Toeplitz operators generated by such symbols is by no means comparable with the knowledge one has in the case of P2 C or even P2 QC symbols. In accordance with 2.89, a function a ∈ L∞ is said to belong to Pn C if its essential range consists of at most n distinct points. Clearly, if T is divided into three pairwise disjoint measurable subsets A, B, C of positive measure and if a, b, c are pairwise distinct complex numbers, then ϕ = aχA + bχB + cχC
(4.43)
belongs to P3 C \ P2 C and every ϕ ∈ P3 C \ P2 C is of this form. Let ∆(a, b, c) denote the triangle which is the closed convex hull of the three points a, b, c. We know from Theorems 2.30 and 2.33 that {a, b, c} ⊂ sp T π (ϕ) ⊂ ∆(a, b, c), and the question is which points of the triangle ∆(a, b, c) are in sp T π (ϕ), the essential spectrum of T (ϕ). There must exist a “suﬃciently large” supply of such points, since sp T π (ϕ) is a connected set (2.35(b)). Using Theorem 2.74 it is easy to produce symbols ϕ ∈ P3 C ∩ P C such that sp T π (ϕ) consists of two or three sides of that triangle. Moreover, the same theorem shows that, for ϕ ∈ P3 C ∩ P C, an interior point of that triangle can never belong to sp T π (ϕ). More exotic symbols in P3 C can be obtained by putting ϕ = p ◦ ω, where p ∈ P3 C ∩ P C and ω ∈ H ∞ is an inner function. Suppose T (p) ∈ Φ(H 2 ). If Ind T (p) = 0, so that T (p) ∈ GL(H 2 ), then T (p ◦ ω) is invertible by Theorem 2.20. However, if Ind T (p) = 0, then by using Theorem 2.64 it is not diﬃcult to show that T (p ◦ ω) is Fredholm if and only if ω is a ﬁnite Blaschke product. In particular, if we let p(eiθ ) be a, b, c for θ ∈ (0, 2π/3), θ ∈ (2π/3, 4π/3), θ ∈ (4π/3, 2π), respectively, and if ω is not a ﬁnite Blaschke product, then sp T (p ◦ ω) = sp T π (p ◦ ω) = ∆(a, b, c). Some suﬃciently interesting Pn C (or even Pn C) functions are contained in the class LCS(T◦ ) we shall deﬁne in the next section. Finally, note that if a ∈ Pn C, then for each τ ∈ T there exists an aτ in Pn C such that aXτ = aτ Xτ . This can be shown as follows. Let τ ∈ T and
4 Toeplitz Operators on H 2
220
a(Xτ ) = {v1 , . . . , vm } (m ≤ n) with vi = vj for i = j. Choose an ε > 0 so that the disks Di with center vi and radius ε (i = 1, . . . , m) are pairwise disjoint. m Di . By virtue of Proposition 2.79(a), there is a U ∈ Uτ such that a(U ) ⊂ i=1
m −1 Put Ui = U ∩ a (Di ). The function aτ := v1 χT\U + i=1 vi χUi belongs to Pn C, and once more using Proposition 2.79(a) it is easy to check that (a − aτ )(Xτ ) = {0}, whence aXτ = aτ Xτ . The preceding observation in conjunction with Theorem 4.12 shows that the determination of the essential spectrum of operators with Pn C symbols can be reduced to the identiﬁcation of the local spectrum of operators with Pn C symbols. 4.72. The class LCS(T◦ ). Let T◦ be the punctured circle T \ {−1} and let LCS(T◦ ) denote the class of all functions a ∈ GL∞ with the following property: there is an ε > 0 such that for each τ ∈ T◦ there are a subarc Uτ of T◦ containing τ and a cτ ∈ C, cτ  = 1, such that Re (cτ a(t)) ≥ ε for almost all t ∈ Uτ . The LCS is for locally Csectorial. An obvious modiﬁcation of the argument used to prove the implication (iii) =⇒ (iv) of Theorem 3.9 shows that every function a ∈ LCS(T◦ ) can be written as a = cs, where c ∈ CU (T◦ ) (recall 2.25) and s ∈ GL∞ is sectorial (on T). Choose any b ∈ CR(T◦ ) satisfying c = eib and deﬁne a# ∈ C(R) by a# (x) = b((i−x)/(i+x)) for x ∈ R. Let a = s1 eib = s2 eib with b1 , b2 ∈ CR(T◦ ) and s1 , s2 sectorial (on T). There are γ1 , γ2 ∈ C such that Re (γ1 s1 ) ≥ ε, Re (γ2 s2 ) ≥ ε a.e. on T, which implies that there is a δ > 0 with the property that + γ s + γ eib1 + + + + + 2 2 2 + arg = + + arg + + < π − δ, ib γ1 e 2 γ1 s1 whence −π + δ < arg
γ 2
γ1
+ b1 − b2 < π − δ
on
T◦ .
Thus, any two functions a# only diﬀer by a function of the form γ + v, where γ ∈ R and vL∞ (R) < π. So it makes a correct sense to say that a# (±∞) is equal to +∞ or −∞ or that a# be bounded from above or below at ±∞ (see 2.25). 4.73. Theorem. Let a ∈ LCS(T◦ ). π (a) If a# (+∞) = +∞ and a# is bounded from above at −∞, then T−1 (a) = is not invertible.
π (a) TX −1
(b) Suppose a# = ϕ + η, where η ∈ L∞ (R), ϕ is monotonuous on (−∞, 0) π π (a) = TX (a) is invertible, we and (0, ∞), and ϕ(±∞) = +∞. Then if T−1 −1 have a# (x) = O(log x) as x → ∞.
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221
Proof. We have a = cs, where c = eib ∈ CU (T◦ ) and s ∈ GL∞ is sectorial. If π (a) is invertible, τ ∈ T◦ , then Tτπ (a) = c(τ )Tτπ (s) is invertible. Hence, if T−1 2 then T (a) ∈ Φ(H ) by Theorem 4.12. In that case there is an n ∈ Z such that T (χn a) ∈ GL(H 2 ) and we deduce from Theorem 2.23 and from 1.48(g) that the argument of χn a/χn a belongs to BM O. But the argument of χn a/χn a is equal to b plus the arguments of χn and s/s, i.e., it diﬀers from b merely by a function in L∞ . Consequently, b ∈ BM O. The proof of 2.26(c) shows that under the hypothesis (a) the function b π (a) is not invertible under this cannot belong to BM O, which proves that T−1 hypothesis. Finally, if the hypothesis (b) is satisﬁed, then the argument of 2.26(d) gives that b((i − x)/(i + x)) = O(log x) as x → ∞, which implies the assertion of part (b) of the present theorem. 4.74. Application to P3 C. The preceding theorem can be applied to get full information about the spectra of Toeplitz operators generated by certain Pn C symbols. To illustrate this it suﬃces to consider the case n = 3. Choose real numbers θn (n ∈ Z) so that −π < . . . < θ−2 < θ−1 < θ0 < θ1 < θ2 < . . . < π and θn → −π, θn → π as n → ∞. Then put tn = eiθn . Let ϕ be of the form (4.43) and suppose each of the sets A, B, C is the union of some subarcs of the form (tn , tn+1 ). By Theorem 4.67, for τ ∈ T◦ the local spectrum sp Tτπ (ϕ) is either one of the points a, b, c (τ = tn ) or one of the line segments [a, b], [b, c], [c, a] (τ = tn ). So the only interesting part of π (ϕ). It is clear that for each λ ∈ C which does not belong sp T π (ϕ) is sp T−1 to the boundary of the triangle ∆(a, b, c) the function ϕ − λ is in LCS(T◦ ). It is also easy to determine the behavior of (ϕ − λ)# . The mapping T◦ → R, t → i(1 − t)/(1 + t) takes the sets A, B, C into certain subsets of R which will be denoted by A, B, C, respectively, too. Theorem 4.73(a) now gives the following. If the location of A, B, C on R is of the type
(in that case (ϕ − λ)# (−∞) = −∞ and (ϕ − λ)# (+∞) = +∞ for λ in the interior of ∆(a, b, c)) or, e.g., of the type
(in that case (ϕ−λ)# is bounded away from above at −∞ and (ϕ−λ)# (+∞) = +∞ for λ in the interior of ∆(a, b, c)), then π sp T−1 (ϕ) = ∆(a, b, c).
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4 Toeplitz Operators on H 2
Theorem 4.73(b) can be applied to the situation where the sets A, B, C are located as follows:
In that case (ϕ−λ)# (±∞) = +∞ and (ϕ−λ)# (x) is monotonically increasing as x → ∞ for each λ in the interior of ∆(a, b, c). If n is suﬃciently large, then the increment of (ϕ − λ)# when t is running through (tn , tn+3 ) or (t−n , t−n−3 ) equals 2π. Thus, in the case under consideration Theorem 4.73(b) together with a simple computation gives the following: if at least one point of the π (ϕ), then there are interior of the triangle ∆(a, b, c) does not belong to sp T−1 constants C > 0 and 0 < q < 1 such that dist(tn , −1) ≤ Cq n for all n ∈ Z. In other words, if, in the case at hand, there do not exist C > 0 and 0 < q < 1 π (ϕ) = ∆(a, b, c). such that dist(tn , −1) ≤ Cq n for all n ∈ Z, then sp T−1 Note that in all the above situations in which we were able to determine the spectrum of a Toeplitz operator generated by a symbol of the form (4.43) that spectrum was either the whole (possibly degenerate) triangle ∆(a, b, c) or consisted of two or three of its sides. A while we conjectured that the same is true for every P3 C symbols; however, we shall see that this is not so in general (Proposition 4.78). 4.75. Proposition. Let ϕ = aχA + bχB + cχC ∈ P3 C. Then either
(a, b) ⊂ sp T π (ϕ)
or
(a, b) ∩ sp T π (ϕ) = ∅.
In other words, a side of the triangle ∆(a, b, c) either entirely belongs to sp T π (ϕ) or, apart from the endpoints, entirely belongs to the complement of sp T π (ϕ). The same is true with sp T π (ϕ) replaced by sp T (ϕ). Proof. We ﬁrst prove the proposition for sp T (ϕ). If ∆(a, b, c) is a line segment, the assertion follows from 2.36. So suppose ∆(a, b, c) is not a line segment and without loss of generality assume a and b are real, a < 0, b > 0, and Im c > 0. Let µ ∈ (a, b). Then, by Proposition 2.19, µ ∈ sp T (ϕ) ⇐⇒ 0 ∈ sp T (ϕ − µ)/ϕ − µ . The essential range of (ϕ − µ)/ϕ − µ consists of the two points −1 and 1 and of a point c(µ) lying on the upper half T+ of the unit circle. Thus, what we must show is the following: if c1 , c2 ∈ T+ and if 0 ∈ sp T (χB − χA + c1 χC ), then 0 ∈ sp T (χB − χA + c2 χC ). So suppose 0 ∈ sp T (χB − χA + c1 χC ). Then Wolﬀ’s result 2.37 implies that there are zn ∈ D such that 2 2 χ2 B (zn ) − χ A (zn ) + c1 χ C (zn ) → 0 as
n → ∞.
(4.44)
2 2 2 Because χ2 A, χ B, χ C are realvalued, we deduce from (4.44) that χ C (zn ) → 0 as n → ∞, and therefore
4.6 Symbols with Speciﬁc Local Range
223
χ2 2 2 2 2 2 2 B (zn )− χ A (zn )+c2 χ C (zn ) ≤ χ B (zn )− χ A (zn )+c1 χ C (zn )+c2 −c1 χ C (zn ) also tends to zero as n → ∞. Again applying 2.37 we see that 0 belongs to sp T (χB − χA + c2 χC ), as desired. To get the assertion for the essential spectrum, let µ ∈ (a, b) and assume µ∈ / sp T π (ϕ). Then T (ϕ − µ) is Fredholm. Let κ := Ind T (ϕ − µ). There is an ε > 0 such that T (ϕ − µ + zχB ) ∈ Φ(H 2 ),
Ind T (ϕ − µ + zχB ) = κ
whenever z ∈ C and z < ε. Since among these z’s there is a z0 such that µ∈ / ∆(a, b +z0 , c), which implies that T (ϕ−µ+z0 χB ) is invertible, we deduce that κ must be zero. Consequently, by Corollary 2.40, T (ϕ − µ) is invertible and hence µ ∈ / sp T (ϕ). From what has been proved above we obtain that (a, b) ∩ sp T (ϕ) = ∅, whence (a, b) ∩ sp T π (ϕ) = ∅. 4.76. Open problem. Is the preceding proposition true with sp T π (ϕ) replaced by sp Tτπ (ϕ) (τ ∈ T)? In this connection it would be interesting to know whether Wolﬀ’s result 2.37 has a local analogue: for a ∈ L∞ , λ belonging to the boundary of conv a(Xτ ), and τ ∈ T, is it true that λ ∈ sp Tτπ (a) ⇐⇒ λ ∈ ClH (a, τ )? 4.77. Harmonic extension of P3 C functions. The harmonic extension of Pn C functions has a very nice geometric interpretation. Let u be the conformal mapping of the unit disk D onto the upper half plane Π given by 1−z . u : D → Π, z → i 1+z Then u−1 is given by u−1 : Π → D,
ζ →
i−ζ . i+ζ
Note that u extends to a continuous function on clos D \ {−1} which maps T◦ = T \ {−1} onto R and −1 into the point at inﬁnity. Given a bounded interval I on R we denote by ωI (ζ) the angle at which I is seen from ζ ∈ Π. Also let ω(x,∞) (ζ) := lim ω(x,y) (ζ) and deﬁne ω(−∞,x) (ζ) similarly. Finally y→∞
set ω(−∞,x)∪(y,∞) (ζ) := ω(−∞,x) (ζ) + ω(y,∞) (ζ). If E is a subarc of T, then u(E) is an interval on R, which is of the form (−∞, x) ∪ (y, ∞) in case E contains the point −1. The harmonic extension of the characteristic function χE at z ∈ D is then given by χ2 E (z) =
1 ωu(E) (u(z)). π
This is well known and can be veriﬁed without diﬃculty, e.g., using the representation of the harmonic extension via Poisson’s integral.
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4 Toeplitz Operators on H 2
Thus, if ϕ is of the form (4.43), where each of the sets A, B, C is a (possibly countable) union of subarcs of T, and if we maintain our convention to denote a set on T and its image on R under the mapping u by the same symbol, then, for z ∈ D, (4.45) π ϕ(z) = aωA (u(z)) + bωB (u(z)) + cωC (u(z)), where, for I = In , each In being an interval and the union being possibly n
countable, ωI (ζ) is deﬁned as n ωIn (ζ); note that the latter series always converges. In the case where {Kλ }λ∈(0,∞) is the approximate identity generated by the Poisson kernel, we write {hr a}r∈(0,1) instead of {kλ a}λ∈(0,∞) . We have seen that Toeplitz operators and the harmonic extension have many features in common. Example: if τ ∈ T, then a(Xτ ) ⊂ sp Tτπ (a) ⊂ conv a(Xτ ),
a(Xτ ) ⊂ sp {hr a}πτ ⊂ conv a(Xτ ).
But we have also seen that there are some decisive diﬀerences (see, e.g., the open problems 4.50, 4.66, 4.68). To study Toeplitz operators we have made frequent and essential use of Propositions 2.19 and 2.32. However, such an argument cannot be applied to harmonic extension: there exist functions ϕ ∈ P3 C ∩ GL∞ such that {hr ϕ} ∈ G(alg H(L∞ ))
but
{hr (ϕ/ϕ)}π ∈ / G(alg Hπ (L∞ )).
(4.46)
A ϕ satisfying (4.46) is ϕ = M χA + e2πi/3 χB + e−2πi/3 χC , where the location of (the images under the mapping u of) the sets A, B, C on R is as follows:
Here the length of an interval belonging to A is 1 and the lengths of the intervals forming B and C are 2. Using the above geometric interpretation of the harmonic extension (we omit the technical details) one can show that for and (ϕ/ϕ)(D) look as follows: suﬃciently large M ∈ R+ the sets ϕ(D)
Thus, Theorem 3.76(a) implies that {hr ϕ} is invertible and that {hr (ϕ/ϕ)} is not invertible. One can show that (ϕ/ϕ)(u−1 (ζ)) → 0 as Im ζ → ∞ (since, for Im ζ suﬃciently large, ωA (ζ) ≈ ωB (ζ) ≈ ωC (ζ) ≈ π/3) and so it follows that even {hr (ϕ/ϕ)}π is not invertible (Theorem 3.76(b)). 4.78. Proposition. There exist ϕ = aχA + bχB + cχC ∈ P3 C such that both (a, c) and (b, c) entirely belong to the complement of sp T (ϕ).
4.6 Symbols with Speciﬁc Local Range
225
Proof. Let a, b, c be any complex numbers such that ∆(a, b, c) is not a line segment and let (the images under the mapping u of) the sets A, B, C be located on R as follows:
that is, C = (−∞, 0), A =
An , B =
n∈Z
Bn , Bn = (22n , 22n+1 ), An =
n∈Z
(22n+1 , 22n+2 ). By virtue of Proposition 4.75, in order to conclude that (a, c)∩ sp T (ϕ) = ∅, it suﬃces to show that µ := (a + c)/2 is not in sp T (ϕ). Assume the contrary, i.e., assume µ ∈ sp T (ϕ). Then, due to 2.37, there are zn ∈ D such that ϕ(z n ) → µ as n → ∞. Let a =: µ + δ, c =: µ − δ, b =: µ + γ. Thus, γ 2 ϕ(z n ) − µ = δ χ2 2 A (zn ) − χ C (zn ) + χ B (zn ) , δ and since χ2 2 2 A, χ B, χ C are realvalued and Im (γ/δ) = 0, we deduce that χ2 B (zn ) → 0,
χ2 2 A (zn ) − χ C (zn ) → 0 (n → ∞).
This and (4.45) show that there is a ζ0 ∈ Π such that ωB (ζ0 ) < ε
(4.47)
and −ε < ωA (ζ0 ) − ωC (ζ0 ) < ε, where tan(2ε) = 1/4 and thus 7◦ < ε < 8◦ . Since ωA + ωB + ωC = 180◦ , we have 90◦ − 2ε < ωC (ζ0 ) < 90◦ + 2ε.
(4.48)
From (4.48) we see that ζ0 lies in the angular sector S := {ζ ∈ Π :  arg ζ − 90◦  < 2ε}. For n ∈ Z, let Tn denote the trapezium Tn := {ζ ∈ S : 22n ≤ Im ζ < 22n+2 }. There is obviously an n ∈ Z such that ζ0 ∈ Tn . We claim that ωBn (ζ0 ) > ε, which is a contradiction to (4.47). Let ζ1 and ζ2 denote the left upper and left lower vertex of the trapezium Tn , respectively. Thus (recall that tan(2ε) = 1/4) ζ1 = 22n + i22n+2 ,
ζ2 = 22n−2 + i22n .
It is clear that ωBn (ζ0 ) > min{ωBn (ζ1 ), ωBn (ζ2 )}. But ωBn (ζ1 ) = α1 − α2 , where tan α1 = 3/4 and tan α2 = 2/4, whence ωBn (ζ1 ) > 9◦ ; also ωBn (ζ2 ) = β1 − β2 , where tan β1 = 9/4 and tan β2 = 5/4, whence ωBn (ζ2 ) > 14◦ . This proves our claim. It can be shown analogously that (b, c) ∩ sp T (ϕ) = ∅, which completes the proof.
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Remark. There are even symbols ϕ = aχA + bχB + cχC ∈ P3 C such that the vertices a, b, c are the only points on the boundary of the triangle ∆(a, b, c) which belong to sp T (ϕ). Thus, in that case sp T (ϕ) \ {a, b, c} (and hence sp T π (ϕ) \ {a, b, c}) is a certain connected set entirely lying in the interior of the triangle ∆(a, b, c). This happens, for instance, if the sets A, B, C are located on R in a Cantorset type position as follows:
The proof is of the same kind as that given for the above proposition, although the technical details are more complicated.
4.7 Toeplitz Algebras 4.79. Theorem. Let B be a C ∗ subalgebra of L∞ satisfying C ⊂ B ⊂ QC. Then the mappings SmbT π , SmbKπ , SmbT KπJ given by (3.34) are isometric π (BN ×N ), restarisomorphisms of alg T π (BN ×N ), alg Kπ (BN ×N ), alg T KJ spectively, onto BN ×N : π alg T π (BN ×N ) ∼ (BN ×N ) ∼ = alg Kπ (BN ×N ) ∼ = alg T KJ = BN ×N .
Proof. Propositions 4.5, 3.51(a), and 4.21 show that 2 QT (BN ×N ) = C∞ (HN ), QK (BN ×N ) = NNB×N , QT K (BN ×N ) = JNB×N , (4.49)
respectively. Let i stand for T , K, or T K. We know that in each case Ker i = {0}. So Corollary 3.44 combined with Theorem 3.52 implies that σi = Smbiπ is a starisomorphism of alg iπ (BN ×N ) onto BN ×N . Finally, from 1.26(e) we deduce that Smbiπ is an isometry. Remark. Thus, for A = BN ×N and B as in the above theorem, the mappings Φπ and Ψ π introduced in 4.19 are isometric starisomorphisms. Note π (A) is isometrically starisomorphic to alg T (A). This follows that alg T KM from the fact that the kernel of the mapping Φ deﬁned in 4.18 coincides with MB N ×N , which, on its hand, follows from combining the representation (4.11), Proposition 4.21(a) and Proposition 4.4(d). The following proposition shows in what a sense the preceding theorem is best possible. 4.80. Proposition. Let B be a C ∗ subalgebra of L∞ . (a) If QT (B) ⊂ C∞ (H 2 ), or QK (B) ⊂ N , or QT K (B) ⊂ J , then necessarily B ⊂ QC. (b) If there exists a bijective linear mapping of B onto alg T π (B), or onto π (B), then necessarily B ⊂ QC. alg Kπ (B), or onto alg T KJ
4.7 Toeplitz Algebras
227
Proof. (a) We ﬁrst show that B ⊂ QC whenever QT (B) ⊂ C∞ (H 2 ). Thus, suppose QT (B) ⊂ C∞ (H 2 ). If a ∈ B, then a ∈ B, and therefore, by (2.18), H(a)H() a) = T (aa) − T (a)T (a) ∈ C∞ (H 2 ).
If a(t) = n∈Z an tn , then a(t) = n∈Z an t−n (t ∈ T), and hence H() a) = H(a)∗ . But if H(a)H(a)∗ ∈ C∞ (H 2 ), then (since L(H 2 )/C∞ (H 2 ) is a C ∗ algebra) H(a)2ess = H(a)H(a)∗ ess = 0, whence H(a) ∈ C∞ (H 2 ). Theorem 2.54 now gives that a ∈ C + H ∞ . The preceding argument applied to a in place of a shows that a ∈ C + H ∞ . Thus, a ∈ QC. That B ⊂ QC if QK (B) ⊂ N is an immediate consequence of Proposition 3.51(c). Finally, if T (kλ a · kλ a) − T (kλ a)T (kλ a) = K + Cλ with K ∈ C∞ (H 2 ) and Cλ → 0 as λ → 0, then passage to the strong limit λ → ∞ gives T (aa) − T (a)T (a) = K, which, by what has been proved above, implies that a ∈ QC. Hence B ⊂ QC if QT K (B) ⊂ J . (b) Let i be T , K, or T K. Corollary 3.44 together with the fact that Ker i = {0} tells us that B is algebraically isomorphic to alg i(B)/Qi (B). Hence, from the hypothesis we deduce that alg i(B)/Qi (B) is isomorphic as a linear space to alg i(B)/J, where J is one of the ideals C∞ (H 2 ), N B , J B . In particular, the zero elements of both spaces must be equal. It follows that Qi (B) = J and part (a) completes the proof. 4.81. Theorem. Let B = C + H ∞ and let the approximate identity be generated by the Poisson kernel. Then the mappings SmbT π , SmbHπ , and SmbT HπJ are isometric isomorphisms of alg T π (BN ×N ), alg Hπ (BN ×N ), and π (BN ×N ), respectively, onto BN ×N . alg T HJ Proof. The three equalities in (4.49) hold with B = C + H ∞ and K = H by virtue of Proposition 4.5, the remark in 3.51, and Proposition 4.21(b), respectively. So we conclude as in the proof of Theorem 4.79 that the corresponding algebras are homeomorphically isomorphic. That the corresponding isomorphisms are actually isometries can now be deduced from 4.10, Proposition 3.54, and 4.19(c), respectively. We ﬁnally show how Theorem 4.79 can be “localized.” 4.82. Theorem. Let B be a C ∗ subalgebra of L∞ satisfying C ⊂ B ⊂ QC, π . Then let F be a closed subset of M (B), and let iπ stand for T π , Kπ , or T KJ π SmbiπF is an isometric starisomorphism of alg iF (BN ×N ) onto (BF )N ×N . Proof. Corollary 3.44 in conjunction with 1.26(e) shows that σiπF is an isometric starisomorphism of alg iπF (BN ×N )/QiπF (BN ×N ) onto BN ×N /Ker iπF . From Theorem 4.79 we deduce that Qiπ (BN ×N ) = {0}, and therefore QiπF (BN ×N ) is also the zero ideal. We saw in 3.60 that Ker iπF equals IF := {b ∈ BN ×N :
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bF = 0} and when proving Theorem 3.61 we established the isometric starisomorphism BN ×N /IF ∼ = (BF )N ×N . Finally, since QiπF (BN ×N ) = {0}, we may identify σiπF and SmbiπF . Our next concern are the algebras generated by P C or P QC symbols. 4.83. Proposition. The algebras alg T π (P QC),
alg Kπ (P QC),
π alg T KJ (P QC)
are commutative. π Proof. The commutativity of alg T KJ (P QC) will follow once we have shown that
{T (kλ a)T (kλ b) − T (kλ b)T (kλ a)} ∈ J
∀ a, b ∈ P QC0 .
(4.50)
In view of Lemma 4.25 it suﬃces to verify (4.50) for a, b ∈ P C0 . Moreover, there is no loss of generality in assuming that a and b have at most one discontinuity, say at the point τ ∈ T. Let χ ∈ P C0 denote any function which is continuous on T \ {τ } and satisﬁes χ(τ − 0) = 0, χ(τ + 0) = 1. Then there are α, β ∈ C and f, g ∈ C such that a = αχ + f , b = βχ + g. Hence, T (kλ a)T (kλ b) − T (kλ b)T (kλ a) = α[T (kλ χ)T (kλ g) − T (kλ g)T (kλ χ)] +β[T (kλ f )T (kλ χ) − T (kλ χ)T (kλ f )] +[T (kλ f )T (kλ g) − T (kλ g)T (kλ f )] and so Lemma 4.25 implies (4.50). The commutativity of alg T π (P QC) can be shown in the same way or π (P QC) by using the fact that Φπ is an can be derived from that of alg T KJ algebraic homomorphism. Finally, there is nothing to prove for alg Kπ (P QC). π , and let B be either C 4.84. Preliminaries. Let iπ stand for T π , Kπ , or T KJ π or QC. The maximal ideal space of alg i (P B) (whose commutativity results from the preceding proposition) will be denoted by NP B ; the possible dependence of NP B on iπ is suppressed in this notation. By the GelfandNaimark theorem 1.26(a), alg iπ (P B) is isometrically starisomorphic to C(NP B ). Because alg iπ (B) is a C ∗ subalgebra of alg iπ (P B) that is isometrically starisomorphic to B ∼ = C(M (B)) (Theorem 4.79), we regard B as a C ∗ subalgebra π B = Mξ (alg iπ (P B)) for of alg i (P B). So the deﬁnition of the ﬁbers NP ξ ξ ∈ M (B) makes acorrect sense. By 1.27(b), these ﬁbers are nonempty and B = NP we have NP B = ξ . For ξ ∈ M (B), put ξ∈M (B)
Jξ := closidalg iπ (P B) iπ (f ) : f ∈ B, f (ξ) = 0 .
(4.51)
So alg iπ (P B)/Jξ coincides with the local algebra alg iπξ (P B) = alg iπXξ (P B) as it was deﬁned in 3.60 and 3.66.
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229
Throughout what follows suppose the index set Λ of the approximate identity {Kλ }λ∈Λ is connected. Also let χτ always denote the characteristic function of the arc (τ, τ eiπ/2 ). 4.85. Proposition. The local algebras alg iπξ (P C) and alg iπξ (P QC) are singly generated. For τ ∈ T, alg iπτ (P C) is generated by iπτ (χτ ), and sp iπτ (χτ ) = [0, 1]. The Gelfand map Γτ : alg iπτ (P C) → C[0, 1] is given by π Γτ iτ (a) (µ) = (1 − µ)a(τ − 0) + µa(τ + 0) for a ∈ P C. If ξ ∈ Mτ (QC) \ Mτ0 (QC), then alg iπξ (P QC) is isometrically isomorphic to the complex ﬁeld C and for a ∈ P QC the isomorphism Γξ is given by Γξ iπξ (a) = a(ξ, 0) and Γξ iπξ (a) = a(ξ, 1) for ξ ∈ Mτ− (QC) and ξ ∈ Mτ+ (QC), respectively. If ξ ∈ Mτ0 (QC), then alg iπξ (P QC) is generated by iπξ (χτ ), and sp iπξ (χτ ) = [0, 1]. In this case the Gelfand map Γξ : alg iπξ (P QC) → C[0, 1] is for a ∈ P QC given by π Γξ iξ (a) (µ) = (1 − µ)a(ξ, 0) + µa(ξ, 1). Proof. Every a ∈ P C is of the form a = αχτ + g, where α ∈ C and g ∈ P C is continuous at τ . Hence, iπτ (a) = αiπτ (χτ ) + g(τ ), and it follows that alg iπτ (P C) is generated by iπτ (χτ ). That the spectrum of iπτ (χτ ) is [0, 1] follows from Theorem 4.67 for iπ = T π , from Corollary 3.78 for iπ = Kπ , and from Theorem 4.24 π . Thus, by 1.19, M (alg iπτ (P C)) can be identiﬁed with [0, 1]. For for iπ = T KJ µ ∈ [0, 1], denote the multiplicative linear functional on alg iπτ (P C) which sends iπτ (χτ ) into µ also by µ. Then π Γτ iτ (a) (µ) = αµ iπτ (χτ ) + g(τ ) = αµ + g(τ ) = (1 − µ)a(τ − 0) + µa(τ + 0), as desired. The algebra alg iπξ (P QC) is generated by the elements iπξ (a), where a
n ranges through P QC0 . Every such a is of the form a = αχτ q + i=1 pi qi , where α ∈ C, q ∈ QC, pi ∈ P C0 is continuous at τ , and qi ∈ QC. Hence, iπξ (a) = αq(ξ)iπξ (χτ ) + i pi (τ )qi (ξ), and what results is that alg iπξ (P QC) is generated by iπξ (χτ ). If ξ ∈ Mτ− (QC) \ Mτ0 (QC), then χτ (Xξ ) = {0} by Theorem 3.36(a), (c), and thus sp iπξ (χτ ) = {0} by Theorem 3.61. So 1.19 specializes to give that alg iπξ (P QC) is isometrically isomorphic to C = C({0}) and that, for a as above, the isomorphism is given by π Γξ iξ (a) (0) = α · 0 · q(ξ) + pi (τ )qi (ξ) pi (ξ, 0)qi (ξ, 0) = a(ξ, 0). = αχτ (ξ, 0)q(ξ, 0) + A simple continuity argument now shows that (Γξ iπξ (a))(0) equals a(ξ, 0) for all a ∈ P QC. The situation is the same for ξ ∈ Mτ+ (QC) \ Mτ0 (QC).
4 Toeplitz Operators on H 2
230
Now suppose ξ ∈ Mτ0 (QC). Then, by Theorem 3.36(a), (d), χτ (Xξ ) is the doubleton {0, 1}. So sp iπξ (χτ ) = [0, 1]: this is Theorem 4.67 for iπ = T π , this π then Theorem 4.24 follows from Theorem 3.79 for iπ = Kπ , and for iπ = T KJ applies. If µ ∈ [0, 1], we let again µ denote the multiplicative linear functional on alg iπξ (P QC) which assumes the value µ at iπξ (χτ ). Thus, for a as above,
pi (τ )qi (ξ) Γξ iπξ (a) (µ) = αµ iπξ (χτ ) q(ξ) + = αµq(ξ) + pi (τ )qi (ξ) = (1 − µ)a(ξ, 0) + µa(ξ, 1),
because a(ξ, 0) = pi (τ )qi (ξ) and a(ξ, 1) = αq(ξ)+ pi (τ )qi (ξ). Since P QC0 is dense in P QC, it follows that (Γξ iπξ (a))(µ) equals (1 − µ)a(ξ, 0) + µa(ξ, 1) for all a ∈ P QC. 4.86. Theorem. The maximal ideal space NP C of alg iπ (P C) is the cylinder T × [0, 1] and the Gelfand map Γ : alg iπ (P C) → C(NP C ) is for a ∈ P C given by π Γ i (a) (τ, µ) = (1 − µ)a(τ − 0) + µa(τ + 0), (τ, µ) ∈ T × [0, 1]. Proof. Let hτ denote the canonical homomorphism of the algebra alg iπ (P C) onto the algebra alg iπτ (P C) = alg iπ (P C)/Jτ . If (τ, µ) ∈ T × [0, 1], then due to the preceding proposition the mapping given for a ∈ P C by iπτ (a) → (1 − µ)a(τ − 0) + µa(τ + 0) extends to a multiplicative linear functional vτ,µ on alg iπτ (P C), and thus vτ,µ ◦ hτ is in NP C . Therefore, if we identify (τ, µ) with vτ,µ ◦ hτ , then π Γ i (a) (τ, µ) = (vτ,µ ◦ hτ )(iπ (a)) = vτ,µ (iπτ (a)) = (1 − µ)a(τ − 0) + µa(τ + 0). C Now suppose v ∈ NP C . Then there is a τ ∈ T such that v ∈ NP (recall τ 4.84). From (4.51) it is obvious that v(Jτ ) = {0}. Hence, the mapping
u : alg iπ (P C)/Jτ → C,
hτ c → v(c)
is well deﬁned (i.e., v(c1 ) = v(c2 ) whenever hτ c1 = hτ c2 ) and is a multiplicative linear functional. Since alg iπ (P C)/Jτ = alg iπτ (P C), it follows from the preceding proposition that there is a µ ∈ [0, 1] such that v(iπ (a)) = u(iπτ (a)) = (1 − µ)a(τ − 0) + µa(τ + 0) ∀ a ∈ P C.
4.87. Theorem. The maximal ideal space NP QC of alg iπ (P QC) can be identiﬁed with a (proper) subset of M (QC) × [0, 1]: 0 0 NP QC = M− (QC) × {0} ∪ M 0 (QC) × [0, 1] ∪ (M+ (QC) × {1} , where
4.7 Toeplitz Algebras
/ 0 Mτ± (QC) \ Mτ0 (QC) , M± (QC) :=
M 0 (QC) :=
τ ∈T
/
231
Mτ0 (QC).
τ ∈T
The Gelfand map Γ : alg iπ (P QC) → C(NP QC ) follows: π Γ i (a) (ξ, 0) = a(ξ, 0) π Γ i (a) (ξ, 1) = a(ξ, 1) π Γ i (a) (ξ, µ) = (1 − µ)a(ξ, 0) + µa(ξ, 1)
is given for a ∈ P QC as for
0 ξ ∈ M− (QC),
for
0 ξ ∈ M+ (QC),
for
ξ ∈ M 0 (QC).
Proof. We denote the canonical homomorphism of the algebra alg iπ (P QC) onto the algebra alg iπξ (P QC) = alg iπ (P QC)/Jξ by hξ . Let ξ ∈ Mτ− (QC) \ Mτ0 (QC). Then, by Proposition 4.85, the mapping given for a ∈ P QC by iπξ (a) → a(ξ, 0) extends to an isometric isomorphism Γξ of alg iπξ (P QC) onto C. It follows that Γξ ◦hξ is in NP QC , and if we identify Γξ ◦ hξ with (ξ, 0), then π Γ i (a) (ξ, 0) = (Γξ ◦ hξ )(iπ (a)) = Γξ iπξ (a) = a(ξ, 0) for a ∈ P QC. We have an analogous situation for ξ ∈ Mτ+ (QC) \ Mτ0 (QC). Now let ξ ∈ Mτ0 (QC). Then, again by Proposition 4.85, the mapping deﬁned for a ∈ P QC by iπξ (a) → (1 − µ)a(ξ, 0) + µa(ξ, 1) extends to a multiplicative linear functional vξ,µ on alg iπξ (P QC). So vξ,µ ◦ hξ is in NP QC , and on identifying vξ,µ ◦ hξ with (ξ, µ) we have π Γ i (a) (ξ, µ) = (vξ,µ ◦ hξ )(iπ (a)) = vξ,µ iπξ (a) = (1 − µ)a(ξ, 0) + µa(ξ, 1). QC Finally, let v ∈ NP , where ξ ∈ M (QC). Then v(Jξ ) = {0} by virtue of ξ (4.51). This implies that the mapping
u : alg iπ (P QC)/Jξ → C,
hξ c → v(c)
is well deﬁned and is a multiplicative linear functional. Taking into account that alg iπ (P QC)/Jξ = alg iπξ (P QC) and applying Proposition 4.85 we con0 (QC) or v(iπ (a)) = a(ξ, 1) clude that either v(iπ (a)) = a(ξ, 0) with ξ ∈ M− 0 π with ξ ∈ M+ (QC) or v(i (a)) = (1 − µ)a(ξ, 0) + µa(ξ, 1) with ξ ∈ M 0 (QC) and µ ∈ [0, 1] for all a ∈ P QC. Remark. The two preceding theorems show that NP B = M (alg iπ (P B)) does not depend on iπ . In particular, the algebras alg T π (P B), alg Kπ (P B), π (P B) are isometrically starisomorphic to each other. alg T KJ
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4.88. The Gelfand topologies on NP C and NP QC . The Gelfand topology on NP B is the coarsest (weakest) topology that makes Γ (iπ (a)) continuous for every a ∈ P B0 . It is therefore standard routine to check that this topology can be described as follows. The space NP C . An open neighborhood base of (τ, µ), where τ ∈ T and 0 < µ < 1, is formed by the sets {τ } × (µ − ε, µ + ε),
0 < ε < min{µ, 1 − µ}.
The sets [τ, τ eiε ) × (1 − ε, 1] ∪ (τ, τ eiε ) × [0, 1 − ε] ,
0 < ε < 1,
form an open neighborhood base of (τ, 1), and the sets −iε (τ e , τ ] × [0, ε) ∪ (τ e−iε , τ ) × [ε, 1] , 0 < ε < 1, form an open neighborhood base of (τ, 0). The space NP QC . First recall Remark 3 of 3.36. For ξ ∈ Mτ± (QC)\Mτ0 (QC), let V ± (ξ) be the family of all sets V ∈ V(ξ) satisfying V = Vτ ∪ Vτ± . Then, for ξ ∈ Mτ+ (QC) \ Mτ0 (QC), the sets 0 1 (Vτ × {1}) ∪ (Vτ+ × [0, 1]) ∩ NP QC , V ∈ V + (ξ), form an open neighborhood base of (ξ, 1). For ξ ∈ Mτ− (QC) \ Mτ0 (QC), the sets 1 0 (Vτ × {0}) ∪ (Vτ− × [0, 1]) ∩ NP QC , V ∈ V − (ξ), form an open neighborhood base of (ξ, 0). The sets {ξ} × (µ − ε, µ + ε),
0 < ε < min{µ, 1 − µ},
form an open neighborhood base of the point (ξ, µ) ∈ Mτ0 (QC) × (0, 1). For ξ ∈ Mτ0 (QC), the sets 0 1 (Vτ × (1 − ε, 1]) ∪ (Vτ+ × [0, 1 − ε]) ∩ NP QC , V ∈ V(ξ), 0 < ε < 1, form an open neighborhood base of (ξ, 1), and the sets 0 1 (Vτ × [0, ε)) ∪ (Vτ− × [ε, 1]) ∩ NP QC , V ∈ V(ξ),
0 < ε < 1,
form an open neighborhood base of (ξ, 0). 4.89. Corollary. Let B = C or B = QC. Then the mappings Φπ and Ψ π are π (P BN ×N ) onto alg T π (P BN ×N ) and isometric starisomorphisms of alg T KJ alg Kπ (P BN ×N ), respectively. Each of these algebras is via ΓN ×N , Γ being the Gelfand map, isometrically starisomorphic to [C(NP B )]N ×N . Thus, π alg T π (P BN ×N ) ∼ (P BN ×N ) ∼ = alg Kπ (P BN ×N ) ∼ = alg T KJ = [C(NP B )]N ×N .
4.7 Toeplitz Algebras
Proof. Immediate from the remark in 4.87.
233
We now describe the structure of Toeplitz algebras generated by C or QC and a characteristic function. 4.90. Deﬁnition. Let E be a measurable subset of T and let χE denote the characteristic function of E. Deﬁne CE := algL∞ (χE , C),
QCE := algL∞ (χE , QC).
Note that both CE and QCE are C ∗ algebras. 4.91. Lemma. Let B = C or B = QC. Then BE = f χE + g : f ∈ B, g ∈ B . Proof. Clearly, it suﬃces to show that f χE + g : f ∈ B, g ∈ B = hχE + gχE c : h ∈ B, g ∈ B is closed (here E c := T \ E). The mapping B → L∞ , b → bχE is an algebraic starisomorphism, and therefore its image, the set χE B := {bχE : b ∈ B}, is closed by 1.26(e). It follows analogously that χE c B is closed. Now let a ∈ L∞ and suppose there are hn , gn such that a − hn χE − gn χE c ∞ → 0 as
n → ∞.
Then aχE − hn χE ∞ → 0 as n → ∞, and since χE B is closed, there is an h ∈ B with aχE = hχE . It can be shown similarly that aχE c = gχE c with some g ∈ B, which completes the proof. π } and B ∈ {C, QC}. Then alg iπ (BE ) 4.92. Lemma. Let iπ ∈ {T π , Kπ , T KJ is commutative.
Proof. In view of the preceding lemma it is enough to verify that iπ (a)iπ (χE ) = iπ (aχE ) for every a ∈ B. But this follows from Proposition 2.14 and Theorem 2.54 for iπ = T π , from Theorem 3.23 for iπ = Kπ , and from Lemma 4.25 π . for iπ = T KJ π } and B ∈ {C, QC}. Then, for 4.93. Proposition Let iπ ∈ {T π , Kπ , T KJ π ξ ∈ M (B), the local algebras alg iξ (BE ) are singly generated by iπξ (χE ). For τ ∈ T, we have sp iπτ (χE ) = conv χE (Xτ ), i.e.,
sp iπτ (χE ) = {1}
if
χE (Xτ ) = {1},
= {0}
if
χE (Xτ ) = {0},
if
χE (Xτ ) = {0, 1}.
sp iπτ (χE )
sp iπτ (χE ) = [0, 1] For ξ ∈ M (QC), we have
sp Tξπ (χE ) = sp T Kξπ (χE ) = conv χE (Xξ ). Remark. We have not been able to prove that sp Kξπ (χE ) = conv χE (Xξ ) for ξ ∈ M (QC). In this connection recall 4.68.
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Proof. Lemma 4.91 implies that alg iπξ (BE ) is generated by iπξ (χE ). The identiﬁcation of sp iπτ (χE ) as conv χE (Xτ ) follows from Theorem 4.67 for iπ = T π , from Corollary 3.78(a) for iπ = Kπ , and then from Theorem 4.24 for π iπ = T KJ . Finally, Theorem 4.67 and subsequent application of Theorem 4.24 give the last assertion of the present proposition. π . Then the 4.94. Theorem. Let B be C or QC and let iπ be T π , Kπ , or T KJ π maximal ideal space M (alg i (BE )) can be identiﬁed with the subset / / 0 1 {ξ} × M alg iπξ (BE ) ∼ {ξ} × sp iπξ (χE ) = ξ∈M (B)
ξ∈M (B)
of M (B) × C, and the Gelfand map / π Γ : alg i (BE ) → C
π {ξ} × M alg iξ (BE )
ξ∈M (B)
is for f χE + g ∈ BE (f, g ∈ B) given by π Γ i (f χE + g) (ξ, w) = f (ξ)w iπξ (χE ) + g(ξ). Proof. By Lemma 4.91, B can be identiﬁed with the subset {0·χE +g : g ∈ B} of BE , and, accordingly, on identifying B with alg iπ (B) we may regard B as a C ∗ subalgebra of alg iπ (BE ). So, as in 4.84, alg iπξ (BE ) = alg iπ (BE )/Jξ with Jξ := closidalg iπ (BE ) iπ (f ) : f ∈ B, f (ξ) = 0 . Now the same reasoning as in the proof of Theorem 4.86 (or 4.87) completes the proof. 4.95. The algebra alg {T (P C), H(P C)}. Given a closed subalgebra A of L∞ let alg T H(A) denote the smallest closed subalgebra of L(H 2 ) containing all Toeplitz and all Hankel operators with symbol in A: alg T H(A) := alg {T (A), H(A)} := alg T (a), H(a) : a ∈ A . If C ⊂ A, then C∞ (H 2 ) ⊂ alg T H(A) (Proposition 4.5). Denote the quotient algebra alg T H(A)/C∞ (H 2 ) by alg T H π (A), and for B ∈ alg T H(A) let B π denote the coset B + C∞ (H 2 ). Since H(c) is compact for every c ∈ C, we have alg T H(C) = alg T (C),
alg T H π (C) = alg T π (C) ∼ = C.
The purpose of what follows is to analyze the C ∗ algebra alg T H(P C). From Proposition 4.51 we know that alg T H(P C) is strictly larger that alg T (P C). c}, where ) c(t) := c(1/t) = c(t) (t ∈ T). It is easily Let Cs := {c ∈ C : c = ) seen that Cs is a C ∗ subalgebra of C and that the maximal ideal space of Cs is homeomorphic to the closed upper halfcircle with its usual topology,
4.7 Toeplitz Algebras
235
M (Cs ) = T+ := {t ∈ T : Im t ≥ 0}. The Gelfand map Γ : Cs → C(T+ ) is of course given by (Γ c)(τ ) = c(τ ). If a ∈ L∞ and c ∈ Cs , then T π (a)T π (c) = T π (c)T π (a) and c) − T π (a)H π () c) = H π (ac) H π (a)T π (c) = H π (a) π π π = H (ca) = T (c)H (a) + H π (c)T π () a) = T π (c)H π (a) (Proposition 2.14). Consequently, if we identify a function c ∈ Cs with the coset (= essential Toeplitz operator) T π (c), then Cs may be viewed as a closed subalgebra of the center of alg T H π (P C). For τ ∈ T+ , let Jτπ denote the smallest closed twosided ideal of the algebra alg T H π (P C) containing the set {T π (c) : c ∈ Cs , c(τ ) = 0}, put alg T Hτπ (P C) = alg T H π (P C)/Jτπ , and for B ∈ alg T H(P C) denote the coset B π + Jτπ by Bτπ . Theorem 1.35(a) implies that, for B ∈ alg T H(P C), / / π spess B = sp Bτπ = sp Bτπ ∪ sp B−1 ∪ sp B1π τ ∈T+
(4.52)
τ ∈T◦ +
where T◦+ := T+ \ {−1, 1}. Let a, b ∈ P C and τ ∈ T◦+ . It can be veriﬁed without diﬃculty that aXτ ∪ Xτ = bXτ ∪ Xτ =⇒ Tτπ (a) = Tτπ (b),
Hτπ (a) = Hτπ (b).
(4.53)
On the other hand, if a, b ∈ P C and τ ∈ {−1, 1}, aXτ = bXτ =⇒ Tτπ (a) = Tτπ (b),
Hτπ (a) = Hτπ (b).
(4.54)
Note that if τ ∈ T◦+ , then Xτ ∪ Xτ is the ﬁber of M (L∞ ) over τ ∈ M (Cs ): Xτ ∪ Xτ = x ∈ M (L∞ ) : f (x) = f (τ ) ∀ f ∈ Cs . 4.96. Lemma. Let τ ∈ T+ and let χE be the characteristic function of any arc E one endpoint of which is τ . Then sp Tτπ (χE ), the spectrum of Tτπ (χE ) in alg T Hτπ (P C), is equal to the segment [0, 1]. Proof. Let θ ∈ P C be any function which coincides with χE on some (sufﬁciently small) arcs Uτ τ and Uτ τ and is continuous and takes values in A := {z ∈ C : z − 1/2 = 1/2, Im z ≥ 0} on T \ (Uτ ∪ Uτ ). We know that the spectrum of T π (θ) in alg T π (P C) is [0, 1] ∪ A and so, by 1.16(b) or 1.26(d), the spectrum of T π (θ) in alg T H π (P C) also equals [0, 1] ∪ A. Using (4.52)–(4.54) it is easily seen that sp Ttπ (θ) = {θ(t), θ(t)} for t ∈ T+ \ {τ }, and (4.53) immediately gives that sp Tτπ (θ) = sp Tτπ (χE ). Hence, by (4.52), {θ(t), θ(t)} ∪ sp Tτπ (χE ), which shows that sp Tτπ (χE ) = [0, 1]. [0, 1] ∪ A =
t =τ
4 Toeplitz Operators on H 2
236
4.97. Theorem. Let τ ∈ {−1, 1} and let χτ be the characteristic function of the arc (τ, τ eiπ/2 ). The algebra alg T Hτπ (P C) is singly generated by Tτπ (χτ ) and the spectrum of Tτπ (χτ ) is the segment [0, 1]. If a ∈ P C and µ ∈ [0, 1], then the Gelfand transform of Tτπ (a) and Hτπ (a) at µ is given by Γτ Tτπ (a) (µ) = a(τ + 0)µ + a(τ − 0)(1 − µ), 6 Γτ Hτπ (a) (µ) = −iτ [a(τ + 0) − a(τ − 0)] µ(1 − µ). Proof. Every function a ∈ P C can be written in the form a = λωτ + c, where λ = a(τ +0)−a(τ −0), c ∈ P C is continuous at τ and satisﬁes c(τ ) = a(τ −0), and ωτ is the characteristic function of the arc (τ, τ eiπ ), i.e., of the halfcircle following the point τ . Consequently, by (4.54), Tτπ (a) = λTτπ (ωτ ) + c(τ ),
Hτπ (a) = λHτπ (ωτ )
(note that Hτπ (c) = c(τ )Hτπ (1) = 0). Proposition 2.14 gives that T (ωτ ) = T (ωτ2 ) = T (ωτ )T (ωτ ) + H(ωτ )H(1 − ωτ ) = T (ωτ )T (ωτ ) − H(ωτ )H(ωτ ), whence [H(ωτ )]2 = −T (ωτ )(I − T (ωτ )).
(4.55)
The nth Fourier coeﬃcient (ω1 )n of ω1 equals (ω1 )n = Hence, if f (t) =
1 2π
j≥0
(H(ω1 )f, f ) =
(
π
e−inθ dθ =
0
1 2πi
1
xn−1 dx −1
(n ≥ 1).
fj tj (t ∈ T) is in PA , then
(ω1 )j+k+1 fj fk =
j,k≥0
=
(
1 2πi
(
( 1 1 fj fk xj+k dx 2πi −1 j,k≥0
1
−1
f j xj
j≥0
fk xk dx.
k≥0
It follows that iH(ω1 ) is a positive operator and therefore we may deduce from (4.55) that 0 11/2 H(ω1 ) = −i T (ω1 ) I − T (ω1 ) . A similar argument yields the equality 0 11/2 . H(ω−1 ) = i T (ω−1 ) I − T (ω−1 ) Since Tτπ (ωτ ) = Tτπ (χτ ) (see (4.54)) and sp Tτπ (χτ ) = [0, 1] (Lemma 4.96), all assertions of the theorem now follow straightforwardly.
4.7 Toeplitz Algebras
237
4.98. Deﬁnitions. (a) Suppose τ ∈ T◦+ . Let ψτ be the characteristic function of the arc (τ, τ ), i.e., of the arc {eiθ : θ0 < θ < 2π − θ0 }, where τ = eiθ0 . .τ . Furthermore, let ϕτ ∈ C be any function such that Note that ψτ2 = ψτ = ψ .τ = 1. Put 0 ≤ ϕτ ≤ 1, ϕτ (τ ) = 1, ϕτ (τ ) = 0, ϕτ + ϕ q := qτ := Tτπ (ψτ ) + Hτπ (ψτ ),
p := pτ := Tτπ (ϕτ )
and let e := eτ denote the identity element of alg T Hτπ (P C). (b) Let A be a C ∗ algebra. Given a ﬁnite subset {a1 , . . . , ak } of A let C (a1 , . . . , ak ) denote the smallest C ∗ subalgebra of A containing the set {a1 , . . . , ak }. An element a ∈ A is called selfadjoint if a = a∗ and is said to be an idempotent if a2 = a. ∗
4.99. Lemma. Let τ ∈ T◦+ . The elements p and q are selfadjoint idempotents and alg T Hτπ (P C) = C ∗ (p, q, e). Moreover, pqp = Tτπ (ψτ ϕτ ),
pq(e − p) = Hτπ (ψτ ϕτ ),
.τ ), (e − p)qp = Hτπ (ψτ ϕ
(e − p)q(e − p) = Tτπ (ψτ ϕ .τ ).
(4.56) (4.57)
The spectrum of pqp in alg T Hτπ (P C) is [0, 1]. Proof. It is clear that p = p∗ and q = q ∗ . Proposition 2.14 shows that T (ψτ ) + H(ψτ ) T (ψτ ) + H(ψτ ) = T (ψτ )T (ψτ ) + H(ψτ )H(ψτ ) + H(ψτ )T (ψτ ) + T (ψτ )H(ψτ ) = T (ψτ ) + H(ψτ ). Thus T (ψτ ) + H(ψτ ) is a projection and therefore q 2 = q. Since Tτπ (ϕτ )Tτπ (ϕτ ) = Tτπ (ϕ2τ ) (Proposition 2.14) and Tτπ (ϕ2τ ) = Tτπ (ϕτ ) (see (4.53)), we deduce that p2 = p. We now prove (4.56), (4.57). A few application of Proposition 2.14 gives T π (ϕτ )T π (ψτ )T π (ϕτ ) = T π (ϕ2τ ψτ ), .τ ) = H π (ϕτ ψτ ϕ .τ ), T π (ϕτ )H π (ψτ )T π (ϕτ ) = T π (ϕτ )H π (ψτ ϕ whence, by (4.53), .τ ) = Tτπ (ϕ2τ ψτ ) = Tτπ (ϕτ ψτ ). pqp = Tτπ (ϕ2τ ψτ ) + Hτπ (ϕτ ψτ ϕ The remaining three equalities can be proved analogously.
4 Toeplitz Operators on H 2
238
From (4.53) we obtain that Tτπ (ϕτ ψτ ) = Tτπ (χE ), where E is some subarc of the arc (τ, −1). So Lemma 4.96 implies that sp Tτπ (ϕτ ψτ ) = [0, 1], i.e., sp (pqp) = [0, 1]. It remains to show that alg T Hτπ (P C) = C ∗ (p, q, e). Let a ∈ P C and write a in the form λψτ + c, where λ = a(τ + 0) − a(τ − 0) and c ∈ P C is continuous at τ and satisﬁes c(τ ) = a(τ − 0). It follows that Tτπ (aϕτ ) = λTτπ (ψτ ϕτ ) + Tτπ (cϕτ ) = pqp + c(τ )p. Writing a = κψτ + d, where κ = a(τ − 0) − a(τ + 0) and d ∈ P C is continuous at τ and satisﬁes d(τ ) = a(τ + 0), we obtain that Tτπ a(1 − ϕτ ) = κTτπ ψτ (1 − ϕτ ) + Tτπ d(1 − ϕτ ) .τ ) + Tτπ d(1 − ϕτ ) = κTτπ (ψτ ϕ = κ(e − p)q(e − p) + d(τ )(e − p). Thus Tτπ (a) = [a(τ + 0) − a(τ − 0)]pqp + [a(τ − 0) − a(τ + 0)](e − p)q(e − p) +a(τ − 0)p + a(τ + 0)(e − p).
(4.58)
It can be shown similarly that Hτπ (a) = [a(τ + 0) − a(τ − 0)]pq(e − p) + [a(τ − 0) − a(τ + 0)](e − p)qp. (4.59) This completes the proof.
4.100. Two projections theorem. Let A be a C ∗ algebra with identity e and let p, q ∈ A be selfadjoint idempotents such that the spectrum of pqp is [0, 1]. Let C2×2 [0, 1] denote the C ∗ algebra of all continuous C2×2 valued functions on [0, 1], let e be the identity element of C2×2 [0, 1], and deﬁne p, q ∈ C2×2 [0, 1] by 6 10 µ µ(1 − µ) 6 p(µ) = , q(µ) = (µ ∈ [0, 1]). 00 µ(1 − µ) 1 − µ Then C ∗ (p, q, e) is isometrically starisomorphic to C ∗ (p, q, e) and the isomorphism takes p, q, e into p, q, e, respectively. Proof. See Halmos [264] (also recall 1.26(b)). 4.101. Theorem. If τ ∈
T◦+ ,
then the mapping
Γτ : Tτπ (a) + Hτπ (a) 6 a(τ + 0)µ + a(τ − 0)(1 − µ) [b(τ + 0) − b(τ − 0)] µ(1 − µ) → 6 [b(τ − 0) − b(τ + 0)] µ(1 − µ) a(τ − 0)(1 − µ) + a(τ + 0)µ
4.7 Toeplitz Algebras
239
extends to an isometric starisomorphism of alg T Hτπ (P C) onto C ∗ (p, q, e), where p, q, e are as in the previous theorem. Proof. Lemma 4.99 and Theorem 4.100 show that the algebras alg T Hτπ (P C) and C ∗ (p, q, e) are isometrically starisomorphic. From (4.58) and (4.59) we deduce that the isomorphism takes Tτπ (a) and Hτπ (b) into
[a(τ + 0) − a(τ − 0)]µ + a(τ − 0)
0
0
[a(τ − 0) − a(τ + 0)](1 − µ) + a(τ + 0)
and
0
6 [b(τ − 0) − b(τ + 0)] µ(1 − µ) respectively.
6 [b(τ + 0) − b(τ − 0)] µ(1 − µ)
,
0
Theorems 4.97 and 4.101 along with (4.52) yield a Fredholm criterion for operators in alg T H(P C). We conﬁne ourselves to stating a consequence of these theorems for the spectral theory of Hankel operators. If c ∈ C, then sp H(c) = spess H(c) = {0}. Indeed, since H(c) is compact, we have spess H(c) = {0}, and if λ = 0, then −1/λ − (1/λ2 )H(c) is the inverse of H(c)−λI (note that (I −P cQ)−1 = I +P cQ). The following result describes the essential spectrum of Hankel operators with P C symbols. 4.102. Corollary (Power). For b ∈ P C and τ ∈ T, put bτ :=
1 [b(τ + 0) − b(τ − 0)]. 2
Then H(b)ess = max bτ , τ ∈T
spess H(b) = [0, ib−1 ] ∪ [0, −ib1 ] ∪
(4.60) /
6 6 − i bτ bτ , i bτ bτ .
(4.61)
τ ∈T◦ +
Proof. Notice that λI − H(b) = λT (1) − H(b), and combine Theorems 4.97 and 4.101 with Theorem 1.35(d) to get (4.60) and with Theorem 1.35(a) (or (4.52)) to get (4.61).
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4 Toeplitz Operators on H 2
4.8 The Role of the Harmonic Extension We conclude this chapter by discussing some questions on the connection between Fredholmness, local sectoriality, harmonic approximation, and stable convergence. 4.103. Special symbol classes. Let {Kλ }λ∈Λ be any approximate identity whose index set Λ is connected. We have proved that for a ∈ P2 C or a ∈ P QC the following implications hold.
Here B = C for a ∈ P2 C and B = QC for a ∈ P QC; “iπ (a) is invertible” means invertibility in alg iπ (L∞ ). The function a = 1/2 − χE , where E is any subarc of T, is in GL∞ but neither {kλ a}π nor T π (a) nor {T (kλ a)}πJ are invertible. In case {Kλ } is generated by the Poisson kernel we write {hr a} in place of {kλ a}. If a ∈ C + H ∞ , then the following implications are valid:
It would be interesting to know a function in G(C + H ∞ ) which is not locally sectorial over C + H ∞ .
4.8 The Role of the Harmonic Extension
241
4.104. L∞ symbols. For a ∈ L∞ , we established the following implications.
We have not been able to show that the implication (*) cannot be reversed. However, it turns out that none of the remaining implications can be reversed. To see this it suﬃces to show that there exist b, c ∈ L∞ such that T π (b) is invertible but {hr b}π is not invertible , {hr c}π is invertible but T π (c) is not invertible.
(4.62) (4.63)
Douglas’ symbol. This is a symbol b satisfying (4.62). It can be constructed as follows. Deﬁne b0 ∈ C by b0 (eiθ ) = e2iθ
(0 < θ < π),
b0 (eiθ ) = e−2iθ
(π < θ < 2π).
(4.64)
So T (b0 ) is invertible and b0 (0), the harmonic extension of b0 at the point 0 (= zeroth Fourier coeﬃcient), is zero. Let ω be an inﬁnite Blaschke product and put b := b0 ◦ ω. Then T (b) is invertible by Proposition 2.20(c). However, (b0 ◦ ω)(z) = b0 (ω(z)) is zero whenever ω(z) is zero, that is, 0 ∈ ClH (b, T). Hence, by Theorem 3.76(b), {hr b}π is not invertible. Wolﬀ ’s symbol. This is a symbol c which satisﬁes (4.63).
Its construction ∞ is as follows. Let M ≥ 4 be an integer. For x ∈ (0, 1) let x = j=1 εj (x)M −j be the base M expansion of x, put p(x) := min{j : εj (x) = 0 or M − 1}, and deﬁne g by 2 ++ θ ++ g(eiθ ) = log 2 − + p + + − 1 log(1 − 2M ), θ ∈ (−π, π). M π *1
∞ Since 0 p(x) dx = n=1 2n(M − 2)n−1 M −n < ∞, we have g ∈ L1 . Then let c = ei)g , where g) refers to the conjugate function of g. Wolﬀ [581] showed that for M suﬃciently large, c satisﬁes (4.63). The proof is rather complicated and therefore omitted here. Neither the Douglas symbol nor the Wolﬀ symbol is locally sectorial over C+H ∞ , since otherwise {hr b}π or T π (c) were invertible. It is easy to construct
242
4 Toeplitz Operators on H 2
symbols which are locally sectorial over QC but not over C: if we let ω be as in 2.80, then a = e2πi Im ω is in GQC and thus locally sectorial over QC, but there is a τ ∈ T such that a(Xτ ) = T (Proposition 2.79(a)), so that a is not locally sectorial over C. Thus, we have a symbol a for which {kλ a}π , T π (a), {T (kλ a)}πJ are invertible but which is not locally sectorial over C. The function ϕ constructed in 4.77 has the following property: {hr ϕ} is invertible, but ϕ is not locally sectorial over QC. Indeed, if ϕ were locally sectorial over QC, then so also were ϕ/ϕ, and this would imply that {hr (ϕ/ϕ)}π is invertible, a contradiction to (4.46). We ﬁnally show that symbols whose harmonic extensions is bounded sufﬁciently far away from zero in D (resp. near T) generate invertible (resp. Fredholm) Toeplitz operators. 4.105. Lemma. If a ∈ L∞ , then distL∞ (a, H ∞ ) ≤ Ba∗ ,
distL∞ (a, H ∞ ) ≤ Ba∗
with some absolute constant B; here a∗ refers to the BMO “norm” introduced in 1.47. Proof. We have distL∞ (a, H ∞ ) = inf ϕ∞ : ϕ ∈ L∞ , ϕ − a ∈ H ∞ ≤ inf ϕ∞ : ϕ ∈ L∞ , P ϕ = P a (because P ϕ = P a =⇒ ϕ − a ∈ H ∞ ) = inf u + v∞ : u, v ∈ L∞ , P (u + v) = P a ≤ inf u∞ + v∞ : u, v ∈ L∞ , P (u + v) = P a ≤ inf u∞ + v∞ : u ∈ H ∞ , v ∈ L∞ , u + P v = P a ≤ inf u∞ + v∞ : u ∈ L∞ , v ∈ L∞ , u + P v = P a , the last inequality resulting from the observation that if u is a function in L∞ and u + P v = P a, then u is in H ∞ . Hence, by 1.48(l) and (k), distL∞ (a, H ∞ ) ≤ β1 P aBM O ≤ β1 β2 aBM O with some absolute constants β1 , β2 . Thus, distL∞ (a, H ∞ ) = distL∞ (a − a0 , H ∞ ) ≤ β1 β2 a − a0 BM O = β1 β2 a − a0 ∗ = β1 β2 a∗ . The proof for distL∞ (a, H ∞ ) is analogous.
4.106. Lemma. Let {Kλ }λ∈Λ be any approximate identity. Then there exist constants D1 and D2 depending only on K and Λ such that for all unimodular a ∈ L∞ ,
4.8 The Role of the Harmonic Extension
243
1/2 a∗ ≤ D1 sup sup 1 − kλ,t a2 , λ∈Λ t∈T
1/2 M0 (a) := lim Mδ (a) ≤ D2 lim sup sup 1 − kµ,t a2 . δ→0
λ→∞ µ>λ t∈T
Proof. In the proof of Proposition 3.51(c) we established the inequality 1/2 kλ,t (a − kλ,t ) ≤ kλ,t (a2 ) − kλ,t a2 . Thus, if a = 1 a.e., then Theorem 3.21 implies that 1/2 a∗ ≤ D1 aK ≤ D1 sup sup 1 − kλ,t a2 . λ∈Λ t∈T
Moreover, when proving Proposition 3.51(c) we also observed that 1/2 M2π/λ (a) ≤ D2 sup sup 1 − kµ,t a2 , µ>λ t∈T
where D2 depends only on K and Λ. This gives the second inequality of the lemma at once. 4.107. Theorem. Let {Kλ }λ∈Λ be any approximate identity. There are constants δG = δG (K, Λ) and δΦ = δΦ (K, Λ) depending only on K and Λ which have the following property: (a) If a ∈ L∞ is unimodular and kλ,t a ≥ δG for all λ ∈ Λ and all t ∈ T, then T (a) ∈ GL(H 2 ). (b) If a ∈ L∞ is unimodular and kλ,t a ≥ δΦ for suﬃciently large λ ∈ Λ and all t ∈ T, then T (a) ∈ Φ(H 2 ). Proof. (a) Let B as in Lemma 4.105 and let D1 be the constant appearing 2 1/2 ) < 1. Lemmas 4.105 and in Lemma 4.106. Choose δG so that BD1 (1 − δG 4.106 then give that 1/2 distL∞ (a, H ∞ ) ≤ BD1 sup sup 1 − kλ,t a2 λ∈Λ t∈T
and so T (a) is rightinvertible by Proposition 2.20(a). The same argument applies to show that T (a) is leftinvertible. (b) Let A2 be the constant occurring in 1.48(f) and let B, D2 be as in Lemmas 4.105 and 4.106. Choose δΦ so that A2 BD2 (1 − δΦ2 )1/2 < 1. Since H(ϕ) is compact for ϕ ∈ QC = L∞ ∩ V M O, we have distL∞ (a, C + H ∞ ) = H(a)ess (Theorem 2.54) ≤ inf H(a − ϕ) : ϕ ∈ L∞ ∩ V M O = inf distL∞ (a − ϕ, H ∞ ) : ϕ ∈ L∞ ∩ V M O (Theorem 2.11) ∞ (Lemma 4.105) ≤ inf Ba − ϕ∗ : ϕ ∈ L ∩ V M O ∞ ≤ inf Ba − ϕBM O : ϕ ∈ L ∩ V M O = B distBM O (a, L∞ ∩ V M O). (4.65)
244
4 Toeplitz Operators on H 2
Let d := distBM O (a, V M O). Thus, there are u, v ∈ C and ϕ ∈ BM O such that a = u+P v +ϕ and ϕBM O < d+ε (1.48(l)). Choose v1 ∈ P and v2 ∈ C so that v = v1 + v2 and P v2 BM O ≤ P L(L∞ ,BM O) v2 ∞ < ε (1.48(k)). So a = u+P v1 +P v2 +ϕ with u+P v1 ∈ L∞ ∩V M O and P v2 +ϕBM O < d+2ε. Hence, (4.66) distBM O (a, L∞ ∩ V M O) ≤ distBM O (a, V M O). Combining (4.65), (4.66), and 1.48(f) we arrive at the inequality distL∞ (a, C + H ∞ ) ≤ BA2 M0 (a). Now Lemma 4.106 implies that distL∞ (a, C + H ∞ ) < 1 and Theorem 2.75 gives that T (a) ∈ Φ− (H 2 ). It can be shown analogously that T (a) ∈ Φ+ (H 2 ). Remark. Tolokonnikov [518], who was the ﬁrst to establish the existence of δG , showed that δG ≤ 45/46 if {Kλ }λ∈(0,∞) is generated by the Poisson kernel. In Nikolski [368], it is shown that even δG ≤ 23/24 in this case.
4.9 Notes and Comments 4.1–4.6. The approach is due to the authors (B¨ ottcher, Silbermann [106], Silbermann [483]), for the results we refer to Nikolski [366] (4.1 for p = 2), Simonenko [492] (4.2), Devinatz, Shinbrot [155] (4.2), Clancey [136] (4.3, 4.4), B¨ottcher, Silbermann [106] (4.3, 4.4), Brown, Halmos [125] (4.4(d)), Douglas [160] (4.6). The proof of 4.5 uses an argument of Coburn [142]. 4.7–4.8. Parts (a) are from Douglas [160], parts (b) are results of the authors and are published here for the ﬁrst time. 4.10–4.13. Local Toeplitz operators are an invention of Douglas [161]. The basic equality (4.8) was established by Axler [12] for N = 1 and by Silbermann [483] in the matrix case. Douglas [160], [161] stated the results of 4.12 and 4.13 for B = QC. Corollary 4.13 for B = C is Simonenko’s [492]. 4.14–4.26. Silbermann [481], [483]. 4.27–4.30. This is the approach of Silbermann [481], [482], [483]. That the index of operators in alg T (P QC) (N = 1) can be computed via the harmonic extension (i.e., as in Theorem 4.28) was ﬁrst shown by Sarason [456]. Corollary 4.30 was obtained by Faour [198] (using other methods) for the following two cases: (i) B = C and N = 1, (ii) a ∈ P2 CN,N . In all these works {Kλ } was the approximate identity generated by the Poisson kernel (i.e., the harmonic extension). For arbitrary approximate identities the results were ﬁrst proved in B¨ ottcher, Silbermann [112]. That the index of a Toeplitz operator can be expressed via the Fej´erCesaro means of the symbol is of importance in connection with Gohberg,
4.9 Notes and Comments
245
Lerer, Rodman [234], where explicit formulas for the partial indices of rational matrix functions are given (also see Chapter 1 of Clancey, Gohberg [138]). In B¨ottcher [70] (see also the remark in 8.53) it is shown that there exist matrix functions a ∈ W2×2 such that the partial indices of hr a (resp. σn a) are zero for all r ∈ (0, 1) (resp. n = 2, 3, . . .) but the partial indices of a itself are not all equal to zero. This shows that even for very smooth a the kernel and cokernel dimensions of the block Toeplitz operator T (a) cannot be recovered from the kernel and cokernel dimensions of T (hr a) and T (σn a). 4.31. These arguments, though originated by the approach of B¨ ottcher [70], are published here for the ﬁrst time. 4.32–4.34. Parrott [376], Power [403], [404]. 4.35–4.39. The results of Pousson [398] and Rabindranathan [409] are, respectively, matrix and operatorvalued analogues of the WidomDevinatz criterion in the form 2.22. The statements of 4.36 and 4.37 can be found in Clancey, Gohberg [138]; also see Devinatz, Shinbrot [155]. Corollary 4.39 is new (but see also B¨ottcher [69]). 4.40. See Sierpinski [472], for example. 4.41–4.49. This is (the matrixcase version of) Axler’s method [12]. 4.50. Result (a) goes back to Sarason [453]. The “if” part of (d) was proved by Axler, Chang, Sarason [15] and the “only if” of (d) is Volberg’s [549]. A discussion and a proof of (d) is also in Nikolski [366]. See also Sarason [457] and Power [404]. A series of new interesting results on compactness of commutators and quasicommutators of Toeplitz and Hankel operators were obtained in recent work by Gorkin, Gu, Guo, and Zheng. In Zheng [589] it is proved that the quasicommutator T (f g) − T (f )T (g) is compact on H 2 if and only if H(f )kz H(g)kz → 0 as z → 1; here kz denotes the normalized reproducing kernel in H 2 for point evaluation at z. Gorkin and and Zheng [241] proved an analogous characterization for the commutator T (f )T (g) − T (g)T (f ). The above two results were extended to the block case in Gu, Zheng [256]. MartinezAvenda˜ no [348] found necessary and suﬃcient conditions guaranteeing that T (f )H(g) = H(g)T (f ). Inspired by this work, Guo and Zheng [259] found criteria for the compactness of T (f )H(g) − H(g)T (f ). Recently Guo and Zheng [260] also established necessary and suﬃcient conditions for a product of Toeplitz operators T (f1 ) . . . T (fm ) (m ≥ 2) to be a compact perturbation of a Toeplitz operator. The approach of all these works is based on LittlewoodPaley theory and some inequalities involving the Luzin area integral and certain maximal functions. In connection with the open problems in 4.50(e) notice that Volberg and Ivanov [550] found necessary conditions in terms of martingales for H(a)H()b) to be in Cp (H 2 ) (p ≥ 2). On the other hand, Rochberg and Semmes [435, Section 7] obtained some estimates for the snumbers of H(a)H()b).
246
4 Toeplitz Operators on H 2
This is perhaps also the right place to mention the paper Gu [254]. An open question by Douglas [162] is as follows: If X is a bounded linear operator on H 2 such that X − T ∗ (θ)XT (θ) is a compact operator for every inner function θ, does it follow that X = T (a) + K with a ∈ L∞ and a compact operator K? Gu’s main result says that the answer is in the aﬃrmative provided the twice occurring “compact” is replaced by “ﬁnite rank.” A theorem by Davidson [146] says that if X is a bounded linear operator on H 2 and XT (h) − T (h)X is a compact operator for every h ∈ H ∞ , then X = T (a) + K with a compact operator K and a function a ∈ L∞ for which H(ψ) is a compact operator. As a corollary of his main result, Gu shows that this theorem remains valid if the thrice occurring “compact” is replaced with “ﬁnite rank.” For commuting ﬁnite Toeplitz matrices, normality of ﬁnite Toeplitz matrices, and ﬁnite Toeplitz matrices with Toeplitz inverses see Gu, Patton [255] and the references cited there. 4.51–4.53. The observation made in 4.51 is due to S. Axler. Theorem 4.53 was stated (but not proved) in Clancey, Gosselin [139]. Douglas [161] proved this theorem for w = 2, that is, for the case where S is a ﬁber Xξ (ξ ∈ M (QC)). The proof given here is based on Douglas’ argument. 4.54–4.65. Clancey, Gosselin [139]. For the ChangMarshall theorem see Sarason [457] and Garnett [211]. 4.67. Clancey [135], Clancey, Morrel [140], Douglas [160], [161] for B = C and Silbermann [483] for B = QC. 4.70. For several classes of symbols (P C, P2 C, P QC) this theorem had been known before it was explicitly stated in this form in Silbermann [483]. We are grateful to I. M. Spitkovsky for pointing out an error in our original proof. 4.71–4.78. B¨ottcher [68]. 4.79–4.94. The derivation of all these results rests on the observation that the corresponding local algebras are singly generated, a fact which was pointed out in Silbermann [483] for the ﬁrst time. Due to this observation the proofs given here are essentially simpler than the original ones. For Theorem 4.79 (i = T, B = C) see Mikhlin [360], Gohberg [217], Coburn [142]. Theorems 4.79 and 4.81 were established by Douglas [162], [160] for i = T . Theorem 4.86 for i = T is due to Gohberg and Krupnik [229], and Theorem 4.87 was obtained by Sarason [456] for i = T and i = H; these authors also described the Gelfand topologies of NP C and NP QC . Theorem 4.94 is new, although results of this type (i = T ) are also in Douglas [161]. It is clear that Theorem 4.94 also holds in the matrix case and that the index of a Fredholm Toeplitz operator with symbol a ∈ (BE )N ×N equals −ind {det(kλ a)}. All statements concerning the cases i = T K and i = K are due to Silbermann [481], [482], [483]. The possibility of such results being valid was suggested by the paper B¨ ottcher, Silbermann [105]. For a detailed discussion of algebras generated by Toeplitz
4.9 Notes and Comments
247
operators and for further results along these lines see also Nikolski [366], [368], [369]. 4.95–4.102. For extensions of the two projections theorem (Theorem 4.100) that are suitable for the study of operators acting on Banach spaces and having massive local spectra see, e.g., Finck, Roch, Silbermann [202], [203], Gohberg, Krupnik [233]. An N projections version was elaborated in B¨ ottcher, Gohberg, Karlovich, Krupnik, Roch, Silbermann, Spitkovsky [79] (many projections required many authors!). The application of this result to singular integral operators on Lebesgue spaces with Muckenhoupt weights over composed Carleson curves is illustrated in B¨ ottcher, Karlovich [92]. For Corollary 4.102 and its generalization to P QC see Power [399], [400], [401]. Theorem 4.101 and the approach presented here are from Silbermann [484]. In this paper, 4.101 is even proved for P QC symbols. Also see Roch, Silbermann [425]. 4.103–4.107. That there are symbols satisfying (4.62) was discovered by Douglas [160, pp. 13–14]. At the same time Douglas asked whether T (c) is invertible whenever {hr c} is bafz. Wolﬀ [581] has shown that the answer is negative by constructing the symbol mentioned in the text. Theorem 4.107(a) was established by Tolokonnikov [518] and independently also in Wolﬀ [581] (where it is attributed to A. Chang) for the case that {Kλ } is generated by the Poisson kernel.
5 Toeplitz Operators on H p
5.1 General Theorems Some questions for Toeplitz operators on H p have already answered in the preceding chapters. In particular, we settled the Fredholm theory for the oper∞ ators in algL(HNp ) T (CN ×N + HN ×N ). Also notice the localization result stated in Theorem 2.96. However, many questions still remain open and it is only a small number of them which will be answered in this chapter. 5.1. Conventions. Throughout the present chapter we suppose that p and q are real numbers satisfying 1 < p < ∞ and 1/p + 1/q = 1. When considering Lp or H p with a weight w, we always suppose that w is (Lebesgue) measurable on T, w ≥ 0 almost everywhere on T, w ∈ Lp , w−1 ∈ Lq , and that w satisﬁes the HuntMuckenhouptWheeden condition (Ap ) (see 1.46). So the Toeplitz operator T (a) is bounded on H p (w) for a ∈ L∞ . 5.2. The HartmanWintner and Coburn theorems. The HartmanWintner spectral inclusion theorem (2.30 and 2.93) continues to hold for the p p spaces H p (w): if a ∈ L∞ N ×N and T (a) ∈ Φ+ (HN (w)) or T (a) ∈ Φ− (HN (w)), ∞ as the one given for the spaces H p . then a ∈ GLN ×N . The proof is the same * Note that under the pairing (f, g) = T f g dm for f ∈ Lp (w) and g ∈ Lq (w−1 ), we may think of Lq (w−1 ) as the dual space of Lp (w); in particular, this shows that U n converges weakly to zero on Lp (w) as n → ∞. Coburn’s theorem (2.38 and 2.40) also remains valid for the spaces H p (w): if a ∈ L∞ , then T (a) ∈ GL(H p (w)) ⇐⇒ T (a) ∈ Φ(H p (w)) and Ind T (a) = 0. This can be proved in the same way as for H p . Note that by virtue of the inequality (
( T
f  dm ≤
T
1/p ( f p wp dm
w−q dm T
1/q
5 Toeplitz Operators on H p
250
H p (w) is contained in H 1 , so that the F. and M. Riesz theorem 1.40(b) is applicable to the spaces H p (w). Finally, notice that Ind T (χk ) = −k (k ∈ Z) in H p (w). 5.3. Rochberg’s invertibility criterion. This is an extension of the WidomDevinatz criterion 2.23 to Toeplitz operators on H p (w). Let a ∈ L∞ . Then T (a) ∈ GL(H p (w)) if and only if a ∈ GL∞ and a/a = ei(c+)y) , where c ∈ R and y(∈ BM O) is a realvalued function with the property that we−y/2 satisﬁes the HuntMuckenhouptWheeden condition (Ap ), i.e., sup I
1 I
(
p −py/2
w e
1/p dm
I
1 I
( w
−q qy/2
e
1/q dm
< ∞.
I
Here y) refers to the conjugate function of y. Proof. For a proof see Rochberg [433]. Let us show that this reduces to the WidomDevinatz criterion in case p = 2 and w = 1. By the HelsonSzeg˝o theorem, e−y/2 satisﬁes (A2 ) if and only if −y/2 = f + g) with f, g ∈ L∞ realvalued and g∞ < π/4. Thus, y = −2f − 2) g , hence c + y) = −2f)+ 2g + const = u ) + v + const with u, v ∈ L∞ realvalued and v∞ = 2g∞ < π/2. We now prove a theorem which provides an alternative criterion for Toeplitz operators on H p (w) in terms of a certain factorization of the symbol. 5.4. Deﬁnition. A function a ∈ GL∞ is said to admit a WienerHopf factorization in Lp (w) if it can be represented in the form a = a− χκ a+ , where κ ∈ Z, a− ∈ Lp− (w),
q −1 a−1 ), − ∈ L− (w
and sup I
a+ ∈ Lq+ (w−1 ),
p a−1 + ∈ L+ (w)
1 −1 a wLp (I) a+ w−1 Lq (I) < ∞, I +
(5.1)
(5.2)
the supremum over all subarcs I of T. In view of the HuntMuckenhouptWheeden theorem 1.46, under the condition that (5.1) holds, (5.2) may be replaced by the requirement that p ∞ and a−1 + P (a+ ϕ) ∈ L (w) for all ϕ ∈ L a−1 + P (a+ ϕ)p,w ≤ cp,w ϕp,w
∀ ϕ ∈ L∞
(5.3)
with some constant cp,w depending only on p and w. Here L∞ may be replaced by any of its subsets which is dense in Lp (w). 5.5. Theorem (Simonenko). Let a ∈ L∞ . Then T (a) ∈ Φ(H p (w)) if and only if a ∈ GL∞ and if a admits a WienerHopf factorization a = a− χκ a+ in Lp (w). In that case Ind T (a) = −κ.
5.1 General Theorems
251
Proof. First suppose a ∈ GL∞ admits the WienerHopf factorization a = a− a+ in Lp (w) (i.e. assume κ = 0). Then Ker T (a) = {0}. Indeed, if T (a)ϕ = 0 1 for ϕ ∈ Lp+ (w), then a− a+ ϕ = g− ∈ Lp− (w), hence a+ ϕ = a−1 − g− ∈ L− , and 1 since a+ ϕ ∈ L+ , it follows that a+ ϕ = 0, which implies that ϕ = 0. We now show that Im T (a) = H p (w). By (5.3), the mapping H ∞ → H p (w),
−1 −1 −1 ϕ → a−1 ϕ + P a− ϕ = a+ P a+ a
extends to a bounded operator A on H p (w). For ϕ ∈ H ∞ , −1 −1 −1 T (a)Aϕ = P a− a+ a−1 + P a− ϕ = P a− P a− ϕ = ϕ − P a− Qa− ϕ = ϕ,
and since both T (a) and A are bounded, it follows that T (a)Aϕ = ϕ for all ϕ ∈ H p (w). Thus Im T (a) = H p (w), and we have proved that T (a) is invertible. If a = a− χκ a+ , then either T (a) = T (a− a+ )T (χκ )
or T (a) = T (χκ )T (a− a+ ),
and because T (a− a+ ) has just been proved to be invertible, we deduce that T (a) ∈ Φ(H p (w)) and that Ind T (a) = Ind T (χκ ) = −κ. Now suppose that T (a) ∈ Φ(H p (w)) and Ind T (a) = −κ. Then, by 5.2, a ∈ GL∞ and T (b) ∈ GL(H p (w)), where b = aχ−κ . Let ϕ+ ∈ Lp+ (w) and ψ+ ∈ Lq+ (w−1 ) denote the solutions of the equations P (bϕ+ ) = 1, P (bψ+ ) = 1. So bϕ+ = 1 + g− and bψ+ = 1 + h+ with g− ∈ Lp− (w) and h+ ∈ Lq+ (w−1 ), hence ψ+ (1 + g− ) = bϕ+ ψ+ = ϕ+ (1 + h+ ) and since H 1 ∩ H 1 = C, we obtain that ψ+ (1 + g− ) = ϕ+ (1 + h+ ) = c = const. By the F. and M. Riesz theorem, ϕ+ = 0 and ψ+ = 0 a.e. on T. In particular, 1 + h+ = bψ+ = 0 a.e. on T, and hence c = 0. Put a+ = ϕ−1 + and a− = 1 − g− . p −1 Then a = a− χκ a+ and a− = 1 − g− ∈ L− (w), a− = c−1 ψ+ ∈ Lq− (w−1 ), p a+ = c−1 (1 + h+ ) ∈ Lq+ (w−1 ), a−1 + = ϕ+ ∈ L+ (w). −1 −1 ∞ If f ∈ H , then P ba+ P a− f = f . Since T (b) is invertible, we deduce that −1 −1 (b) f p,w ∀ f ∈ H ∞ . a−1 + P a− f p,w ≤ T ∞ Moreover, because P a−1 − f = 0 for f ∈ H , we even have −1 −1 a−1 (b) P f p,w + P a− f p,w ≤ T
and thus
∀f ∈P
−1 (b) P b∞ gp,w a−1 + P a+ gp,w ≤ T
for all g ∈ b−1 P := {h ∈ Lp (w) : bh ∈ P}. But b−1 P is clearly dense in Lp (w) and so (5.2) holds by what was said at the end of Section 5.4.
252
5 Toeplitz Operators on H p
Remark. The preceding theorem extends to the case of matrixvalued symbols as follows. p p Let a ∈ L∞ N ×N . Then T (a) ∈ GL(HN (w)) (resp. Φ(HN (w))) if and only if a admits a factorization a = a− a+ (resp. a = a− da+ , d being of the form d = diag (χκ1 , . . . , χκN ) with κj ∈ Z for all j), where
a− ∈ [Lp− (w)]N ×N ,
q −1 a−1 )]N ×N , − ∈ [L− (w
a+ ∈ [Lq+ (w−1 )]N ×N ,
p a−1 + ∈ [L+ (w)]N ×N
p and the following holds: a−1 + P a+ ϕ ∈ HN (w) for all ϕ ∈ PN and
a−1 + P (a+ ϕ)p,w ≤ cp,w ϕp,w
∀ ϕ ∈ PN .
p If T (a) ∈ Φ(HN (w)), then
dim Ker T (a) = −
κj ,
dim Coker T (a) =
κj <0
κj .
κj >0
For a proof see Simonenko [496], Clancey, Gohberg [137], or Litvinchuk, Spitkovsky [340]. Note that the above proof with only minor modiﬁcations can be used to prove the part of this result concerned with invertibility. 5.6. Convention. We have always assumed that the norm on L2N is deﬁned by ( ( N 2 2 f L2 := fj (t) dm = f (t)2CN dm, N
T
T
j=1
but so far we have not speciﬁed a norm on LpN or LpN (w). Henceforth suppose the norm on LpN (w) is given by f pLp (w) := N
( N T
p/2 fj (t)2
( wp dm =
j=1
T
f (t)pCN wp dm.
(5.4)
∞ Recall that the norm on L∞ N ×N was deﬁned as aLN ×N := M (a)L(L2N ) and that = ess sup a(t)L(CN ) = max a(x)L(CN ) . aL∞ N ×N
t∈T
x∈X
The choice of the norm (5.4) is motivated by the following proposition. 5.7. Proposition. If the norm on LpN (w) is given by (5.4), then M (a)L(LpN (w)) = aL∞ N ×N
∀ a ∈ L∞ N ×N .
(5.5)
5.2 Khvedelidze Weights
253
Proof. In (5.5) the inequality “≥” holds for every norm on LpN (w) that is equivalent to the norm (5.4). Indeed, we then have aL∞ ≤ CM (a)L(LpN (w)) N ×N with some constant C > 0 (see, e.g., (4.3)), thus a = an 1/n ≤ C 1/n M (an )1/n ≤ C 1/n M (a), and letting n go to inﬁnity, we get the “≥” in (5.5). The reverse inequality is a consequence of the particular choice of the p norm: for a ∈ L∞ N ×N and f ∈ LN (w), one has ( ( p p p af LN (w) = a(t)f (t)CN w dm ≤ a(t)pL(CN ) f (t)pCN wp dm T T ( p p p f (t)CN w dm = apL∞ f pLp (w) . ≤ aL∞ N ×N
N ×N
T
N
5.2 Khvedelidze Weights 5.8. Deﬁnition. A Khvedelidze weight is a function on T of the form (t) =
n
t − tj µj
(t ∈ T),
(5.6)
j=1
where t1 , . . . , tn are pairwise distinct points on T and µ1 , . . . , µn are real numbers. 5.9. Theorem. Let 1 < p < ∞ and 1/p + 1/q = 1. Suppose is a weight of the form (5.6). Then P is bounded on Lp () if and only if −1/p < µj < 1/q for j = 1, . . . , n. Proof. Since −1/p < µj ∀ j ⇐⇒ ∈ Lp and µj < 1/q ∀ j ⇐⇒ −1 ∈ Lq , the “only if” part results from the HuntMuckenhouptWheeden theorem 1.46. However, there is a simple direct argument to prove this part of the above (and of the HuntMuckenhouptWheeden) theorem: if P ∈ L(Lp ()), then S = 2P − I ∈ L(Lp ()), where S is the Cauchy singular integral operator (1.15), hence A := SM (χ1 ) − M (χ1 )S ∈ L(Lp ()), and thus B := A−1 belongs to L(Lp ); but ( (t) −1 (τ )ϕ(τ ) dτ, (Bϕ)(t) = πi T whence ∈ Lp and −1 ∈ Lq . Now suppose that ∈ Lp and −1 ∈ Lq . The boundedness of P will follow once we have shown that the weight satisﬁes the HuntMuckenhouptWheeden condition (Ap ): there is a constant M independent of I such that Lp (I) −1 Lq (I) ≤ M I for all arcs I of length I ≤ δ.
254
5 Toeplitz Operators on H p
Choose δ so small that the arcs (tj − 2δ, tj + 2δ) (j = 1, . . . , n) are pairwise disjoint and then choose M1 , M2 so that (t) ≤ M1 ,

−1
(t) ≤ M1
if
t∈T\
n /
(tj − δ, tj + δ),
j=1
(t) t − tj −µj ≤ M2 ,
−1 (t) t − tj µj ≤ M2
t ∈ (tj − 2δ, tj + 2δ).
if
If I ∩ (tj − δ, tj + δ) = ∅ for all j, then Lp (I) −1 Lq (I) ≤ M12 I1/p+1/q = M12 I, and if I overlaps with (tj − δ, tj + δ), then Lp (I) −1 Lq (I) ≤ M22
1/p (
( t − tj pµj dm
I 2 M2 M3 Iµj +1/p I−µj +1/q
t − tj −qµj dm
1/q
I
≤ = M22 M3 I, * with M3 arising when I t − tj α dm is replaced by an integral of the type *b x − x0 α dx. a 5.10. Convention. In what follows the letter always denotes a Khvedelidze weight satisfying the conditions of the preceding theorem. Sometimes we shall say “let be a Khvedelidze weight on Lp ” to mean that is a weight of the form (5.6) which satisﬁes −1/p < µj < 1/q for j = 1, . . . , n. 5.11. The norm of the Cauchy singular integral operator. Let t ∈ T, 1 < p < ∞, −1/p < µ < 1/q, N ≥ 1, suppose the norm in LpN (t − τ µ ) is given by (5.4), and let S = diag (S, . . . , S) for N > 1. Then SL(LpN (t−τ µ )) = cot
π , 2r
where r = max{p, q, (1/p+µ)−1 , (1/q −µ)−1 }. A proof is in in Krupnik’s book [329].
5.3 Locally p, Sectorial Symbols 5.12. Deﬁnitions. Let a be a matrix function in GL∞ N ×N , let F be a closed subset of X = M (L∞ ), and let 2 ≤ r < ∞. For r > 2, Sr will denote the sector {z ∈ C : Im z < (tan π/r)Re z}, and S2 will refer to the right open halfplane {z ∈ C : Re z > 0}. The unit sphere in CN will be denoted by SCN , i.e., SCN = {z ∈ CN : zCN = 1}. The numerical range of a matrix d ∈ CN ×N (operator d ∈ L(CN )) is deﬁned by W (d) := {(dz, z) : z ∈ SCN }. If f, g ∈ CN ×N are selfadjoint (f = f ∗ , g = g ∗ ), then f ≥ g (resp. f > g) will mean that (f z, z) ≥ (gz, z) (resp. (f z, z) > (gz, z)) for all z ∈ SCN . In
5.3 Locally p, Sectorial Symbols
255
case g = αI (α ∈ R), we shall write f ≥ α and f > α instead of f ≥ αI and f > αI, respectively. The matrix function a is said to be rsectorial on F if there are c, d ∈ GCN ×N such that W (ca(x)d) ⊂ Sr for all x ∈ F . Note that W (ca(x)d) ⊂ Sr (r ≥ 2) if and only if π Re (ca(x)dz, z) > cos (ca(x)dz, z) ∀ z ∈ SCN . (5.7) r It is easily seen that each of the two conditions π ca(x)d Re (a(x)d) > cos r and max I − ca(x)dL(CN ) < sin x∈F
(5.8)
π r
(5.9)
is suﬃcient for (5.7) to hold. Also notice that, for r > 2, (5.7) is equivalent to π Im (ca(x)dz, z) < tan Re (ca(x)dz, z) ∀ z ∈ SCN . (5.10) r Since {(ca(x)dz, z) : x ∈ F, z ∈ SCN } is compact, the matrix function a is 2sectorial on F if and only if there are c, d ∈ GCN ×N and ε > 0 such that Re (ca(x)d) ≥ ε for all x ∈ F , i.e., if and only if a is (analytically) sectorial on F in the sense of Deﬁnition 3.1. It is clear that a scalarvalued function a ∈ GL∞ is rsectorial on F if and only if a(F ) is contained in some open angular sector spanned by an angle whose vertex is the origin and whose size is 2π/r. Furthermore, if r = 2 or if N = 1, then (5.7) and (5.8) are equivalent. A connection between (5.7) and (5.9) will be established in Lemma 5.14. Now allow r to take values in (1, ∞). Given two points z1 , z2 in the complex plane, let Ar (z1 , z2 ) denote the circular arc from the points of which the line segment [z1 , z2 ] is seen at the angle 2π/ max{r, s} (1/r + 1/s = 1) and which lies on the right (resp. left) of the straight line passing ﬁrst z1 and then z2 if 2 < r < ∞ (resp. 1 < r < 2). For r = 2, Ar (z1 , z2 ) is nothing but the line segment [z1 , z2 ] itself. Ar (z1 , z2 ) is thought of as being oriented from z1 to z2 . Note that Ar (z1 , z2 ) has the parametric representation z(µ) = z1 + (z2 − z1 )σr (µ),
0 ≤ µ ≤ 1,
where σr (µ) = µ for r = 2 and sin(θµ) exp(iθµ) σr (µ) = , sin θ exp(iθ)
θ := π
1 1 − s r
for r = 2. In what follows let [r, s] refer to the segment [min{r, s}, max{r, s}]. Finally, let Or (z1 , z2 ) denote the (closed) lentiform domain between Ar (z1 , z2 ) and As (z1 , z2 ):
256
5 Toeplitz Operators on H p
Or (z1 , z2 ) =
/
Aν (z1 , z2 ).
ν∈[r,s]
If a ∈ L∞ and conv a(F ) is the line segment [z1 , z2 ], then it is clear that a is rsectorial on F (r ≥ 2) if and only if 0 ∈ / Or (z1 , z2 ). It is not too diﬃcult to show that a ∈ GL∞ N ×N is rsectorial on a ﬁber Xτ = Mτ (L∞ ) (τ ∈ T) if and only if there are a neighborhood U ∈ Uτ and matrices c, d ∈ GCN ×N such that W (ca(t)d) ⊂ Sr for almost all t ∈ U . Matrix functions which are rsectorial on the whole space X will be called rsectorial on T. Now let be a Khvedelidze weight on Lp of the form (5.6) and let B be a closed subalgebra of L∞ containing C. A matrix function a ∈ GL∞ N ×N is said to be locally p, sectorial over B if it has the following property: for each τ ∈ T, a is rτ sectorial on each maximal antisymmetric set for B which is contained in the ﬁber Xτ , and rτ is given by ! max{p, q} for τ ∈ T \ {t1 , . . . , tn }, rτ = max{p, q, (1/p + µj )−1 , (1/q − µj )−1 } for τ = tj . Note that a matrix function is locally 2, 1sectorial (i.e., p = 2, ≡ 1) over B if and only if it is (analytically) locally sectorial over B in the sense of Deﬁnition 3.1. 5.13. Lemma. Let v ∈ L∞ N ×N and let F be a closed subset of X. Suppose v is positive deﬁnite on F , that is v(x) = v ∗ (x) ≥ ε > 0 for all x ∈ F . Then ∞ ∗ there is an h ∈ GHN ×N such that v(x) = h (x)h(x) for all x ∈ F . Proof. The mapping X × SCN → C, (x, z) → Re (v(x)z, z) is continuous. Hence, there is a clopen neighborhood U ⊂ X such that Re v(x) ≥ ε/2 for all ∗ x ∈ U . Put f = χU (Re v) + (1 − χU )I. Then f ∈ GL∞ N ×N and f = f ≥ ε/2 2 ∗ on X. Therefore T (f ) ∈ GL(HN ) (Corollary 4.2) and thus f = g k with 2 ∗ g ±1 , k±1 ∈ HN ×N (see the remark after Theorem 5.5). Since f = f , we have ∗ ∗ ∗ −1 ∗ −1 = c ∈ CN ×N , that is, g = ck and so g k = k g, hence (k ) g = gk f = k∗ ck. Clearly, c = c∗ . Because f ∈ GL∞ N ×N , there is a τ ∈ T such that k(τ ) ∈ GCN ×N . Consequently, for z ∈ CN and y = k−1 (τ )z, (cz, z) = (ck(τ )y, k(τ )y) = (f (τ )y, y) ≥
ε ε y2 ≥ k(τ )−2 z2 . 2 2
Thus, c is positive deﬁnite, and so c = e∗ e for some e ∈ GCN ×N . If we put h = ek, then f = h∗ h. Consideration of the diagonal entries of f and h∗ h ∞ −1 = h−1 (h−1 )∗ , it follows shows that actually h ∈ HN ×N . Similarly, since f −1 ∞ that h ∈ HN ×N . Because f (x) = v(x) for x ∈ F , we are done. 5.14. Lemma. Let a ∈ GL∞ N ×N be rsectorial on a closed subset F of X. Then ∞ ∞ a can be represented in the form a = f bg, where f, g ∈ GHN ×N , b ∈ GLN ×N , and π (5.11) max I − b(x)L(CN ) < sin . x∈F r
5.3 Locally p, Sectorial Symbols
257
Proof. If a matrix function is rsectorial on a closed subset of X, then it is ssectorial for all s ∈ [2, r + ε), where ε is suﬃciently small. Thus, we may suppose that r > 2. Choose c, d ∈ GCN ×N so that (5.10) is fulﬁlled for all x ∈ F , and put v := Re (cad). Then v satisﬁes the hypothesis of the preceding ∞ lemma, and hence v(x) = h∗ (x)h(x) for all x ∈ F with some h ∈ GHN ×N . ∗ −1 −1 Let ω := (h ) cadh . Clearly Re ω(x) = (h∗ (x))−1 v(x)h−1 (x) = I
∀ x ∈ F.
If z ∈ CN and x ∈ F , then + + + + + Im ω(x)z, z + = +Im ca(x)dh−1 (x)z, h−1 (x)z + π < tan Re ca(x)dh−1 (x)z, h−1 (x)z r π π −1 z2 . (h (x))∗ v(x)h−1 (x)z, z = tan = tan r r Consequently, for x ∈ F , the spectrum of the normal matrix I + iIm ω(x) is contained in the interior of the line segment whose endpoints are 1−i tan(π/r) and 1+i tan(π/r). Put b := (cos2 (π/r))ω. Then, again for x ∈ F , the spectrum of b(x) = (cos2 (π/r))(I + iIm ω(x)) is a subset of the open disk with center 1 and radius sin(π/r). Hence, the spectral radius and thus the norm of the normal matrix I − b(x) is less than sin(π/r). From the compactness of F we deduce that (5.11) holds. If we put f := (cos2 (π/r))−1 c−1 h∗ and g = hd−1 , then a = f bg is the desired representation. 5.15. Lemma. Let B ∈ L(CN ). For s ∈ (2, ∞), put ω = (cos(π/s))−1 . Then I − BL(CN ) ≤ sin
π s
(5.12)
if and only if I + ωB ∈ GL(CN )
and
(I + ωB)−1 (I − ωB)L(CN ) ≤ tan
π . 2s
(5.13)
Proof. If (5.12) is satisﬁed, then ω(1 + ω)−1 (I − B) ≤ tan(π/(2s)) < 1, and since I + ωB = (1 + ω)[I − ω(1 + ω)−1 (I − B)], (5.12) implies the invertibility of I + ωB. Now note that for A ∈ L(CN ) the equality A2 = sup{(AA∗ z, z) : z ∈ CN , z = 1} holds and that therefore A2 ≤ M 2 if and only if AA∗ ≤ M 2 . Thus, π ⇐⇒ ω −2 + BB ∗ ≤ B + B ∗ s ⇐⇒ (ω + 1)(I − ωB)(I − ωB ∗ ) ≤ (ω − 1)(I + ωB)(I + ωB ∗ ) π ω−1 = tan2 ⇐⇒ (I + ωB)−1 (I − ωB)(I − ωB ∗ )(I + ωB ∗ )−1 ≤ ω+1 2s ⇐⇒ (5.13). (5.12) ⇐⇒ (I − B)(I − B ∗ ) ≤ sin2
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5 Toeplitz Operators on H p
5.16. Theorem. Let 1 < p < ∞, −1/p < µ < 1/q, t ∈ T, (t) := t − τ µ , and put r = max{p, q, (1/p + µ)−1 , (1/q − µ)−1 }. p ∞ (a) If b ∈ L∞ N ×N and I − bLN ×N < sin(π/r), then T (b) ∈ GL(HN ()). p (b) If a ∈ GL∞ N ×N is rsectorial on T, then T (a) ∈ GL(HN ()). p Proof. (a) By formula (2.29), the invertibility of T (b) on HN () is equivalent p to the invertibility of bP + Q on LN () (the multiplication operator M (ϕ) will be simply denoted by ϕ). There is an s > r such that I −b ≤ sin(π/s). Put ω = (cos(π/s))−1 . Then I +ωb(x) ∈ GL(CN ) for all x ∈ X by Lemma 5.15, and thus I +ωb ∈ GL∞ N ×N . This in turn implies that M (I + ωb) ∈ GL(LpN ()). Now write bP + Q in the form
bP + Q =
1 (I + ωb)[I − (I + ωb)−1 (I − ωb)S](ω −1 P + Q), 2
where S = I − 2Q is the singular integral operator on LpN (). Since ωP + Q is the inverse of ω −1 P +Q, it remains to show that D := I −(I +ωb)−1 (I −ωb)S is invertible. Due to 5.11, SL(LpN ()) = cot(π/(2r)). Because I−b ≤ sin(π/s), we deduce from Lemma 5.15 that " " "(I + ωb)−1 (I − ωb)" ∞ ≤ tan π . LN ×N 2s Consequently, by Proposition 5.7, I − DL(LpN ()) ≤ tan
π π cot < 1. 2s 2r
(b) Immediate from Lemma 5.14 and part (a). 5.17. Theorem. Let be a Khvedelidze weight on Lp and let a ∈ GL∞ N ×N be locally p, sectorial over a C ∗ subalgebra B between C and QC. Then T (a) p ()). is in Φ(HN Proof. First note that it suﬃces to consider the case B = QC. Then notice that p p replaced by HN (); the proof is almost Theorem 2.96 remains true with HN literally the same. Hence, it is enough to show that for each ξ ∈ M (QC) there p exists an aξ ∈ L∞ N ×N such that aξ Xξ = aXξ and T (aξ ) ∈ Φ(HN ()). Let ξ ∈ Mτ (QC) (τ ∈ T). Put τ (t) := 1 if τ ∈ T \ {t1 , . . . , tn } and τ (t) := t − tj µj if τ = tj . Since a is rsectorial on Xξ (r depends on the τ ∞ above which ξ lies as in 5.12), we have a = f bg, where f, g ∈ GHN ×N , b ∈ , and I − b(x) < sin(π/r) for x ∈ X (Lemma 5.14). Let U GL∞ ξ 1 ⊂ X be N ×N a (suﬃciently small) clopen neighborhood of Xξ . A little thought shows that U1 ⊂ X can be chosen so that Vτ := {t ∈ T : U1 ∩ Xt = ∅} has the following property: the restriction Lp (Vτ , τ ) of Lp (τ ) to Vτ is equal to the restriction Lp (Vτ , ) of Lp () to Vτ . Then let U2 ⊂ U1 be a clopen neighborhood of Xξ
5.3 Locally p, Sectorial Symbols
259
such that I − b(x) < sin(π/r) for x ∈ U2 , and put bξ := χU2 b + (1 − χU2 )I. ∞ Clearly, bξ ∈ GL∞ N ×N , bξ Xξ = bXξ , and I − bξ LN ×N < sin(π/r). If we set p ()). aξ = f bξ g, then aξ Xξ = aXξ . So it remains to show that T (aξ ) ∈ Φ(HN p This will follow once we have proved that T (bξ ) ∈ Φ(HN ()). p (τ )). By construcFrom Theorem 5.16(a) we know that T (bξ ) ∈ GL(HN tion, we have the following direct sums: ·
LpN (τ ) = LpN (Vτ , τ ) + LpN (Vτc , τ ), ·
LpN () = LpN (Vτ , τ ) + LpN (Vτc , ) (Vτc := T \ Vτ ). Let R1 denote the projection of LpN () onto LpN (Vτ , τ ) parallel to LpN (Vτc , ) and let R2 := I − R1 . Put A := bξ P + Q. Then A = R1 A + R2 A = R1 A + R2 bξ P + R2 Q = R1 A + R2 P + R2 Q (since bξ Vτc = I) = R1 A + R2 = (R1 A + R2 )(R1 + R2 ) = R1 AR1 + R1 AR2 + R2 = (I + R1 AR2 )(R1 AR1 + R2 ).
(5.14)
Since A ∈ GL(LpN (τ )) and I + R1 AR2 ∈ GL(LpN (τ )) (the inverse is I − R1 AR2 ), it follows from (5.14) that R1 AR2 + R2 ∈ GL(LpN (τ )), hence R1 AR1 is invertible on R1 LpN (τ ) = R1 LpN (), and thus R1 AR1 + R2 is in GL(LpN ()). Again by (5.14), this implies that A ∈ GL(LpN ()), whence p ()). T (bξ ) ∈ GL(HN 5.18. Proposition. Let be a Khvedelidze weight on Lp and let a ∈ GL∞ N ×N be locally p, sectorial over C. In addition, suppose at least one of the following three conditions is satisﬁed: (a)
∞ a ∈ CN ×N + HN ×N ,
(b)
≡ 1,
(c)
N = 1.
Then Indp, T (a) = Ind2 T (a), where Indp, T (a) and Ind2 T (a) refer to the p 2 () and HN , respectively. index of T (a) as an operator on HN Remark. A matrix function which is locally p, sectorial over C is necessarily 2 locally sectorial over C in the sense of Deﬁnition 3.1. Thus, T (a) ∈ Φ(HN ). Also recall that Ind2 T (a) = −ind {kλ a} (Corollary 4.30). 2 ∞ Proof. (a) Because T (a) ∈ Φ(HN ), it follows that a−1 ∈ CN ×N + HN ×N and so the argument of the proof of Theorem 2.94 can be applied.
(b) By the hypothesis, a is rsectorial on each ﬁber Xτ (τ ∈ T), where r = max{p, q}. In a similar way as this was done in the proof of the implication (iii) =⇒ (iv) of Theorem 3.9, one can show that a = ϕs, where ϕ is in GCN ×N and s ∈ GL∞ N ×N is rsectorial on T. Hence T (a) = T (ϕ)T (s) + K, where K p 2 ) and C∞ (HN ). Thus, is in both C∞ (HN
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5 Toeplitz Operators on H p
Indp T (a) = Indp T (ϕ) + Indp T (s) = Indp T (ϕ) (Theorem 5.16(b)) = Ind2 T (ϕ) (Theorem 2.94) = Ind2 T (ϕ) + Ind2 T (s) = Ind2 T (a).
(Corollary 4.2)
(c) A moment’s thought reveals that T (a) is homotopic to a Toeplitz operator T (a0 ) with piecewise constant symbol a0 through Toeplitz operators the symbols of which are locally p, sectorial (and in particular 2, 1sectorial) over C. So Indp, T (a) = Indp, T (a0 ) and Ind2 T (a) = Ind2 T (a0 ) and the assertion follows from the P Ctheory (Proposition 5.39). 5.19. Remark. I. M. Spitkovsky turned our attention to the following fact: p if a ∈ L∞ N ×N , then T (a) ∈ Φ(HN ()) and Indp, T (a) = κ if and only if p T (aϕ) ∈ Φ(HN ) and Indp T (aϕ) = κ, where ϕ is a certain appropriately chosen function in P C (see 5.60–5.62). The results of Shneiberg [471] imply r ) for all r ∈ [p, 2]. that Indp T (aϕ) = Ind2 T (aϕ) whenever T (aϕ) is in Φ(HN This provides a possibility of computing Indp, T (a) in case assumption (b) of Proposition 5.18 is not satisﬁed. The attempt of applying a result like Theorem 4.28 leads to the following problem. Let Bp, be the Banach algebra of all (generalized) sequences p p ()) such that there exists an A ∈ L(HN ()) {Aλ }λ∈Λ of operators Aλ ∈ L(HN p q ∗ ∗ −1 with Aλ → A strongly on HN () and Aλ → A strongly on HN ( ) as λ → ∞. Let Ip, denote the closed twosided ideal of Bp, consisting of the p ()) and Cλ L(HNp ()) → 0 sequences of the form {L+Cλ }, where L ∈ C∞ (HN as λ → ∞. Finally, let {Kλ }λ∈Λ be an approximate identity. Is it true that {T (kλ a)}+Ip, is in G(Bp, /Ip, ) whenever a ∈ GL∞ N ×N is locally p, sectorial over C (or over QC)? 5.20. Theorem. Let u ∈ GL∞ N ×N be unitaryvalued, and let and r be as in Theorem 5.16. ∞ (a) If distL∞ (u, HN ×N ) < sin(π/r), then the operator T (u) is leftN ×N p invertible on HN (). ∞ (b) If distL∞ (u, CN ×N + HN ×N ) < sin(π/r), then the operator T (u) is N ×N p leftFredholm on HN ().
(c) The assertions (a) and (b) remain true if H ∞ is replaced by H ∞ and “left” is replaced by “right.” ∞ Proof. (a) Choose h ∈ HN ×N so that u − h < sin(π/r). Then
I − h∗ u = u∗ u − h∗ u ≤ u∗ − h∗ = u − h < sin
π . r
Thus, by Theorem 5.16(a), T (h∗ u) = T (h∗ )T (u) is invertible, which implies that T (u) is leftinvertible.
5.3 Locally p, Sectorial Symbols
261
∞ (b) There are an n ≥ 0 and h ∈ HN ×N such that uχn I − h < sin(π/r). Hence, by virtue of part (a), T (uχn I) = T (u)T (χn I) is leftinvertible, and because T (χn I) is Fredholm, it follows that T (u) is leftFredholm.
(c) Take adjoints.
We conclude with two theorems on scalar Toeplitz operators which can be viewed as H p analogues of Theorem 2.85 and Corollary 2.22, respectively. 5.21. Theorem. Let be a Khvedelidze weight on Lp and let a ∈ L∞ be locally p, sectorial over C + H ∞ . Then T (a) ∈ Φ(H p ()). Proof. Fix τ ∈ T and consider the C ∗ algebra L∞ Xτ ∼ = C(Xτ ). This algebra contains (C + H ∞ )Xτ = H ∞ Xτ as a closed subalgebra (see 2.81). It can be checked straightforwardly that each antisymmetric set for C + H ∞ that is contained in Xτ is an antisymmetric set for (C + H ∞ )Xτ (as subalgebra of C(Xτ )) and that, conversely, each antisymmetric set for (C + H ∞ )Xτ is an antisymmetric set for C +H ∞ . Consequently, the maximal antisymmetric sets for C + H ∞ which are contained in Xτ are just the maximal antisymmetric sets for (C + H ∞ )Xτ . Suppose S is any maximal antisymmetric set for (C + H ∞ )Xτ . If a is rτ sectorial on S, then so also is ϕ := a/a, and it is readily seen that then distXτ (ϕ, C) < sin(π/rτ ). Now Theorem 1.22 (in the setting Y = Xτ and B = (C + H ∞ )Xτ ) can be applied to see that distS (ϕ, H ∞ ) < sin(π/rτ ). Thus, there is an hτ ∈ H ∞ such that ϕ(x)−hτ (x) < sin(π/rτ ) for all x ∈ Xτ , and using Proposition 2.79 we conclude that there is an open neighborhood Uτ ⊂ T of τ such that ϕ(t) − hτ (t) < sin(π/rτ ) a.e. on Uτ . Let τ (t) := 1 if τ ∈ T \ {t1 , . . . , tn } and let τ (t) = t − tj µj if τ = tj . Assume Uτ is small enough, so that Lp (Uτ , ) = Lp (Uτ , τ ). Deﬁne bτ ∈ L∞ by bτ (t) = ϕ−1 (t)hτ (t) for t ∈ Uτ and bτ (t) = 1 for t ∈ T \ Uτ . Since 1 − bτ (t) = ϕ(t) − hτ (t) < sin
π rτ
for
t ∈ Uτ ,
is rτ sectorial on T, too. So T (b−1 bτ is rτ sectorial on T, and hence b−1 τ τ ) p is in GL(H (τ )) by Theorem 5.16(b). The argument used in the proof of p Theorem 5.17 shows that T (b−1 τ ) is even in GL(H ()). ∞ Now choose any gτ ∈ GL so that gτ Uτ = hτ Uτ , and let fτ ∈ C be any function such that supp fτ ⊂ Uτ and fτ ≡ 1 in some open neighborhood of τ . Then −1 −1 −1 T (gτ−1 )T −1 (b−1 (bτ )T (ϕτ fτ ) + K1 τ )T (ϕτ )T (fτ ) = T (gτ )T −1 −1 = T (gτ )T −1 (b−1 τ )T (bτ hτ fτ ) + K1 −1 = T (gτ−1 )T −1 (b−1 τ )T (bτ )T (hτ )T (fτ ) + K2
= T (gτ−1 hτ )T (fτ ) + K2 = T (gτ−1 hτ fτ ) + K3 = T (fτ ) + K3 ,
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5 Toeplitz Operators on H p
where K1 , K2 , K3 ∈ C∞ (H p ()). The conclusion is that T (ϕτ ) + C∞ (H p ()) is Mπτ invertible from the left in L(H p ())/C∞ (H p ()) (recall 1.30), with Mπτ deﬁned similarly as in the proof of Theorem 2.68. Now it is an easy matter to apply Theorem 1.32 to obtain that T (ϕ) is leftFredholm on H p (). It can be shown analogously that T (ϕ) is rightFredholm on H p (). Thus, T (ϕ) = T (a/a) is in Φ(H p ()), and Proposition 2.32 (for H p () in place of H p ) completes the proof. 5.22. Theorem (Krupnik). Let 1 < p < ∞ and 1/p + 1/q = 1, and let a ∈ L∞ . Then the following are equivalent: (i) T (a) ∈ GL(H p ) and T (a) ∈ GL(H q ); (ii) T (a) ∈ GL(H r ) for all r ∈ [p, q]; (iii) a = heu+iv , where h ∈ GH ∞ , u and v are realvalued functions in L , and v∞ < π/ max{p, q}. ∞
Proof. (ii) =⇒ (i). Trivial. (iii) =⇒ (ii). In view of Propositions 2.31 and 2.32 it suﬃces to show that T (eiv ) ∈ GL(H r ) for all r ∈ [p, q]. But eiv is obviously rsectorial on T and so Theorem 5.16 gives the assertion. (i) =⇒ (iii). Without loss of generality assume p ≥ 2 and a = 1. Theorem 5.5 shows that a = a− a+ = b− b+ with p a− , b−1 − ∈ L− ;
q a−1 − , b− ∈ L− ;
q a+ , b−1 + ∈ L+ ;
p a−1 + , b+ ∈ L+
−1 −1 −1 q 1 1 and P ∈ L(Lp (a−1 + )) ∩ L(L (b+ )). Since b+ a+ = b− a− is in L+ ∩ L− = C −1 −1 and so equals a constant γ ( = 0, by 1.40(b)), we have b+ = γa+ , hence P is in L(Lq (a−1 + )), and thus, by 1.46, w+) y , a−1 + =e
w, y ∈ L∞ realvalued,
y∞ <
π . 2p
(5.15)
−1 1 1 Because a+ a+ a− a− = 1, it follows that a+ a− = a−1 − a+ is in L+ ∩ L− = C and so equals a constant γ1 of modulus 1. Thus (if necessary, replace a− by γ2 a− and a+ by γ2−1 a+ , where γ22 = γ1 ), we may assume that a− = a−1 + . We know from 1.41 that a+ = g0 b and a−1 = h d, where g and h0 are 0 0 + inner and ( 2π iθ e +z b(z) = exp 1 log a+ (eiθ ) dθ, 2π 0 eiθ − z ( 2π iθ e +z iθ = exp 1 log a−1 d(z) + (e ) dθ (z ∈ D). 2π 0 eiθ − z
Consequently, bd = 1. This implies that g0 h0 = 1, and since g0 and h0 are inner, these functions must be constants. The conclusion is that both a+ and a−1 + are outer. Thus by 1.41(e), a+ has an analytic logarithm in D.
5.3 Locally p, Sectorial Symbols
If ϕ ∈ L1 , then (sign n)ϕn χn , ϕ ) = −i
Sϕ = ϕ0 +
n =0
263
(sign n)ϕn χn .
n =0
Consequently, if ϕ ∈ L1 is realvalued, then ϕ + iϕ ) = 2P ϕ + const,
Sϕ + Sϕ = const.
(5.16)
In particular, a+ = exp log a+ = exp(log a+  + i(log a+ )) + const) = λ exp(2P log a+ ) = λ exp(−2P log a−1 + ) with some λ ∈ C \ {0}. Taking into account (5.15) we get a+ = λ exp − 2P (w − iSy + const) = µ exp(−w − Sw + iSy + iy) with some µ ∈ C \ {0} (note that 2P S = S + S 2 = S + I), thus a+ = ν exp(−w + Sw + iSy − iy) with some ν ∈ C \ {0} (recall (5.16)). Since a− = a−1 + , we ﬁnally have a = µν −1 exp(−2Sw + 2iy) = µν −1 exp(−4P w) exp(2w + 2iy). The functions u = 2w and v = 2y possess the properties required. The functions exp(±4P w) are clearly analytic and in L∞ (because a ∈ GL∞ and exp(2w + 2iy) ∈ GL∞ ). Thus h = µν −1 exp(−4P w) is in GH ∞ . Remark. Also notice that the following is true. r1 r2 If a ∈ L∞ N ×N , 1 < r1 < r2 < ∞, T (a) ∈ GL(HN ), and T (a) ∈ GL(HN ), r then T (a) ∈ GL(HN ) for all r ∈ [r1 , r2 ].
To see this note ﬁrst that, by the remark to Theorem 5.5, a admits WienerHopf factorizations a = a− a+ in Lr1 and a = b− b+ in Lr2 . It is easily seen that −1 2 b+ a−1 + = b− a− is in LN ×N . Consequently, a− and a+ are constant multiples s of b− and b+ , respectively. This implies that a− ∈ [Lr− ]N ×N , a−1 − ∈ [L− ]N ×N , −1 s r a+ ∈ [L+ ]N ×N , a+ ∈ [L+ ]N ×N for all r ∈ [r1 , r2 ] (1/r + 1/s = 1). The RieszThorin interpolation theorem shows that ϕ → a−1 + P (a+ ϕ) is bounded on Lr for r ∈ [r1 , r2 ]. Hence, a = a− a+ is a WienerHopf factorization of a in r ) Lr , and thus, again by the remark following Theorem 5.5, T (a) ∈ GL(HN for all r ∈ [r1 , r2 ]. Note that the result just proved can be generalized to certain other situations, for example to spaces with weight. In the scalar case an immediate consequence reads as follows. If a ∈ L∞ , 1 < r1 < r2 < ∞, T (a) ∈ Φ(H r1 ), T (a) ∈ Φ(H r2 ), and T (a) has the same index κ on H r1 and H r2 , then T (a) ∈ Φ(H r ) for all r ∈ [r1 , r2 ] and the index of T (a) on H r equals κ.
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5 Toeplitz Operators on H p
5.23. Open problems. Extend Theorems 5.21 and 5.22 to the case of matrix symbols. Establish “if and only if” versions of Theorem 5.20 (recall 4.36 and 4.37). Generalize Theorem 2.83 to the spaces H p .
5.4 Localization p p 5.24. Deﬁnitions. The Calkin algebra L(HN (w))/C∞ (HN (w)) will be denoted by (L/C∞ )p,w . Here and in what follows the “parameter” N is suppressed, since its value is always known from the context. The coset of p (w)) will be denoted by Ap,w . Given a closed (L/C∞ )p,w containing A ∈ L(HN ∞ subalgebra A of LN ×N which contains CN ×N let algp,w T (A) denote the smallp (w)) containing the set {T (a) : a ∈ A}. It can est closed subalgebra of L(HN p (w)) is a subset of be shown as in the proof of Proposition 4.5 that C∞ (HN p algp,w T (A). The quotient algebra algp,w T (A)/C∞ (HN (w)) is easily seen to be p (w)) with a ∈ A and will generated by the cosets Tp,w (a) := T (a) + C∞ (HN therefore be denoted by alg Tp,w (A). Let B be a C ∗ subalgebra of L∞ such that C ⊂ B ⊂ QC and BN ×N ⊂ A. For ξ ∈ M (B), deﬁne the localizing class Mξp,w in (L/C∞ )p,w as in the proof of Theorem 2.96: Mξp,w := Tp,w (ϕ) : ϕ = diag (f, . . . , f ), f ∈ B, 0 ≤ f ≤ 1, f is identically 1 in some neighborhood Uξ ⊂ M (B) of ξ . ξ ξ Put FB p,w := {Mp,w : ξ ∈ M (B)}. It is clear that {Mp,w }ξ∈M (B) is a covering and overlapping system of localizing classes in (L/C∞ )p,w . The commutant of B FB p,w in (L/C∞ )p,w will be denoted by Com Fp,w (recall 1.30). Since Hankel operators with QC symbols are compact on H p (w) we have B alg Tp,w (A) ⊂ alg Tp,w (L∞ N ×N ) ⊂ Com Fp,w .
(5.17)
ξ If ξ ∈ M (B), then Zp,w will refer to the set of all Ap,w ∈ Com FB p,w which ξ ξ is a are Mp,w equivalent form the left and the right to zero. Note that Zp,w B ξ closed twosided ideal of Com Fp,w . We let Jp,w denote the smallest closed twosided ideal of alg Tp,w (A) containing the set Tp,w (ϕ) : ϕ = diag (f, . . . , f ), f ∈ B, f (ξ) = 0 . ξ ξ It is easy to see that the quotient algebra alg Tp,w (A) := alg Tp,w (A)/Jp,w is ξ ξ generated by the cosets Tp,w (a) := Tp,w (a) + Jp,w with a ∈ A. ξ ξ ) and spA (Ap,w + Jp,w ) refer to For A ∈ alg Tp,w (A), we let sp(Ap,w + Zp,w ξ ξ B ξ ξ (A), the spectrum of Ap,w +Zp,w and Ap,w +Jp,w in Com Fp,w /Zp,w and alg Tp,w ξ ξ respectively. Henceforth we write Ap,w in place of Ap,w + Jp,w . Finally, given a ∈ L∞ N ×N deﬁne $ spΦ(HNp (w)) T (f ). sp ξp,w T (a) := f Xξ =aXξ
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265
5.25. Proposition. Let a, b ∈ L∞ N ×N and suppose aXξ = bXξ . Then ξ ξ ξ ξ = Tp,w (b) + Zp,w , Tp,w (a) = Tp,w (b), sp ξp,w T (a) = sp ξp,w T (b). Tp,w (a) + Zp,w
Proof. When proving Theorem 2.96 we showed that Tp,w (a) and Tp,w (b) are Mξp,w equivalent from both the left and the right if aXξ = bXξ . This gives the ﬁrst equality. Lemma 3.65 implies that f IN ×N : f ∈ B, f (ξ) = 0 a − b ∈ closidL∞ N ×N whenever (a − b)Xξ = 0. This proves the second equality. The third equality is trivial. 5.26. Corollary. If f ∈ BN ×N and ξ ∈ M (B), then ξ ξ = spL∞ Tp,w (f ) = sp ξp,w T (f ) = {det f (ξ)}. sp Tp,w (f ) + Zp,w Proof. Immediate from the preceding proposition.
ξ coincides with the ideal Jξπ For p = 2 and w identically 1 the ideal Jp,w ξ introduced in 4.11 and Tp,w (a) is nothing else but the local Toeplitz operator π which was denoted by Tξ (a) in Chapter 4.
5.27. Proposition. If a ∈ L∞ N ×N , then ξ ξ ξ ⊂ spL∞ Tp,w (a) + Zp,w (a), sp Tp,w
ξ ξ ⊂ sp ξp,w T (a). sp Tp,w (a) + Zp,w
If p = 2, w is identically 1, N = 1, B ∈ {C, QC}, and a ∈ L∞ , then ξ ξ ξ = spL∞ Tp,w (a) + Zp,w (a) = sp ξp,w T (a). sp Tp,w ξ (a) be invertible. Then there is an operator D in the algebra Proof. Let Tp,w ∞ algp,w T (LN ×N ) such that ξ Dp,w Tp,w (a) − Ip,w ∈ Jp,w ,
ξ Tp,w (a)Dp,w − Ip,w ∈ Jp,w .
ξ ξ ξ ξ ⊂ Zp,w , it follows that Tp,w (a) + Zp,w is invertible Since obviously Jp,w (recall (5.17)), and this proves the ﬁrst spectral inclusion. Now suppose 0 ∈ / sp ξp,w T (a). So there exists a b ∈ L∞ N ×N such that bXξ = aXξ and Tp,w (b) ∈ G(L/C∞ )p,w . Because Tp,w (b) ∈ Com FB p,w , we actually have ξ ξ ). Therefore T (b)+Z is invertible, and from PropoTp,w (b) ∈ G(Com FB p,w p,w p,w ξ ξ (a) + Zp,w is invertible, too. This completes sition 5.25 we conclude that Tp,w the proof of the second spectral inclusion. ξ ξ is invertible in Com FB Now let p = 2 and assume T2,w (a) + Z2,w 2,w /Z2,w . Abbreviate T2,w (a) and I2,w to A and I, respectively, and put alg := ξ alg T2,w (L∞ N ×N ) and Z := Z2,w . From 1.26(g) we deduce that alg + Z is a ∗ ∗ C subalgebra of the C algebra Com Fξ2,w . Hence, by virtue of 1.26(d), A + Z
5 Toeplitz Operators on H p
266
is invertible in (alg + Z)/Z. Once more using 1.26(g) we obtain that there is a D ∈ alg such that DA − I ∈ alg ∩ Z. Choose Φ := T2,w (ϕ) ∈ Mξ2,w so that (DA − I)Φ < 1 and let Ψ := T2,w (ψ) ∈ Mξ2,w satisfy ΦΨ = Ψ (recall 1.30). We have DAΨ = Ψ + (DA − I)Ψ = Ψ + (DA − I)ΦΨ = (I + U )Ψ, where U := (DA − I)Φ ∈ alg has norm less than 1. Therefore (I + U )−1 is in alg and so EAΨ = Ψ with E := (I + U )−1 D ∈ alg . It follows that ξ ξ ξ ξ ξ ξ E2,w Aξ2,w Ψ2,w = Ψ2,w , and since Ψ2,w = T2,w (ψ) = I2,w , we conclude that ξ ξ ξ ξ ∈ alg T2,w (L∞ ) is a left inverse of A = T E2,w 2,w 2,w (a). As right inN ×N vertibility can be treated analogously, we arrive at the spectral inclusion ξ ξ (a) ⊂ sp(T2,w (a) + Z2,w ). sp T2,w Finally, if p = 2, w ≡ 1, N = 1, and B ∈ {C, QC}, then, due to Corolξ lary 4.65, spL∞ T2,1 (a) = sp ξ2,1 T (a), which completes the proof. 5.28. Corollary. Let be a Khvedelidze weight on Lp , let B be a C ∗ algebra between C and QC, let ξ ∈ Mτ (B) (τ ∈ T), and suppose a ∈ L∞ N ×N is / sp ξp, T (a) and rτ sectorial on Xξ , where rτ is given as in 5.12. Then 0 ∈ ξ is invertible. Tp, (a) + Zp, Proof. That the origin does not belong to sp ξp, T (a) follows from the proof of Theorem 5.17, where we constructed an aξ ∈ L∞ N ×N such that aξ Xξ = aXξ p ξ and T (aξ ) ∈ Φ(HN ()). The invertibility of Tp, (a) + Zp, now results from the second spectral inclusion of the preceding proposition. 5.29. Theorem. Let A ∈ algp,w T (A) and a ∈ L∞ N ×N . Then / ξ sp (Ap,w + Zp,w ), (a) spΦ(HNp (w)) A = ξ∈M (B)
(b) spalg Tp,w (A) Ap,w =
/
spA Aξp,w ,
ξ∈M (B)
(c) spΦ(HNp (w)) T (a) =
/
sp ξp,w T (a).
ξ∈M (B)
Proof. / (a) Theorem 1.32(b). (b) Theorem 1.35(a). (c) It is clear that the set sp ξ T (a) is a subset of spess T (a). To show the reverse inclusion, suppose ξ∈M (B)
µ is not in
/
sp ξ T (a). Then for each ξ ∈ M (B) there is an aξ ∈ L∞ N ×N
ξ∈M (B) p such that aξ Xξ = aXξ and T (aξ −µI) ∈ Φ(HN (w)). So Theorem 2.96 (whose extension to spaces with weight can be proved in the same way as for spaces p (w)), and thus µ ∈ / spess T (a). without weight) implies that T (a − µI) ∈ Φ(HN
5.5 P C Symbols
267
5.30. Open problem. Clarify the connection between the three “local spectra” ξ ξ ), sp Tp,w (a), spξp,w T (a). sp(Tp,w (a) + Zp,w Under what restrictions on the nature of the “local range” of a two of them ξ (or all three) are equal to each other? In particular, is sp(Tp,w (a)+Zp,w ) equal ξ ∞ to spp,w T (a) for every a ∈ LN ×N ?
5.5 P C Symbols 5.31. Theorem. Let B be a C ∗ algebra between C and QC. (a) Every element in alg Tp,w (BN ×N ) is of the form Tp,w (f ) with f in BN ×N , and the mapping Smb : alg Tp,w (BN ×N ) → BN ×N ,
Tp,w (f ) → f
is a homeomorphic algebraisomorphism. (b) Let f ∈ BN ×N . Then p T (f ) ∈ Φ(HN (w)) ⇐⇒ f ∈ GBN ×N ⇐⇒ det f ∈ GB. p If T (f ) ∈ Φ(HN (w)) and {Kλ }λ∈Λ is any approximate identity, then {kλ f } is bounded away from zero and Ind T (f ) = −ind {kλ det f }.
Proof. (a) The same arguments as in the proofs of Propositions 4.1 and 4.4(d) apply to show that c1 aL∞ ≤ Tp,w (a)(L/C∞ )p,w ≤ T (a)L(HNp (w)) ≤ c2 aL∞ , N ×N N ×N
(5.18)
where c1 and c2 are certain constants depending only on p, w, N . Since B ⊂ QC, the set Tp,w (BN ×N ) := {Tp,w (f ) : f ∈ BN ×N } is an algebra, and from the ﬁrst “≤” in (5.18) we deduce that this algebra is closed. Therefore alg Tp,w (BN ×N ) = Tp,w (BN ×N ). Due to (5.18) the mapping Smb is continuous, and since it is an algebraic homomorphism which is onto (obvious) and onetoone (by (5.18)), it follows that it is actually a homeomorphism. (b) The equivalences “⇐⇒” result from the HartmanWintner spectral inclusion theorem 5.2 and from 1.26(d). Corollary 3.69(b) shows that {kλ f } is bounded away from zero. The same reasoning as in the proof of Theorem 2.94 gives that Indp,w T (f ) = −ind {hr det f }. Since −ind {hr det f } = Ind2 T (f ) (again Theorem 2.94) and Ind2 T (f ) = −ind {kλ det f } (Corollary 4.30), we obtain that Indp,w T (f ) is equal to −ind {kλ det f }. 5.32. Theorem. Let a, b ∈ L∞ and suppose for each ξ ∈ M (QC) either aXξ ∈ H ∞ Xξ or bXξ ∈ H ∞ Xξ . Then T (ab) − T (a)T (b) ∈ C∞ (H p (w)).
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5 Toeplitz Operators on H p
Proof. Choose any ε > 0, and let A and B denote the set of all points ξ in M (QC) such that distXξ (a, H ∞ ) ≥ ε and distXξ (b, H ∞ ) ≥ ε, respectively. The sets A and B are disjoint (by the hypothesis) and closed (due to the upper semicontinuity of the mapping M (QC) → R, ξ → distXξ (a, H ∞ )). Hence, there is a ϕ ∈ QC such that 0 ≤ ϕ ≤ 1, ϕA = 0, and ϕB = 1. Put ψ := 1 − ϕ. Then, by (2.18), T (ab) − T (a)T (b) = T (aϕb) − T (a)T (ϕ)T (b) + T (aψb) − T (a)T (ψ)T (b) ) + T (a)H(ψ)H()b) = H(aϕ)H()b) + H(a)H(ϕ)T ) (b) + H(a)H(ψb) = H(aϕ)H()b) + H(a)H(ψ))b) + K, where K ∈ C∞ (H p (w)), since H(ϕ) ) and H(ψ) are compact on H p (w) (note ∞ that ϕ, ) ψ ∈ C + H ). Because distXξ (aϕ, H ∞ ) < ε for all ξ ∈ M (QC), it follows from Theorem 1.22 (with B = C + H ∞ ) that distL∞ (aϕ, C + H ∞ ) < ε. As in the proof of Theorem 2.54 one can see that there is a constant cp,w such that H(f )Φ(H p (w)) ≤ cp,w dist(f, C + H ∞ ) ∀ f ∈ L∞ . Consequently, H(aϕ)Φ(H p (w)) ≤ εcp,w . It can be shown analogously that H(ψ))b)Φ(H p (w)) ≤ εcp,w . Thus H(aϕ)H()b) + H(a)H(ψ))b)Φ(H p (w)) ≤ εcp,w cp,w (b∞ + a∞ ), where cp,w is from the estimate H(f )L(H p (w)) ≤ cp,w f ∞ . As ε > 0 can be chosen arbitrarily small, it follows that T (ab) − T (a)T (b) − K is compact. 5.33. Corollary. If a, b ∈ P C have no common points of discontinuity on T, then T (ab) − T (a)T (b) is compact on H p (w). Proof. If τ ∈ T, then either aXτ ∈ CXτ = CXτ or bXτ ∈ CXτ = CXτ , so that the previous theorem applies. 5.34. Theorem. The algebra alg Tp,w (P QC) is commutative. Proof. It suﬃces to prove that T (aϕ)T (bψ) − T (bψ)T (aϕ) ∈ C∞ (H p (w)) whenever a, b ∈ P C0 and ϕ, ψ ∈ QC. Because T (aϕ)T (bψ) − T (ϕ)T (ψ)T (a)T (b),
T (bψ)T (aϕ) − T (ϕ)T (ψ)T (b)T (a)
are compact, it remains to show that T (a)T (b) − T (b)T (a) ∈ C∞ (H p (w))
5.5 P C Symbols
269
for every a, b ∈ P C0 . We may clearly assume that a and b have at most one discontinuity, and in view of the preceding corollary it can be assumed that a and b have the jump at the same point of T. Then a = λb + c with λ ∈ C and c ∈ C, and hence T (a)T (b) − T (b)T (a) = λ[T (c)T (b) − T (b)T (c)] ∈ C∞ (H p (w)).
5.35. Deﬁnitions. Henceforth the argument arg z of a number z ∈ C \ {0} will be always chosen so that arg z ∈ (−π, π]. For β ∈ C and τ ∈ T, deﬁne ϕβ,τ ∈ P C0 as ϕβ,τ (t) := exp{ i β arg(−t/τ )}
(t ∈ T).
The dependence of ϕβ,τ on τ will be usually suppressed, that is, we shall write ϕβ in place of ϕβ,τ . It is readily seen that ϕβ has at most one discontinuity, namely a jump at τ , and that ϕβ (τ + 0) = e−πiβ and ϕ(τ − 0) = eπiβ . Let a ∈ P C0 and denote the points of discontinuity of a by t1 , . . . , tm . If a(tj ± 0) = 0 for all j = 1, . . . , m, then there are βj ∈ C such that a(tj − 0) = exp(2πiβj ) a(tj + 0) and thus a = bϕβ1 ,t1 . . . ϕβm ,tm
(5.19)
with some continuous function b ∈ C. Next, for t ∈ T \ {τ }, deﬁne + τ τ ++ τ β + ξβ (t) := ξβ,τ (t) := 1 − , := exp β log +1 − + + iβ arg 1 − t t t + t t ++ t β + ηβ (t) := ηβ,τ (t) := 1 − := exp β log +1 − + + iβ arg 1 − . τ τ τ The following basic identity can be veriﬁed straightforwardly: ϕβ (t) = ξ−β (t)ηβ (t) ∀ t ∈ T \ {τ }.
(5.20)
Note that ξβ (resp. ηβ ) is the limit on the unit circle T of that branch of the function (1 − τ /z)β (resp. (1 − z/τ )β ) which is analytic for z > 1 (resp. z < 1) and takes the value 1 at z = ∞ (resp. z = 0). Also notice that obviously ξα (t)ξβ (t) = ξα+β (t),
ηα (t)ηβ (t) = ηα+β (t) ∀ t ∈ T \ {τ }.
We have for t ∈ T \ {t0 }, + t0 t0 ++ + ξβ,t0 (t) = exp Re β log +1 − + − i Im β arg 1 − = t − t0 Re β b(t), t t
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5 Toeplitz Operators on H p
where b ∈ GL∞ , and therefore, if (t) = weight, ξβ ∈ Lp () ⇐⇒ ηβ ∈ Lp () ⇐⇒ −
5n j=0
t − tj µj is a Khvedelidze
1 1 < Re β + µ0 < . p q
(5.21)
Clearly, ξβ ∈ Lp () ⇐⇒ ξβ ∈ Lp− (), ηβ ∈ Lp () ⇐⇒ ηβ ∈ Lp+ (). (5.22) 5n 5.36. Lemma. Let (t) = j=0 t − tj µj be a Khvedelidze weight and let β ∈ C. Then the following are equivalent: (i) T (ϕβ,t0 ) ∈ Φ(H p ()) and Ind T (ϕβ,t0 ) = −κ; (ii) κ − 1/q < Re β − µ0 < κ + 1/p; (iii) 0 ∈ / Ar (ϕβ (t0 − 0), ϕβ (t0 + 0)), where r = (1/p + µ0 )−1 , and the index of the closed continuous and naturally oriented curve obtained from the range of ϕβ by ﬁlling in the arc Ar (ϕβ (t0 − 0), ϕβ (t0 + 0)) is equal to κ. Proof. (ii) ⇐⇒ (iii). Straightforward. (ii) =⇒ (i). Put γ = β − κ. Then −1/q < Re γ − µ0 < 1/p. Hence, by (5.21), (5.22), ξ−γ ∈ Lp− (),
−1 ξ−γ ∈ Lq− (−1 ),
ηγ ∈ Lq+ (−1 ),
ηγ−1 ∈ Lp+ (),
and since ηγ−1  is also a Khvedelidze weight, it follows from Theorem 5.9 that P ∈ L(Lp (ηγ−1 )). Thus, by (5.20) and Theorem 5.5, T (ϕγ ) = T (ξ−γ ηγ ) is in GL(H p ()). Because T (ϕβ ) equals T (ϕγ )T (χκ ) or T (χκ )T (ϕγ ), we conclude that T (ϕβ ) ∈ Φ(H p ()) and Ind T (ϕβ ) = Ind T (χκ ) = −κ. (i) =⇒ (ii). There is a k ∈ Z such that k − 1/q < Re β − µ0 ≤ k + 1/p. If Re β − µ0 < k + 1/p, then k = κ by what has just been proved and we are done. So assume Re β − µ0 = k + 1/p. The hypothesis (i) implies that T (ϕβ1 ) and T (ϕβ2 ) are Fredholm of index κ whenever β1 , β2 ∈ C are suﬃciently close to β. But if we let k−
1 1 1 < Re β1 − µ0 < k + < Re β2 − µ0 < k + 1 + , q p p
then, again, by what has already been proved, Ind T (ϕβ1 ) = k, Ind T (ϕβ2 ) = k + 1, which is a contradiction. 5.37. Deﬁnitions. Let 1 < p < ∞ and let be a Khvedelidze weight of the form (5.6). For a ∈ P CN ×N , deﬁne ap, : T × [0, 1] → CN ×N by ap, (t, µ) := 1 − σ(t, µ) a(t − 0) + σ(t, µ)a(t + 0), (t, µ) ∈ T × [0, 1],
5.5 P C Symbols
271
where σ(t, µ) := σp (µ) for t ∈ T \ {t1 , . . . , tn } and σ(t, µ) := σ(1/p+µj )−1 (µ) for t = tj , and σr (µ) is deﬁned as in 5.12. The range of det(ap, ) is a continuous closed and naturally oriented curve. If N = 1, it is obtained from the (essential) range of a by ﬁlling in the arcs Ar(τ ) (a(τ − 0), a(τ + 0)) for each τ ∈ T at which a has a jump; here r(τ ) := p for τ ∈ T \ {t1 , . . . , tn } and r(τ ) := (1/p + µj )−1 for τ = tj . If the curve does not pass through the origin, its winding number with respect to the origin will be denoted by ind det(ap, ). Note that, in general, det(ap, ) = (det a)p, and ind det(ap, ) = ind (det a)p, . For a ∈ (P C0 )N ×N , we have ind det(ap, ) = ind (det(ap, ) ◦ ωS(det a) ), where ωS(det a) is deﬁned as in 2.73 and the latter “ind ” refers to the index as it was deﬁned in 2.41. Finally, if a ∈ P CN ×N and det(ap, )(t, µ) = 0 for all (t, µ) ∈ T × [0, 1], then ind det(ap, ) = lim ind det(anp, ), where {an } n→∞
is any sequence of functions in (P C0 )N ×N such that anp, (t, µ) = 0 for all → 0 as n → ∞. (t, µ) ∈ T × [0, 1] and a − an L∞ N ×N 5.38. Lemma. Let a, b ∈ P C0 have no common points of discontinuity on T and suppose ap, and bp, do not vanish on T × [0, 1]. Then (ab)p, = ap, bp, and ind (ab)p, = ind ap, + ind bp, . Proof. The equality (ab)p, = ap, bp, is obvious and the index formula can be shown as follows: ind (ab)p, = ind (ab)p, ◦ ωS(ab) 0 1 = ind (ap, bp, ) ◦ ωS(ab) = ind ap, ◦ ωS(ab) bp, ◦ ωS(ab) = ind ap, ◦ ωS(ab) + ind bp, ◦ ωS(ab) = ind ap, ◦ ωS(a) + ind bp, ◦ ωS(a) = ind ap, + ind bp, . 5.39. Proposition. Let a ∈ P C0 and let be a Khvedelidze weight. Then T (a) ∈ Φ(H p ()) ⇐⇒ ap, (t, µ) = 0 ∀ (t, µ) ∈ T × [0, 1]. If T (a) ∈ Φ(H p ()), then Ind T (a) = −ind ap, . Proof. Suppose ap, does not vanish on T × [0, 1]. Then a can be written in the form (5.19). In view of Corollary 5.33 and Lemma 5.38 we have T (a) − T (ϕβ1 ) . . . T (ϕβm )T (b) ∈ C∞ (H p ()),
(5.23)
ap, = (ϕβ1 )p, . . . (ϕβm )p, bp, ,
(5.24)
ind ap, = ind b +
m
ind (ϕβj )p, .
(5.25)
j=1
From (5.24) we deduce that (ϕβj )p, (t, µ) = 0 for all (t, µ) ∈ T × [0, 1]. Thus, by Lemma 5.36, T (ϕβj ) ∈ Φ(H p ()) and Ind T (ϕβj ) = −ind (ϕβj )p, . Theorem 5.31(b) implies that T (b) ∈ Φ(H p ()) and that Ind T (b) = −ind b. So
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5 Toeplitz Operators on H p
(5.23) shows that T (a) ∈ Φ(H p ()) and Atkinson’s theorem combined with (5.25) gives the index formula Ind T (a) = −ind ap, . Once the index formula has been proved, the usual perturbation argument (see the proof of Theorem 2.74) shows that ap, (t, µ) = 0 for all (t, µ) in T × [0, 1] if T (a) ∈ Φ(H p ()). 5.40. Proposition. Let be a Khvedelidze weight, let a ∈ P C, let τ ∈ T, and deﬁne r(τ ) as in 5.37. Then τ τ τ = spP C Tp, (a) = spL∞ Tp, (a) sp Tp, (a) + Zp, = sp τp, T (a) = Ar(τ ) a(τ − 0), a(τ + 0) . Proof. Let A and B denote the arcs (τ e−iπ/2 , τ ) and (τ, τ eiπ/2 ), respectively. Choose fτ , gτ ∈ P C0 so that fτ A = gτ A = 0, fτ B = gτ B = 1, fτ and gτ are continuous on T \ {τ }, R(fτ ) ⊂ Ar(τ ) (0, 1), and R(gτ ) ⊂ As (0, 1), where s = r(τ ). The preceding proposition gives that spΦ(H p ()) T (fτ ) = Ar(τ ) (0, 1),
spΦ(H p ()) T (gτ ) = Ar(τ ) (0, 1) ∪ As (0, 1).
From 1.16(b) we deduce that spalg Tp, (P C) Tp, (fτ ) = spalg Tp, (L∞ ) Tp, (fτ ) = Ar(τ ) (0, 1). Since each of the “local” spectra is obviously contained in the corresponding τ ), “global” spectrum, it follows that each of the spectra sp(Tp, (fτ ) + Zp, τ τ τ spP C Tp, (fτ ), spL∞ Tp, (fτ ), sp p, T (fτ ) is a subset of Ar(τ ) (0, 1). τ (fτ ) contains Ar(τ ) (0, 1). Theorem 5.29(b) imLet us show that spP C Tp, plies that / t spΦ(H p ()) T (gτ ) ⊂ spalg Tp, (P C) T (gτ ) = spP C Tp, (gτ ) =
/
t∈T
t τ spP C Tp, (gτ ) ∪ spP C Tp, (gτ ).
(5.26)
t =τ
Since gτ is continuous on T \ {τ }, the ﬁrst union in (5.26) equals As (0, 1) τ (gτ ), and (Corollary 5.26). Hence Ar(τ ) (0, 1) must be contained in spP C Tp, τ τ because Tp, (gτ ) = Tp, (fτ ) (Proposition 5.25), it results that Ar(τ ) (0, 1) ⊂ τ (fτ ). spP C Tp, τ (fτ ), It can be shown analogously that Ar(τ ) (0, 1) is contained in spL∞ Tp, τ τ sp(Tp, (fτ ) + Zp, ), sp p, T (fτ ). Thus, for a = fτ the proposition is proved. If a is any function in P C, then a = cfτ + g with fτ as above, c ∈ C, and g ∈ P C continuous at τ . Therefore, by Proposition 5.25 and Corollary 5.26, τ τ (a) = c spP C Tp, (fτ ) + g(τ ) spP C Tp, = cAr(τ ) (0, 1) + g(τ ) = Ar(τ ) (a(τ − 0), a(τ + 0)),
and equally for the other three spectra.
5.5 P C Symbols
273
5.41. Proposition. Let be a Khvedelidze weight, let bjk be ﬁnitely many m 5n functions in P C, and put A = j=1 k=1 T (bjk ). Then if τ ∈ T, τ ) = spP C Aτp, = spL∞ Aτp, sp (Ap, + Zp, = (bjk )p, (τ, λ) : λ ∈ [0, 1] . j
k
Proof. Each bjk can be written in the form bjk = cjk χτ +gjk , where cjk ∈ C, χτ is the characteristic function of the arc (τ, τ eiπ/2 ), and gjk ∈ P C is continuous at τ . Hence, by Proposition 5.25 and Corollary 5.26, τ τ cjk Tp, (χτ ) + gjk (τ ) + Zp, sp (Ap, + Zp, ) = sp =
j
j
k
τ cjk sp Tp, (χτ ) + Zp, + gjk (τ )
k
(here we applied the spectral mapping theorem) cjk Ar(τ ) (0, 1) + gjk (τ ) (Proposition 5.40) = j
k
cjk σr(τ ) (λ) + gjk (τ ) : λ ∈ [0, 1] = = =
j
k
j
k
j
k
0
1 1 − σr(τ ) (λ) bjk (τ − 0) + σr(τ ) (λ)bjk (τ + 0) : λ ∈ [0, 1]
(bjk )p, (τ, λ) : λ ∈ [0, 1] .
The same argument applies to spP C Aτp, and spL∞ Aτp, . 5.42. Deﬁnition. For A ∈ algp,w T (P CN ×N ), let det A ∈ algp,w T (P C) denote the determinant deﬁned by (1.6). Since alg Tp,w (P C) is commutative (Theorem 5.34), any determinant of A resulting by reordering the factors in the terms of the sum (1.6) diﬀers from that one only by a compact operator. 5.43. Theorem. Let be a Khvedelidze weight on Lp . Let A=
r s
T (ajk ),
ajk ∈ P CN ×N
j=1 k=1
and det A =
m n
T (bjk ),
bjk ∈ P C.
j=1 k=1 p ()) if and only if Then A ∈ Φ(HN m n
(bjk )p, (t, λ) = 0 ∀ (t, λ) ∈ T × [0, 1].
j=1 k=1
274
5 Toeplitz Operators on H p
p Proof. Theorems 5.34 and 1.14(c) show that A ∈ Φ(HN ()) if and only if p det A ∈ Φ(H ()). So it remains to apply Theorem 5.29(a) and Proposition 5.41.
5.44. Proposition. Let be a Khvedelidze weight on Lp and let τ ∈ T. τ τ (a) The algebra alg Tp, (P C) is singly generated by Tp, (χτ ), where χτ is iπ/2 the characteristic function of the arc (τ, τ e ). τ (b) The maximal ideal space M (alg Tp, (P C)) is homeomorphic to the segment [0, 1] (equipped with the topology inherited from the Euclidean R) and the τ (P C) → C([0, 1]) is for a ∈ P C given by Gelfand map Γ : alg Tp, τ Γ Tp, (a) (λ) = 1 − σr(τ ) (λ) a(τ − 0) + σr(τ ) (λ)a(τ + 0),
where r(τ ) and σr(τ ) are deﬁned as in 5.37 and 5.12. Proof. (a) If a ∈ P C, then a = cχτ + g, where c ∈ C and g ∈ P C is continuous τ τ (a) = cTp, (χτ )+g(τ ), at τ . Hence, by Proposition 5.25 and Corollary 5.26, Tp, τ τ and it follows that alg Tp, (P C) is generated by Tp, (χτ ). τ (b) From Proposition 5.40 we know that spP C Tp, (χτ ) = Ar(τ ) (0, 1) and τ (P C)) is homeomorphic to Ar(τ ) (0, 1) (see 1.19) and thus therefore M (alg Tp, to [0, 1]. For λ ∈ [0, 1] let λ denote the multiplicative linear functional on τ τ (P C) which sends Tp, (χτ ) into (1 − σr(τ ) (λ)) · 0 + σr(τ ) (λ) · 1. Since alg Tp, every a ∈ P C can be written in the form cχτ + g as in (a), we have τ τ Γ Tp, (a) (λ) = λ(Tp, (a)) = 1 − σr(τ ) (λ) a(τ − 0) + σr(τ ) (λ)a(τ + 0)
for every a ∈ P C.
5.45. Theorem. Suppose is a Khvedelidze weight on Lp . Then the maximal ideal space M (alg Tp, (P C)) is homeomorphic to the cylinder T × [0, 1] equipped with an exotic topology. For a function a ∈ P C the Gelfand map Γ : alg Tp, (P C) → C(T × [0, 1]) is given by Γ Tp, (a) (τ, λ) = ap, (τ, λ). (5.27) Proof. Let πτ denote the canonical homomorphism of the algebra alg Tp, (P C) τ (P C). onto the algebra alg Tp, τ (P C), then vτ ◦ πτ is If vτ is a multiplicative linear functional on alg Tp, clearly in M (alg Tp, (P C)). Thus, by the preceding proposition, T × [0, 1] can be identiﬁed with a subset of M (algp, (P C)) and (5.27) holds. Now let v ∈ M (alg Tp, (P C)). Since alg Tp, (C) is a closed subalgebra of alg Tp, (P C) and M (alg Tp, (C)) = T (Theorem 5.31), there must exist a τ ∈ T such that v(Tp, (f )) = f (τ ) for all f ∈ C. This implies that τ ) = {0}. Consequently, there exists a multiplicative linear functional vτ v(Jp, τ τ on alg Tp, (P C) = alg Tp, (P C)/Jp, such that v = vτ ◦ πτ . The conclusion is that T × [0, 1] equals M (alg Tp, (P C)).
5.5 P C Symbols
275
Remark. The Gelfand topology on M (alg Tp, (P C)) = T × [0, 1] coincides with the topology on T × [0, 1] described in 4.88. 5.46. Theorem. Let be a Khvedelidze weight on Lp . Then the Shilov boundary of the maximal ideal space M (alg Tp, (P C)) coincides with the whole maximal ideal space M (alg Tp, (P C)). Proof. According to 1.20(b) we must show that for each (τ0 , λ0 ) ∈ T × [0, 1] and each open neighborhood U ⊂ T × [0, 1] of (τ0 , λ0 ) there exists an operator A ∈ algp, T (P C) such that sup (Γ Ap, )(t, λ) > sup (Γ Ap, )(t, λ). (t,λ)∈U
(5.28)
(t,λ)∈U /
First suppose λ0 ∈ (0, 1). Then U can be assumed to be of the form U = {(τ0 , λ) : λ − λ0  < ε}, where ε < min{λ0 , 1 − λ0 }. Recall that r(τ0 ) := p / {t1 , . . . , tn } and r(τ0 ) := (1/p + µj )−1 for τ0 = tj . In either case for τ0 ∈ there is a ν ∈ (−1/2, 1/2) satisfying (1/2 + ν)−1 = r(τ0 ). Let 0 denote the Khvedelidze weight t − τ0 ν . Since alg T2,0 (P C) is a C ∗ algebra, there is an 2, 5 A ∈ alg2,0 T (P C) which satisﬁes (5.28) with
0 in place of p, . We may clearly assume that A is a ﬁnite productsum j k T (ajk ) with ajk ∈ P C0 and thus that A is in algp, T (P C). Because for (τ0 , λ) ∈ U both (Γ Ap, )(τ0 , λ) and (Γ A2,0 )(τ0 , λ) are equal to 1 − σr(τ0 ) (λ) ajk (τ0 − 0) + σr(τ0 ) (λ)ajk (τ0 + 0) , j
k
it follows that A fulﬁlls (5.28). Now suppose λ0 = 0. Choose a z0 ∈ D so that Ar(τ0 ) (1, z0 ) ∩ T = {1} and let a ∈ P C0 be any function with the following properties: a(τ0 − 0) = 1, a(τ0 + 0) = z0 , a is continuous on T \ {τ0 }, and a(t) < 1 for t = τ0 . It is easily seen that for each neighborhood U of (τ0 , 0) the inequality + + + + sup + Γ Tp, (a) (t, λ)+ = 1 > sup + Γ Tp, (a) (t, λ)+ (t,λ)∈U
(t,λ)∈U /
holds. The situation is analogous for λ0 = 1. 5.47. Theorem. Let be a Khvedelidze weight on Lp and let A belong to algp, T (P CN ×N ). Then p A ∈ Φ(HN ()) ⇐⇒ Γ (det A)p, (t, λ) = 0 ∀ (t, λ) ∈ T × [0, 1]. Proof. In view of Theorems 5.34 and 1.14(c) we have p ()) ⇐⇒ det A ∈ Φ(H p ()). A ∈ Φ(HN
So the implication “⇐=” of the theorem is immediate from Theorem 5.45. To get the opposite implication assume (Γ (det A)p, )(t0 , λ0 ) = 0 for some
5 Toeplitz Operators on H p
276
(t0 , λ0 ) ∈ T × [0, 1]. Then Theorem 5.46 in conjunction with 1.20(c) gives that (det A)p, is a topological divisor of zero in alg Tp, (P C) and thus in the Calkin algebra L(H p ())/C∞ (H p ()). Consequently, det A cannot be in Φ(H p ()). 5.48. Index computation. (a) Proposition 5.39 says that Ind T (a) equals −ind ap, whenever a ∈ P C0 and T (a) ∈ Φ(H p ()). The same is true for a ∈ P C. Indeed, if a ∈ P C and T (a) ∈ Φ(H p ()), then ap, does not vanish on T × [0, 1] (Theorem 5.43), so there is an ε > 0 such that T (b) ∈ Φ(H p ()), Ind T (b) = Ind T (a), and ind bp, = ind ap, whenever b ∈ P C0 , a − b∞ < ε (the equality Ind T (b) = Ind T (a) is a consequence of 1.12(d)), and since it is already known that Ind T (b) = −ind bp, , it follows that Ind T (a) = −ind ap, . p ()), then Ind T (a) = −ind det(ap, ) (b) If a ∈ (P C0 )N ×N , T (a) ∈ Φ(HN (note that det(ap, ) = 0 on T × [0, 1] by Theorem 5.43). To see this recall ﬁrst the following well known fact:
If a ∈ (P C0 )N ×N ∩ GL∞ N ×N , then a can be written in the form a = ϕbψ, where ϕ and ψ are in GCN ×N and b ∈ (P C0 )N ×N ∩ GL∞ N ×N is an uppertriangular matrix function. A proof is, e.g., in Clancey, Gohberg [138, Chap. VIII, Lemma 2.2]. Now let a ∈ (P C0 )N ×N and suppose det(ap, ) = 0 on T × [0, 1]. Write a in the form a = ϕbψ as above and note that ap, = ϕbp, ψ. Hence det(bp, ) and thus bjp, (b1 , . . . , bN are the diagonal entries of b) do not vanish on T × [0, 1]. It follows that T (b) and T (bj ) are Fredholm, and since Ind T (ϕ) = −ind det ϕ,
Ind T (ψ) = −ind det ψ,
and Ind T (b) =
N
Ind T (b ) = − j
j=1
N
ind bjp,
j=1
= −ind
N
bjp,
(because ind fp, + ind gp, = ind fp, gp, )
j=1
= −ind det(bp, ), we have Ind T (a) = Ind (T (ϕ)T (b)T (ψ) + compact operator) = Ind T (ϕ) + Ind T (b) + Ind T (ψ) (Atkinson) = −ind (det ϕ · det(bp, ) · det ψ) = −ind det(ϕbp, ψ) = −ind det(ap, ). (c) The same argument as in (a) shows that Ind T (a) = −ind det(ap, ) p ()). whenever a ∈ P CN ×N and T (a) ∈ Φ(HN
5.6 P2 C Symbols
277
r 5 s
m 5n (d) Now let A = i=1 k=1 T (ajk ) and det A = i=1 k=1 T (bjk ) with ajk ∈ P CN ×N and bjk ∈ P C. One can show that there exists a c ∈ P CM ×M , where M = N (mn + n + 1), such that p bjk = det c, IndHNp () A = IndHM () T (c) j
k
(see Krupnik [329, Theorems 1.7 and 2.4]). This and (c) imply that Ind A = −ind det(cp, ) = −ind (bjk )p, . j
k
(e) Finally, if A is any operator in algp, T (P CN ×N ) which is Fredholm p () and An ∈ algp, T (P CN ×N ) are any ﬁnite productsums such that on HN p ()) for all n large enough A−An L(HNp ()) → 0, then, by 1.12(d), An ∈ Φ(HN and Ind A = −ind Γ (det A)p, := − lim ind Γ (det An )p, . n→∞
5.6 P2 C Symbols 5.49. Deﬁnitions. Let a ∈ (P2 C)N,N . Denote by Ψ (a) the set of all points τ ∈ T such that aXτ = bXτ for some b ∈ P CN ×N and set a(τ, 0) := b(τ − 0), a(τ, 1) := b(τ + 0) in that case. Put Ψ c (a) = T \ Ψ (a), and for τ ∈ Ψ c (a) let a(τ, 0) and a(τ, 1) simply denote the two elements of the set a(Xτ ). Given α, β ∈ CN ×N deﬁne 0 1 AN r (α, β) := λ ∈ C : det (1 − σr (µ))α + σr (µ)β − λI = 0 ∀ µ ∈ [0, 1] , where σr is as in 5.12. For 1 < p < ∞ put / OpN (α, β) := AN r (α, β) r∈[p,q]
1 p
+
1 =1 . q
N In the case N = 1 the sets AN r (α, β) and Op (α, β) are nothing else than the arcs and lentiform domains introduced in 5.12. Finally if ≡ 1 we denote spτp, T (a) by spτp T (a).
5.50. Theorem (Spitkovsky). If a ∈ (P2 C)N,N , then , N Ap (a(τ, 0), a(τ, 1)) for τ ∈ Ψ (a), τ spp T (a) = OpN (a(τ, 0), a(τ, 1)) for τ ∈ Ψ c (a).
(5.29)
Proof. A proof of this theorem is in Spitkovsky [503]. We conﬁne ourselves to the proof of the above theorem for a special but suﬃciently large class of scalarvalued P2 C functions. Note that the inclusion spτp T (a) ⊂ Op (a(τ, 0), a(τ, 1)) (scalar case!) immediately results from Corollary 5.28.
278
5 Toeplitz Operators on H p
5.51. Deﬁnition. For p > 0 let H p denote the set of all functions f which are analytic in D and satisfy ( 2π f (reiθ )p dθ < ∞. sup r∈(0,1)
0
For 1 ≤ p < ∞ this deﬁnition agrees with Deﬁnition 1.39. If f ∈ H p (p > 0), then the nontangential limits f (eiθ ) exists almost everywhere and deﬁne a * 2π function in Lp , that is, 0 f (eiθ )p dθ < ∞ (see Duren [178, Theorem 2.2]). Therefore functions in H p may be identiﬁed with their boundary values on T. 5.52. Lemma. Let f ∈ H 1/2 and suppose that f (z) = 0 for all z ∈ D and that f is realvalued and nonnegative on T. Then f is constant in D. Proof. Since f does not vanish in D, there is a function g which is analytic in D and satisﬁes g 2 = f . Because g is in H 1 and is realvalued on T, it follows that g is constant in D. Hence f is constant in D, too. Note that every function a ∈ P2 C can be written in the form a = αχE + βχE c , where α, β ∈ C, E is a measurable subset of T, and E c := T \ E. 5.53. Proposition. If a = αχE + βχE c ∈ P2 C \ C, then spL(H p ) T (a) = Op (α, β). Proof. Theorem 5.16(b) shows that spL(H p ) T (a) ⊂ Op (α, β). So let us prove the reverse inclusion. Without loss of generality suppose that 1 < p < 2. Assume 0 ∈ Op (α, β) and T (a) ∈ GL(H p ). Then 0 ∈ Op (α/α, β/β) and T (a/a) ∈ GL(H p ), and therefore it can be assumed that α = 1 and β = eiθ , where 2π/q ≤ θ < π. From Theorem 5.5 we deduce that a = bc−1 , where b ∈ H q , b−1 ∈ H p and c ∈ H q , c−1 ∈ H p (note that actually b and c are outer: if b = hω with h ∈ H q outer and ω inner then ω = hb−1 ∈ H 1 , whence ω = const). Because bc = bc−1 c2 , the argument of bc takes only the values 2kπ and θ+2kπ (k ∈ Z) on T. Let v be the realvalued function on T given by 0 ≤ v ≤ θ,
v ≡ arg(bc) mod 2π,
let v) be the conjugate function of v, and put ϕ := e−)v+i v . It is clear that ϕ(z) = 0 for z ∈ D, and Theorem V, D, 1◦ in Koosis [316] implies that ϕ and ϕ−1 are in H r for all r < π/θ. Put ψ := (bc)ϕ−1 . Obviously, ψ(z) = 0 for z ∈ D. By the choice of v, ψ is realvalued and nonnegative on T. Since b, c ∈ H q ⊂ H 2π/θ , we have bc ∈ H π/θ ⊂ H 1 . Finally, because ϕ−1 ∈ H 1 , it follows that ψ ∈ H 1/2 . So Lemma 5.52 shows that ψ is some positive constant, ψ(0) say, in D. Put := 2π/θ. We have bc = ϕψ and eiv = 1 a.e. on T, hence b(t)c(t) = ψ(0) ϕ(t) = ψ(0) e−)v(t) a.e. on T.
5.6 P2 C Symbols
279
Thus, (bc) is nonnegative on T. Clearly, (b(z)c(z)) = 0 for z ∈ D, and since bc ∈ H π/θ , the function (bc) belongs to H 1/2 . Therefore again Lemma 5.52 can be applied to deduce that (bc) is constant in D, and hence bc itself must be constant in D. The conclusion is that the argument of bc and thus the argument of bc−1 is constant on T, which contradicts our assumption that a be not constant. 5.54. Deﬁnition. A function a = αχE +βχE c ∈ P2 C will be said to be regular if for each open subarc U of T at least one of the sets E ∩ U and E c ∩ U has a nonempty interior. For instance, if both E and E c are (possibly countable) unions of subarcs of T, then a is regular. Also notice that a is regular if E or E c is a Cantor set. A function b ∈ P2 C will be called locally regular if for each τ ∈ T there exists a regular function aτ ∈ P2 C such that bXτ = aτ Xτ . p 5.55. Lemma. Let 1 < p < 2, a ∈ L∞ N ×N , and suppose T (a) ∈ Φ(HN ) and 2 T (a) ∈ Φ(HN ). Then Indp T (a) ≥ Ind2 T (a). p q 2 Proof. Since HN ⊃ HN ⊃ HN , we conclude that αp (T (a)) ≥ α2 (T (a)) and α2 (T (a∗ )) ≥ αq (T (a∗ )), which gives the assertion at once.
5.56. Proposition. Let a = αχE + βχE c be a regular function in P2 C and suppose Ψ c (a) is not empty. Then spΦ(H p ) T (a) = Op (α, β). Proof. From Proposition 5.53 we deduce that spΦ(H p ) T (a) ⊂ Op (α, β). To prove the opposite inclusion suppose 1 < p < 2, 0 ∈ Op (α, β) \ [α, β] and / [α, β], we have T (a) ∈ GL(H 2 ). The preceding T (a) ∈ Φ(H p ). Since 0 ∈ lemma therefore shows that Ind T (a) = m ≥ 0. Let τ ∈ Ψ c (a). In view of the hypothesis that a be regular we may assume that there are points t1 , t2 , . . . ∈ T such that tn ≺ tn+1 for all n, tn → τ as n → ∞, a is identically α on the arcs (t2k , t2k+1 ) and takes the value β on a subset of positive measure of each of the arcs (t2k+1 , t2k+2 ). Deﬁne b1 ∈ P2 C / (t3 , t4 ). Put q := a/b1 . by b1 (t) = α for t ∈ (t3 , t4 ) and b1 (t) = a(t) for t ∈ Due to Theorem 5.32 we have T (a) = T (b1 )T (q) + K = T (q)T (b1 ) + L with K, L ∈ C∞ (H p ). Therefore T (b1 ) and T (q) belong to Φ(H p ) and Ind T (b1 ) = Ind T (a) − Ind T (q). Lemma 5.55 together with Proposition 5.53 gives that Ind T (q) ≥ 1, and thus Ind T (b1 ) ≤ m − 1. On repeating this construction we ﬁnally arrive at a function bm+1 ∈ P2 C \ C whose essential range is the set {α, β} and which has the property that T (bm+1 ) ∈ Φ(H p ) and Ind T (bm+1 ) < 0. By Lemma 5.55 this is impossible. Thus Op (α, β) \ [α, β] is a subset of spΦ(H p ) T (a), and therefore spΦ(H p ) T (a) = Op (α, β). 5.57. Theorem Let a ∈ P2 C be locally regular. Then (5.29) holds.
280
5 Toeplitz Operators on H p
Proof. For τ ∈ Ψ (a) this follows from Propositions 5.25 and 5.40. So let τ ∈ Ψ c (a) and let b ∈ L∞ be any function satisfying bXτ = aXτ . We must show that Op (a(τ, 0), a(τ, 1)) is a subset of spΦ(H p ) T (b). This shows that Op (a(τ, 0), a(τ, 1)) is contained in sp τp T (a), and as the reverse inclusion results from Corollary 5.28, the assertion of the theorem follows. Thus, assume T (b) ∈ Φ(H p ) but let the origin lie in the interior of Op (a(τ, 0), a(τ + 0)). Choose a regular function c ∈ P2 C so that cXτ = τ ) is invertible (PropoaXτ = bXτ . Then Tp (c) + Zpτ (:= Tp,1 (c) + Zp,1 sition 5.25 and Theorem 5.29(a)). From Theorem 1.32(c) we deduce that Tp (c) + Zpτ is invertible for all t in some open arc U := (τ − δ, τ + δ). Because c is regular, there are open arcs V := (τ − δ1 , τ − δ2 ) ⊂ (τ − δ, τ ) and W := (τ + δ3 , τ + δ4 ) ⊂ (τ, τ + δ) such that cV and cW are constant. Deﬁne d ∈ P2 C by d(t) = cV for t ∈ (−τ, τ − δ2 ), d(t) = cW for t ∈ (τ + δ3 , −τ ), d(t) = c(t) for t ∈ (τ − δ2 , τ + δ3 ). Then Tp (d) + Zpτ is invertible for all t ∈ T (recall that the origin has been supposed to be not on the boundary of Op (a(τ, 0), a(τ + 0))). Theorem 5.29(a) now implies that T (d) ∈ Φ(H p ), and therefore, by Proposition 5.56, 0∈ / Op (d(τ, 0), d(τ, 1)) = Op (a(τ, 0), a(τ, 1)). This contradicts our assumption that the origin be an inner point of the set Op (a(τ, 0), a(τ, 1)). Thus, all inner points of Op (a(τ, 0), a(τ + 0)) belong to spΦ(H p ) T (b) and hence this spectrum contains Op (a(τ, 0), a(τ, 1)). 5.58. Index computation. Let a ∈ P2 C and T (a) ∈ Φ(H p ). We claim that the set Ψ0 (a) := τ ∈ Ψ (a) : 0 ∈ Op (a(τ, 0), a(τ, 1)) is ﬁnite. Indeed, if τ ∈ Ψ0 (a), then τ ∈ Ψ (a) and therefore the (essential) values taken by a on the right (resp. left) of τ are close to a(τ, 1) (resp. a(τ, 0)) and thus close to each other. It follows that there is an open subarc U (τ ) of T containing τ such that U (τ ) \ {τ } ⊂ T \ Ψ0 (a). In other words, the points in Ψ0 (a) are isolated points of T. Theorem 5.50 implies that a is psectorial on / Ψ0 (a). So the “psectorial version” of Theorem 3.9 (see 5.12) Xt for each t ∈ implies that T \ Ψ0 (a) is open and thus that Ψ0 (a) is closed. It results that Ψ0 (a) is ﬁnite, as desired. We now construct b ∈ P2 C as follows. Choose an ε > 0 so that for each τ ∈ Ψ0 (a) the union of (τ − ε, τ ) and (τ, τ + ε) is a subset of T \ Ψ0 (a) and that the origin is in Op (τ1 , τ2 ) whenever τ1 ∈ (τ − ε, τ ), τ2 ∈ (τ, τ + ε), τ ∈ Ψ0 (a). Then let b ∈ P2 C be any function which is continuous on [τ − ε, τ + ε] and satisﬁes b(τ − ε) = a(τ, 0), b(τ + ε) = a(τ, 1), b [τ − ε, τ + ε] ⊂ Ap (a(τ, 0), a(τ, 1)) for each τ ∈ Ψ0 (a) and equals a on T \ [τ − ε, τ + ε]. Theorem 5.50 τ ∈ Ψ0 (a)
shows that b is locally p, 1sectorial (i.e., is identically 1) over C, and thus
5.7 FisherHartwig Symbols
281
Indp T (b) = Ind2 T (b) = −ind {kλ b} (Proposition 5.18) One can show that Indp T (a) = Indp T (b). Thus, Indp T (a) = −ind {kλ b}, a formula which is not very good but better than nothing. 5.59. Open problem. Establish an index formula for (block) Toeplitz operators on H p (or H p ()) with P QC symbols. In this connection (and in connection with 5.58, too) it would be interesting to know whether the harmonic extension hr a can be replaced by something, hr,p a say, so that hr,p a ∈ C and Ind T (a) = −ind {hr,p a} for every a ∈ P C (P2 C, P QC, P2 QC) generating a Fredholm operator T (a).
5.7 FisherHartwig Symbols 5.60. Deﬁnitions. A FisherHartwig symbol is a function of the form a(t) = b(t)
m
t − tj 2αj ϕβj ,tj (t)
(t ∈ T),
(5.30)
j=1
where (i) b ∈ C and b(t) = 0 for t ∈ T; (ii) t1 , . . . , tm are pairwise distinct points on T; (iii) αj ∈ C and Re αj > −1/2 for j = 1, . . . , m; (iv) βj ∈ C for j = 1, . . . , m and ϕβj ,tj is given as in 5.35. Condition (iii) ensures that a is a function in L1 . Denote the function t−tj 2αj by ωαj = ωαj ,tj . Then due to 5.35, ωαj = ξαj ηαj ,
ϕβj = ξ−βj ηβj ,
and therefore the function (5.30) can be written in the form a(t) = b(t)
m j=1
tj 1− t
δj
t 1− tj
γj
(t ∈ T),
(5.31)
where δj = αj − βj and γj = αj + βj . Note that ωαj = ωRe αj ωi Im αj , and that ωRe αj has a zero (Re αj > 0) or a pole (Re αj < 0) at t = tj , while ωi Im αj has a discontinuity of oscillating type at t = tj if Re αj = 0 and Im αj = 0. For βj = 0, ϕβj has a jump discontinuity at t = tj and ϕβj (tj − 0) = exp(2πiβj ). ϕβj (tj + 0) H p spaces with Khvedelidze weight are a natural terrain for the study of Toeplitz operators with FisherHartwig symbols.
282
5 Toeplitz Operators on H p
5.61. Lemma. Let ν ∈ C, τ ∈ T, and let be a Khvedelidze weight on Lp . If t − τ Re ν (t) is also a Khvedelidze weight on Lp , then T (ξν,τ ) : H p (t − τ Re ν (t)) → H p ((t)) is bounded and invertible and the inverse is T (ξ−ν,τ ). The same is true with ξ±ν,τ replaced by η±ν,τ . * Proof. If ϕ ∈ H p (t − τ Re ν (t)), then T ξν ϕp p dm equals ( τ exp − p Im ν arg 1 − t − τ pRe ν ϕ(t)p p (t) dm t T ( ≤ exp(πp Im ν) ϕ(t)p (t − τ Re ν (t))p dm, T
which gives the boundedness of T (ξν ). It can be shown similarly that T (ξ−ν ) : H p ((t)) → H p (t − τ Re ν (t)) is bounded. Since, for ∈ H p (t − τ Re ν (t)), P (ξ−ν P (ξν ϕ)) = P (ξ−ν ξν ϕ) − P (ξ−ν Q(ξν ϕ)) = ϕ, it follows that T (ξ−ν ) is the inverse of T (ξν ). The proof for T (ην ) is analogous. 5.62. Theorem. Let a be given by (5.30) and suppose that, in addition to (iii), Re αj < 1/2 for all j = 1, . . . , n. Choose µj ∈ R and κj ∈ Z so that µj  <
1 − Re αj , 2
for j = 1, . . . , n. If 0 (t) =
Re βj − κj − µj  < m
1 2
t − τk λk
k=1
is a Khvedelidze weight on L2 and the sets {t1 , . . . , tn } and {τ1 , . . . , τm } are disjoint, then 1 (t) := 0 (t)
n
t − tj µj +Re αj ,
2 (t) := (t)
j=1
n
t − tj µj −Re αj
j=1
are Khvedelidze weights on L and T (a) : H (1 ) → H 2 (2 ) is a bounded Fredholm operator whose index is 2
Ind T (a) = −
2
n
κj − ind b.
j=1
If Ind T (a) = 0, then T (a) is invertible. Remark. The consideration of T (a) on H p () (p = 2) does not remove the extra assumption that Re αj < 1/2 for all j = 1, . . . , n.
5.8 Notes and Comments
283
Proof. That 1 and 2 are Khvedelidze weights on L2 follows from the choice of the µj ’s. Moreover, 3 (t) :=
n
t − tj µj
j=1
m
t − τk λk
k=1
is also a Khvedelidze weight. The diagram
is commutative. By virtue of Lemma 5.61, the vertical arrows represent bounded and invertible operators. Corollary 5.33 and Lemma 5.36 show that the lower horizontal arrow represents a bounded Fredholm operator of index n − j=1 κj − ind b, which, by 5.2 is invertible if its index is zero. This yields the assertion of the theorem.
5.8 Notes and Comments 5.2. Simonenko [492], [496], Gohberg, Krupnik [232], Krupnik [326]. 5.3. Rochberg [433]. 5.4–5.5. This theorem goes back to Simonenko [492], [496], but see also Widom [556], [559] and Krupnik [326]. For more about this topic we refer to Gohberg, Krupnik [232] (scalar case) and Clancey, Gohberg [138], Krupnik [329], Litvinchuk, Spitkovsky [340], and Gohberg, Kaashoek, Spitkovsky [226] (matrix case). We remark that instead of (5.1) it suﬃces to require only 1 that a±1 ± ∈ L± ; the argument of the ﬁrst part of the proof of Theorem 5.9 shows that the boundedness of P on Lp (a−1 + w) automatically implies that a− ∈ Lp− (w) etc. Ehrhardt [192] and Basor and Ehrhardt [32], [34] introduced a socalled asymmetric factorization in Lp (1 < p < ∞). This factorization does for ToeplitzplusHankel operators what WienerHopf factorization does for Toeplitz operators. They proved that R(a) := T (a) + H(a) is Fredholm on H p if and only if a ∈ L∞ admits an asymmetric factorization in Lp and that in this case dim Ker R(a) = max{0, −κ}, dim Ker R∗ (a) = max{0, κ}, where κ is the index of the asymmetric factorization of a. 5.6–5.7. The norm (5.4) has been used by many authors, but we do not know who was the ﬁrst to observe that it is the choice of just this norm which guarantees equality in (5.5). 5.8–5.9. Hardy, Littlewood, and K. I. Babenko established Theorem 5.9 for n = 1, and Khvedelidze [312] proved it for general n. Of course, they did this
284
5 Toeplitz Operators on H p
without using 1.46. We learned the proof of the “only if” part given here from Krupnik [328], [329]. For a proof of the “if” portion which does not invoke 1.46 see also Gohberg, Krupnik [232]. 5.11. The norm of the Cauchy singular integral operator on LpN (t − τ µ ) was ﬁrst computed by Verbitsky and Krupnik. For more about this and for the history of the problem we refer to the books by Krupnik [329] and by Gohberg, Krupnik [232, Chap. 13] and also to the recent paper by Hollenbeck and Verbitsky [286]. The norm of the Riesz projection on Lp (p = 2) was intractable a long time, although there was an old conjecture: P L(Lp ) = 1/ sin(π/r) (r = max{p, q}) (see Verbitsky, Krupnik [544]). Only recently Hollenbeck and Verbitsky [286] were able to prove this conjecture. This paper also contains a detailed history of this diﬃcult problem. For further results in this direction see also Hollenbeck, Kalton, Verbitsky [287]. 5.12–5.23. There is no problem in deﬁning rsectoriality on a closed subset of X in the scalar case, but there are several possibilities of deﬁning rsectoriality in the matrix case. Any such deﬁnition for N × N matrix functions should meet the following three requirements. (i) For N = 1 and general r as well as for r = 2 and general N it should go over into the “canonical” deﬁnitions for these two cases. (ii) Theorem 5.16(b) should hold. (iii) The deﬁnition should be as geometrical as possible. The deﬁnition given in 5.12 meets (i) and (ii), and we leave the reader with the judgement whether it meets (iii). Theorem 5.16(a) was proved by Spitkovsky [501], [502] for µ = 0 and by Krupnik [329] for spaces with weight. The proof given here and also Lemma 5.15 are Krupnik’s. Theorem 5.16(b) for µ = 0 was established by Spitkovsky [501], [502] under the hypothesis (5.10), and Krupnik [329] derived Theorem 5.16(b) for the case that a satisﬁes either (5.8) or (5.9). A key role in the proof of this result is of course played by Lemma 5.14. Krupnik [329] proved this lemma under the assumption (5.8), our contribution is that we prove it under the weaker assumption (5.10). We ﬁnally remark that Krupnik [329] even established Theorem 5.16 for functions with values in L(H), where H is an arbitrary separable Hilbert space. The scalar version of Theorem 5.17 (for B = C) is due to Frolov [206]. Again it is not at all clear how to deﬁne local p, sectoriality in the matrix case. Spitkovsky [501], [502] calls a function a ∈ GL∞ N ×N psectorial if for each τ ∈ T there exists a complex number γτ of modulus 1 such that W (γτ a(x)) ⊂ p ) whenever Sr (r = max{p, q}) for all x ∈ Xτ , and he proved that T (a) ∈ Φ(HN a is psectorial in this sense. The (serious) disadvantage of this deﬁnition is that functions in GCN ×N need not be locally psectorial. Krupnik [327], [329] proved Theorem 5.17 (for B = C) under the hypothesis that for each τ ∈ T
5.8 Notes and Comments
285
∞ ∞ there are fτ , gτ ∈ GHN ×N and bτ ∈ GLN ×N such that aXτ = fτ bτ gτ Xτ and I − bτ < sin(π/rτ ), where rτ is as in 5.12. Theorem 5.17 for B = QC is due to the authors. The proof of Theorem 5.17 given in the text makes use of an argument of Gohberg, Krupnik [232, Section 7.1]. Theorem 5.21 was established in B¨ ottcher [69]. Theorem 5.22 is Krupnik’s [328], [329].
5.24–5.30. This is due to the authors (also see B¨ ottcher, Roch, Silbermann [100]). 5.31. This theorem is known, but we know no reference where it is stated in the form presented here. We refer to the notes to Theorem 4.79 and add the article Gohberg, Krupnik [230]. 5.32–5.34. Corollary 5.33 goes back to Gohberg, Krupnik [229], [230] and Theorem 5.34 to Sarason [456]. The proof of Theorem 5.32 makes essential use of the argument of Sarason [453]. 5.36–5.39. See Gohberg, Krupnik [229], [230], [232]. The latter reference also contains a history of this topic. 5.40–5.48. Theorems 5.43, 5.45, 5.47, and the results of 5.48 were obtained by Gohberg and Krupnik [231], [232]. The present method of deriving these results (except for those of 5.48) is due to the authors. Propositions 5.40 and 5.44 are new. Note that their formulation requires an appropriate deﬁnition of local Toeplitz operators and local spectra, for which see B¨ottcher, Roch, Silbermann [100]. A result of the type of Theorem 5.46 ﬁrst appeared in B¨ottcher [64], [67], and this observation will play a crucial role in the bilocal theory of Toeplitz operators over the quarterplane (see Chapter 8). A Fredholm criterion for Toeplitz operators with P QC symbols on Hardy spaces with Khvedelidze weights was obtained by B¨ ottcher and Spitkovsky [120]. The theories of locally sectorial and piecewise quasicontinuous symbols were united in B¨ ottcher, Silbermann, Spitkovsky [117]. In that paper we introduced the notion of a symbol that is piecewise p, sectorial over QC and proved that Toeplitz operators with such symbols are Fredholm on H p (). In this book we restrict ourselves to Toeplitz operators on H p over the unit circle T with a Khvedelidze weight w. One may replace T by a more general Jordan curve Γ (= a curve that is homeomorphic to T) and one can take w from a more general class of weights. In a sense, the most general setting is that where Γ belongs to the class of Carleson curves (which are frequently also referred to as AhlforsDavid curves) and w is a Muckenhoupt weight. In this case, every a ∈ L∞ (Γ ) induces a bounded Toeplitz operator on the weighted Hardy space H p (Γ, w) := P Lp (Γ, w). Coburn’s theorem remains true in this context: T (a) is invertible if and only if it is Fredholm of index zero. For continuous symbols, a ∈ C(Γ ), the operator T (a) is Fredholm on H p (Γ, w) if and only if a has no zeros Γ , in which case the index of T (a) is minus the winding number of a about the origin. Surprises emerge in the case of piecewise continuous symbols, a ∈ P C(Γ ). Spitkovsky [506] discovered that
286
5 Toeplitz Operators on H p
if Γ is a nice curve (for instance, if Γ = T) but w is allowed to be an arbitrary Muckenhoupt weight, then the circular arcs Ar (z1 , z2 ) may blow up to socalled horns. Yu.I. Karlovich and one of the authors subsequently elaborated the Fredholm theory of Toeplitz operators with piecewise continuous symbols on H p over arbitrary Carleson curves with arbitrary Muckenhoupt weights. The metamorphosis of the (local) spectra in this most general situation is extremely fascinating: the circular arcs and horns may change into logarithmic doublespirals and logarithmic horns and these may in turn become socalled leaves. The metamorphosis ends up with socalled leaves with a halo. For a ﬁrst acquaintance with these beautiful issues we refer to the articles B¨ ottcher, Karlovich [93] and B¨ ottcher, Karlovich, Rabinovich [94]. A comprehensive treatment of the entire matter is the monograph B¨ ottcher, Karlovich [92]. 5.49–5.59. Theorem 5.50 and the results of 5.58 were established by Spitkovsky [503]. Spitkovsky’s proof is rather complicated. The proof presented here does not give Theorem 5.50 in full generality, but is essentially simpler than the one of Spitkovsky. It is due to the authors; the proof of Proposition 5.53 was worked out by the authors together with I. M. Spitkovsky. For further results on Toeplitz operators with Pn C and Pn C symbols we refer to Shargorodsky [469], [470]. 5.60–5.62. Symbols of the form (5.19) were ﬁrst considered by Fisher and Hartwig [204], [205] in connection with the asymptotic behavior of Toeplitz determinants. Lemma 5.61 and Theorem 5.62 were established in B¨ottcher, Silbermann [106], [108], [110].
6 Toeplitz Operators on p
We have already settled the Fredholm theory of the operators in the algebra algL(pN ) T ((Cp + Hp∞ )N ×N ) (Corollaries 4.7 and 4.8) and stated a localization result for Toeplitz operators on pN (Theorem 2.95). This chapter is devoted to some more delicate questions of the p theory of Toeplitz operators. The basic diﬃculty in the theory of Toeplitz operators on p is the multiplier problem: while a Toeplitz operator is bounded on H p or 2 if and only if its symbol is in L∞ , it is diﬃcult to describe those functions which generate bounded Toeplitz operators on p .
6.1 Multipliers on Weighted p Spaces 6.1. Deﬁnitions. Let 1 < p < ∞ and µ ∈ R. We let Mµp denote the collection of all functions in L1 such that M (a)x ∈ pµ (Z) for each x ∈ 0 (Z) and ! # M (a)xp,µ aMµp := sup : x ∈ 0 (Z), x = 0 < ∞. xp,µ If µ = 0, then Mµp is just the class (Banach algebra) M p deﬁned in 2.3. Given a ∈ L1 we shall write T (a) ∈ L(pµ ) if T (a)x ∈ pµ for each x ∈ 0 and ! # T (a)xp,µ 0 T (a)L(pµ ) := sup : x ∈ , x = 0 < ∞. xp,µ Since the discrete Riesz projection is bounded (and has norm 1) on every space pµ (Z) (recall 1.49), it is clear that T (a) ∈ L(pµ ) whenever a ∈ Mµp and that T (a)L(pµ ) ≤ aMµp . 6.2. Basic properties of the classes Mµp . Throughout this chapter we suppose that 1 < p < ∞ and we let q satisfy 1/p + 1/q = 1. q , and if a ∈ Mµp then a ∈ Mµp and (a) Mµp = M−µ q q . = aM−µ aMµp = aMµp = aM−µ
6 Toeplitz Operators on p
288
Proof. The proof is the same as the one for the case µ = 0 (for which see 2.5(a), (b)). (b) If T (a) ∈ L(pµ ), then a ∈ M p ( = M0p ) and aM p = T (a)L(p ) ≤ T (a)L(pµ ) . Proof. By virtue of 2.5(a), (b) we may assume that p ≥ 2. The mapping Λ given by Λ : pµ (Z) → p (Z),
{xn }n∈Z → {xn (n + 1)µ }n∈Z
is an isometric isomorphism of pµ (Z) onto p (Z). If T (a) ∈ L(pµ ), then clearly P M (a)P ∈ L(pµ (Z)) and P M (a)P L(pµ (Z)) ≤ T (a)L(pµ ) . Hence, ΛP M (a)P Λ−1 xp ≤ T (a)L(pµ ) xp
∀ x ∈ 0 (Z),
and since U ±n are isometries on p (Z), it follows that U −1 ΛP M (a)P Λ−1 U n xp ≤ T (a)L(pµ ) xp
∀ x ∈ 0 (Z).
(6.1)
We claim that M (a)x ∈ p (Z) for all x ∈ 0 (Z) and that the elements An x := U −n ΛP M (a)P Λ−1 U n x converge to M (a)x as n → ∞ in the norm of p (Z) for each x ∈ p (Z). Once this has been proved passage to the limit n → ∞ in (6.1) yields the boundedness of M (a) on p (Z) and the inequality aM p ≤ T (a)L(pµ ) , so that (2.9) completes the proof. Because M (a)ej = U j M (a)e0 and An ej = U j An+j e0 , it suﬃces to prove that M (a)e0 ∈ p and An e0 − M (a)e0 p → 0 as n → ∞. First let µ ≥ 0. Then {ak }k∈Z+ = T (a)e0 ∈ pµ ⊂ p . To see that {a−k }k∈Z+ is in p denote T (a)pL(pµ ) by C and observe that ∞
ak p (k + j + 1)pµ = T (a)ej pp,µ ≤ Cej pp,µ = C(j + 1)pµ ,
k=−j
whence
j k=0
a−k 
p
j+1−k j+1
pµ ≤C
for all j ∈ Z+ . If N ∈ Z+ , then there is a j ≥ N such that pµ j+1−N 1 > , j+1 2 and it results that
6.1 Multipliers on Weighted p Spaces N
a−k p ≤ 2
k=0
N
a−k p
k=0
j+1−k j+1
289
pµ ≤ 2C,
i.e., {a−k }k∈Z+ ∈ p , as desired. A simple computation shows that An e0 − M (a)e0 pp =
+ µ +p + n + 1 − k ++ a−k p ++1 − + n+1 k=1 + µ +p ∞ + n + 1 + k ++ p+ + ak  +1 − + . n+1 n
(6.2)
k=0
Let ε > 0 be arbitrarily given. Because + µ +p + + +1 − n + 1 − k + ≤ 1 + + n+1 and
+ µ +p µ p + + +1 − n + 1 + k + ≤ 1 + n + 1 + k + + n+1 n+1 ≤ (1 + (1 + k)µ )p ≤ 2p (k + 1)pµ
and {an }n∈Z ∈ p,p 0,µ , there is an N = N (ε) such that (6.2) is not greater than + + µ +p µ +p N N + + n + 1 − k ++ n + 1 + k ++ ε p+ 1 − + a−k p ++1 − + a  k + + + (6.3) 2 n+1 n+1 k=1
k=0
for all n ≥ N . But if n ∈ Z+ is large enough, then (6.3) is smaller than ε, and this ﬁnally shows that (6.2) goes to zero as n → ∞. Now let µ < 0. Since T (a) ∈ L(qµ ), we have {a−k }k∈Z+ = T (a)e0 ∈ qµ ⊂ pµ (recall that p ≥ 2). Put C := T (a)pL(pµ ) and notice that ∞
ak p (k + j + 1)−pµ = T (a)ej pp,µ ≤ Cej pp,µ = C(j + 1)−pµ
k=−j
and therefore
∞ k=0
ak p
j+1 j+k+1
pµ ≤C
for all j ∈ Z+ . If there would exist an N ∈ Z+ such that N k=0
ak p > 4C,
6 Toeplitz Operators on p
290
then for all j ≥ N satisfying
j+1 N +j+1
pµ >
1 2
we would have N k=0
ak p
j+1 j+k+1
pµ
1 ak p > 2C, 2 N
≥
k=0
which is a contradiction. Thus, {ak }k∈Z+ ∈ . Since An e0 − M (a)e0 pp again equals (6.2) and + µ +p + + +1 − n + 1 + k + ≤ 1 + + n+1 and + µ +p µ p + + n+1 +1 − n + 1 − k + ≤ 1 + + + n+1 n+1−k p
≤ (1 + (1 + k)µ )p ≤ 2(k + 1)pµ , and because we have seen that {an }n∈Z ∈ p,p µ,0 , the same argument as for µ ≥ 0 can be applied to show that (6.2) converges to zero as n → ∞. (c) If T (a) ∈ L(pµ ), then aM p ≤ T (a)Φ(pµ ) . Proof. Without loss of generality assume µ ≥ 0. Let K be any operator in C∞ (p ). The restriction of the mapping Λ deﬁned in the preceding proof to pµ is an isometric isomorphism of pµ onto p and will be denoted by Λ, too. Since V (±n) L(p ) = 1, we have V (−n) Λ(T (a) + K)Λ−1 V n xp ≤ T (a) + KL(pµ ) xp
∀x ∈ 0 .
(6.4)
If x ∈ 0 , then T (a)x ∈ pµ ⊂ p . A straightforward computation gives that V (−n) ΛT (a)Λ−1 V n ej − T (a)ej pp,µ (j = 0, 1, 2, . . .) equals j k=0
+ + µ +p µ +p ∞ + + n + j + 1 − k ++ n + j + 1 + k ++ p+ + a−k  +1 − ak  +1 − + + + n+j+1 n+j+1 p
k=0
and a similar reasoning as in the previous proof shows that this converges to zero as n → ∞. Because V n → 0 weakly on p , we deduce from 1.1(f) that V (−n) ΛKΛ−1 V n → 0 strongly on p . Thus, passage to the limit n → ∞ in (6.4) leads to T (a)xp ≤ T (a) + KL(pµ ) xp
∀ x ∈ 0 .
It follows that T (a)L(p ) ≤ T (a) + KL(pµ ) , and as K can be chosen arbitrarily, we obtain that T (a)L(p ) ≤ T (a) + KΦ(pµ ) . Equality (2.9) completes the proof.
6.1 Multipliers on Weighted p Spaces
291
(d) If µ > 1/q, then Mµp = F pµ ⊂ W . If µ < −1/p, then Mµp = F q−µ ⊂ W . Proof. Let µ > 1/q. It is not diﬃcult to see that then −qµ (k + 1)(n − k + 1) sup (n + 1)qµ =: Ap,µ < ∞. n∈Z
k∈Z
If a ∈ Mµp and x ∈ pµ (Z), then, by H¨ older’s inequality, + +p + ++ + p M (a)xp,µ = an−k xk + (n + 1)pµ + + + n k p p pµ pµ ≤ an−k  xk  (n − k + 1) (k + 1) n
k
×
p/q −qµ
(n − k + 1)
−qµ
(k + 1)
(n + 1)pµ
k
≤
Ap−1 µ,p
n
an−k p xk p (n − k + 1)pµ (k + 1)pµ
k
p p = Ap−1 (6.5) p,µ ap,µ xp,µ .
Also note that M (a)e0 pp,µ = k ak p (k + 1)pµ . This and (6.5) give
ap,µ ≤ aMµp ≤ A1/q p,µ ap,µ ,
(6.6)
which shows that Mµp = F pµ . From property (a) we now deduce that Mµp = F q−µ in case µ < −1/p. Finally, the inclusions F pµ ⊂ W (µ > 1/q) and older’s inequality. F q−µ ⊂ W (µ < −1/p) can be easily veriﬁed using H¨ (e) Suppose µ = 1/q or µ = −1/p and let a ∈ Mµp . Then a cannot have jump discontinuities, i.e., if τ ∈ T and the ﬁnite limits a(τ − 0) and a(τ + 0) exist, then a(τ − 0) = a(τ + 0). Proof. Let µ = 1/q. Assume the ﬁnite limits a(τ ±0) exist. Then (see Zygmund [591, Chap. II, Theorem 8.13]) lim
n→∞
a(τ − 0) − a(τ + 0) (Sn ) a)(τ ) = , log n π
where (Sn ) a)(τ ) denotes the nth partial sum of the Fourier series of the conjugate function ) a at τ . But n 1/p n 1/q n a)(τ ) ≤ ak  ≤ ak p (k + 1)pµ (k + 1)−qµ (Sn ) k=−n
≤ M (a)e0 p,µ
k=−n
n
1/q (k + 1)−qµ
k=−n
k=−n
≤ const aMµp (log n)1/q ,
292
6 Toeplitz Operators on p
whence a(τ − 0) = a(τ + 0). Property (a) gives the assertion for µ = −1/p. (f) Let −1/p < µ < 1/q, and suppose a is in L∞ and has ﬁnite total variation V1 (a). Then a ∈ Mµp and there is a constant cp,µ depending only on p and µ such that aMµp ≤ cp,µ aL∞ + V1 (a) . Proof. This can be proved in the same way as for µ = 0 (see 2.5(f)). Note that the restriction −1/p < µ < 1/q comes from the discrete HuntMuckenhouptWheeden theorem 1.49. (g) Mµp is a Banach algebra under the norm aMµp := M (a)L(pµ (Z)) . Proof. The same arguments as in the proof of 2.5 apply. 6.3. Remark. A decisive distinction between Toeplitz operators on p spaces with and without weight is the failure of the BrownHalmos theorem 2.7 for weighted spaces: If µ = 0 and the Toeplitz operator T (a) is bounded on pµ , then a need not belong to Mµp . Indeed, if µ > 0 and a(t) =
∞
a−n t−n (t ∈ T),
n=0
∞
a−n  < ∞,
n=0
∞ then T (a) = n=0 a−n V (−n) ∈ L(pµ ) because V (−n) L(pµ (Z)) = 1; on the other hand, since ∞
M (a)e0 pp,µ =
a−n p (n + 1)pµ ,
n=0
L(pµ (Z))
we have M (a) ∈ / whenever a ∈ / F pµ ; ﬁnally, if m ∈ Z satisﬁes the inequality m > (1/µ)(1 + 1/q), then a(t) :=
∞ −km t k=1
k2
(t ∈ T)
belongs to W \ F pµ . 6.4. Open problem. Is it true that T (a) ∈ L(pµ ) ⇐⇒ M (a) ∈ L(p,p 0,µ )? Note that the implication “⇐=” is trivial. 6.5. Theorem. Let 1 < p < ∞ and µ ∈ R. If T (a) ∈ L(pµ ) and T (a) ∈ Φ(pµ ), then a ∈ GM p .
6.2 Continuous Symbols
293
Proof. In view of 2.5(b) we may assume that p ≥ 2. Let Λ be deﬁned as in the proof of 6.2(b). If T (a) ∈ L(pµ ) and T (a) ∈ Φ(pµ ), then a ∈ M p (see 6.2(b)) and clearly ΛP M (a)P Λ−1 ∈ Φ(p ). As in the proof of Theorem 2.30 one can see that there are K ∈ C∞ (p ) and δ > 0 such that U −n ΛP M (a)P Λ−1 U n xp + P KP U n xp + δU −n QU n xp ≥ δxp (6.7) for all x ∈ p (Z). The arguments of the proof of 6.2(b) and Theorem 2.30 show that passage to the limit n → ∞ in (6.7) gives that M (a)xp ≥ δxp for all x ∈ p (Z). Now Proposition 2.29(b) implies that M (a) ∈ GL(p (Z)) and Proposition 2.28(c) ﬁnally shows that a ∈ GM p . Remark. It is clear that the above proof also works in the matrix case. Thus, p if T (a) ∈ L((pµ )N ) and T (a) ∈ Φ((pµ )N ), then a ∈ GMN ×N . 6.6. Theorem. Let 1 < p < ∞ and µ ∈ R. (a) If a ∈ Mµp ∩ GM p , then the kernel of T (a) in pµ or the kernel of T (a) in q−µ is trivial. (b) If a ∈ Mµp , T (a) ∈ Φ(pµ ), and Ind T (a) = 0, then T (a) ∈ GL(pµ ). Proof. (a) By 6.2(a), we may assume that µ ≥ 0. Let T (a)x+ = 0 and T (a)y+ = 0, where x+ ∈ pµ , y+ ∈ q−µ , and y+ = 0. A similar reasoning as in the proof of Theorem 2.38(b) leads to the equality M (a)x+ = 0. Because x+ ∈ pµ ⊂ p and a−1 ∈ M p , it follows that x+ = M (a−1 )M (a)x+ = 0. (b) Immediate from part (a) and Theorem 6.5.
6.2 Continuous Symbols 6.7. Deﬁnitions. For 1 < p < ∞ and µ ∈ R, let Cp,µ denote the closure of the Laurent polynomials in Mµp , i.e., Cp,µ := closMµp P. It is clear that Cp,µ is a closed subalgebra of Mµp . Inequality (6.6) implies that Cp,µ = F pµ ( = Mµp ) for µ > 1/q or µ < −1/p. Also notice that Cp,µ ⊂ Cp ⊂ C due to 6.2(b). Deﬁne M p,µ (p = 2 and µ = 0) as the collection of all a ∈ L∞ which belong to Mµ)p) for all p) and µ ) in some neighborhood of p and µ (depending on 2,µ a), respectively. We let M and M p,0 (p = 2 and µ = 0) refer to the set ∞ 2 ) and p) in some neighborhood of of all a ∈ L which are in Mµ) and M p) for µ µ and p, respectively. Finally, let M 2,0 := L∞ . p,µ p,µ p,µ p In what follows we shall write MN ×N , CN ×N , N instead of (Mµ )N ×N , p (Cp,µ )N ×N , and (µ )N . 6.8. Proposition. Let 1 < p < ∞ and µ ∈ R. (a) The maximal ideal space M (Cp,µ ) of Cp,µ is T and the Gelfand map is given by Γ : Cp,µ → C(T), (Γ a)(τ ) = a(τ ). (b) Cp,µ = closMµp (C ∩ M p,µ ).
294
6 Toeplitz Operators on p
Proof. (a) See the proof of Proposition 2.46(a). (b) First note that Lemma 2.44 remains valid for spaces with weight: if a ∈ Mµp , then σn aMµp ≤ aMµp for all n ≥ 0; the proof is the same as the one for spaces without weight. Now let a ∈ C ∩ M p,µ and suppose p = 2 and µ = 0. Then, by 6.2(b), a ∈ M0p+ε , where ε > 0 (resp. ε < 0) for p > 2 (resp. p < 2). The RieszThorin interpolation theorem gives γ M (a − σn a)M p ≤ M (a − σn a)1−γ M p+ε M (a − σn a)M 2 ,
(6.8)
where γ ∈ (0, 1) is some constant. By what was said above, the ﬁrst factor in (6.8) remains bounded as n → ∞, and since M 2 = L∞ and a ∈ C, the second p , where ε > 0 (resp. factor in (6.8) goes to zero as n → ∞. Because a ∈ Mµ+ε ε < 0) for µ > 0 (resp. µ < 0), application of the SteinWeiss interpolation theorem leads to M (a − σn a)δM p , M (a − σn a)Mµp ≤ M (a − σn a)1−δ Mp µ+ε
(6.9)
where δ ∈ (0, 1) is some constant. The ﬁrst factor in (6.9) remains bounded by what was said above and the second converges to zero by what has already been proved. The conclusion is that a ∈ Cp,µ . The proof can now be ﬁnished as in 2.45. 6.9. Deﬁnition. Let 1 < p < ∞ and µ ∈ R. Deﬁne p,µ p,µ p,µ alg T (CN ×N ) := algL(N ) T (f ) : f ∈ CN ×N , T (f ) : f ∈ PN ×N . algp,µ T (P) := algL(p,µ N ) It is clear that the second algebra is contained in the ﬁrst algebra, and from the deﬁnition of Cp,µ it is immediately seen that the two algebras are actually p equal to each other. In the case µ = 0 we write alg T (CN ×N ) instead of p,0 alg T (CN ×N ). If f ∈ Cp,µ , then H(f ) ∈ C∞ (pµ ). This can be proved as for µ = 0 (Theorem 2.47(a)). Consequently, if fjk is a ﬁnite collection of functions in Cp,µ , then, by (2.18), T (fjk ) − T fjk ∈ C∞ (pµ ). j
k
j
k
The same reasoning as in the proof of Proposition 4.5 shows that C∞ (p,µ N ) is p,µ p,µ ). Therefore, alg T (C ) equals a subset of alg T (CN ×N N ×N p,µ p,µ T (f ) + K : f ∈ CN closL(p,µ ×N , K ∈ C∞ (N ) N ) T (f ) + K : f ∈ PN ×N , K ∈ C∞ (p,µ = closL(p,µ N ) . N )
(6.10)
6.2 Continuous Symbols
295
6.10. Remark. Let a be the function constructed in Remark 6.3. Then n " " " " a−k χ−k " "T (a) − T
L(p µ)
k=0
≤
∞
a−k  = o(1)
(n → ∞),
k=n+1
hence T (a) ∈ algp,µ T (P), and thus T (a) ∈ alg T (Cp,µ ), although a is not in Cp,µ (µ > 0)! 6.11. Deﬁnition. Put p,µ p,µ p,µ alg T π (CN ×N ) := alg T (CN ×N )/C∞ (N ), p,µ π and for A ∈ alg T (CN ×N ) let A denote the coset of the quotient algebra containing A. It is readily veriﬁed that π p,µ p,µ p,µ alg T π (CN ×N ) = closL(N )/C∞ (N ) T (g) : g ∈ PN ×N
and that alg T π (Cp,µ ) is commutative. 6.12. Theorem. Let 1 < p < ∞ and µ ∈ R. (a) The maximal ideal space M (alg T π (Cp,µ )) is T and for T (a) in the algebra alg T (Cp,µ ) the Gelfand transform is given by (Γ T π (a))(τ ) = a(τ ). (b) The Shilov boundary of M (alg T π (Cp,µ )) coincides with the whole space M (alg T π (Cp,µ )). p,µ (c) If A ∈ alg T (CN ×N ), then π A ∈ Φ(p,µ N ) ⇐⇒ (Γ (det A) )(τ ) = 0
∀ τ ∈ T.
In that case Ind A = −ind (Γ (det A)π ). Proof. (a) Let v ∈ M (alg T π (Cp,µ )) and put τ = v(T π (χ1 )). Theorem 6.5 implies that T ⊂ sp T π (χ1 ) and from Proposition 6.8(a) we deduce that / T, then T π ((χ1 − λ)−1 ) is the inverse of T π (χ1 − λ)). sp T π (χ1 ) ⊂ T (if λ ∈ π Hence sp T (χ1 ) = T and we have τ ∈ T. It follows that v(T π (g)) = g(τ ) for all g ∈ P. If T (a) ∈ alg T (Cp,µ ), then there are gn ∈ P such that T (a) − T (gn )L(pµ ) → 0 as n → ∞, whence a − gn L∞ → 0 (6.2(b)) and so v(T π (a)) = lim v(T π (gn )) = lim gn (τ ) = a(τ ). n→∞
n→∞
On the other hand, if τ ∈ T, then, by 6.2(c), g(τ ) ≤ g∞ ≤ T π (g) for all g ∈ P, and therefore (recall what was said at the end of Section 6.9) the mapping vτ : T π (g) → g(τ ) (g ∈ P) extends to a multiplicative linear functional on alg T π (Cp,µ ). That the Gelfand topology on T is the topology inherited from the inclusion T ⊂ C can be checked straightforwardly.
296
6 Toeplitz Operators on p
(b) This follows from 1.20(a) along with the observation that T (f ) is in alg T (Cp,µ ) for every f ∈ C ∞ . (c) The commutativity of alg T π (Cp,µ ) and Theorem 1.14(c) imply that p A ∈ Φ(p,µ N ) ⇐⇒ det A ∈ Φ(µ ).
If Γ (det A)π = 0 on T, then (det A)π ∈ G(alg T π (Cp,µ )) and thus det A belongs to Φ(pµ ). If (Γ (det A)π )(τ0 ) = 0 for some τ ∈ T, then, by (b) and 1.20(c), / Φ(pµ ). det Aπ is a topological divisor of zero and therefore det A ∈ We are left with the index formula. Without loss of generality assume µ ≥ 0 (otherwise consider adjoints). Again from what was said at the end of Section 6.9 we deduce that there are gn ∈ PN ×N and Kn ∈ C∞ (p,µ N ) such that → 0 (n → ∞). (6.11) A − T (gn ) − Kn L(p,µ N ) Hence, there is an n0 such that T (gn ) ∈ Φ(p,µ N ) and Indp,µ A = Indp,µ T (gn ) for all n ≥ n0 . Clearly, T (gn ) ∈ Φ(pN ). We claim that Indp,µ T (gn ) = Indp T (gn ). Because pµ ⊂ p and q−µ ⊃ q , we have Indp,µ T (gn ) = αp,µ (T (gn )) − αq,−µ (T (gn∗ )) ≤ αp (T (gn )) − αq (T (gn∗ )) = Indp T (gn ). The operator T (gn−1 ) is a regularizer of T (gn ). Thus, analogously, Indp,µ T (gn−1 ) ≤ Indp T (gn−1 ). Because Ind T (gn−1 ) = −Ind T (gn ) (see 1.12(c)), we arrive at the desired equality Indp,µ T (gn ) = Indp T (gn ). This and Corollary 4.8(c) give that Indp,µ A = −ind (det gn ) (n ≥ n0 ). But it is easily seen from (6.11) that Γ (det A)π − det gn L∞ → 0 (n → ∞), which implies the asserted index formula. Remark. Part (c) is true for p = 1 and Γ (det A)π replaced by det SmbT (A). This can be proved using an index perturbation argument. We ﬁnally mention some (trivial) consequences of a (deep) result of Zafran which show that, to put it mildly, the theory of Toeplitz operators with symbols in C ∩ M p is substantially more complicated than the corresponding theory for symbols in Cp (no weight is involved!). Note that C ∩ M p is obviously a closed subalgebra of M p . 6.13. Deﬁnition. A function F : [−1, 1] → C is said to operate from C ∩ M p into M p if F ◦ a ∈ M p whenever a ∈ C ∩ M p and a(T) ⊂ [−1, 1]. 6.14. Theorem (Zafran). If p = 2 and F : [−1, 1] → C operates from C∩M p into M p , then F is the restriction of an entire function to [−1, 1].
6.3 Piecewise Continuous Symbols
297
Proof. For a proof see Zafran [585]. 6.15. Corollary. By identifying τ ∈ T with the multiplicative linear functional vτ : C ∩ M p → C,
a → a(τ ),
T may be viewed as a subset of the maximal ideal space M (C ∩M p ) of C ∩M p , but if p = 2 then M (C ∩ M p ) is strictly larger than T. Proof. The functional vτ is clearly linear and multiplicative on C ∩ M p . Since a∞ ≤ aM p for a ∈ M p , we have a(τ ) ≤ a∞ ≤ aM p for every a ∈ C ∩ M p , which implies that vτ is continuous. Now assume p = 2 and M (C ∩ M p ) = T. Then a − i is in G(C ∩ M p ) whenever a ∈ C ∩ M p is realvalued, and hence the function F (x) := 1/(x − i) operates from C ∩ M p into M p . But this contradicts Theorem 6.14, since F (z) := 1/(z − i) (z ∈ C) is not entire. In connection with the following corollary see Proposition 2.32 and Theorems 2.42 and 2.47 (and also the remark in 2.29). 6.16. Corollary. Let p = 2. / M p. (a) There exist a ∈ C ∩ M p such that a ∈ (b) There exist realvalued a ∈ C ∩ M p such that the spectrum of a in M p , i.e. the spectrum of M (a) in L(p (Z)), is not contained in R. Proof. (a) It is immediate from Theorem 6.14 that F (x) := x does not operate from C ∩ M p into M p . (b) If spM p a would be a subset of R for every realvalued a ∈ C ∩M p , then F (x) := 1/(x − i) would operate from C ∩ M p into M p , which is impossible by virtue of Theorem 6.14.
6.3 Piecewise Continuous Symbols 6.17. Deﬁnitions. Recall how ϕβ = ϕβ,τ , ξδ = ξδ,τ , ηγ = ηγ,τ were deﬁned in 5.35. The functions ξδ and ηγ are in L1 if and only if Re δ > −1 and Re γ > −1. In that case their Fourier coeﬃcients are δ n (6.12) ξδ,−n = (−τ ) (n ≥ 0), ξδ,n = 0 (n > 0), n γ (6.13) ηγ,n = (−1/τ )n (n ≥ 0), ηγ,−n = 0 (n > 0). n On deﬁning T (ξδ ) and T (ηγ ) as the Toeplitz matrices (ξδ,j−k )∞ j,k=0 and with ξ and η given by (6.12) and (6.13), respectively, T (ξδ ) (ηγ,j−k )∞ δ,n γ,n j,k=0 and T (ηγ ) make sense for all δ, γ ∈ C.
298
6 Toeplitz Operators on p
The function ξδ ηγ is in L1 if and only if Re (γ + δ) > −1. In that case we let T (ξδ ηγ ) denote the Toeplitz matrix ((ξδ ηγ )j−k )∞ j.k=0 , where (ξδ ηγ )n refers to the nth Fourier coeﬃcient of ξδ ηγ . The computation of these Fourier coeﬃcients is our ﬁrst concern. 6.18. Lemma. If Re (γ + δ) > −1, then the nth Fourier coeﬃcient of ξδ ηγ is equal to Γ (1 + γ + δ) (6.14) (−1/τ )n Γ (γ − n + 1)Γ (δ + n + 1) in case neither γ − n + 1 nor δ + n + 1 is a nonpositive integer and is equal to zero in case γ − n + 1 or δ + n + 1 is a nonpositive integer. Proof. Suppose ﬁrst that neither γ nor δ is an integer. Choose an integer κ so that −1 < Re γ + κ ≤ 0. Then Re δ − κ > −1. Write ξδ (t)ηγ (t) as ξ(t)η(t)(−τ /t)κ with ξ(t) := (1 − τ /t)δ−κ and η(t) := (1 − t/τ )γ+κ (t ∈ T). The Fourier coeﬃcients of ξ and η are δ−κ γ+κ , ηn = (−1/τ )n (n ≥ 0), ξ−n = (−τ )n n n ξn = η−n = 0 (n > 0). There exist p, q ∈ (1, ∞) such that 1/p + 1/q = 1, Re γ + κ > −1/p, and Re δ − κ > −1/q. Consequently, ξ ∈ Lq and η ∈ Lp , and this ensures (see Zygmund [591, Chap. IV, Theorem 8.7]) that for n ≥ 0 the nth Fourier coeﬃcient of ξη is n
(ξη)n = (−1/τ )
∞ γ+κ δ−κ
n+j j ⎤ ⎡ ∞ (−γ + κ + n) (−δ + κ) γ + κ j j ⎦, ⎣1 + = (−1/τ )n n j!(n + 1)j j=0
j=1
where (x)j := x(x + 1) . . . (x + j − 1). The sum in square brackets is nothing else than the hypergeometric series F (−γ − κ + n, −δ + κ; n + 1; 1), which converges just for Re (n + 1 − (−γ − κ + n) − (−δ + κ)) = Re (γ + δ + 1) > 0 and has the sum Γ (n + 1)Γ (1 + γ + δ) Γ (1 + γ + κ)Γ (δ + n − κ + 1) (see Whittaker, Watson [554, 14.11]). From Γ (1 + γ + κ) γ+κ = n Γ (n + 1)Γ (γ − n + κ + 1) we obtain that (ξη)n is (−1/τ )n times
6.3 Piecewise Continuous Symbols
Γ (1 + γ + δ) . Γ (γ − n + κ + 1)Γ (δ + n − κ + 1)
299
(6.15)
Repeating these arguments one can see that (ξη)n is (−1/τ )n times the expression (6.15) for n < 0, too. Finally, since (ξδ ηγ )n = (−τ )κ (ξη)n+κ , we arrive at (6.14). Now let γ be an integer. Then ∞ τ δ+γ τ γ δ + γ = ξδ (t)ηγ (t) = 1 − (−τ )j−γ tγ−j . − j t t j=0
δ+γ Hence (ξδ ηγ )n = 0 for n ≥ γ + 1 and (ξδ ηγ )n = (−1/τ ) for n ≤ γ, γ−n which coincides with (6.14) for δ ∈ Z as well as for δ ∈ / Z. The case where δ is an integer can be treated analogously. n
6.19. Deﬁnition. For α ∈ C, put n µ(α) n := (−1)
−1 − α n
and let Mα denote the diagonal matrix (operator) (α)
(α)
(α)
Mα = diag (µ0 , µ1 , µ2 , . . .). 6.20. Theorem (Duduchava/Roch). Let γ, δ ∈ C \ {−1, −2, . . .} and suppose that Re (γ + δ) > −1. Then T (ηγ )Mγ+δ T (ξδ ) = Γγ,δ Mδ T (ξδ ηγ )Mγ ,
(6.16)
where Γγ,δ := Γ (1 + γ)Γ (1 + δ)/Γ (1 + γ + δ). Proof. By computing the ik entry of both sides of (6.16) (using Lemma 6.18 for the righthand side) one sees that (6.16) will follow as soon as one has shown that
min{i,k}
j=0
j
(−1)
γ i−j
−1 − γ − δ j
δ k−j
=
δ+i k
γ+k i
for all i, k ∈ Z+ . Without loss of generality assume i ≥ k. Then i = k + m with m ≥ 0 and what we must prove is that k j (−1) j=0
γ k+m−j
−1 − γ − δ j
δ k−j
=
Let F (a, b; c; x) denote the hypergeometric function
δ+k+m k
γ+k k+m
.
300
6 Toeplitz Operators on p
F (a, b; c; x) := 1 +
∞ (a)j (b)j j=1
j!(c)j
xj .
A well known formula of Gauss (see Whittaker, Watson [554, 14.4]) says that F (a, b; c, x) = (1 − x)c−a−b F (c − a, c − b; c; x).
(6.17)
After putting a = γ+ 1,b = δ + m + 1, c = m + 1 and multiplying both sides γ of this formula by we get m γ γ 1−γ−δ F (γ+1, δ+m+1; m+1; x) = (1−x) F (−γ+m, −δ; m+1; x). m m Now expand both sides of this equality into apower series to x. with respect γ+k δ+m+k k The coeﬃcient of x on the lefthand side is . Taking n+k k into account that ∞ −1 − γ − δ −1−γ−δ j (1 − x) = (−1) xj , j j=0 ∞ γ δ γ xl , F (−γ + m, −δ; m + 1; x) = m+l l m l=0
the coeﬃcient of xk on the righthand side is seen to be k −1 − γ − δ γ δ j (−1) j m+k−j k−j j=0
and so the proof is complete. 6.21. Lemma. Let K be a compact subset of C\{−1, −2, −3, . . .}. Then there exists a constant cK depending only on K such that Re α Re α c−1 ≤ µ(α) ∀ n ∈ Z+ ∀ α ∈ K. n  ≤ cK (n + 1) K (n + 1) 5 (α) n Proof. We have µn  = k=1 1 + α/k. It is well known that the inﬁnite 5∞ −α/k converges uniformly on K to e−Cα /Γ (α + 1), product k=1 (1 + α/k)e where C := 0.577 . . . is Euler’s constant. Hence, if n0 ∈ Z+ is large enough, then + + −Cα + + + + n + e + α ++ ++ −α(1+ 11 +...+ n1 ) ++ 1 ++ e−Cα ++ ++ + + ≤ 1 + ≤ 2 e + + ++ + + + + 2 Γ (α + 1) k + Γ (α + 1) + k=1
for all n ≥ n0 and all α ∈ K. Taking into account that 1 1 + . . . + = C + log(n + 1) + o(1) (n → ∞) 1 n it is now not diﬃcult to complete the proof.
6.3 Piecewise Continuous Symbols
301
6.22. Corollary. If α ∈ C \ {−1, −2, −3, . . .}, 1 < p < ∞, and µ ∈ R, then Mα is a bounded and (boundedly) invertible operator from pµ onto p−µ−Re α . Proof. Immediate from the preceding lemma.
We are now prepared to begin with the study of Toeplitz operators with P C symbols on pµ . 6.23. Proposition. Let β ∈ C \ Z, 1 < p < ∞, µ ∈ R. Then T (ϕβ ) ∈ L(pµ ) ⇐⇒ −
1 1 <µ< . p q
Proof. The implication “⇐=” follows from 6.2(f). So let T (ϕβ ) ∈ L(pµ ). Then T (ϕβ ) ∈ L(q−µ ). From Lemma 6.18 or by a direct computation it is easily seen that the nth Fourier coeﬃcient of ϕβ = ξ−β ηβ equals π sin(πβ)τ −n /(β − n). Therefore, T (ϕβ )e0 ∈ pµ and T (ϕβ )e0 ∈ q−µ if and only if ∞
β − n−p (n + 1)pµ < ∞,
n=0
∞
β − n−q (n + 1)−qµ < ∞,
n=0
that is, if and only if µ < 1/q and µ > −1/p.
6.24. Proposition. Let β ∈ C, 1 < p < ∞, −1/p < µ < 1/q. Then the following are equivalent: (i) T (ϕβ ) ∈ Φ(pµ ) and Ind T (ϕβ ) = −κ. (ii) κ − 1/p < Re β + µ < κ + 1/q. (iii) 0 ∈ / Ar (ϕβ (τ − 0), ϕβ (τ + 0)), where r := (1/q − µ)−1 , and the index of the closed continuous and naturally oriented curve obtained from the range of ϕβ by ﬁlling in the arc Ar (ϕβ (τ − 0), ϕβ (τ + 0)) is equal to κ. Proof. (ii) ⇐⇒ (iii). Straightforward. (ii) =⇒ (i). Put α = β − κ. Then Re α < 1. There is nothing to prove for α = 0; so assume α = 0. Let Aα denote the matrix T (η−α )T (ξα ) and let ajk (α) and ϕj−k (α) denote the jk entry of Aα and T (ϕα ), respectively. If Re α = 0, then ξ±α ∈ H ∞ and η±α ∈ H ∞ . Therefore, since ϕα = ξ−α ηα , we have for x ∈ 2 ∼ = H 2, T (ϕα )Aα x = T (ξ−α )T (ηα )T (η−α )T (ξα )x = x, Aα T (ϕα )x = T (η−α )T (ξα )T (ξ−α )T (ηα )x = x. This implies that for Re α = 0 both T (ϕα )Aα and Aα T (ϕα ) are equal to the identity matrix. We want to show that the same is true for all α’s in question. It is easily seen that each ajk (α) and each ϕj−k (α) is an analytic function in the punctured stripe S := {α ∈ C : Re α < 1, α = 0}. We claim that for each j ∈ Z+ and each k ∈ Z+ the series
302
6 Toeplitz Operators on p ∞
ϕj−n (α)ank (α)
∞
and
n=0
ajn (α)ϕn−k (α)
(6.18)
n=0
converge uniformly on compact subsets of S. This will imply that the entries of T (ϕα )Aα and Aα T (ϕα ) are analytic in S, which together with the above result for Re α = 0 shows that T (ϕα )Aα = I and Aα T (ϕα ) = I for all α ∈ S. To prove our claim choose r ∈ (1, ∞) so that Re α < 1 − 1/r an let s satisfy 1/r + 1/s = 1. From Theorem 6.20 (with δ = α, γ = −α) we deduce that (−α) . ank (α) = Γ−α,α µ(α) n ϕn−k (−α)µk Hence, n
≤
(−α)
ϕj−n (α)ank (α) = Γ−α,α µk
ϕj−n (α)µ(α) n ϕn−k (−α)
n
(−α) Γ−α,α µk 

n
1/s ϕj−n (α)
s
1/r r µ(α) n ϕn−k (−α)
,
n
and since ϕl (β) = π sin(πβ) τ −l /(β − l), Lemma 6.22 gives our claim. We now prove that the matrix Aα generates a bounded operator on pµ : by virtue of Theorem 6.20, Aα = Γ−α,α Mα T (ϕ−α )M−α , we have M−α ∈ L(pµ , pµ+Re α ), Mα ∈ L(pµ+Re α , pµ ) (Corollary 6.22) and T (ϕ−α ) ∈ L(pµ+Re α ) (Proposition 6.23 and hypothesis (ii)), whence Aα ∈ L(pµ ). Thus, we have proved that Aα ∈ L(pµ ) and that Aα T (ϕα ) = T (ϕα )Aα = I. The conclusion is that T (ϕα ) ∈ GL(pµ ). Since T (ϕβ ) = T (ϕα )T (χκ ) (κ ≥ 0) or T (ϕβ ) = T (χκ )T (ϕα ) (κ ≤ 0) and since T (χκ ) is Fredholm with index −κ, we deduce that T (ϕβ ) ∈ Φ(pµ ) and Ind T (ϕβ ) = −κ. (i) =⇒ (ii). This can be proved using the same perturbation argument as in the proof of Lemma 5.36. 6.25. Deﬁnitions. Let P K denote the collection of all piecewise constant functions on T having only a ﬁnite number of jumps. For 1 < p < ∞ and −1/p < µ < 1/q, deﬁne P Cp,µ as the closure in Mµp of P K, that is, P Cp,µ := closMµp P K (that P K is a subset of Mµp follows from 6.2(f)). Note that obviously T (a) ∈ L(pµ ) for a ∈ P Cp,µ . It is clear that P Cp,µ is a closed subalgebra of Mµp . Also notice that P C2,0 = P C. The following proposition shows that P Cp,µ contains suﬃciently many interesting functions.
m 6.26. Proposition. If ϕ = i=1 gi fi with gi ∈ P K and fi ∈ Cp,µ , then ϕ is in P Cp,µ . Proof. It suﬃces to show that χ1 ∈ P Cp,µ , where χ1 (t) = t (t ∈ T). One can easily see that there are functions gn ∈ P K such that χ1 − gn L∞ → 0 as n → ∞ and V1 (χ1 − gn ) ≤ M with some constant M > 0. Both χ1 and all
6.3 Piecewise Continuous Symbols
303
the functions gn belong to Mµp for all p ∈ (1, ∞) and µ ∈ (−1/p, 1/q). The RieszThorin interpolation theorem gives γ χ1 − gn M p ≤ χ1 − gn 1−γ M p±ε χ1 − gn L∞
(6.19)
(we identify M (f ) with f ), where ε > 0 (resp. ε < 0) for p > 2 (resp. p < 2) and γ ∈ (0, 1) is some constant. The inequality in 6.2(f) shows that the ﬁrst factor in (6.19) remains bounded while the second factor goes to zero as n → ∞. Now we apply the SteinWeiss interpolation theorem to get χ1 − gn δM p , χ1 − gn Mµp ≤ χ1 − gn 1−δ Mp µ±ε
(6.20)
where ε > 0 (resp. ε < 0) for µ > 0 (resp. µ < 0) and δ ∈ (0, 1) is some constant. The ﬁrst factor in (6.20) again remains bounded (by 6.2(f)) and the second factor has just been shown to converge to zero as n → ∞. Thus, χ1 − gn Mµp → 0 as n → ∞ and so χ1 ∈ P Cp,µ . 6.27. Open problem. Once Proposition 6.8(b) has been proved it is not diﬃcult to show that P C0 ∩ M p,µ is contained in P Cp,µ , and this gives that P Cp,µ = closMµp (P C0 ∩ M p,µ ). We have not been able to show that P C ∩ M p,µ is a subset of P Cp,µ and therefore we must raise the inclusion P C ∩M p,µ ⊂ P Cp,µ and the resulting equality P Cp,µ = closMµp (P C ∩M p,µ ) as an open question (even for µ = 0). 6.28. Proposition. Let 1 < p < ∞ and −1/p < µ < 1/q. The maximal ideal space of P Cp,µ is T × {0, 1} and the Gelfand map Γ : P Cp,µ → C(T × {0, 1}) is given by (Γ f )(τ, 0) = f (τ − 0),
(Γ f )(τ, 1) = f (τ + 0).
Proof. From 6.2(b) and 2.5(d) we deduce that (τ, 0) and (τ, 1) belong to M (P Cp,µ ). Conversely, let v ∈ M (P Cp,µ ). The preceding proposition shows that Cp,µ is a closed subalgebra of P Cp,µ , and since M (Cp,µ ) = T (Proposition 6.8(a)), v belongs to some ﬁber Mτ (P Cp,µ ) over τ ∈ T. Every function f ∈ P K can be written as f = cχτ +g, where c ∈ C, χτ is the characteristic function of the arc (τ, τ eiπ/2 ), and g ∈ P K is constant on some arc (τ e−iδ , τ eiδ ). The spectrum of χτ in P Cp,µ is clearly the doubleton {0, 1}. To see that v(g) = g(τ ), let ϕ ∈ C ∞ ⊂ Cp,µ be any function satisfying ϕ(τ ) = 1 and ϕg ∈ C ∞ and note that v(g) = v(g)ϕ(τ ) = v(g)v(ϕ) = v(gϕ) = (gϕ)(τ ) = g(τ ). Hence, either v(f ) = f (τ −0) or v(f ) = f (τ +0) for every f ∈ P K and thus for all f ∈ P Cp,µ , which implies that Mτ (P Cp,µ ) is the doubleton {(τ, 0), (τ, 1)}. Remark. The Gelfand topology on M (P Cp,µ ) coincides with that of M (P C) described in Proposition 3.24.
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6 Toeplitz Operators on p
6.29. Proposition. Let a, b ∈ Mµp (1 < p < ∞, µ ∈ R) and suppose for each τ ∈ T there are an open neighborhood Uτ ⊂ T and a function fτ ∈ Cp,µ such that aUτ = fτ Uτ or bUτ = fτ Uτ . Then T (ab) − T (a)T (b) is in C∞ (pµ ). Proof. Choose a ﬁnite collection {Uτi } of neighborhoods with the property required in the proposition such that their union is T. Let i ψi = 1 be a subordinate smooth partition of unity. It is obvious that each ψi can be written as ψi = ϕ2i where ϕi is also smooth. Then, by (2.18), 0 1 T (aϕ2i b) − T (a)T (ϕ2i )T (b) T (ab) − T (a)T (b) = i
0 1 T (aϕ2i b) − T (a)T (ϕi )T (ϕi )T (b) − T (a)H(ϕi )H(ϕ)i )T (b) = i
0 T (aϕ2i b) − T (aϕi )T (ϕi b) + T (aϕi )H(ϕi )H()b) = i
1 +H(a)H(ϕ)i )T (ϕi b) − H(a)H(ϕ)i )H(ϕi )H(b) − T (a)H(ϕi )H(ϕ)i )T (b) . Since H(ϕi ) and H(ϕ)i ) are in C∞ (pµ ), we have T (ab) − T (a)T (b) −
0 1 T (aϕ2i b) − T (aϕi )T (ϕi b) ∈ C∞ (pµ ). i
But
T (aϕ2i b) − T (aϕi )T (ϕi b) = H(aϕi )H(ϕ)i)b) ∈ C∞ (pµ ),
because by our hypothesis at least one of the functions aϕi and ϕ)i)b belongs to Cp,µ and thus generates a compact Hankel operator on pµ . 6.30. Proposition. If a, b ∈ P Cp,µ (1 < p < ∞, −1/p < µ < 1/q), then T (a)T (b) − T (b)T (a) ∈ C∞ (pµ ). Proof. Since a and b can be approximated by piecewise constant functions in the norm of Mµp as closely as desired, we may suppose that a, b ∈ P K. Moreover, because every function in P K can be written as a ﬁnite sum of piecewise smooth (C ∞ ) functions each of which has at most one jump discontinuity, we may assume that a and b are piecewise smooth and have at most one jump. In view of the preceding proposition it is enough to consider the case that a and b have the jump at the same point of T. Then a = λb + c with λ ∈ C and c ∈ Cp,µ . So 0 1 T (a)T (b) − T (b)T (a) = λ T (c)T (b) − T (b)T (c) 0 1 0 1 = λ T (c)T (b) − T (cb) + λ T (bc) − T (b)T (c) , which is in C∞ (pµ ) by the preceding proposition.
6.3 Piecewise Continuous Symbols
305
6.31. Deﬁnition. Let 1 < p < ∞ and −1/p < µ < 1/q. For a ∈ P C, deﬁne ap,µ : T × [0, 1] → C by ap,µ (t, λ) := 1 − σr (λ) a(t − 0) + σr (λ)a(t + 0) (t ∈ T, λ ∈ [0, 1]), where r := (1/q − µ)−1 and σr is as in 5.12. The range of ap,µ is a continuous closed curve with a natural orientation; it is obtained from the (essential) range of a by ﬁlling in the arcs Ar (a(τ − 0), a(τ + 0)), r = (1/q − µ)−1 , for each τ ∈ T at which a has a jump. If the curve does not pass through the origin, its winding number with respect to the origin will be denoted by ind ap,µ .
n 6.32. Proposition. Let a = i=1 gi fi , where the functions gi are piecewise constant and the functions fi are continuously diﬀerentiable on T. Then T (a) ∈ Φ(pµ ) ⇐⇒ ap,µ (t, λ) = 0
∀ (t, λ) ∈ T × [0, 1].
If T (a) ∈ Φ(pµ ), then Ind T (a) = −ind ap,µ . Proof. If ap,µ does not vanish on T × [0, 1], then a can be written in the form (5.19): a = ϕβ1 . . . ϕβm b. Since continuously diﬀerentiable functions have ﬁnite −1 total variation, 6.2(f) implies that a ∈ M p,µ . Hence, b = aϕ−1 β1 . . . ϕβm is in p,µ C ∩M and thus, by Proposition 6.8(b), in Cp,µ . Now one can proceed as in the proof of Proposition 5.39. 6.33. Remark. Let 2 < p < ∞. If a is as in the preceding proposition, then T (a) ∈ L(p ) and T (a) ∈ L(H p ). For deciding of whether T (a) is Fredholm on p or H p we must ﬁll in an arc Ar (a(τ − 0), a(τ + 0)) for each τ ∈ T at which a has a jump and then look whether the curve obtained in this way does pass through the origin or not. In the H p case the r is p and in the p case r equals q. Thus, although the angle at which the line segment [a(τ − 0), a(τ + 0)] is seen from that arc equals 2π/p in both cases, in the H p case the arc must be drawn on the right whereas in the p case it must be drawn on the left of the segment [a(τ − 0), a(τ + 0)]. Naturally, a similar observation can be made for 1 < p < 2. To express this circumstance analytically, note that ap,0 = aq,1 (6.31 and 5.37). 6.34. Deﬁnitions. Let 1 < p < ∞ and −1/p < µ < 1/q. Put p,µ p,µ p,µ alg T (P CN ×N ) := algL(N ) T (f ) : f ∈ P CN ×N , T (g) : g ∈ P KN ×N . algp,µ T (P KN ×N ) := algL(p,µ N ) These two algebras are easily seen to be equal to each other. The compact p,µ operators on p,µ N belong to alg T (P CN ×N ) (see 6.9). Deﬁne p,µ p,µ p,µ alg T π (P CN ×N ) := alg T (P CN ×N )/C∞ (N ), p,µ π denote the coset containing A ∈ alg T (P CN ×N ) by A , and notice that
306
6 Toeplitz Operators on p
π p,µ p,µ p,µ alg T π (P CN ×N ) = algL(N )/C∞ (N ) T (g) : g ∈ RN ×N , where R can be P K or P Cp,µ . By Proposition 6.30, the algebra alg T π (P Cp,µ ) is commutative. For τ ∈ T, let Jτπ denote the smallest closed twosided ideal p,µ of alg T π (P CN ×N ) containing the set π T (f ) : f = diag (ϕ, . . . , ϕ), ϕ ∈ B, ϕ(τ ) = 0 , where B may be P, C ∞ , or Cp,µ . It is clear that Jτπ does not depend on the particular choice of B. Finally, deﬁne p,µ p,µ π π alg Tτπ (P CN ×N ) := alg T (P CN ×N )/Jτ , p,µ let Aπτ refer to the coset containing Aπ ∈ alg T π (P CN ×N ), and observe that p,µ π the algebra alg Tτ (P CN ×N ) is generated by the set {Tτπ (g) : g ∈ RN ×N }, p,µ π π where R is P K or P Cp,µ . For A in alg T (P CN ×N ), let spp,µ A and spp,µ Aτ p,µ p,µ π π π π denote the spectrum of A and Aτ in alg T (P CN ×N ) and alg Tτ (P CN ×N ), respectively. p,µ 6.35. Proposition. If a, b ∈ P CN ×N and aXτ = bXτ for some τ ∈ T, then p,µ π π Tτ (a) = Tτ (b). If A ∈ alg T (P CN ×N ), then / spp,µ Aπτ . (6.21) spp,µ Aπ = τ ∈T
Proof. To prove the ﬁrst assertion we must show that T π (a) ∈ Jτπ whenever a ∈ P Cp,µ and aXτ = 0. Take such an a, let ε > 0 be an arbitrarily given number, and choose f ∈ P K so that a − f Mµp < ε. There is an open arc U = (τ e−iδ , τ eiδ ) such that a(t) < ε a.e. on U (Proposition 2.79) and f has at most one jump discontinuity in U . So 6.2(b) and 2.5(d) give that f (t) < 2ε on U . Now choose ϕ ∈ C ∞ so that ϕ(τ ) = 1, supp ϕ ⊂ U , and ϕ is monotonous on (τ e−iδ , τ ) and (τ, τ eiδ ). Then, by Proposition 6.29, T π (f ) − T π (f ϕ) = T π (f )T π (1 − ϕ) ∈ Jτπ . Since f ϕL∞ < 2ε and V1 (f ϕ) < 4ε, we deduce from 6.2(f) that T π (f ϕ) < 6cp,µ ε. Because dist(T π (a), Jτπ ) < dist(T π (f ), Jτπ ) + ε ≤ T π (f ϕ) + ε, it follows that dist(T π (a), Jτπ ) = 0, i.e., T π (a) ∈ Jτπ , as desired. p,µ Now let us prove (6.21). Put A := alg T π (P CN ×N ) and π B := D = diag (Aπ , . . . , Aπ ) : A ∈ alg T (Cp,µ ) . The algebra B is a closed subalgebra of the center of A (Proposition 6.30). From Theorem 6.13(a) we know that M (B) = T. Since, by (6.10), closidA Dπ ∈ B : (Γ Dπ )(τ ) = 0 = closidA T π (a) : f = diag (ϕ, . . . , ϕ), ϕ ∈ Cp,µ , ϕ(τ ) = 0 and this is nothing else than Jτπ , equality (6.21) can be immediately derived from Theorem 1.35(a).
6.3 Piecewise Continuous Symbols
307
6.36. Proposition. If a ∈ P Cp,µ , then
spp,µ Tτπ (a) = A(1/q−µ)−1 a(τ − 0), a(τ + 0) .
Proof. This can be proved in the same way as Proposition 5.40; note that the functions f and g occurring there can be chosen continuously diﬀerentiable on T \ {τ }, so that Proposition 6.32 can occupy the place of Proposition 5.39. Remark. There are pµ versions of the deﬁnitions and results in 5.24, 5.25, 5.26, 5.27, 5.28 and of Theorem 5.29 (although some troublesome but immaterial technical complications arise). In particular, pµ analogues of Propositions 5.40 and 5.41 (B = C) can be established. We want not to tire the reader with these things. 6.37. Proposition. (a) The algebra alg Tτπ (P Cp,µ ) is singly generated by Tτπ (χτ ), where χτ is the characteristic function of the arc (τ, τ eiπ/2 ). (b) The maximal ideal space M (alg Tτπ (P Cp,µ )) is homeomorphic to the segment [0, 1] (with its usual topology) and for a ∈ P Cp,µ the Gelfand map Γ : alg Tτπ (P Cp,µ ) → C([0, 1]) is given by π Γ Tτ (a) (λ) = 1 − σr (λ) a(τ − 0) + σr (λ)a(τ + 0), where r := (1/q − µ)−1 and σr is as in 5.12. Proof. See the proof of Proposition 5.44. 6.38. Theorem. (a) The maximal ideal space M (alg T π (P Cp,µ )) is homeomorphic to the cylinder T × [0, 1] equipped with the topology described in 4.88 and the Gelfand map Γ : alg T π (P Cp,µ ) → C(T × [0, 1]) is for a ∈ P Cp,µ given by (Γ T π (a))(τ, λ) = ap,µ (τ, λ), where ap,µ is as in 6.31. (b) The Shilov boundary of M (alg T π (P Cp,µ )) coincides with the whole maximal ideal space M (alg T π (P Cp,µ )). Proof. The proofs are the same as those of Theorems 5.45 and 5.46.
p,µ alg T (P CN ×N ).
Then 6.39. Theorem. Let A ∈ p,µ π A ∈ Φ(N ) ⇐⇒ Γ (det A) (t, λ) = 0 ∀ (t, λ) ∈ T × [0, 1]. Proof. The same arguments as in the proof of Theorem 5.47 apply.
6.40. Index computation. It can be shown that the index of a Fredholm p,µ π operator A ∈ alg T (P CN ×N ) is given by Ind A = −ind Γ (det A) . To verify this one can proceed as in 5.48. Finally notice that Theorem 6.6(a) and Proposition 6.28 give that dim Kerp,µ T (a) = max{Indp,µ T (a), 0}, dim Cokerp,µ T (a) = max{−Indp,µ T (a), 0} for every a ∈ P Cp,µ such that T (a) ∈ Φ(pµ ).
308
6 Toeplitz Operators on p
6.4 Analytic Symbols In Chapter 5 we saw that H p spaces with Khvedelidze weight can be advantageously used to study Toeplitz generated by FisherHartwig sym5operators m bols, i.e., symbols of the form b j=1 ξδj ,tj ηγj ,tj . However, Theorem 5.9 forced us into restricting ourselves to symbols with small “size” of the singularities (that is, with small real parts of the exponents δj and γj ). The spaces p with weight enable us to treat Toeplitz operators generated by symbols with only one FisherHartwig singularity but with large “size” of the singularity. One of the purposes of what follows is to establish an invertibility theory for the Toeplitz operators T (ξδ ηγ b) with Re γ + Re δ ≥ 0, Re γ > −1, and Re δ > −1. The nth Fourier coeﬃcient of ηα,τ (recall 5.35) is α (−1/τ )n = O(n−1−Re α ) (n → ∞) = τ −n µ(−1−α) n n (Lemma 6.21), hence ηα ∈ W and thus ηα ∈ Hp∞ := H ∞ ∩ M p (1 ≤ p < ∞) for all α ∈ C with Re α > 0. Our ﬁrst concern is to show that ηα is in Hp∞ even in the case Re α = 0. 6.41. Theorem (Vinogradov). Let a ∈ L∞ and let {an }n∈Z denote its Fourier coeﬃcients sequence. If there exists a constant A > 0 such that an − an−k  ≤ A, an − an+k  ≤ A n≥2k
n≥2k
for all k ∈ N, then a ∈ M p for all p ∈ (1, ∞). This result was established by Vinogradov [545]. He ﬁrst showed the discrete analogue of Lemma 2.2 of H¨ormander [288] (the proof in the discrete case is the same as the one in the continuous situation) and observed that then the above theorem can be proved in an analogous fashion as its continuous version (Theorem 2.2 of H¨ ormander [288]). 6.42. Theorem (Vinogradov). If Re α = 0 then ηα ∈ Hp∞ for all values p ∈ (1, ∞). Proof. Because ηα ∈ H ∞ , it suﬃces by the previous theorem to show that there is a constant A > 0 such that n≥2k ηα,n − ηα,n−k  ≤ A for all k ∈ N. α n Using that ηα,n = (−1/τ ) it is easy to see that n ηα,n − ηα,n−k = (−1/τ )n−k
α n−k
3 1−
1+α n−k+1
4 1+α ... 1 − −1 . n
If z1 , . . . , zk are complex numbers and zj  ≤ r for all j, then
6.4 Analytic Symbols
309
(1 + z1 ) . . . (1 + zk ) − 1 ≤ z1 + . . . + zk  + z1 z2 + z1 z3 + . . . + zk−1 zk  + . . . + z1 z2 . . . zk  k k rk = (1 + r)k − 1 ≤ k(1 + r)k−1 r. ≤ kr + r2 + . . . + k 2 Thus n≥2k
k−1 ++ α ++ 1 + α 1 + α + + , ηα,n − ηα,n−k  ≤ + n − k +k 1 + n − k + 1 n−k+1 n≥2k
+ + + + cα α + + + n−k +≤ n−k+1
and since
(Lemma 6.21) and k−1 k−1 1 + α 1 + α ≤ 1+ ≤ e1+α 1+ n−k+1 k−1
it follows that n≥2k ηα,n − ηα,n−k  is not greater than cα e1+α k
n≥2k
1 ≤ cα e1+α k (n − k + 1)2
(
∞ k
(n ≥ 2k)
dx = cα e1+α . x2
6.43. Further results about analytic multipliers. All results listed in this section are essentially due to Vinogradov and Verbitsky. (a) Theorem 6.42 is also an immediate consequence of the following much more general theorem. Let a be a function which is analytic and bounded in some region of the form π z ∈ C : z < r,  arg(z − 1) > δ r > 1, 0 ≤ δ < . (6.22) 2 Then the restriction aT is in Hp∞ for all p ∈ (1, ∞). (b) Let b be a Blaschke product of the form (1.13). If the sequence {αn } of its zeros can be divided into a ﬁnite number of subsequences {αnk } having the property that 1 − αnk  < 1, (6.23) sup 1 − αnk+1  k then b ∈ Hp∞ for all p ∈ (1, ∞). Note that the zeros αn are allowed to approach 1 tangentially. (c) Let b be a Blaschke product of the form (1.13). Then the following are equivalent:
6 Toeplitz Operators on p
310
(i) The sequence of the zeros {αn } is contained in some Stolz angle {z ∈ D : 1 − z ≤ c(1 − z)}
(c > 1)
and can be divided into a ﬁnite number of subsequences {αnk } each of which satisﬁes (6.23). (ii) b is the restriction to D (or T) of a function which is analytic and bounded in some region of the form (6.22). (iii)
sup
θb (eiθ ) < ∞.
θ∈[−π,π]
(d) Let b be a Blaschke product of the form (1.13). Then the following are equivalent: (iv) The sequence {αn } of the zeroth of b can be divided into a ﬁnite number of subsequences {αnk } which satisfy (6.23). (v) The variation of b(eiθ ) on the sets {π/2k+1 ≤ θ ≤ π/2k } is uniformly bounded with respect to k ∈ Z+ . Notice that each of the conditions (i)–(v) implies that b ∈ Hp∞ for all p ∈ (1, ∞). (e) That no conditions relating only to the moduli of the zeros of a Blaschke product b can guarantee that b is in Hp∞ (p = 2) is seen from the following result. Let {rn }∞ n=1 be any sequence of real numbers such that 0 < r1 ≤ r 2 ≤ . . . < 1
and
∞
(1 − rn ) < ∞.
n=1
Then there exists a sequence {ξn }∞ T, such that the Blaschke product n=1 , ξn ∈ M p. with the zeros {rn ξn } does not belong to 1
(f) There exists a nonnegative absolutely continuous function ψ on T satisfying ess inf ψ > 0 pon T such that the outer function given by (1.12) does M . not belong to 1
z+1 (g) The singular inner function Sa (z) = exp a z−1 M p.
(a > 0) is not in
1
6.44. Proposition. Let Re α ≥ 0. (a) If µ > −1/p, then T (ξα ) ∈ L(pµ ) and Ker T (ξα ) = {0}. (b) If µ < 1/q, then T (ηα ) ∈ L(pµ ) and Ker T (ηα ) = {0}.
6.4 Analytic Symbols
311
∞ Proof. First let Re α > 0. Then T (ξα ) = n=0 ξα,−n V (−n) , and since ξα,−n = −1−Re α (−n) ) and V L(pµ ) = 1 for µ ≥ 0, it follows that T (ξα ) ∈ L(pµ ) O(n in case µ ≥ 0. Taking adjoints we conclude that T (ηα ) ∈ L(pµ ) for all µ ≤ 0. This together with the representation T (ξα ) = T (ξα )T (η−α )T (ηα ) = T (ϕ−α )T (ηα )
(6.24)
and Proposition 6.23 implies that T (ξα ) ∈ L(pµ ) for all −1/p < µ < 0. Again passing to the adjoint we obtain the boundedness of T (ηα ) on pµ for 0 < µ < 1/q. That T (ηα ) has a trivial kernel is obvious. Let us prove that Ker T (ξα ) is trivial, too. Suppose T (ξα )x = 0 for some x ∈ pµ (−1/p < µ < 0). Then (6.24) shows that T (ϕ−α )T (ηα )x = 0. If 0 < Re α < 1/p + µ, then T (ϕ−α ) is in GL(pµ ) (Proposition 6.24) and since Ker T (ηα ) = {0}, we get x = 0. If Re α ≥ 1/p + µ, choose an n ∈ Z+ so that Re α/n < 1/p + µ. Because T (ξα ) = T (ξα/n ) . . . T (ξα/n ), it follows again that Ker T (ξα ) = {0}. Now let Re α = 0. From Theorem 6.42 we know that T (ξα ) ∈ L(p ) and so we are left with the case µ = 0. Suppose we had already proved that T (ξα ) ∈ L(pj/p ), where j is a nonnegative integer. We show that then T (ξα ) is in L(p(j+1)/p ). To this end notice ﬁrst that xk :=
∞ n+k
n=0
is an equivalent norm in
pk/p
k
1/p xn p
(k = 0, 1, 2 . . .). Further, since
xpj+1 =
j+1 j+2 j+3 x0 p + x1 p + x2 p + . . . j+1 j+1 j+1 j+1 j+2 j x1 p + x2 p + . . . = x0 p + j j j j+1 j x2 p + . . . + x1 p + j j j + x2 p + . . . j
= xpj + V (−1) xpj + V (−2) xpj + . . . , it results that
xk :=
∞
1/p V
(−m)
xpk−1
m=0
is also an equivalent norm in pk/p (k = 0, 1, 2, . . .). Hence, if T (ξα ) has the (ﬁnite) norm c on pj/p equipped with the norm norm  · j then
6 Toeplitz Operators on p
312
T (ξα )xpj+1 = =
∞ m=0 ∞
V (−m) T (ξα )xpj T (ξα )V (−m) xpj ≤ c
m=0
∞
V (−m) xpj = cxpj+1
m=0
and therefore T (ξα ) ∈ L(p(j+1)/p ). Thus we have proved that T (ξα ) ∈ L(pk/p ) for k ∈ Z+ . The SteinWeiss interpolation theorem now implies that T (ξα ) ∈ L(pµ ) for all µ ≥ 0. Passage to the adjoint gives T (ηα ) ∈ L(pµ ) for all µ ≤ 0. Finally, from (6.24) and Proposition 6.23 we deduce that T (ξα ) is in L(pµ ) for −1/p < µ < 0, and once more taking adjoints we see that T (ηα ) ∈ L(pµ ) for 0 < µ < 1/q. Since for Re α = 0 the operators T (ξα ) and T (ηα ) are invertible on pµ if they are bounded (by Theorem 6.42, the inverses are T (ξ−α ) and T (η−α )), the asserted triviality of their kernels follows immediately. 6.45. Proposition (Pomp). Let Re α ≥ 0. (a) If µ > −1/p, then T (ξ−α ) ∈ L(pµ+Re α , pµ ). (b) If µ < 1/q, then T (η−α ) ∈ L(pµ , pµ−Re α ). Remark. Because V (−n) L(pµ+δ ,pµ ) = O(1/nδ ) for µ ≥ 0 and δ ≥ 0 and because ξ−α,n = O(1/n1−Re α ), it is obvious that T (ξ−α ) ∈ L(pµ+Re α+ε , pµ ) (µ ≥ 0), where ε is any positive number which can be chosen as small as desired. A long time it had been open whether the ε can be removed or not. Pomp’s result shows that it can be removed. Proof. Let µ ≥ 0, 0 < α < 1, x ∈ pµ+α , y = T (ξ−α )x. Deﬁne an auxiliary function by f (s, t) := tα (s − 1)α−1 x[st]−2  (s, t ≥ 1), where [z] denotes the integral part of z and where x−1 := 0. Then (with certain possibly diﬀerent constants c independent of k and j) ( ∞ ( ∞ ( ∞ α−1 f (s, t) ds = z x[z+t−2]  dz ≥ z α x[z+t−2]  dz 1
0
=
∞ (
1+[t]−t j+2+[t]−t
z α−1 xj+[t]−1  dz
j=0 j+1+[t]−t ∞ α−1
≥c
(j + 1)
xj+[t]−1  ≥ cy[t]−1 .
j=0
Hence
(
∞
(
p
∞
µ
f (s, t)t ds 1
1
Now apply the well known inequality
dt ≥ cyppµ .
6.4 Analytic Symbols
( 1
∞
(
∞
p f (s, t)tµ ds
1/p
( ≤
dt
1
1
and notice that ( ( ∞ p f (s, t)tµ dt = (s − 1)p(α−1)
∞
(
∞
p f (s, t)tµ dt
313
1/p ds
1
∞
tp(α+µ) x[st−2] 2 dt 1 ( ∞ p(α−1) −1−p(α+µ) s up(α+µ) x[u]−2 p du = (s − 1)
1
s p(α−1) −1−p(α+µ)
≤ c(s − 1) What results is that
s
(
ypµ ≤ cxpα+µ
∞
xpp
α+µ
.
(s − 1)α−1 s−1/p−α−µ ds
1
and since the integral exists and is ﬁnite, the assertion for µ ≥ 0 and 0 < α < 1 follows. If α ≥ 1, choose an n so that 0 < α/n < 1 and take into consideration that T (ξ−α ) = T (ξ−α/n ) . . . T (ξ−α/n ). Since T (ξ−α ) = T (ξ−i Im α )T (ξ−Re α ), we deduce from Proposition 6.44 that T (ξ−α ) is in L(pµ+Re α , pµ ) for Re α ≥ 0 and µ ≥ 0. Passage to the adjoint implies that T (η−α ) ∈ L(pµ , µ−Re α ) for Re α ≥ 0 and µ ≤ 0. Finally, (6.24) (with α replaced by −α) proves the assertion for T (ξ−α ) and −1/p < µ < 0. Once again taking adjoints we get it for T (η−α ) and 0 < µ < 1/q. 6.46. Deﬁnitions. The preceding proposition can be stated in another language. Let Re α > 0. Denote the image of pµ (µ > −1/p) under the operator T (ξα ) by Rµp (α). On deﬁning a norm in Rµp (α) by yRµp (α) := T (ξ−α )ypµ (note that, by Proposition 6.44, T (ξα ) is onetoone on pµ ) we make Rµp (α) become a Banach space and T (ξα ) become an isometric isomorphism of pµ onto Rµ (α). Let Dµp (α) denote the linear set of all sequences x = {xn }∞ n=0 of complex numbers such that T (ηα )x ∈ pµ (µ < 1/q). Since T (ηα )x = 0 can only occur if x = 0, through xDµp (α) := T (ηα )xpµ a norm is deﬁned in Dµp (α) and T (ηα ) is an isometric isomorphism of the Banach space Dµp (α) onto pµ . Finally, in case Re α = 0 put Dµp (α) = Rµp (α) = pµ . Now Proposition 6.45 can be formulated as follows. 6.47. Proposition. Let Re α ≥ 0. (a) If µ > −1/p, then the space pµ+Re α is continuously and densely embedded in the space Rµp (α). (b) If µ < 1/q, then the space Dµp (α) is continuously and densely embedded in the space pµ−Re α . Proof. The continuity of the embeddings is equivalent to Proposition 6.45 and their density can be veriﬁed straightforwardly.
314
6 Toeplitz Operators on p
6.48. Theorem. Let Re δ ≥ 0, Re γ ≥ 0, and −1/p < µ < 1/q. Suppose b ∈ Mµp and T (b) belongs to GL(pµ ). Then the operator T (η−γ )T −1 (b)T (ξ−δ ) belongs to L(pµ+Re δ , pµ−Re γ ), the operator T (ξδ ηγ b) is a (boundedly) invertible operator in L(Dµp (γ), Rµp (δ)), and the restriction of its inverse T −1 (ξδ ηγ b) to pµ+Re γ coincides with T (η−γ )T −1 (b)T (ξ−δ ). Proof. Because T (ξδ ηγ b) = T (ξδ )T (b)T (ηγ ), all assertions follow from the def inition of Dµp (γ) and Rµp (δ) and from Proposition 6.45. 6.49. Theorem. (a) Let Re γ + Re δ ≥ 0 and −1 < Re γ < 0. If Re γ −
1 1 <µ< p q
and b ∈ Cp,µ does not vanish on T and ind b = 0, then T −1 (ϕγ b)T (ξ−γ−δ ) is an operator in L(pµ+Re γ+Re δ , pµ ), the operator T (ξδ ηγ b) is a (boundedly) invertible operator in L(pµ , Rµp (γ + δ)), and the restriction of its inverse T −1 (ξδ ηγ b) to pµ+Re γ+Re δ coincides with T −1 (ϕδ b)T (ξ−γ−δ ). (b) Let Re γ + Re δ ≥ 0 and −1 < Re γ < 0. If −
1 1 < µ < − Re δ p q
and b ∈ Cp,µ does not vanish on T and ind b = 0, then T (η−γ−δ )T −1 (ϕ−δ b) belongs to L(pµ , pµ−Re γ−Re δ ), the operator T (ξδ ηγ b) is a (boundedly) invertible operator in L(Dµp (γ + δ), pµ ), and its inverse T −1 (ξδ ηγ b) coincides with T (η−γ−δ )T −1 (ϕ−δ b). Proof. (a) Put β = −γ and ν = γ + δ. Then Re ν ≥ 0, 0 < Re β < 1, and ξδ ηγ = ξν ϕ−β . Since T (ξδ ηγ b) = T (ξν )T (ϕ−β b) and T (ξν ) is bounded and invertible from pµ to Rµp (ν) while T (ϕ−β b) ∈ GL(pµ ), the assertion follows from Proposition 6.45. (b) The proof is analogous.
Sometimes the necessity arises to study Toeplitz operators on the Banach spaces Rµp (α) and Dµp (α) (see 7.84 and 7.85). We therefore state some results pertaining to this set of problems. r r,r Recall how F r,s α,β was deﬁned in 1.49 and that F α refers to F α,α . 6.50. Lemma. Let 1 < p < ∞ and 1/p + 1/q = 1. Each of the following conditions is suﬃcient for the Hankel operator H(f ) to be a compact operator from pµ into qλ : (i)
−1/q < λ, µ < 1/q, f ∈ F q1/q+λ−µ ;
(ii)
−1/q > λ, µ < 1/q, f ∈ F q−µ ;
(iii) −1/q < λ, µ > 1/q, f ∈ F qλ ; (iv) −1/q > λ, µ > 1/q, f ∈ F qmax{λ,−µ} .
6.4 Analytic Symbols
315
If p = q = 2, then these conditions ensure that H(f ) is HilbertSchmidt from 2µ into 2λ . Proof. The operator H(f ) is compact from pµ into qλ (HilbertSchmidt in case p = q = 2) if ∞ n=0
H(f )en qq (n λ
−µq
+ 1)
=
∞
fn 
n=0
q
n
(n − k + 1)λq (k + 1)−µq < ∞,
k=0
and so the assertion can be easily proved by straightforward estimation of the n sums k=0 (n − k + 1)λq (k + 1)−µq . 6.51. Lemma. Let 1 < p < ∞, 1/p + 1/q = 1, let γ and δ be any complex numbers such that σ := Re γ + Re δ > −1, and let ε > 0 be a real number which can be chosen as small as desired. (a) Each of the following conditions is suﬃcient for H(ξδ ηγ )H(f ) to be compact from pµ into pλ : (i)
µ ≤ 1/q, −1/p ≤ λ − σ < 1/q, f ∈ F q1/p+λ−σ−µ+ε ;
(ii)
µ ≤ 1/q, −1/p > λ − σ,
f ∈ F q−µ+ε ;
(iii) µ > 1/q, −1/p < λ − σ < 1/q, f ∈ F q1/p+λ−σ−1/q+ε . (b) Each of the following conditions is suﬃcient for H(f )H(ξδ ηγ ) to be compact from pµ into pλ : (i)
λ ≥ −1/p, −1/p < µ + σ ≤ 1/q, f ∈ F p1/q−µ−σ+λ+ε ;
(ii)
λ ≥ −1/p,
µ + σ > 1/q, f ∈ F pλ+ε ;
(iii) λ < −1/p, −1/p < µ + σ < 1/q, f ∈ F p1/q−µ−σ+1/p+ε . (c) If p = q = 2, then in (a) and (b) “compact” can be replaced by “trace class.” Proof. To prove (a) and (c) for the operator H(ξδ ηγ )H(f ) it suﬃces to choose an appropriate number τ and then to apply Lemma 6.50 and the fact that (ξδ ηγ )n = O(1/n1+Re γ+Re δ ) (n → ∞) (resulting from Lemma 6.18) to H(f ) : pµ → qτ and H(ξδ ηγ ) : qτ → pλ . The assertion for H(f )H(ξδ ηγ ) follows by taking adjoints. 6.52. Deﬁnition. Given a real number x deﬁne (x)◦ as (x)◦ := max{0, x}.
316
6 Toeplitz Operators on p
6.53. Proposition. Let Re α > 0, b+ ∈ H ∞ , b− ∈ H ∞ . (a) If µ > −1/p and b− ∈ F 1(−µ)◦ , then T (b− ) ∈ L(Rµp (α)). (b) If µ < 1/q and b+ ∈ F 1(µ)◦ , then T (b+ ) ∈ L(Dµp (α)). (c) If µ > −1/p and b+ ∈ F 1(µ)◦ ∩ F ps , where s > max{1/q, Re α + µ}, then T (b+ ) ∈ L(Rµp (α)). (d) If µ < 1/q and b− ∈ F 1(−µ)◦ ∩ F pr , where r > max{1/p, Re α − µ}, then T (b− ) ∈ L(Dµp (α)). Proof. First notice that T (b) ∈ L(Rµp (α)) ⇐⇒ T (ξ−α )T (b)T (ξα ) ∈ L(pµ ), T (b) ∈ L(Dµp (α)) ⇐⇒ T (ηα )T (b)T (η−α ) ∈ L(pµ ). If b− ∈ F 1(−µ)◦ , then T (b− ) ∈ L(pµ ), since the norms of V (−n) on pµ are O(1) for µ ≥ 0 and O(nµ ) for µ ≤ 0. This and the equality T (b− )T (ξα ) = T (ξα )T (b− ) give (a). Let us prove (c). Due to Proposition 2.14, T (ξ−α )T (b+ )T (ξα ) = T (b+ ) − T (ξ−α )H(b+ )H(ξ. α ). If b+ ∈ F 1(µ)◦ , then T (b+ ) ∈ L(pµ ). Using Lemma 6.51 one can show that p p p H(b+ )H(ξ. α ) ∈ L(µ , µ+Re α ) if b+ ∈ F s with s given as in the propop p sition. So Proposition 6.47(a) gives H(b+ )H(ξ. α ) ∈ L(µ , Rµ (α)) and thus p T (ξ−α )H(b+ )H(ξ. α ) ∈ L(µ ), as desired. Finally, (b) and (d) follow from (a) and (c) by passing to the adjoint: T (ηα )T (b± )T (η−α ) ∈ L(pµ ) ⇐⇒ T (ξ−α )T (b± )T (ξα ) ∈ L(q−µ ). Before proceeding to Toeplitz operators on Rµp (α) and Dµp (α) whose symbols are not necessarily analytic or antianalytic we state a result which is 1,1 . also of interest by itself. Recall that W denotes the Wiener algebra F 0,0 6.54. Theorem (D. Horbach). If 1 ≤ r < ∞, 1 ≤ s < ∞, 0 ≤ α < ∞, 0 ≤ β < ∞, then W ∩ F r,s α,β is an algebra under pointwise multiplication. Proof. Because f g = (1/2)[(f + g)2 − f 2 − g 2 ], it suﬃces to show that a2 is r,s in F r,s α,β whenever a is in W ∩ F α,β . Let {an } be the Fourier coeﬃcients sequence of a and put bn = an . We have, for n ≥ 0, + ∞ + ∞ s + + + + aj an−j + ≤ bj bn−j + + + j=−∞ j=−∞ −1 s ∞ n = bj bn−j + bj bn−j + bj bn−j j=−∞
j=n+1
j=0
6.4 Analytic Symbols
∞
≤
b−j bn+j +
j=1
=2
s
n−[n/2]
bn+j b−j + 2
j=1
∞
s
∞
317
bj bn−j
j=0
s
n−[n/2]
b−j bn+j +
j=1
bj bn−j
.
(6.25)
j=0
If yj , zj are nonnegative real numbers and s ≥ 1, then
s
≤
yj zj
j
yj
s−1
j
yj zjs .
j
Indeed, the function f (x) = xs is convex and so Jensen’s inequality gives
s
yj zjs y z y z y f (zj )
j j
j j ≤
j =f =
. yj yj yj yj Thus, (6.25) is not greater than 2s
∞
s−1
n−[n/2]
b−j +
j=1
≤ 2s as−1 W
bj
j=0 ∞
∞
b−j bsn+j +
j=1
b−j bsn+j +
j=1
n−[n/2]
n
bj bsn−j
j=0
bn−j bsj .
j=[n/2]
It follows that N n=0
+ ∞ +s + + + (n + 1) + aj an−j + ≤ 2s as−1 W (σ1 + σ2 ), + + βs +
j=−∞
where σ1 :=
N
(n + 1)βs
n=0
∞
b−j bsn+j ,
σ2 :=
N
(n + 1)βs
n=0
j=1
n j=[n/2]
For σ1 we have σ1 ≤
N ∞
b−j (n + j + 1)βs bsn+j
n=0 j=1
=
∞ j=1
b−j
N
(n + j + 1)βs bsn+j ≤ aW asF r,s
n=0
and σ2 can be estimated as follows:
α,β
bn−j bsj .
318
6 Toeplitz Operators on p
σ2 ≤
N n
bn−j 2βs (j + 1)βs bsj
(since n + 1 ≤ 2(j + 1))
n=0 j=[n/2]
≤
∞ N
bn−j 2βs (j + 1)βs bsj
n=0 j=0
= 2βs
∞
(j + 1)βs bsj
N
bn−j ≤ 2βs asF r,s aW .
n=0
j=0
α,β
As N can be chosen arbitrarily, we arrive at the inequality ∞
(n + 1)βs (a2 )n s ≤ 2s (1 + 2βs )asW asF r,s . α,β
n=0
It can be shown analogously that ∞
(n + 1)αr (a2 )−n r ≤ 2r (1 + 2αr )arW arF r,s . α,β
n=0
6.55. Corollary. Let r, s, α, β be as in the preceding theorem. Suppose a function b ∈ W ∩F r,s α,β does not vanish on T and ind b = 0. Then b has a logarithm r,s in W ∩ F α,β and if we let G(b) := exp(log b)0 and ,∞ ±n (log b)±n t (t ∈ T), (6.26) b± (t) := exp n=1
then b = G(b)b− b+ and
b±1 ±
∈ W ∩ F r,s α,β .
Proof. Theorem 6.54 shows that W ∩ F r,s α,β is a Banach algebra with the ), where a constant c can be chosen so that norm a := c(aW + aF r,s α,β ab ≤ a b. It is easy to see that the maximal ideal space of this algebra is T. Hence 2.41(e) implies that b has a logarithm in W ∩ F r,s α,β and this gives the remaining assertions of the corollary immediately. 6.56. Theorem. Let Re α > 0 and suppose b ∈ C does not vanish on T and ind b = 0. 1,p (a) If µ > −1/p and b ∈ F 1,1 (−µ)◦ ,(µ)◦ ∩ F 0,s , where s is as in Proposition 6.53(c), then T (b) ∈ GL(Rµp (α)). 1,1 p,1 (b) If µ < 1/q and b ∈ F (µ) ◦ ,(−µ)◦ ∩ F r,0 , where r is as in Proposition 6.53(d), then T (b) ∈ GL(Dµp (α)). ±1 Proof. (a) Corollary 6.55 implies that b = b− b+ , where b±1 − and b+ satisfy the hypotheses of Proposition 6.53(a) and (c), respectively. This shows that −1 p T (b) = T (b− )T (b+ ) is in L(Rµp (α)) and that T (b−1 + )T (b− ) ∈ L(Rµ (α)) is the inverse of T (b).
(b) The proof is analogous.
6.4 Analytic Symbols
319
6.57. Toeplitz operators on p,±∞ . Let p,+∞ (1 ≤ p < ∞) denote the linear space
∞
pµ . In this section µ stands for an integer. It is clear that
µ=0
p,+∞ regarded as a set does not depend on the value p ∈ [1, ∞). Deﬁne a metric on p,+∞ by d(x, y) :=
∞ 1 x − ypµ 2µ 1 + x − ypµ µ=0
(x, y ∈ p,+∞ ).
(6.27)
This metric makes p,+∞ into a Fr´echet space and for diﬀerent values of p the spaces p,+∞ are homeomorphic to each other. We let p,−∞ refer to the dual space of p,+∞ and think of p,−∞ as being provided with the strong ∞ topology. Note that p,−∞ = p−µ . The topologies on the corresponding µ=0
spaces of vectorvalued sequences p,+∞ and p,−∞ can be introduced in a N N natural way. into p,±∞ is said to be A linear and bounded operator A of p,±∞ N N /Im A are ﬁniteFredholm if its image is closed and both Ker A and p,±∞ N dimensional. In that case the index Ind±∞ A is deﬁned by Ind±∞ A := dim Ker A − dim(p,±∞ /Im A). N ∞ If a ∈ CN ×N :=
∞ µ=0
F [pµ (Z)]N ×N , then T (a) is obviously bounded on
p,±∞ . Proposition 6.45(a) and Corollary 6.55 imply that if a ∈ C ∞ is of the N form a = bξα1 ,τ1 . . . ξαm ,τm with τj ∈ T, αj ∈ N, b ∈ C ∞ , b(t) = 0 for t ∈ T, and ind b = 0, then 1 −1 T (b−1 + )T (b− )T (ξ−α1 ,τ1 ) . . . T (ξ−αm ,τm ) G(b) is bounded on p,+∞ and is an (the) inverse of T (a). It turns out that these Toeplitz operators are the only invertible ones on p,+∞ . This is the consequence of the following result, whose suﬃciency part is due to Pr¨ ossdorf [405] and whose necessity portion was proved in Silbermann [473]. p,±∞ ∞ Let a ∈ CN ×N . Then T (a) is Fredholm on N ×N if and only if det a has at most ﬁnitely many zeros of integral order on T.
If det a ∈ C ∞ has at most ﬁnitely many zeros of integral order on T, then det a can be written in the form γj m t τj δj det a(t) = b(t) (t ∈ T), (6.28) 1− 1− τj t j=1 where τ1 , . . . , τm are pairwise distinct points on T, γj ∈ Z+ , δj ∈ Z+ , and b ∈ C ∞ does not vanish on T. Notice that such a representation is not unique
320
6 Toeplitz Operators on p
(the sums γj + δj , however, are determined uniquely). If det a is of the form (6.28), then Ind+∞ T (a) = −
m
γj − ind b,
Ind−∞ T (a) = −
j=1
m
δj − ind b.
j=1
Moreover, one can show that T (a) is invertible on p,±∞ (scalar case!) if and only if T (a) is Fredholm in p,±∞ and has index zero. In particular, if a ∈ C ∞ , then T (a) is invertible on p,+∞ resp. p,−∞ if and only if a(t) = b(t)
m τj γj 1− t j=1
resp. a(t) = b(t)
δ m t j , 1− τj j=1
where τj ∈ T, γj resp. δj are nonnegative integers, b ∈ C ∞ does not vanish on T, and ind b = 0.
6.5 Notes and Comments 6.2. Almost all these results were established by Verbitsky [538]. 6.5–6.6. Under somewhat stronger hypotheses, such results are known from the work of Duduchava, Verbitsky, and Krupnik. 6.7–6.12. The theory of Toeplitz operators with continuous symbols on weighted p spaces was developed in the work of Duduchava [167], [169], Gohberg, Krupnik [232], Verbitsky, Krupnik [543]. The approach presented here is the authors’. 6.13–6.16. Zafran [585]. See also Nikolski [365] and Peller, Khrushchev [390, Section 3.4]. A naturally arising question is as follows: is T dense in M (C∩M p ) or has T a “corona”? 6.20. This formula was established by Duduchava [169] for the case γ + δ = 0. Because there is no p analogue of Theorem 5.5, the theory of Toeplitz operators on the spaces p with piecewise continuous symbols diﬀers from the corresponding theory for the spaces H p signiﬁcantly (although, curiously, the ﬁnal results are almost the same in both cases). Duduchava’s formula T (ηβ )T (ξβ ) = Γβ,−β M−β T (ϕβ )Mβ was just the discovery on the basis of which the p theory of Toeplitz operators with P C symbols could be developed. When studying Toeplitz determinants with FisherHartwig symbols we were led to the problem of proving that the ﬁnite section method (on 2 ) is applicable to the operator T −1 (ϕα )T (ϕβ )T −1 (ϕ−α ) with Re α < 1/2, Re β < 1/2. Since we had been unable to prove this in full generality, we asked Steﬀen Roch to try his hands. He proved what we wanted and, as a byproduct or, more precisely, as the key observation for his proof, he discovered
6.5 Notes and Comments
321
formula (6.16). It was published in B¨ ottcher, Silbermann [108], [110] for the ﬁrst time. The original proofs of Duduchava and Roch were very complicated; the proof given here and in our paper [110] is due to the authors. 6.23–6.40. Propositions 6.29 and 6.30, Theorems 6.38(a) and 6.39, and the results of 6.40 go back to Duduchava [167], [169], Gohberg, Krupnik [232], and Verbitsky, Krupnik [543]. The approach presented here is new. This concerns in particular Propositions 6.36 and 6.37. The proof of Proposition 6.29 is the one given in B¨ ottcher, Silbermann [106], and Theorem 6.38(b) was established in B¨ ottcher [64]. The theory of Toeplitz operators with piecewise continuous symbols on p with general (discrete) Muckenhoupt weight is elaborated in B¨ottcher, Seybold [101]. 6.41–6.43. Theorems 6.41 and 6.42 are Vinogradov’s [545] (the luxurious proof of Theorem 6.42 given here is the author’s). 6.43(a) and 6.43(b) for the case that αn → 1 nontangentially are in Vinogradov [545], 6.43(b), (c), (d) can be found in Verbitsky [541], [542], 6.43(e), (f) were established in Vinogradov [547], and 6.43(g) is contained in Verbitsky [541], [542]. 6.44–6.49. A major part of these results is well known (see Pr¨ ossdorf [406] and Pr¨ ossdorf, Silbermann [407]), the presentation follows B¨ ottcher, Silbermann [111]. The diﬃcult part of Proposition 6.44 (Re α = 0) and Theorems 6.48 and 6.49 were ﬁrst obtained in the latter paper. Both the result and the proof of Proposition 6.45 are Pomp’s [397]; we have already remarked in the text that Pomp’s result is very nontrivial. 6.50–6.53. The tedious Lemmas 6.50 and 6.51 were established in B¨ottcher, Silbermann [111]. Results like Proposition 6.53 are in Pr¨ ossdorf [406] and Pr¨ ossdorf, Silbermann [407]. 6.54. The ﬁrst result along these lines go back to Hirschman [281] and Krein 2,2 ∞ ∩ F 1/2,1/2 are [323], who showed, respectively, that W ∩ F 2,2 1/2,1/2 and L algebras. Since that time it has been noticed (but as far as we know not published) by several people that W ∩ F r,s α,β is an algebra for certain values of α, β, r, s. In particular, in B¨ ottcher, Silbermann [106] we pointed out that this is so if either r = s = 2, α ≥ 0, β ≥ 0 or r > 1, s > 1, 1/r + 1/s = 1, α > 1/s, β > 1/r. In 1984, during an examination in mathematical analysis, we asked Detlef Horbach, a gifted second year student of ours, the question of whether W ∩ F r,s α,β is an algebra. A few weeks later he reported to us the proof given in the text. It is certainly typical that such “naive” proofs can only be found by students but not by “professionals.” A. Karlovich [300] generalized Theorem 6.54 to the setting of Orlicz spaces with weights wn satisfying w2n ≤ Cwn . 6.56. B¨ottcher, Silbermann [111]. 6.57. The study of Toeplitz operators on spaces of generalized functions was originated by Cherski [133]. Dybin and Karapetyants [181] were the ﬁrst to
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6 Toeplitz Operators on p
show that a Toeplitz operator is Fredholm on p,±∞ if its symbol is in C ∞ and has at most ﬁnitely many zeros of integral order on T. Independently, Pr¨ ossdorf [405] established this result (for the matrix case) and developed a systematic theory of singular integral equations and convolution equations with “degenerate” symbols on locally convex vector spaces. Silbermann [473] ﬁnally showed that the symbol of a Toeplitz operator which is Fredholm on p,±∞ N ×N can have at most ﬁnitely many zeros of integral order. For more about this topic see Pr¨ ossdorf [406]. Some developments of the matter (e.g. symbols with countably many zeros) are illuminated in the books by Dybin [179] and Dybin, Grudsky [180].
7 Finite Section Method
7.1 Basic Facts 7.1. Projection methods. Let X and Y be Banach spaces and A ∈ L(X, Y ). A projection method is a method for the approximate solution of the equation Ax = y
(7.1)
which can be described as follows. Let {Pn } and {Rn } be sequences of projections Pn ∈ L(X) and Rn ∈ L(Y ) with the property Pn → IX and Rn → IY strongly, and let An : Pn X → Rn Y be certain given bounded operators. For example, one can take An = Rn APn Pn X. We shall frequently identify An with An Pn and may therefore regard An as an element of L(X, Y ). Now consider the equation An xn = Rn y,
xn ∈ Pn X.
(7.2)
We write A ∈ Π{X, Y ; An } if (i) there exists an n0 such that for each y ∈ Y equation (7.2) has a unique solution xn ∈ Pn X for all n ≥ n0 ; (ii) xn converges in the norm of X to a solution x ∈ X of equation (7.1). If X = Y and Pn = Rn , then Π{X, Y ; An } will be abbreviated to Π{X; An } and if there is no fear of confusion even to Π{An }. In case An = Pn APn Pn X we shall write Π{X, Y ; Pn } and Π{X; Pn } (or even Π{Pn }) in place of Π{X, Y ; Pn APn } and Π{X, X; Pn APn }, respectively. 7.2. Algebraization and essentialization. Let X be a Banach space and ∞ = D∞ (X) let {Pn }∞ n=0 be a sequence of projections in L(X). We let D ∞ denote the collection of all sequences {An }n=0 of operators An : Pn X → Pn X such that sup An Pn L(X) < ∞. On deﬁning n≥0
α{An } := {αAn },
{An } + {Bn } := {An + Bn },
{An }{Bn } := {An Bn },
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7 Finite Section Method
and {An } := sup An Pn L(X) we make D∞ into a Banach algebra. If X is n≥0
a Hilbert space, D∞ is a C ∗ algebra with the involution {An }∗ := {A∗n }. Let D = D(X) denote the (closed) subalgebra of D∞ consisting of all sequences {An } ∈ D∞ for which there exists an A ∈ L(X) such that An → A strongly on X. Let G = G(X) refer to the set of all sequences {An } ∈ D∞ with An Pn L(X) → ∞ as n → ∞. It is easy to see that G is a closed twosided ideal of both D∞ and D. For {An } ∈ D∞ , let {An }πG stand for the coset {An } + G. If A ∈ L(X), if the projections Pn converge strongly to I on X and if Pn = 1 for all n, then {Pn APn }πG = AL(X) .
(7.3)
It is clear that in (7.3) “≤” holds. On the other hand, if {Cn } is any element of G, then, by 1.1(e), A ≤ lim inf Pn APn + Cn ≤ sup Pn APn + Cn = {Pn APn } + {Cn }, n→∞
n
which implies the “≥” in (7.3). 7.3. Proposition. Suppose Pn → I and An Pn → A strongly as n → ∞. Then the following are equivalent. (i) A ∈ Π{X; An }. (ii) A ∈ GL(X), An ∈ GL(Pn X) for all suﬃciently large n (n ≥ n0 , say), and sup A−1 n Pn L(X) < ∞. n≥n0
(iii) A ∈ GL(X), An ∈ GL(Pn X) for all suﬃciently large n, and A−1 n Pn converges to A−1 strongly on X as n → ∞. (iv) {An }πG ∈ G(D/G). (v) A ∈ GL(X) and {An }πG ∈ G(D∞ /G). Proof. (i) =⇒ (ii). It is clear that An ∈ GL(Pn X) for all n ≥ n0 . From 1.1(e) we deduce that sup A−1 n Pn =: C < ∞. Hence n≥n0
Pn x ≤ CAn Pn x ∀ x ∈ X
∀ n ≥ n0 .
(7.4)
Passage to the limit n → ∞ in (7.4) gives the inequality x ≤ CAx. Thus, Ker A = {0} and since the deﬁnition of Π{X; An } involves that Im A = X, we have A ∈ GL(X). (ii) =⇒ (iii). Again (7.4) holds and so, for n ≥ n0 and y ∈ X, −1 −1 A−1 y ≤ A−1 y + Pn A−1 y − A−1 y n Pn y − A n Pn y − Pn A −1 ≤ CPn y − An Pn A y + Pn A−1 y − A−1 y. (7.5)
7.1 Basic Facts
325
−1 Since Pn → I and An Pn → A strongly, it follows that A−1 y for n Pn y → A each y ∈ X.
(iii) =⇒ (iv). Suppose An ∈ GL(Pn X) for n ≥ n0 . Put Bn = Pn for n < n0 ∞ and Bn = A−1 n Pn for n ≥ n0 . From 1.1(e) we obtain that {Bn } ∈ D , and −1 −1 since An Pn converges strongly to A , we actually have {Bn } ∈ D. Because {Bn }{An } − {Pn } ∈ G and {An }{Bn } − {Pn } ∈ G, it results that {Bn }πG is the inverse of {An }πG . (iv) =⇒ (v). Obvious. (v) =⇒ (i). Let {Bn }πG be the inverse of {An }πG . Then Bn An = Pn + Cn and An Bn = Pn + Dn with certain {Cn } and {Dn } in G. There is an n0 such that Cn < 1/2 and Dn < 1/2 for all n ≥ n0 . Thus Pn + Cn and Pn + Dn are in GL(Pn X) for all n ≥ n0 and the norms of their inverses are uniformly bounded by 1/(1 − 1/2) = 2. This implies that An ∈ GL(Pn X) for all n ≥ n0 and that sup A−1 n Pn ≤ 2 sup Bn =: C < ∞. Consequently, again (7.4) n≥n0
n≥n0
and thus (7.5) holds, which shows that A ∈ Π{X; An }. 7.4. Corollary. If Pn converges strongly to I, then Π{X; Pn } is an open subset of L(X). Proof. This is immediate from the equivalence (i) ⇐⇒ (iv) of the preceding proposition applied to An = Pn APn Im Pn . 7.5. The ﬁnite section method. Let Pn (n = 0, 1, 2, . . .) denote the projections acting on p,µ N by the rule Pn : {x0 , x1 , x2 , , . . .} → {x0 , x2 , . . . , xn , 0, 0, . . .}.
(7.6)
Here xk ∈ CN . It is clear that Pn L(p,µ = 1 and that Pn converges strongly N ) p,µ to I on N for all p ∈ [1, ∞) and µ ∈ R. p (w) by Deﬁne Pn (n = 0, 1, 2, . . .) on HN Pn :
∞ k=0
ϕk χk →
n
ϕk χk
(ϕk ∈ CN ).
(7.7)
k=0
Here and throughout what follows we suppose that the assumptions 5.1 are p (w)) satisﬁed. Because Pn ϕ = ϕ−χn+1 P (χ−n−1 ϕ), it follows that Pn ∈ L(HN and that sup Pn L(HNp (w)) < ∞. Moreover, since Pn ϕ − ϕHNp (w) → 0 as n≥0
n → ∞ for each polynomial ϕ ∈ PA , we deduce from 1.1(d) that Pn → I p (w). strongly on HN p Let X be p,µ N or HN (w), let the projections Pn be given by (7.6) or (7.7), and let A ∈ L(X). If A ∈ Π{X; Pn }, then the ﬁnite section method is said to be applicable to A on X. We are mainly interested in the case where A is a bounded Toeplitz operator T (a) on X. Note that the operators Pn T (a)Pn : Pn X → Pn X may
326
7 Finite Section Method
be identiﬁed with the ﬁnite block Toeplitz matrices Tn (a) := (aj−k )nj,k=0 (where ai is the ith matrix Fourier coeﬃcient of a). Thus, by Proposition 7.3, T (a) ∈ Π{X; Pn } if and only if T (a) is invertible on X, if the matrices Tn (a) are invertible for all suﬃciently large n, and if Tn−1 (a)Pn → T −1 (a) strongly on X as n → ∞. Throughout the rest of the chapter suppose 1 < p < ∞ (if it is not explicitly stated otherwise). Our ﬁrst concern is to establish a formula which plays the same role in the theory of the ﬁnite section method for Toeplitz operators as identity (2.18) in their Fredholm theory. 7.6. The operator Wn . Deﬁne the linear operators Wn (n = 0, 1, 2, . . .) on p p,µ N and HN (w) by Wn : {x0 , x1 , x2 , . . .} → {xn , xn−1 , . . . , x0 , 0, 0, . . .}, ∞ n Wn : ϕk χk → ϕn−k χk , k=0
k=0
p respectively. It is clear that Wn ∈ L(p,µ N ). We have Wn L(N ) = 1 and it is p,µ easily seen that sup Wn L(N ) < ∞ if and only if µ = 0. Because (Wn ϕ)(t)
n≥0
equals tn (Pn ϕ)(1/t) (t ∈ T), the equality p Wn L(HNp (w)) = Pn L(HNp (w),H ) N (w))
(7.8)
holds; here w(t) ) := w(1/t). Since Pn ϕHNp (w) ≤ M (n, p, N, w)Pn ϕHNp (w) )
p ∀ ϕ ∈ HN (w) )
(note that any two norms on a ﬁnitedimensional space are equivalent to each p p other) and Pn is bounded on HN (w) ) (by 7.5), we obtain that Wn ∈ L(HN (w)). Using (7.8) it is not diﬃcult to see that sup Wn L(HNp (w)) < ∞ if and only if n
H p (w) ) is continuously embedded in H p (w). In particular, sup Wn L(HNp ) < ∞. n
The following identities are extremely important and can be veriﬁed straightforwardly: Wn2 = Pn ,
Wn Pn = Pn Wn = Wn ,
Wn Tn (a)Wn = Tn () a),
where, as usual, ) a(t) = a(1/t) (t ∈ T). Also notice that Wn converges weakly p (1 < p < ∞). to zero on pN as well as HN Finally, throughout what follows let Qn := I − Pn , where I is the identity p operator on p,µ N or HN (w).
7.1 Basic Facts
327
p,µ 7.7. Proposition. Let a, b ∈ L∞ N ×N resp. a, b ∈ MN ×N . Then
Pn T (a)Qn T (b)Pn = Wn H() a)H(b)Wn , Tn (ab) = Tn (a)Tn (b) + Pn H(a)H()b)Pn + Wn H() a)H(b)Wn .
(7.9) (7.10)
Proof. We have Pn T (a)Qn T (b)Pn = Wn (Wn T (a)Qn )(Qn T (b)Wn )Wn and since, with V (±n) := T (χ±n I), Wn T (a)Qn = Pn H() a)V (−n−1) ,
Qn T (b)Wn = V n+1 H(b)Pn ,
(7.11)
identity (7.9) follows. Formula (2.18) gives Tn (ab) = Pn T (ab)Pn = Pn T (a)T (b)Pn + Pn H(a)H()b)Pn = Pn T (a)Pn T (b)Pn + Pn T (a)Qn T (b)Pn + Pn H(a)H()b)Pn and now (7.10) results from (7.9). 7.8. Toeplitzadapted algebraization and essentialization. For a Toeplitz operator to be Fredholm means to have an inverse modulo the ideal C∞ of compact operators. According to (2.18) we have T (a)T (b) = T (ab) − H(a)H()b),
(7.12)
and there are many cases in which the product of Hankel operators in (7.12) is known to be compact. Then T (a)T (b) equals T (ab) modulo C∞ and from this fact at long last all what we know about the Fredholm theory of Toeplitz operators follows. In particular, the applicability of local principles essentially rests on (7.12) and the circumstance that H(f ) ∈ C∞ for f ∈ C + H ∞ . The analogue of (7.12) for ﬁnite Toeplitz matrices is formula (7.10): Tn (a)Tn (b) = Tn (ab) − Pn H(a)H()b)Pn − Wn H() a)H(b)Wn .
(7.13)
In view of Proposition 7.3 the applicability of the ﬁnite section method to T (a) is equivalent to the existence of the inverse {An } ∈ D of {Tn (a)} ∈ D modulo the ideal G. Looking for a connection between G and (7.13) we observe that the ideal G is, in a sense, too small: sequences of the form {Pn KPn + Wn LWn }
(K, L ∈ C∞ )
(7.14)
do, in general, not belong to G. Thus, in order to develop a theory of the ﬁnite section method in analogy to the Fredholm theory, it would be desirable to have an ideal J that contains all sequences of the form (7.14). But there is no such ideal in D. This algebra is, again in a certain sense, too large. We therefore shall construct a smaller algebra possessing, on the one hand,
328
7 Finite Section Method
such an ideal and containing, on the other hand, suﬃciently many interesting elements, in particular, all elements of the form {Tn (a)}. p (note that no weight is allowed), let Pn and Wn Let X be either pN or HN be as in 7.5 and 7.6, and deﬁne D∞ , D, G as in 7.2. Given An ∈ L(Pn X) deﬁne )n ∈ L(Pn X) as A )n := Wn An Wn . Let S = S(X) denote the subset of D∞ A ) ∈ L(X) consisting of all {An } ∈ D∞ for which there exist A ∈ L(X) and A such that An Pn → A,
A∗n Pn → A∗ ,
)n Pn → A, ) A
)∗n Pn → A∗ . A
Here “→” denotes strong convergence and the asterisk refers to the adjoint p with the operator whose operator on X ∗ . On identifying an operator B on HN ◦
·
p matrix representation with respect to the decomposition LpN = (H pN )− + HN I 0 p ∗ q is , we may identify (HN ) with HN (recall 2.39). It is easy to see that 0B S is a closed subalgebra of D, i.e., S itself is a Banach algebra. Obviously, G ⊂ S. If K ∈ C∞ (X), then, by virtue of 1.1(f), {Pn KPn } and {Wn KWn } are in S(X) (note that Pn∗ = Pn , Wn∗ = Wn ). The identity a) implies that {Tn (a)} ∈ S(X) whenever T (a) ∈ L(X). Wn Tn (a)Wn = Tn () Let J = J (X) denote the collection of all elements {An } ∈ D∞ (X) of the form {An } = {Pn KPn + Wn LWn + Cn }, where K ∈ C∞ (X), L ∈ C∞ (X), {Cn } ∈ G(X). Clearly, J is a subset of S.
7.9. Proposition. J is a closed twosided ideal of S. Proof. We ﬁrst show that J is closed. If {Bn } = {Pn KPn +Wn LWn +Cn } ∈ J then, by 1.1(e), K ≤ lim inf Bn and L ≤ lim inf Bn . Thus, if n→∞
n→∞
∞ (j) {A(j) Pn + Wn L(j) Wn + Cn(j) n }j=1 = Pn K
∞ j=1
⊂J
is a Cauchy sequence, then {K (j) } and {L(j) } are Cauchy sequences in C∞ . Consequently, there exist K and L in C∞ such that K − K (j) → 0 and L − L(j) → 0 as j → ∞. But if ai = ki + li + ci and {ai }, {ki }, {li } are (j) Cauchy sequences, then so also is {ci }. Hence, {Cn }∞ j=1 is a Cauchy sequence (j)
in G and thus, {Cn } − {Cn } → 0 as j → ∞ for some {Cn } ∈ G. It follows that as j → ∞, {A(j) n } → {Pn KPn + Wn LWn + Cn } ∈ J which proves that J is closed. Now let {An } = {Pn KPn + Wn LWn + Cn } ∈ J and {Bn } ∈ S. Then Bn An = Bn Pn KPn + Bn Wn LWn + Bn Cn )n Pn LWn + Bn Cn = Bn Pn KPn + Wn B ) = Pn BKPn + Wn BLW n + Pn (Bn Pn − B)KPn )n Pn − B)LW ) +Wn (B n + Bn Cn
7.1 Basic Facts
329
and the Banach space version of 1.3(d) with p = ∞ implies that {Bn An } ∈ J . It can be shown similarly that {An Bn } ∈ J . p , put 7.10. Deﬁnitions. For X = pN or X = HN
SJπ = SJπ (X) := S(X)/J (X) and denote the coset {An } + J by {An }πJ . If A ∈ L(X), {An } ∈ S(X), and An Pn → A strongly on X, then the strong limit s lim Wn An Wn will be n→∞
denoted by W{An } (A). If An = Pn APn , then W{An } (A) will be abbreviated to W(A). 7.11. Theorem. Let A ∈ L(X) and let {An } ∈ S(X) be any sequence such that An Pn → A strongly on X. Then A ∈ Π{X; An } ⇐⇒ A ∈ GL(X), W{An } (A) ∈ GL(X), {An }πJ ∈ GSJπ (X). If A ∈ Π{X; An } and if {Bn }πJ ∈ SJπ (X) is the inverse of {An }πJ in SJπ (X), then π Bn + Pn (A−1 − B)Pn + Wn [W{An } (A)]−1 − W{Bn } (B) Wn G (7.15) is the inverse of {An }πG in D(X)/G(X); here B := s lim Bn . n→∞
Proof. Suppose A ∈ Π{An }. Then A ∈ GL(X) by Proposition 7.3. Let us ) := W{A } (A) ∈ GL(X). Again from Proposition 7.3 we deduce prove that A n ) that An := Wn An Wn is invertible for all suﬃciently large n (n ≥ n0 ) and that −1 )−1 sup A n Pn = sup Wn An Wn
n≥n0
n≥n0
≤ sup Wn 2 A−1 n Pn =: C < ∞.
(7.16)
n≥n0
)n Pn ϕ for all ϕ ∈ X, and passing to the limit n → ∞ Hence Pn ϕ ≤ CA ) we obtain ϕ ≤ CAϕ for all ϕ ∈ X. It can be shown analogously that ∗ ) ϕ for all ϕ ∈ X ∗ . This implies that A ) ∈ GL(X). ϕ ≤ CA ) ∈ Π{A )n }, A∗ ∈ Π{A )∗n }, Using (7.16) and Proposition 7.3 we see that A ∗ ∗ ) ) and A ∈ Π{An }. Let {Bn } ∈ D be the inverse of {An } modulo G. Then Bn An = Pn + Cn with {Cn } ∈ G and thus, )n = (Pn + Wn Cn Wn )A )−1 )−1 , B n Pn → A Bn∗ = (A∗n )−1 (Pn + Cn∗ )Pn → (A∗ )−1 , )n∗ = (A )∗n )−1 (Pn + Wn Cn∗ Wn )Pn → (A )∗ )−1 B (strong convergence). Therefore {Bn } ∈ S and it follows that {Bn }πJ is the inverse of {An }πJ , that is, {An }πJ ∈ GSJπ .
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7 Finite Section Method
) are invertible on X and {An }π is invertible in S π . Now suppose A and A J J Then there is a sequence {Bn } ∈ S such that An Bn = Pn + Pn KPn + Wn LWn + Cn ,
(7.17)
where K and L are in C∞ (X) and {Cn } ∈ G. Passage to the limit n → ∞ gives AB = I + K, and if we multiply (7.17) by Wn from the left and the right and )B ) = I + L, where then pass to the limit n → ∞, we arrive at the equality A −1 −1 ) ) ) =: T ∈ C∞ (X). B := W{An } (B). Hence A − B =: R ∈ C∞ (X) and A − B Put Bn := Bn + Pn RPn + Wn T Wn . Then {Bn } ∈ S and An Bn = Pn + Pn (K + An Pn R)Pn + Wn (L + An Pn T )Wn + Cn ) )Wn + Cn = Pn + Cn , = Pn + Pn (K + AR)Pn + Wn (L + AT where {Cn } ∈ G. It can be shown analogously that {Bn An } − {Pn } ∈ G. This gives the remaining assertion of the theorem. Our next objective is to show that {Pn APn } is in S(H p ) (resp. S(p )) if A is in algL(H p ) T (L∞ ) (resp. algL(p ) T (M p )). We also want to compute W(A) = s lim Wn AWn for these cases. n→∞
p or pN and let LT (X) denote the collection 7.12. Deﬁnitions. Let X be HN of all operators A ∈ L(X) for which the four strong limits
T (A) := s lim V (−n−1) AV n+1 ,
W(A) := s lim Wn AWn ,
H(A) := s lim V (−n−1) AWn ,
K(A) := s lim Wn AV n+1
n→∞
n→∞
n→∞
n→∞
exist and are in L(X). The “LT ” is for “like Toeplitz.” 7.13. Theorem. LT (X) is a closed subalgebra of L(X). If A, B ∈ LT (X), then T (AB) = T (A)T (B) + H(A)K(B),
(7.18)
W(AB) = W(A)W(B) + K(A)H(B),
(7.19)
H(AB) = H(A)W(B) + T (A)H(B),
(7.20)
K(AB) = W(A)K(B) + K(A)T (B).
(7.21)
Proof. It is clear that LT (X) is a linear set. To see that LT (X) is an algebra, it suﬃces to verify identities (7.18)–(7.21). But these follow from the obvious equality I = Pn + Qn = Wn2 + V n+1 V (−n−1) : V (−n−1) ABV n+1 = V (−n−1) AWn Wn BV n+1 +V (−n−1) AV n+1 V (−n−1) BV n+1 → H(A)K(B) + T (A)T (B) strongly as n → ∞,
7.1 Basic Facts
331
and analogously for (7.19)–(7.21). It remains to show that LT (X) is closed. From 7.5 and 7.6 we know that M := sup V (−n) , V n , Pn , Wn < ∞. n≥0
Hence, by 1.1(e), T (A), W(A), H(A), K(A) ≤ M 2 A ∀ A ∈ LT (X).
(7.22)
Now let Ak ∈ LT (X), A ∈ L(X), and suppose A−Ak → 0 as k → ∞. From (7.22) we deduce that {T (Ak )} is a Cauchy sequence in L(X). Consequently, there is a B ∈ L(X) such that B − T (Ak ) → 0 as k → ∞. Thus, if x ∈ X, then V (−n) AV n x − Bx ≤ V (−n) AV n x − V (−n) Ak V n x +V (−n) Ak V n x − T (Ak )x + T (Ak )x − Bx,
(7.23)
and given ε > 0 there is a k0 = k0 (ε) such that the ﬁrst and the third terms on the right of (7.23) are smaller than ε/2 for k = k0 and all n ∈ Z+ , and then one can ﬁnd an n0 = n0 (k0 , ε) such that the second term on the right of (7.23) becomes smaller than ε/2 for k = k0 and n ≥ n0 . Hence, s lim V (−n) AV n n→∞ exists and equals B. It can be shown similarly that the remaining three limits occurring in 7.12 also exist for A. In accordance with 3.43 (also recall Proposition 4.1 and Corollary 4.3) we let SmbT denote the continuous extension of the mapping given by SmbT :
n m
T (ajk ) →
j=1 k=1
n m
ajk
j=1 k=1
p p to algL(HNp ) T (L∞ N ×N ) (resp. algL(N ) T (MN ×N )). Note that SmbT is a continp ∞ uous algebraic homomorphism onto LN ×N (resp. MN ×N ).
7.14. Corollary. We have p algL(HNp ) T (L∞ N ×N ) ⊂ LT (HN ),
p p algL(pN ) T (MN ×N ) ⊂ LT (N ).
(7.24)
p p If A is in algL(HNp ) T (L∞ N ×N ) resp. algL(N ) T (MN ×N ), then {Pn APn } belongs p p to S(HN ) resp. S(N ) and
T (A) = T (SmbT A),
W(A) = T ((SmbT A))),
(7.25)
H(A) = H(SmbT A),
K(A) = H((SmbT A))).
(7.26)
Proof. Because V (−n) T (a)V n = T (a), V
(−n−1)
Wn T (a)Wn = Pn T () a)Pn → T () a),
T (a)Wn = H(a)Pn → H(a),
Wn T (a)V n+1 = Pn H() a) → H() a),
332
7 Finite Section Method
it follows that every bounded Toeplitz operator belongs to LT (X), and since LT (X) is a closed algebra, we arrive at the inclusions (7.24). If A is in ∗ ∞ q algL(HNp ) T (L∞ N ×N ), then A belongs to algL(HN ) T (LN ×N ). So (7.24) shows p q that the limits W(A) and W(A∗ ) exist and belong to L(HN ) and L(HN ), p respectively, which implies that {Pn APn } ∈ S(HN ). The proof is analogous for pN . From the computations at the beginning of this proof we see that (7.25) and (7.26) hold for A = T (a). Identities (7.18)–(7.21)
and (2.18)–(2.19) then 5m n show that (7.25) and (7.26) are true for A of the form j=1 k=1 T (ajk ), and from the continuity of SmbT , T , W, H, K (recall (7.22)) we obtain (7.25) and (7.26) for the general case. The following proposition provides a further important tool for the study of the ﬁnite section method. 7.15. Proposition. Let X1 and X2 be linear spaces, A : X1 → X2 a linear and invertible operator, P1 : X1 → X1 and P2 : X2 → X2 linear projections, and put Q1 = I − P1 and Q2 = I − P2 . Then P2 AP1 : Im P1 → Im P2 is invertible if and only if Q1 A−1 Q2 : Im Q2 → Im Q1 is invertible. In that case P1 (P2 AP1 )−1 P2 = P1 A−1 P2 − P1 A−1 Q2 (Q1 A−1 Q2 )−1 Q1 A−1 P2 .
(7.27)
Proof. (Kozak). P2 AP1 P1 A−1 P2 − P1 A−1 Q2 (Q1 A−1 Q2 )−1 Q1 A−1 P2 = P2 AP1 A−1 P2 − P2 AP1 A−1 Q2 (Q1 A−1 Q2 )−1 Q1 A−1 P2 = P2 AP1 A−1 P2 − P2 AA−1 Q2 (Q1 A−1 Q2 )−1 Q1 A−1 P2 +P2 AQ1 A−1 Q2 (Q1 A−1 Q2 )−1 Q1 A−1 P2 = P2 AP1 A−1 P2 − 0 + P2 AQ1 A−1 P2 = P2 . 7.16. Corollary. Suppose X1 , X2 are Banach spaces and A ∈ L(X1 , X2 ) is (boundedly) invertible. Let Pn1 ∈ L(X1 ) and Pn2 ∈ L(X2 ) be projections converging strongly to the identity operator on X1 and X2 , respectively, and put Q1n = I − Pn1 , Q2n = I − Pn2 . Then A ∈ Π{X1 , X2 ; Pn2 APn1 } if and only if Q1n A−1 Q2n : Q2n X2 → Q1n X1 is invertible for all suﬃciently large n (n ≥ n0 , say) and if sup (Q1n A−1 Q2n )−1 Q1n L(X1 ,X2 ) < ∞. n≥n0
Proof. Immediate from Propositions 7.3 and 7.15. 7.17. Corollary. Let X be a Banach space, let Pn ∈ L(X) be projections, and suppose Pn → I strongly on X. If A ∈ Π{X; Pn }, K ∈ C∞ (X), and A + K ∈ GL(X), then A + K ∈ Π{X; Pn }. Proof. A ∈ Π{Pn } implies that A ∈ GL(X). We have (A + K)−1 − A−1 = −(A + K)−1 KA−1 =: L ∈ C∞ (X).
7.1 Basic Facts
333
By the preceding corollary, Qn A−1 Qn : Qn X → Qn X is invertible for all suﬃciently large n and the norms of the inverses are uniformly bounded. Hence Qn (A + K)−1 Qn = Qn A−1 Qn I + (Qn A−1 Qn )−1 Qn LQn , and because Qn LQn converges uniformly to zero as n → ∞, it follows that Qn (A + K)−1 Qn : Qn X → Qn X is invertible for all n large enough and that the norms of the inverses are uniformly bounded (by Neumann’s series expansion). Once more applying the previous corollary we get the assertion. Note that the hypotheses of the preceding two corollaries are satisﬁed for p (w) and X = X1 = X2 = p,µ X = X1 = X2 = H N N and Pn as in 7.5. The following corollary is another curious consequence of Corollary 7.16. 7.18. Corollary. We have p p −1 (a) ∈ Π{HN (w); Pn }, a ∈ L∞ N ×N , T (a) ∈ GL(HN (w)) =⇒ T p p −1 a ∈ MN (a) ∈ Π{pN ; Pn }. ×N , T (a) ∈ GL(N ) =⇒ T
Proof. Because V (−n−1) V n+1 = I,
V n+1 V (−n−1) = Qn ,
V (−n−1) T (a)V n+1 = T (a),
we have Qn T (a)Qn V n+1 T −1 (a)V (−n−1) = Qn , V n+1 T −1 (a)V (−n−1) Qn T (a)Qn = Qn , whence (Qn T (a)Qn )−1 Qn = V n+1 T −1 (a)V (−n−1) , and since sup V (±n) L(X) < ∞
n≥0
p (w) or X = pN , the assertion follows from Corollary 7.16. for X = HN
Open problem. Is the above result true for the space p,µ N ? We ﬁnally establish some connection between the Toeplitz operators generated by the matrix functions a, ) a, a∗ , a−1 . Recall that ) a(t) := a(1/t),
a∗ (t) := a(t)
(t ∈ T),
∗ where
denotes transposition. For N = 1, a is also denoted by a. Thus, if a = n∈Z an χn , then ) a= a−n χn , a∗ = a∗−n χn . n∈Z
n∈Z
334
7 Finite Section Method
p ∞ 7.19. Proposition. Let X = HN (w) (resp. p,µ N ) and let a ∈ GLN ×N (resp. p,µ q ∗ −1 a ∈ GMN ) (resp. X ∗ = q,−µ ). ×N ). Put X = HN (w N
(a) T (a) ∈ GL(X) ⇐⇒ T (a∗ ) ∈ GL(X ∗ ). (b) T () a) ∈ GL(X) ⇐⇒ T (a−1 ) ∈ GL(X ∗ ). (c) If X = H p (N = 1, w = 1) or X = pµ (N = 1), then T () a) ∈ GL(X) ⇐⇒ T (a) ∈ GL(X). (d) There exist a ∈ P2×2 such that T (a) ∈ GL(H22 ) but T () a) ∈ / GL(H22 ). (e) T (a) ∈ Φ(X) ⇐⇒ T (a∗ ) ∈ Φ(X ∗ ).
IndX T (a) = −IndX ∗ T (a∗ ).
(f) T () a) ∈ Φ(X) ⇐⇒ T (a−1 ) ∈ Φ(X).
IndX T () a) = IndX T (a−1 ).
(g) If X = H p (N = 1, w = 1) or X = pµ (N = 1), then T () a) ∈ Φ(X) ⇐⇒ T (a) ∈ Φ(X).
IndX T () a) = IndX T (a).
Proof. (a), (e) This is obvious for X = p,µ N , and the arguments of the proof p (w). of Lemma 2.39 can be used for X = HN (b) If T () a) ∈ GL(X) resp. T (a−1 ) ∈ GL(X), then T (a)−H(a)T −1 () a)H() a) −1 a−1 )T −1 (a−1 )H(a−1 ) is the inverse of T (a−1 ) resp. T () a). resp. T () a ) − H() This can be easily veriﬁed using Proposition 2.14. (c) This follows from the identity T () a) = W T (a)W , where W is given by (W ϕ)(t) = ϕ(1/t) for ϕ ∈ H p and (W x)n = xn for {xn } ∈ pµ . (d) Let a = (aij )2i,j=1 , where a11 = χ1 , a12 = 1, a21 = 0, a22 = χ−1 . Then a = g− dg+ , where a = h− h+ and ) 1 0 t 1 h− (t) = , , h+ (t) = t−1 1 −1 0 −1 t 1 01 t 0 g− (t) = (t) = , , d(t) = , g + 10 1 0 0 t−1 ∞ and h , g ∈ GH ∞ , we deduce that Since h− , g− ∈ GH2×2 + + 2×2
T (a) ∈ GL(H22 ),
dim Ker T () a) = dim Coker T () a) = 1.
(f) If R and S are regularizers of T () a) and T (a−1 ), respectively, then, a−1 )SH(a−1 ) are by Proposition 2.14, T (a) − H(a)RH() a) and T () a−1 ) − H() −1 a), respectively. Let us prove the index equality. regularizers of T (a ) and T () By 1.12(a), there is a regularizer R of T () a) such that I − T () a)R ∈ C0 (X),
I − RT () a) ∈ C0 (X).
7.2 C + H ∞ Symbols
335
Hence, by Proposition 2.14, T (a−1 ) T (a) − H(a)RH() a) = I − H(a−1 )H() a) + H(a−1 )T () a)RH() a) −1 = I − H(a ) I − T () a)R H() a) ∈ I + C0 (X), −1 −1 T (a) − H(a)RH() a) T (a ) = I − H(a)H() a ) + H(a)RT () a)H() a−1 ) = I − H(a) I − RT () a) H() a−1 ) ∈ I + C0 (X). So 1.12(b) gives that Ind T (a−1 ) equals 0 1 0 1 tr H(a) I − RT () a) H() a−1 ) − tr H(a−1 ) I − T () a)R H() a) 1 0 1 0 a)R H() a)H(a−1 ) (by 1.4(b)) = tr I − RT () a) H() a−1 )H(a) − tr I − T () 0 1 0 1 = tr I − RT () a) I − T () a−1 )T () a) − tr I − T () a)R I − T () a)T () a−1 ) 1 0 1 0 = tr T () a)R − RT () a) + tr T () a) I − RT () a) T () a−1 ) 1 0 a) + tr T () a)T () a−1 ) − tr T () a−1 )T () a) −tr I − RT () a) T () a−1 )T () = tr [T () a)R − RT () a)] which is equal to Ind T () a), again by 1.12(b). (g) If T () a) resp. T (a) is in Φ(X), then there is an n ∈ Z such that T () aχn ) resp. T (aχn ) is in GL(X). So the assertion can be derived from (c). Remark. Note that there is a close connection between (b) and Proposition 7.15.
7.2 C + H ∞ Symbols p ∞ 7.20. Theorem. (a) Let a ∈ CN ×N + HN ×N and K ∈ C∞ (HN ). Then p p p T (a) + K ∈ Π{HN ; Pn } ⇐⇒ T (a) + K ∈ GL(HN ), T () a) ∈ GL(HN ).
(b) Let a ∈ (Cp + Hp∞ )N ×N and K ∈ C∞ (pN ). Then T (a) + K ∈ Π{pN ; Pn } ⇐⇒ T (a) + K ∈ GL(pN ), T () a) ∈ GL(pN ). (c) Under the hypothesis of (a) or (c), if T (a) + K ∈ Π{Pn } then π Pn (T (a) + K)−1 Pn + Wn T −1 () a) − T () a−1 ) Wn G (7.28) is the inverse of {Pn (T (a) + K)Pn }πG . Proof. We apply Theorem 7.11 with A = T (a)+K and An = Pn (T (a)+K)Pn . Thus, if A ∈ Π{Pn }, then both A and W(A) = s lim Wn (T (a) + K)Wn = T () a) n→∞
336
7 Finite Section Method
must be invertible. Conversely, suppose A and W(A) are invertible. To get that A ∈ Π{Pn }, it remains to show that {An }πJ is in GSJπ . Theorem 2.94 implies that a is invertible in (C + H ∞ )N ×N resp. (Cp + Hp∞ )N ×N , and therefore H() a) and H() a−1 ) are compact. Hence, by (7.10), π π π π Pn (T (a) + K)Pn J Pn T (a−1 )Pn J = Pn T (a)Pn J Pn T (a−1 )Pn J π = Pn − Pn H(a)H() a−1 )Pn − Wn H() a)H(a−1 )Wn J = {Pn }πJ and it can be shown equally that the coset {Pn T (a−1 )Pn }πJ is a left inverse of the coset {Pn (T (a) + K)Pn }πJ . Thus, {An }πJ ∈ GSJπ . Finally, (c) results from the fact that (7.15) is the inverse of {An }πJ . p 7.21. Remark. Let a ∈ L∞ a) N ×N (resp. MN ×N ) and suppose T (a) and T () are invertible. It can be shown that then 0 1 Pn T −1 (a)Pn + Wn T −1 () a) − T () a−1 ) Wn Pn T (a)Pn = Pn − Pn T −1 (a) − T (a−1 ) Qn T (a)Pn − Wn T −1 () a) − T () a−1 ) Qn T () a)Wn .
In the C + H ∞ case the operators T −1 (a) − T (a−1 ) and T −1 () a) − T () a−1 ) are compact and so this identity immediately gives the implications “⇐=” of the above theorem for K = 0. Using Corollary 7.16 one can treat Toeplitz operators with QCN ×N symbols on H p with weight. p (w)). Then 7.22. Theorem. Let a ∈ QCN ×N and K ∈ C∞ (HN p p p T (a) + K ∈ Π{HN (w); Pn } ⇐⇒ T (a) + K ∈ GL(HN (w)), T () a) ∈ GL(HN (w)).
Proof. Suppose T (a) + K ∈ Π{Pn }. Then T (a) + K is invertible (Proposition 7.3), and Theorem 5.31(b) implies that a ∈ GQCN ×N . Hence, by (2.18) p (w), and the compactness of Hankel operators with QC symbols on HN (T (a) + K)−1 = T (a−1 ) + L with
p L ∈ C∞ (HN (w)).
(7.29)
p p By virtue of Corollary 7.16, Qn (T (a) + K)−1 Qn : Qn HN (w) → Qn HN (w) is invertible for all suﬃciently large n and the norms of the inverses are uniformly bounded. Since 0 −1 1 Qn T (a−1 )Qn = Qn (T (a) + K)−1 Qn I − Qn (T (a) + K)−1 Qn Qn LQn p and Qn LQn → 0 as n → ∞, we deduce that Qn T (a−1 )Qn : Qn HN (w) → p p n+1 : HN (w) → Qn HN (w) is invertible for all suﬃciently large n. Because V p (w) is an isometric isomorphism, it follows that T (a−1 ) is invertible Q n HN and Proposition 7.19(b) yields the invertibility of T () a).
7.2 C + H ∞ Symbols
337
Now suppose T (a) + K and T () a) are invertible. Then T (a−1 ) is invertible p p −1 (w) → Qn HN (w) is invertible for all and, consequently, Qn T (a )Qn : Qn HN −1 −1 n ≥ 0 and the norms of the inverses are equal to T (a )L(HNp (w)) . Again (7.29) holds and hence, 0 1 Qn (T (a) + K)−1 Qn = Qn T (a−1 )Qn I + (Qn T (a−1 )Qn )−1 Qn LQn = Qn T (a−1 )Qn (I + Cn ) with Cn L(HNp (w)) → 0 as n → ∞. It follows that Qn (T (a) + K)−1 Qn : p p (w) → Qn HN (w) is invertible for all suﬃciently large n and that the Q n HN norms of the inverses are uniformly bounded. Corollary 7.16 completes the proof. p,µ 7.23. Lemma. Let 1 < p < ∞ and µ ≥ 0. If a ∈ MN ×N and T (a) is in p,µ a) is an operator of regular type on pN , i.e., Π{N ; Pn }, then T ()
T () a)xpN ≥ mxpN
∀ x ∈ pN
(7.30)
with some m > 0 independent of x ∈ pN . Proof. Let Λ be as in the proof of 6.2(c) and denote ΛIN ×N by Λ, too. It is eas−1 ∈ Π{pN ; Pn }. ily seen that T (a) ∈ Π{p,µ N ; Pn } if and only if A := ΛT (a)Λ p a) on N . This will imply We claim that Wn AWn converges strongly to T () (7.30), since if A ∈ Π{pN ; Pn }, we have for every x ∈ pN , Pn x = (Wn AWn )−1 Wn AWn x = Wn (Pn APn )−1 Wn Wn AWn x ≤ sup (Pn APn )−1 Pn Wn AWn x, n
which gives (7.30) as n → ∞. To prove that Wn AWn → T () a) strongly on pN we may assume that N = 1. Since the operators Wn AWn are uniformly bounded, it suﬃces to show that (Pn T () a)Pn − Wn AWn )ej pp → 0 (n → ∞)
(7.31)
for all j ∈ Z+ . A simple computation shows that for n > j the lefthand side
n−j (n) of (7.31) is equal to k=−j bk , where (n)
bk
:= a−k p 1 − (n + 1 − j − k)µ (n + 1 − j)−µ p . (n)
For k = 0, . . . , n − j we have bk ≤ a−k p (1 + 1µ )p = 2p a−k p . Hence,
n−j (n) given any ε > 0 there is an L = L(ε) such that < ε/2 for k=n+1 bk all n ≥ j + L + 1, and then one can ﬁnd an M = M (j, L, ε) such that
L (n) < ε/2 for all n > M . As ε > 0 can be chosen arbitrarily we get k=−j bk (7.31).
338
7 Finite Section Method
7.24. Proposition. Let 1 ≤ p < ∞ and let µ, λ ≥ 0. Suppose A is an operator p,λ π which belongs to both GL(p,µ N ) and GL(N ). If {Pn APn }G is invertible in p,µ p,λ π ∞ p,λ D∞ (p,µ N )/G(N ), then {Pn APn }G is invertible in D (N )/G(N ). Proof. For simplicity let N = 1; it is easily seen that the following proof also works for N > 1. We ﬁrst state two simple estimates: if δ, γ are real numbers such that 0 ≤ δ ≤ γ, then Pn xp,γ ≤ (n + 1)γ−δ Pn xp,δ
∀ x ∈ pδ ,
(7.32)
Qn xp,δ ≤ (n + 1)
∀x∈
(7.33)
δ−γ
Qn xp,γ
pγ .
Indeed, Pn xpp,γ = ≤ Qn xpp,δ = ≤
n
(k + 1)γp xk p =
k=0 n k=0 ∞
k+1 n+1
γp n k+1 k=0
δp
k=n+1
k+1 n+1
(n + 1)γp xk p
(n + 1)γp xk p = (n + 1)(γ−δ)p Pn xpp,δ ,
(k + 1)δp xk p =
k=n+1 ∞
n+1
γp
δp ∞ k+1 (n + 1)δp xk p n+1
k=n+1
(n + 1)δp xk p = (n + 1)(δ−γ)p Qn xpp,γ .
Put An = Pn APn Im Pn and suppose {An }πG is invertible in the quotient algebra D∞ (pµ )/G(pµ ). Then, for x ∈ pλ and all suﬃciently large n, −1 −1 Pn x + Pn A−1 Pn xp,λ A−1 n Pn xp,λ = An Pn x − Pn A −1 ≤ A−1 Pn xp,λ + A−1 L(pλ ) Pn xp,λ . n Pn x − Pn A
(7.34)
First assume µ ≤ λ. Then −1 A−1 Pn xp,λ n Pn x − Pn A −1 ≤ (n + 1)λ−µ A−1 Pn xp,µ n Pn x − Pn A
(by (7.32))
≤ C(n + 1)λ−µ Pn x − Pn APn A−1 Pn xp,µ p (because sup A−1 n Pn L(µ ) < ∞)
n
= C(n + 1)λ−µ Pn AQn A−1 Pn xp,µ = C(n + 1)λ−µ AL(pµ ) (n + 1)µ−λ A−1 L(pµ ) Pn xp,λ
(by (7.33)).
From this and (7.34) we get the invertibility of {An }πG in D∞ (pλ )/G(pλ ). Now let µ ≥ λ. Then
7.2 C + H ∞ Symbols
339
−1 A−1 Pn xp,λ n Pn x − Pn A
= A−1 (APn − Pn APn )A−1 n Pn xp,λ = A−1 Qn APn A−1 n Pn xp,λ ≤ A−1 L(pµ ) (n + 1)λ−µ AL(pµ ) A−1 n Pn xp,µ
(by (7.33))
and since A−1 n Pn xp,µ ≤ CPn xp,µ
(because {An }πG ∈ G(D∞ (pµ )/G(pµ )))
≤ C(n + 1)µ−λ Pn xp,λ
(by (7.32)),
we deduce from (7.34) that {An }πG is invertible in D∞ (pλ )/G(pλ ).
Remark. The above proposition together with Proposition 7.3 gives the following implication: p,µ p,λ p,λ A ∈ GL(p,µ N ), A ∈ Π{N ; Pn }, A ∈ GL(N ) =⇒ A ∈ Π{N ; Pn }. p,µ p,µ 7.25. Theorem. Let 1 < p < ∞, µ ∈ R, a ∈ CN ×N , and K ∈ C∞ (N ). Then the following are equivalent:
(i) T (a) + K ∈ Π{p,µ N ; Pn }; (ii) T (a) + K ∈ GL(p,µ a) ∈ GL(p,µ N ), T () N ); (iii) T (a) + K ∈ GL(p,µ a) ∈ GL(pN ). N ), T () Proof. (i) =⇒ (iii). Proposition 7.3 gives the invertibility of T (a) + K on p,µ N . Hence det a does not vanish on T and ind det a = 0 by 6.12(c). From 6.2(b) and 6.12(c) we deduce that T () a) is Fredholm of index zero on pN , and Lemma 7.23 then gives the invertibility of T () a) on pN . (ii) ⇐⇒ (iii). Without loss of generality assume µ ≥ 0. Theorem 6.12(c) p implies that T () a) is Fredholm of index zero on both p,µ N and N . Since the p,µ p a) in N , it follows that kernel of T () a) in N is contained in the kernel of T () p if it is invertible on . Analogously, because the T () a) is invertible on p,µ N N a) in qN is a subset of the kernel of T ∗ () a) in q,−µ , the invertibility kernel of T ∗ () N p implies the invertibility of T () a ) on . of T () a) on p,µ N N (ii)+(iii) =⇒ (i). It suﬃces again to consider the case µ ≥ 0. From 7.19(b) we deduce that T (a−1 ) is invertible on both pN and p,µ N . Corollary 7.18 shows that A := T −1 (a−1 ) ∈ Π{pN ; Pn }. Combining this with Propositions 7.3 and 7.24 we obtain that A ∈ Π{p,µ N ; Pn }. Finally, since T (a) + K = T −1 (a−1 ) − H(a)H() a−1 )T −1 (a−1 ) + K, we see that T (a) + K diﬀers from A only by a compact operator, and so Corollary 7.17 implies that T (a) + K ∈ Π{p,µ N ; Pn }.
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7 Finite Section Method
Remark. Theorem 7.25 also holds for p = 1. Open problem. Extend Theorem 7.20 to spaces with weight. 7.26. alg T F (AN ×N ). Let A be a closed subalgebra of L∞ resp. M p containing C resp. Cp and put A = AN ×N . The norm of a ∈ A is deﬁned as the norm of the multiplication operator M (a) on L2N resp. pN (Z). (a) The mapping given by p T F : (A ⊂ L∞ N ×N ) → S(HN ),
a → {Tn (a)}
is a submultiplicative embedding, and the mapping p p T F : (A ⊂ MN ×N ) → S(N ),
a → {Tn (a)}
is a 1submultiplicative isometry. To see this for the ﬁrst mapping notice that aL∞ ≤ c1 T (a)L(HNp ) N ×N
(by (4.6))
≤ c1 lim inf Tn (a)L(HNp )
(by 1.1(e))
n→∞
≤ c1 sup Tn (a)L(HNp ) = c1 {Tn (a)}S(HNp ) ,
(7.35)
n≥0
which implies that Im T F is closed and that Ker T F = {0}, and also take into account that " " " " " " " " ajk " ≤ c a " Tn "T " 2 jk p p j
k
" " " " ≤ c3 " T (ajk )" j
k
S(HN )
p L(HN )
" " " " Tn (ajk )" ≤ c3 lim inf " n→∞
j
k
" " " " Tn (ajk )" ≤ c3 sup " n≥0
j
k
j
k
L(HN )
(by Proposition 4.1) (by 1.1(e))
p L(HN )
p L(HN )
" " " " =" {Tn (ajk )}" j
k
p S(HN )
,
which gives the submultiplicativity. The proof for the p case is analogous (note that in this case one can take c1 = 1 in (7.35) and that (7.35) can be continued by “≤ aMNp ×N ”). (b) Thus Theorem 3.42 gives that ·
alg T F (A) = T F (A) + QT F (A). Deﬁne the mapping LD by p p LD : D(HN or pN ) → L(HN or pN ),
{An } → s lim An . n→∞
7.2 C + H ∞ Symbols
341
It is clear that LD = 1 and that LD (alg T F (A)) is a subset of alg T (A). The mappings ST F and SmbT F (recall 3.43) which are given at ﬁnite productsums (correctly) by ST F : {Tn (ajk )} → Tn ajk , j
SmbT F :
j
k
j
{Tn (ajk )} →
k
j
k
ajk
k
can be easily veriﬁed to be representable as ST F = T F ◦ SmbT ◦ LD alg T F (A),
(7.36)
SmbT F = SmbT ◦ LD alg T F (A).
(7.37)
p Sometimes, in order to indicate that the underlying space is HN resp. pN , we shall write algS(HNp ) T F (A) resp. algS(pN ) T F (A) instead of alg T F (A).
(c) Let ∆ denote the restriction of LD to alg T F (A). The analogue of the “upper half” of the diagram 4.18(f) looks as follows:
Unfortunately we do not know any analogue of the “lower half” of that diagram. It should be mentioned here that it was the attempt to search for this analogue which led us ﬁrst to the study of the algebra alg K(A), where {Kλ }λ∈Z+ is generated by the Fej´er kernel, and then to the observation that some important results on Toeplitz operators which can be expressed in terms of the harmonic extension remain valid if the AbelPoisson means are replaced by an arbitrary approximate identity. 7.27. Proposition. (a) If B is a closed subalgebra of C + H ∞ containing C, then p ). QT F (BN ×N ) = J (HN (b) If B is a closed subalgebra of Cp + Hp∞ containing Cp , then QT F (BN ×N ) = J (pN ). Proof. It suﬃces to consider the case N = 1. If a, b ∈ B, then H() a) and H()b) are compact and therefore, by Proposition 7.7, a)H(b)Wn } ∈ J , {Tn (ab) − Tn (a)Tn (b)} = {Pn H(a)H()b)Pn + Wn H() which shows that QT F (B) ⊂ J . So we are left with the opposite inclusion.
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7 Finite Section Method
Suppose we had shown that J is a subset of the algebra alg T F (B). Then if {An } = {Pn KPn + Wn LWn + Cn } (K and L compact, {Cn } in G), we have, by (7.36)–(7.37) and Proposition 4.5, ST F {An } = (T F ◦ SmbT ◦ LD ){An } = (T F ◦ SmbT )(K) = 0, and it results that J ⊂ QT F (B). Thus let us prove that J ⊂ alg T F (B). To see that {Pn KPn } ∈ alg T F (B) for every compact operator K, it is suﬃcient to show that {Pn T (a1 ) . . . T (am )Pn } is in alg T F (B) for every ﬁnite collection a1 , . . . , am of functions from C resp. Cp (Proposition 4.5). This is + trivial for m = 1. Let m = 2. If a1 is a Laurent polynomial, then a1 = a− 1 + a1 − + ∞ ∞ with a1 ∈ H and a1 ∈ H , hence + {Pn T (a1 )T (a2 )Pn } = {Pn T (a− 1 )T (a2 )Pn + Pn T (a1 )T (a2 )Pn } + = {Tn (a− 1 a2 ) + Tn (a1 )Tn (a2 )} ∈ alg T F (B).
Since every function a1 in C resp. Cp can be approximated as closely as desired in the norm of L∞ reps. M p by Laurent polynomials, it follows that {Pn T (a1 )T (a2 )Pn } is in alg T F (B) for every a1 , a2 ∈ C resp. Cp . The assertion for general m ≥ 1 can be proved analogously by induction. In a similar fashion one can show that {Wn LWn } belongs to T F (B) for every L ∈ C∞ . It remains to prove that {Cn } ∈ alg T F (B) for every {Cn } ∈ G. For i, j ∈ Z+ , let Kij denote the ﬁniterank operators on H p resp. p whose ma∞ trix representation with respect to the standard bases {χn }∞ n=0 resp. {en }n=0 ∞ is given by (δir δjs )r,s=0 (δkl the Kronecker delta). A little thought shows that the inclusion G ⊂ alg T F (B) will follow as soon as we have proved that {Cn } belongs to alg T F (B), where {Cn } is any sequence with Cn = 0 for n = n0 and Cn0 = Pn0 Kij Pn0 (0 ≤ i, j ≤ n0 ). Since the operators Kij and Kn0 −j,n0 −j have ﬁnite rank, from what has already been proved we deduce that {Pn Kij Pn } and {Wn Kn0 −j,n0 −j Wn } are in alg T F (B). Because Pn Kij Pn · Wn Kn0 −j,n0 −j Wn equals Pn0 Kij Pn0 for n = n0 and 0 for n = n0 , we arrive at the conclusion that {Cn } is also in alg T F (B). 7.28. alg J FGπ (AN ×N ) and alg T FGπ (AN ×N ). Let A and A be as in 7.26. (a) The preceding proposition implies that both G and J are closed twosided ideals of alg T F (A) and that ST F (G) = ST F (J ) = {0} ⊂ G ⊂ J . Hence, by Theorem 3.52, the mappings T FGπ : A → alg T F (A)/G, a → {Tn (a)}πG := {Tn (a)} + G, T FJπ : A → alg T F (A)/J , a → {Tn (a)}πJ := {Tn (a)} + J are submultiplicative quasiembeddings and ·
alg T F (A)/G = alg T FGπ (A) = T FGπ (A) + QT FGπ (A), ·
alg T F (A)/J = alg T FJπ (A) = T FJπ (A) + QT FJπ (A).
7.2 C + H ∞ Symbols
343
If {Tn (a)} ∈ J , then T (a) = s lim Tn (a) must be compact and so a = 0. It n→∞ follows that the mappings T FGπ and T FJπ are actually embeddings. (b) We have {Tn (a)}πJ SJπ (pN ) = aMNp ×N
p ∀a ∈ MN ×N .
(7.38)
The inequality “≤” in (7.38) is obvious. If K and L are compact and {Cn } is in G, then {Tn (a) + Pn KPn + Wn LWn + Cn } = sup Tn (a) + Pn KPn + Wn LWn + Cn n
≥ lim inf Tn (a) + Pn KPn + Wn LWn + Cn n→∞
≥ T (a) + K ≥ T (a)Φ(pN ) = aMNp ×N , where the last equality results from Propositions 4.4(d) and 4.1(b). This gives “≥” in (7.38). Hence, in the p case T FJπ is an isometry. (c) Since ∆ (recall 7.26(c)) maps J into (even onto) the set of all compact operators, the quotient mapping ∆π : alg T FJπ (A) → alg T π (A) is well deﬁned. So we arrive at the following analogue of the “upper half” of the diagram 4.19(g):
Theorem 7.91 will imply that ∆π is not oneto one in case A = L∞ . However, the following theorem (which may be viewed as the “ﬁnite section analogue” of Corollary 4.7 and Theorems 4.79 and 4.81) shows that ∆π is an isomorphism if A is a closed algebra between C and C + H ∞ . 7.29. Theorem. (a) Let B be a closed subalgebra of C + H ∞ containing C. Then the mapping SmbT FJπ given by (3.34) is a homeomorphic algebraic isomorphism of the algebra algSJπ (HNp ) T FJπ (BN ×N ) onto BN ×N . (b) If B is a closed subalgebra of Cp + Hp∞ containing Cp , then the mapping SmbT FJπ given by (3.34) is an isometric algebraic isomorphism of algSJπ (pN ) T FJπ (BN ×N ) onto BN ×N .
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7 Finite Section Method
Proof. Combining Corollary 3.44 and Proposition 7.27 we see that SmbT FJπ is a homeomorphic isomorphism. That SmbT FJπ is an isometry in the case (b) results from (7.38). 7.30. Corollary. Let A=
r s
T (ajk ) + K,
An =
j=1 k=1
r s
Tn (ajk ) + Pn KPn ,
j=1 k=1
p where ajk belong to (C + H ∞ )N ×N resp. (Cp + Hp∞ )N ×N and K ∈ C∞ (HN ) p resp. K ∈ C∞ (N ). Then A ∈ Π{An } if and only if both A and
W{An } (A) :=
r s
T (a. jk )
j=1 k=1
are invertible. Proof. Clearly, An Pn → A and Wn Tn (ajk ) Wn + Wn KWn Wn An Wn = j
k
Wn Tn (ajk )Wn + Wn KWn = j
=
k
j
Tn (a. jk ) + Wn KWn →
k
j
T (a. jk ).
k
So Theorem 7.11 gives 5 the “only if” part. On the other hand, if A is invertible, then SmbT A = j k ajk is invertible in (C +H ∞ )N ×N resp. (Cp +Hp∞ )N ×N (Corollary 4.8), hence {An }πJ is invertible in SJπ (Theorem 7.29), and Theorem 7.11 completes the proof.
7.3 Locally Sectorial Symbols 7.31. Localization. Let F be a closed subset of X = M (L∞ ), let A be a closed subalgebra of L∞ containing C, and put A = AN ×N . In this section 2 . we always assume that the underlying space is HN (a) In accordance with 3.58 deﬁne JF0 := closidalg T FJπ (A) {Tn (a)}πJ : a ∈ A, aF = 0 . Lemma 3.59 and Theorem 3.52 imply that T FFπ : A → alg T FFπ (A) := alg T FJπ (A)/JF0 , a → {Tn (a)}πF := {Tn (a)}πJ + JF0
7.3 Locally Sectorial Symbols
345
is a 1submultiplicative quasiembedding whose kernel is {a ∈ A : aF = 0}. If F is a ﬁber Xβ (β ∈ M (B), B being a C ∗ subalgebra of L∞ with identity), π and {An }πXβ will be abbreviated to T Fβπ and {An }πβ , respectively. then T FX β (b) It is not diﬃcult to see that ∆π (recall 7.28(c)) maps JF0 into closidalg T π (A) T π (a) : a ∈ A, aF = 0 . So the quotient mapping ∆πF of ∆π can be naturally deﬁned and we arrive at the following analogue of the “upper half” of the diagram 4.23(c):
(c) The arguments used to prove Theorem 4.24 also yield the following (local) spectral inclusions: sp (aF ) ⊂ sp TFπ (a) ⊂ sp {Tn (a)}πF , sp {Tn (a)}πF ⊂ conv a(F ) if N = 1. (d) Theorem 3.61 specializes to give that {Tn (a)}πF = aF for all a ∈ L∞ N ×N . (e) Let B be a C ∗ algebra between C and QC. By virtue of Theorem 7.29(b) (p = 2), B is isometrically starisomorphic to alg T FJπ (BIN ×N ) and may therefore be identiﬁed with a C ∗ subalgebra of alg T FJπ (L∞ N ×N ). If c ∈ QC and a ∈ L∞ , then, by (7.10), {Tn (ca) − Tn (c)Tn (a)} = {Pn H(c)H() a)Pn + Wn H() c)H(a)Wn } ∈ J (7.39) and hence Theorem 3.67 gives sp {An }πJ = sp {An }πβ for every {An } in alg T F (L∞ N ×N ).
β∈M (B)
7.32. Theorem. Let B be a C ∗ algebra between C and QC and let K be 2 ). Suppose a ∈ L∞ in C∞ (HN N ×N satisﬁes at least one of the following three conditions: (i) For each β ∈ M (B) there exists a bβ ∈ L∞ N ×N such that aXβ = bβ Xβ 2 and T (bβ ) ∈ Π{HN ; Pn }; (ii) a is locally sectorial over B; (iii) for each β ∈ M (B) the set a(Xβ ) is contained in some straight line segment (which may depend on β). 2 2 ; Pn } if and only if T (a) + K ∈ GL(HN ) and Then T (a) + K ∈ Π{HN 2 ). T () a) ∈ GL(HN
346
7 Finite Section Method
Proof. Let a satisfy (i). From Theorem 7.11 we deduce that {Tn (bβ )}πJ and thus {Tn (bβ )}πβ is invertible for each β ∈ M (B). In view of 7.31(d) we have {Tn (bβ )}πβ = {Tn (a)}πβ and so 7.31(e) implies the invertibility of {Tn (a)}πJ . The assertion now follows from Theorem 7.11. If a satisﬁes (ii), then Corollary 3.62, 7.31(e), and Theorem 7.11 give the assertion. Finally, let a have the property (iii). If T (a) + K is invertible, then T (a) is Fredholm, and so Theorem 4.70 shows that a must be locally sectorial over QC. This reduces the things to the case that a satisﬁes (ii). π π (P C) and alg T FJ (P QC). Again suppose the underlying 7.33. alg T FJ 2 space is H . Once (7.39) has been established the same arguments as in the proof of Proposition 4.83 show that alg T FJπ (P QC) is commutative. From the spectral inclusions 7.31(c) and Theorem 4.67 we obtain that sp {Tn (χτ )}πβ = sp Tβπ (χτ ), where τ ∈ T, χτ is the characteristic function of the arc (τ, τ eiπ/2 ), and β ∈ M (C) or β ∈ M (QC). This implies the following.
(a) Proposition 4.85 and Theorems 4.86 and 4.87 are true with iπ = T FJπ . (b) If B is C or QC, then ∆π is an isometric starisomorphism of the algebra alg T FJπ (P BN ×N ) onto the algebra alg T π (P BN ×N ) and the algebra alg T FJπ (P BN ×N ) is isometrically starisomorphic to the algebra [C(NP B )]N ×N . (c) Lemma 4.92, Proposition 4.93, and Theorem 4.94 are true with iπ = T FJπ . Since alg T FJπ (P QCN ×N ) ∼ = alg T π (P QCN ×N ), the same reasoning as in the proof of Corollary 7.30 gives the following result.
r 5 s (d) Let A = j=1 k=1 T (ajk ) + K, where ajk ∈ P QCN ×N and K be in
r 5 s 2 2 ). Put An = j=1 k=1 Tn (ajk ) + Pn KPn . Then A ∈ Π{HN ; An } if C∞ (HN
r 5 s and only if both A and W{An } (A) := j=1 k=1 T (a. jk ) are invertible on the 2 space HN . 7.34. Open problems. (a) Is T (a) in Π{H p ; Pn } if a ∈ L∞ is locally psectorial (p > 2) over a C ∗ algebra B between C and QC and both T (a) and T () a) are invertible on H p ? We conjecture that the answer is yes. Note that we have not been able to answer the question even for B = C, i.e., for symbols which are (globally) psectorial on T (in this case T (a) and T () a) are automatically invertible). (b) What can be said about the applicability of the ﬁnite section method to Toeplitz operators on H 2 generated by symbols that are locally sectorial over C + H ∞ ?
7.4 P C Symbols: p Theory
347
7.4 P C Symbols: p Theory In the following we assume that 1 < p < ∞, 1/p + 1/q = 1, −1/p < µ < 1/q. 7.35. Lemma. Let a ∈ M p , suppose {Tn (a)}πJ is invertible in SJπ (p ), and assume at least one of the conditions (i) T (a) and T () a) are leftinvertible, (ii) T (a) and T () a) are rightinvertible is satisﬁed. Then T (a) ∈ Π{p ; Pn }. Proof. For the sake of deﬁniteness, assume (i) is fulﬁlled. Because {Tn (a)}πJ is invertible, there are {Bn } ∈ S(p ), K and L in C∞ (p ), and {Cn } ∈ G(p ) such that (7.40) Bn Tn (a) = Pn + Pn KPn + Wn LWn + Cn . Let X and Y be left inverses of T (a) and T () a), respectively, and put Bn := Bn − Pn KXPn − Wn LY Wn . A computation similar to that in the proof of Theorem 7.11 shows that Bn Tn (a) = Pn + Cn with {Cn } ∈ G(p ). It follows that Tn (a) is invertible for all suﬃciently large n and that Bn (Pn + Cn )−1 is the inverse. Since Bn (Pn + Cn )−1 converges strongly to B, we conclude that {Tn (a)}πG is in G(D/G), and so Proposition 7.3 gives the assertion. 7.36. Proposition. (a) If a ∈ M p and T (a) ∈ Π{p ; Pn }, then T (a) ∈ Π{r ; Pn } for all r ∈ [p, q] and, in particular, T (a) ∈ GL(r ) for all r ∈ [p, q]. (b) If a ∈ M p and {Tn (a)}πJ is invertible in SJπ (p ), then {Tn (a)}πJ is in for all r ∈ [p, q], T (a) is in Φ(r ) for all r ∈ [p, q], and the index of T (a) on r does not depend on r ∈ [p, q]. GSJπ (r )
Proof. (a) Let C denote the mapping given on r by C : {xn } → {xn }. If A is in L(r ), then CAC is also in L(r ) and we have CAC = A. Let T (a) ∈ Π{p ; Pn }. From Proposition 7.3 we know that there is an M such that Tn−1 (a)Pn L(p ) ≤ M for all n ≥ n0 . Hence, Tn−1 (a)Pn L(q ) = Tn−1 (a)Pn L(p ) = CWn Tn−1 (a)Wn CL(p ) ≤ Tn−1 (a)Pn L(p ) ≤ M for all n ≥ n0 , and the RieszThorin interpolation theorem gives Tn−1 (a)Pn L(r ) ≤ M
∀ n ≥ n0
∀ r ∈ [p, q].
(7.41)
In view of Proposition 7.3 it remains to show that T (a) ∈ GL(r ) for all r ∈ [p, q]. From (7.41) we deduce that Tn−1 (a)Pn L(s ) ≤ M for all n ≥ n0 and s ∈ [p, q]. Thus, if x ∈ r , y ∈ s (1/r + 1/s = 1), and n ≥ n0 , then Pn xr ≤ M T (a)Pn xr ,
Pn ys ≤ M T (a)Pn ys
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7 Finite Section Method
and passage to the limit n → ∞ gives xr ≤ M T (a)xr ,
ys ≤ M T (a)ys ,
from which we infer that T (a) ∈ GL(r ). (b) Passage to the strong limit n → ∞ in (7.40) implies that T (a) ∈ Φ(p ). After multiplying (7.40) from both sides by Wn and then passing to the strong limit n → ∞ we see that T () a) ∈ Φ(p ). Choose m ∈ Z so that p π T (aχm ) ∈ GL( ). Since {Tn (aχm )}J equals {Tn (a)}πJ {Tn (χm )}πJ , it results that {Tn (aχm )}πJ is also invertible. By virtue of Theorem 2.38 the kernel or the cokernel of the (Fredholm) operator T () aχ−m ) is trivial. Hence, T (aχm ) and T () aχ−m ) are either simultaneously leftinvertible or simultaneously rightinvertible. So Lemma 7.35 can be applied to deduce that T (aχm ) ∈ Π{p ; Pn }. Now part (a) gives that T (aχm ) ∈ Π{r ; Pn } for all r ∈ [p, q], which implies that T (a) ∈ Φ(r ) and Ind T (a) = m for all r ∈ [p, q] and that {Tn (a)}πJ = {Tn (aχm }πJ {Tn (χ−m )}πJ is invertible in SJπ (r ) for all r ∈ [p, q] (Theorem 7.11). 7.37. Theorem (Verbitsky/Krupnik). Let β ∈ C. Then the following are equivalent: (i) T (ϕβ ) ∈ Π{pµ ; Pn }; (ii) T (ϕβ ) ∈ GL(pµ ), T (. ϕβ ) ∈ GL(p ); (iii) −1/p < Re β + µ < 1/q, −1/q < Re β < 1/p. Proof. (ii) ⇐⇒ (iii): Proposition 6.24. (i) =⇒ (ii). Without loss of generality assume 0 ≤ µ < 1/q. Let Λ be as in the proof of 6.2(c) (see also 7.23) and put An := Pn ΛT (ϕβ )Λ−1 Pn . We shall prove that {An } ∈ S(p ). In the proof of Lemma 7.23 we established that ϕβ ) strongly on p , so that Theorem 7.11 can be applied to Wn An Wn → T (. deduce that T (. ϕβ ) ∈ GL(p ). To prove that {An } ∈ S(p ), it remains to show that Wn A∗n Wn → T ϕ .β strongly on q . As in the proof of Lemma 7.23, we have n−j (n) "q " ∗ " "(Pn T ϕ .β Pn − Wn An Wn )ej q = ck ,
(7.42)
k=−j
+ µ +q + +q + n+1−j + . + +1 − + −k n+1−j−k + + +
0 (n) It is clear that k=−j ck → 0 as n → ∞. Since + ϕ .β −k + is not larger than
n−j (n) a constant times 1/k, the sum k=1 ck can be estimated by the integral 4q 3 µ ( n−j n+1−j 1 − 1 dx, xq n+1−j−x 1 (n) ck
+ =+ ϕ .β
7.4 P C Symbols: p Theory
349
and substituting x = (n + 1 − j)y one sees that this integral can be estimated by q ( 1 1 − (1 − y)µ 1 (n + 1 − j)1−q dy. (7.43) y (1 − y)µq 0 If µq < 1, the integral in (7.43) is ﬁnite, and hence (7.43) and thus (7.42) goes to zero as n → ∞. (iii) =⇒ (i). Because −1/p < Re β + µ < 1/q, the operator T (ϕβ ) is invertible on pµ . Theorem 6.20 (δ = −β, γ = β) gives Γ−β,β M−β T (ϕβ )Mβ = T (ηβ )T (ξ−β ).
(7.44)
Taking into account that Pn T (ηβ ) = Pn T (ηβ )Pn , T (ξ−β )Pn = Pn T (ξ−β )Pn we obtain from (7.44) that Γ−β,β Pn M−β Pn Tn (ϕβ )Pn Mβ Pn = Tn (ηβ )Tn (ξ−β ), and it follows that det Tn (ϕβ ) = 0 for all n ∈ Z+ . Hence, by Proposition 7.15, Qn T −1 (ϕβ )Qn is invertible on Qn pµ for all n ∈ Z+ . But (7.44) implies that Qn T −1 (ϕβ )Qn = Γ−β,β Qn Mβ T (ϕ−β )M−β Qn , (α)
(α)
and therefore, if we put Mα,n := diag (µn+1 , µn+2 , . . .), Mβ,n T (ϕ−β )M−β,n is invertible on the weighted p space ∞ p pµ p pn := x = {xk }∞ : x := (n + k + 1) x  < ∞ k k=0 p,n k=0
for all n ∈ Z+ . Moreover, we have −1 −1 −1 (Qn T −1 (ϕβ )Qn )−1 Qn L(pµ ) = Γ−β,β M−β,n T −1 (ϕ−β )Mβ,n L(pn ) −1 −1 = M−β,n M−β T (ϕβ )Mβ Mβ,n L(pn ) (by (7.42)) + + + µ(β) µ(−β) + + k k + −1 −1 M−β )T (ϕβ )(M−β,n M−β )−1 L(pn ) ≤ sup + (β) + (M−β,n (−β) + + k µn+k µn+k −1 −1 ≤ c4 (M−β,n M−β )T (ϕβ )(M−β,n M−β )−1 L(pn )
(Lemma 6.21)
≤ c6 T (ϕβ )L(prn ) , where prn is the weighted p space ∞ (n + k + 1)(µ+α)p p p x  < ∞ , prn := x = {xk }∞ k k=0 : xp,rn := (k + 1)αp k=0
with α := Re β (again use Lemma 6.21). Without loss of generality assume µ + α ≥ 0; otherwise consider adjoints. Then
350
7 Finite Section Method
(n + k + 1)(µ+α)p ≤ d (n + 1)(µ+α)p + (k + 1)(µ+α)p and hence, for x ∈ prn ,
T (ϕβ )xpp,rn ≤ d T (ϕβ )xpp,−α (n + 1)(µ+α)p + T (ϕβ )xpp,µ ≤ d T (ϕβ )pL(p ) xpp,−α (n + 1)(µ+α)p + T (ϕβ )pL(pµ ) xpp,µ −α p ≤ d T (ϕβ )L(p ) xpp,rn + T (ϕβ )pL(pµ ) xpp,rn . −α
Since −1/q < α < 1/p and −1/p < µ < 1/q, we deduce from Proposition 6.23 that T (ϕβ )L(p−α ) < ∞, T (ϕβ )L(pµ ) < ∞, and thus sup (Qn T −1 (ϕβ )Qn )−1 Qn L(pµ ) < ∞. Corollary 7.16 completes the proof.
n
Remark. Notice that in the case µ = 0 T (ϕβ ) ∈ Π{p ; Pn } ⇐⇒ T (ϕβ ) ∈ GL(p ), T (. ϕβ ) ∈ GL(p ) ⇐⇒ T (ϕβ ) ∈ GL(p ), T (ϕβ ) ∈ GL(q ) ⇐⇒ T (ϕβ ) ∈ GL(r ) ∀ r ∈ [p, q] ⇐⇒ Re β < min{1/p, 1/q}. 7.38. Lemma. (a) If a, b ∈ M p and if for each τ ∈ T there are an open arc Uτ ⊂ T and a function fτ ∈ Cp such that aUτ = fτ Uτ or bUτ = fτ Uτ , then {Tn (ab)}πJ = {Tn (a)}πJ {Tn (b)}πJ . (b) If a, b ∈ P Cp , then {Tn (a)}πJ {Tn (b)}πJ = {Tn (b)}πJ {Tn (a)}πJ . Proof. Using formula (7.10) this can be shown in the same way as Propositions 6.29 and 6.30.
r 7.39. Proposition. Let a = i=1 gi fi , where the functions gi are piecewise constant and the functions fi are continuously diﬀerentiable on T. Then T (a) ∈ Π{p ; Pn } ⇐⇒ T (a) ∈ GL(p ), T () a) ∈ GL(p ). Proof. The implication “=⇒” is an immediate consequence of Theorem 7.11. On the other hand, if T (a) and T () a) are invertible on p , then a can be written in the form (5.19), a = ϕβ1 . . . ϕβm b, and one has Re βi  < min{1/p, 1/q} for all i and b ∈ Cp (see the proof of Proposition 6.32). Hence, by Theorems 7.37, 7.20(a), and 7.11, {Tn (ϕβi )}πJ and {Tn (b)}πJ are invertible in SJπ (p ). Lemma 7.38(a) implies that {Tn (a)}πJ = {Tn (ϕβ1 )}πJ . . . {Tn (ϕβm )}πJ {Tn (b)}πJ , and therefore {Tn (a)}πJ ∈ GSJπ (p ). It remains to apply Theorem 7.11.
7.4 P C Symbols: p Theory
351
7.40. Localization. For τ ∈ T, let Jτ0 denote the smallest closed twosided p ideal of the algebra algSJπ (pN ) T FJπ (P CN ×N ) containing the set {Tn (f )}πJ : f = diag (ϕ, . . . , ϕ), ϕ ∈ B, ϕ(τ ) = 0 , where B may be P, C ∞ , or Cp (Jτ0 does not depend on the particular choice of B, see also 6.34). Deﬁne p p π 0 alg T Fτπ (P CN π (p ) T FJ (P C ×N ) := algSJ N ×N )/Jτ , N
(7.45)
denote the coset of this algebra containing {An }πJ by {An }πτ , and for {An } in p π π π alg T F (P CN ×N ) let spp {An }τ and spp {An }J refer to the spectrum of {An }τ π and {An }J as element of the algebra (7.45) and as element of the algebra p algSJπ (pN ) T FJπ (P CN ×N ), respectively. Lemma 7.38(b) implies that alg T FJπ (P Cp ) is commutative. A similar reasoning as in the proof of Proposition 6.35 shows that p π π a, b ∈ P CN ×N , aXτ = bXτ =⇒ {Tn (a)}τ = {Tn (b)}τ
and that spp {An }πJ =
/
spp {An }πτ
(7.46) (7.47)
τ ∈T p for every {An } ∈ alg T F (P CN ×N ).
7.41. Proposition. If a ∈ P Cp , then spp {Tn (a)}πτ = Op (a(τ − 0), a(τ + 0)), / spp {Tn (a)}πJ = Op (a(τ − 0), a(τ + 0)).
(7.48) (7.49)
τ ∈T
Proof. By virtue of (7.46) we may assume that τ is the only point of discontinuity of a, that a is as in Proposition 7.39, and that R(a) is the arc Ap (a(τ − 0), a(τ + 0)). Propositions 6.32 and 7.25 together with Theorem 7.11 give the inclusion spSJπ (p ) {Tn (a)}πJ ⊂ Op (a(τ − 0), a(τ + 0)),
(7.50)
and Propositions 7.36(b) and 6.32 yield that the reverse inclusion in (7.50) also holds. Since alg T FJπ (P Cp ) is a closed subalgebra of SJπ (p ), we can apply 1.16(b) to deduce that the spectrum of {Tn (a)}πJ in alg T FJπ (P Cp ) equals Op (a(τ − 0), a(τ + 0)). Again using (7.46) we obtain that spp {Tn (a)}πt = {a(t)} ∈ Ap (a(τ − 0), a(τ + 0)) for t = τ . Thus, by (7.47), Op (a(τ − 0), a(τ + 0)) ⊃ spp {Tn (a)}πτ ⊃
/
Ar (a(τ − 0), a(τ + 0))
r∈(p,q)
and since a spectrum is always closed, equality (7.48) follows. Equality (7.49) results from (7.48) and (7.47).
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7 Finite Section Method
p p 7.42. Theorem. Let a ∈ P CN ×N and K ∈ C∞ (N ). Then
a) ∈ GL(pN ). T (a) + K ∈ Π{pN ; Pn } ⇐⇒ T (a) + K ∈ GL(pN ), T () Proof. The implication “=⇒” is a consequence of Theorem 7.11. So we are left with the opposite implication. First let N = 1. If T (a) + K and T () a) are invertible on p , then T (a) ∈ p q Φ( ), T (a) ∈ Φ( ), and Indp T (a) = Indq T (a) = 0. From 6.39 and 6.40 (with Op (a(τ − 0), a(τ + 0)), and so (7.49) and A = T (a)) we infer that 0 ∈ / τ ∈T
Theorem 7.11 give the assertion. Now let N > 1 and suppose T (a) + K and T () a) are invertible on pN . We π π p must show that {Tn (a)}J is invertible in SJ (N ). In view of (7.47) it suﬃces p to show that {Tn (a)}πτ is invertible in alg T Fτπ (P CN ×N ) for each τ ∈ T. Let aτ denote the matrix function which equals a(τ + 0) on (τ, τ eiπ ) and a(τ − 0) on (τ e−iπ , τ ). Using 6.39 and 6.40 it is not diﬃcult to see that T (aτ ) ∈ Φ(pN ),
T (aτ ) ∈ Φ(qN ),
Indp T (aτ ) = Indq T (aτ ) = 0.
{Tn (aτ )}πJ
is invertible; then 7.40 will imply the invertibilWe shall prove that ity of {Tn (a)}πτ . Write aτ in the form ϕbψ as in 5.48(b). Due to Lemma 7.38(b) it is enough to show that {Tn (b)}πτ is invertible. Since b = ϕ−1 aτ ψ −1 , we have T (b) ∈ Φ(pN ),
T (b) ∈ Φ(qN ), Indp T (b) = Indq T (b).
N Consequently, as Ind T (b) = j=1 Ind T (bj ) (bj are the diagonal entries of b) and Indp T (b) ≤ Indq T (b) (resp. Indp T (b) ≥ Indq T (b)) if p ≤ q and thus pN ⊂ qN (resp. p ≥ q and thus pN ⊃ qN ), it follows that Indp T (bj ) = Indq T (bj ) for each j. Hence, T (bj ) ∈ Φ(r ) for all r ∈ [p, q] and all j = 1, . . . , N . Choose integers mj so that T (bj χmj ) ∈ GL(r ) for all r ∈ [p, q]. From (7.49) we obtain the invertibility of {Tn (bj χmj )}πJ and thus the invertibility of {Tn (bj )}πJ itself in alg T FJπ (P Cp ). Thus π p {Bn }πJ := diag Tn (b1 ), . . . , Tn (bN ) J ∈ G(alg T FJπ (P CN ×N )). Therefore {Tn (b)}πJ may be written in the form {Bn }πJ {I + Xn }πJ , where p N π π {Xn }πJ in alg T FJπ (P CN ×N ) has the property that {Xn }J = {0}J . But this implies that {I + Xn }πJ is invertible, since if xN = 0, then N N I = (I + x − x)N = (−1)N xN + (I + x)k (−x)N −k k = (I + x) =
N
k=1
N k=1
N k
N k
k=1
(I + x)k−1 (−x)N −k
k−1
(I + x)
N −k
(−x)
which shows that I + x is invertible.
(I + x),
7.5 P C Symbols: H p Theory
353
7.43. Proposition. (a) The algebra alg T Fτπ (P Cp ) is singly generated by {Tn (χτ )}πτ , where χτ is the characteristic function of the arc (τ, τ eiπ/2 ). (b) The maximal ideal space of alg T Fτπ (P Cp ) is homeomorphic to Op (0, 1) (with the topology inherited from C) and the Gelfand map Γ : alg T Fτπ (P Cp ) → C(Op (0, 1)) is for a ∈ P Cp given by Γ {Tn (a)}πτ (x) = (1 − x)a(τ − 0) + xa(τ + 0)
x ∈ Op (0, 1) .
Proof. Similar arguments as in the proof of Proposition 5.44 apply. 7.44. Theorem. The maximal ideal space of alg T FJπ (P Cp ) can be identiﬁed with T×Op (0, 1) (equipped with an exotic topology). For a ∈ P Cp , the Gelfand transform Γ : alg T FJπ (P Cp ) → C(T × Op (0, 1)) is given by Γ {Tn (a)}πJ (τ, x) = (1 − x)a(τ − 0) + xa(τ + 0) τ ∈ T, x ∈ Op (0, 1) . Proof. Proceed as in the proof of Theorem 5.45. 7.45. Toeplitz operators on p with weight. The following result is proved in detail in Roch, Silbermann [426, Part I]. If a ∈ (M p,µ ∩ P C)N ×N and K ∈ C∞ (p,µ N ), then p,µ T (a) + K ∈ Π{p,µ a) ∈ GL(pN ). N ; Pn } ⇐⇒ T (a) + K ∈ GL(N ), T ()
7.5 P C Symbols: H p Theory In what follows we always assume that 1 < p < ∞, 1/p + 1/q = 1. 7.46. Proposition. (a) If a ∈ L∞ and T (a) ∈ Π{H p ; Pn }, then T (a) is in Π{H r ; Pn } and thus T (a) is in GL(H r ) for all r ∈ [p, q]. (b) If a ∈ L∞ and {Tn (a)}πJ ∈ GSJπ (H p ), then {Tn (a)}πJ is in GSJπ (H r ) for all r ∈ [p, q], T (a) is in Φ(H r ) for all r ∈ [p, q], and the index of T (a) on H r does not depend on r ∈ [p, q]. Proof. The same reasoning as in the proof of Proposition 7.36 can be applied. 7.47. Marcinkiewicz’ multiplier theorem. Let λ0 , λ1 , λ2 , . . . be complex numbers such that λn  ≤ M,
2n+1 −1 k=2n
λk − λk+1  ≤ M
(n = 0, 1, 2, . . .).
354
7 Finite Section Method
Then the operator Λ:
∞ k=0
fk χk →
∞
λ k f k χk
k=0
belongs to L(H p ) and ΛL(H p ) ≤ Ap M , where Ap is some constant depending only on p. Proof. For a proof see Zygmund [591, Vol. II, Chap. XV, Theorem 4.14]. 7.48. Theorem (Verbitsky). Let β ∈ C. Then the following are equivalent: (i) T (ϕβ ) ∈ Π{H p ; Pn }; (ii) T (ϕβ ) ∈ GL(H p ), T (. ϕβ ) ∈ GL(H p ); (iii) Re β < min{1/p, 1/q}. Proof. (ii) ⇐⇒ (iii): Lemma 5.36. (i) =⇒ (ii): Theorem 7.11. (iii) =⇒ (i). Without loss of generality assume α := Re β ≥ 0. As in the proof of Theorem 7.37 we arrive at the equality (Qn T −1 (ϕβ )Qn )−1 Qn L(H p ) −1 −1 −1 = Γ−β,β M−β,n T −1 (ϕ−β )M−β,n L(H p )
(7.51)
(note that Qn H p is isometrically isomorphic to H p ). Let Λn denote the operator diag ((n + 1)α , (n + 2)α , . . .) and write −1 −1 −1 Γ−β,β M−β,n T −1 (ϕ−β )M−β,n −1 −1 −1 −1 = Γ−β,β M−β,n Λ−1 (ϕ−β )M−β,n n (Λn − Λ0 )T −1 −1 +M−β,n Λ−1 n Λ0 M−β T (ϕβ )Mβ M−β,n .
(7.52)
Using the Marcinkiewicz multiplier theorem we ﬁrst show that −1 sup M−β,n Λ−1 n L(H p ) < ∞. n
In what follows cβ (resp. cβ,p ) denotes a constant depending only on β (resp. β and p) but not necessarily the same at each occurrence. From Lemma 6.21 + (−β) −1 + we obtain that sup + µn+k (n+k +1)−α + ≤ cβ < ∞. So, by 7.47, it remains n,k
to verify that σmn :=
2m+1 −1
+ (−β) −1 + (−β) −1 + µ (n + k + 1)−α − µ (n + k + 2)−α + ≤ cβ n+k
n+k+1
k=2m
for all m, n ∈ Z+ . Taking into account Lemma 6.21 we get (the sums from k = 2m to k = 2m+1 − 1)
7.5 P C Symbols: H p Theory
σmn
355
+ (−β) + (−β) α α+ +µ ≤ cβ n+k+1 (n + k + 2) − µn+k+1 (n + k + 1) + + (−β) + ++ + β α α+ ++ 1 − +µ = cβ − (n + k + 1) (n + k + 2) n+k + + n+k+1 0 1 ≤ cβ (n + k + 1)−α (n + k + 2)α − (n + k + 1)α (n + k + 1)−α (n + k + 2)α (n + k + 1)−1 +cβ
and since (n + k + 2)α − (n + k + 1)α ≤ α(n + k + 1)α−1 , it follows that σmn ≤ cβ
m+1
2 1 1 ≤ cβ = cβ < ∞. n+k+1 k m k=2
Thus, sup σmn ≤ cβ , as desired. It can be shown analogously that m,n
Λ0 M−β L(H p ) ≤ cβ,p , Since, by Lemma 6.21, 2m+1 −1
(β) (µn+k )−1
k=2m
−
−1 sup Mβ Mβ,n L(H p ) ≤ cβ,p . n
(β) sup (µn+k )−1  k (β) (µn+k+1 )−1 
≤ cβ n−α and
≤ cβ
2m+1 −1
(n + k + 1)−α−1 ≤ cβ n−α ,
k=2m
−1 we conclude from 7.47 that Mβ,n L(H p ) ≤ cβ,p n−α . Finally consider ∞ Λn − Λ0 = diag (n + k + 1)α − (k + 1)α k=0 .
Because, for ﬁxed n, the sequence {(n+k +1)α −(k +1)α }∞ k=0 is monotonically decreasing, we have (n + k + 1)α − (k + 1)α ≤ cα nα and 2m+1 −1
(n + k + 1)α − (k + 1)α − (n + k + 2)α + (k + 2)α 
k=2m
= (n + 2m + 1)α − (2m + 1)α − [(n + 2m+1 + 1)α − (2m+1 + 1)α ] ≤ (n + 2m + 1)α − (2m + 1)α ≤ cα,p nα . Hence, again by 7.47, Λn − Λ0 L(H p ) ≤ cα,p nα . The above estimates together with the fact that T (ϕ−β ) ∈ GL(H p ) and T (ϕβ ) ∈ L(H p ) show that (7.52) and thus (7.51) is uniformly bounded for n ≥ 0, which by virtue of Corollary 7.16 gives the assertion. 7.49. Deﬁnition. For τ ∈ T, let Jτ0 be the smallest closed twosided ideal of algSJπ (H p ) T FJπ (P C) containing the set {{Tn (f )}πJ : f ∈ C, f (τ ) = 0}, put algp T Fτπ (P C) := algSJπ (H p ) T FJπ (P C)/Jτ0 , let {Tn (a)}πτ denote the coset {Tn (a)}πJ + Jτ0 , and let spp {Tn (a)}πτ and spp {Tn (a)}πJ refer to the spectrum of {Tn (a)}πτ and {Tn (a)}πJ in algp T Fτπ (P C) and algSJπ (H p ) T FJπ (P C), respectively.
356
7 Finite Section Method
7.50. Theorem. (a) If a ∈ P C and τ ∈ T, then spp {Tn (a)}πτ = Op (a(τ − 0), a(τ + 0)), / spp {Tn (a)}πJ = Op (a(τ − 0), a(τ + 0)). τ ∈T
algp T Fτπ (P C)
(b) The algebra is singly generated by {Tn (χτ )}πτ , where χτ is the characteristic function of the arc (τ, τ eiπ/2 ). The maximal ideal space of algp T Fτπ (P C) is homeomorphic to Op (0, 1) (with the topology inherited from C) and for a ∈ P C the Gelfand transform is given by Γ {Tn (a)}πτ (x) = (1 − x)a(τ − 0) + xa(τ + 0) (x ∈ Op (0, 1)). (c) The algebra algSJπ (H p ) T FJπ (P C) is commutative. Its maximal ideal space can be identiﬁed with T×Op (0, 1) (the topology is exotic) and for a ∈ P C the Gelfand transform is given by Γ {Tn (a)}πJ (τ, x) = (1 − x)a(τ − 0) + xa(τ + 0) τ ∈ T, x ∈ Op (0, 1) . p 7.51. Theorem. Let a ∈ P CN ×N and K ∈ C∞ (HN ). Then p p p T (a) + K ∈ Π{HN ; Pn } ⇐⇒ T (a) + K ∈ GL(HN ), T () a) ∈ GL(HN ).
The two preceding theorems can be proved similarly as their p analogues and their “Fredholm counterparts.” 7.52. Open problems. (a) Establish a criterion for the applicability of the ﬁnite section method to Toeplitz operators with P C symbols on H p with Khvedelidze weight. The case p = 2 will be settled by Corollary 7.75. (b) Is Theorem 7.51 true for a ∈ P QCN ×N ? Even the case N = 1 is of interest. For p = 2 the answer is known to be aﬃrmative (Theorem 7.32(iii)). (c) Extend Propositions 7.36 and 7.46 to the matrix case. 7.53. GohbergKrupnik localization. For τ ∈ T, let Mτp := {Tn (ϕ)}πJ ∈ SJπ (H p ) : ϕ ∈ C, 0 ≤ ϕ ≤ 1,
ϕ is identically 1 in some open neighborhood of τ ,
put Fp :=
τ ∈T
Mτp , and denote the commutant of Fp in SJπ (H p ) by Com Fp .
Note that Com Fp is a closed subalgebra of SJπ (H p ) containing alg T FJπ (L∞ ). Theorem 7.20 implies that {Tn (ϕ)}πJ is invertible in SJπ (H p ) if ϕ ∈ C and ϕ ≥ ε > 0 on T. This can be used to prove that {Mτp }τ ∈T is a covering system of bounded localizing classes in SJπ (H p ). Let Zτp denote the collection of all elements in Com Fp which are Mτp equivalent to zero from the left and the right, and observe that Zτp is a closed twosided ideal in Com Fp . For
7.6 Operators from algL(H 2 ) T (P C)
357
{An }πJ ∈ Com Fp let spp,τ {An }πJ refer to the spectrum of {An }πJ + Zτp as element of Com Fp /Zτp . It is not diﬃcult to see that a, b ∈ L∞ , aXτ = bXτ =⇒ {Tn (a)}πJ + Zτp = {Tn (b)}πJ + Zτp , and Theorem 1.32(b) gives that / spSJπ (HNp ) {An }πJ = spp,τ {An }πJ
∀ {An }πJ ∈ Com Fp .
(7.53)
(7.54)
τ ∈T
7.54. Theorem. Let A=
r s
T (ajk ),
An =
j=1 k=1
r s
Tn (ajk ),
j=1 k=1
W{An } (a) =
r s
T (a. jk ),
j=1 k=1
where ajk ∈ P C, and let K ∈ C∞ (H p ). Then A + K ∈ Π{H p ; An + Pn KPn } if and only if A + K ∈ GL(H p ), W{An } ∈ GL(H p ), and A ∈ Φ(H r ) for all r ∈ [p, q]. Proof. First suppose A + K ∈ Π{H p ; An + Pn KPn }. Theorem 7.11 along with the computation in the proof of Corollary 7.30 shows that A + K and W{An } (A) are invertible on H p . Theorem 7.11 also implies that {An }πJ is in / spp,τ {An }πJ for each τ ∈ T. Using (7.53), GSJπ (H p ). Hence, by (7.54), 0 ∈ (7.54) one can verify as in the proof of Proposition 7.41 that, if a ∈ P C, spp,τ {Tn (a)}πJ is Op (a(τ − 0), a(τ + 0)), and then the same argument as in the proof of Proposition 5.41 shows that spp,τ {An }πJ is equal to the spectrum of {An }πτ in algp T Fτπ (P C) (recall 7.49 and 7.50). The conclusion is that {An }πτ is invertible and so Theorems 7.50 and 5.43 can be combined to obtain that A ∈ Φ(H r ) for all r ∈ [p, q]. To get the “if” part of the present theorem it suﬃces in view of Theorem 7.11 to show that {An }πJ is in GSJπ (H p ) if A ∈ Φ(H r ) for all r ∈ [p, q]. But this follows from Theorems 7.50 and 5.43. Remark. Also recall 7.33(d).
7.6 Operators from algL(H2 ) T (P C) Theorem 7.20 solves the problem of the applicability of the ﬁnite section ∞ method to operators in algL(HN2 ) T (CN ×N + HN ×N ), since every operator in ∞ this algebra can be written in the form T (a) + K with a ∈ CN ×N + HN ×N 2 and K ∈ C∞ (HN ) (Corollary 4.7). However, we shall arrive at situations (e.g., when investigating the ﬁnite section method for T (a) with a ∈ P C on H 2 ()) in which the necessity emerges to check whether the ﬁnite section method is applicable to operators belonging to algL(H 2 ) T (P C). Note that not every
358
7 Finite Section Method
operator in this algebra is the sum of a Toeplitz operator and a compact operator.
r 5s For simplicity, let A = j=1 k=1 T (ajk ), where ajk ∈ P C. The question we are interested in reads: When is A ∈ Π{2 ; Pn APn }? Notice that this the following question: When is A ∈ Π{2 ; An }, where An =
r is5not s j=1 k=1 Tn (ajk )? The latter question is answered by 7.33(d) (and for ajk ∈ P C and H p as underlying space by Theorem 7.54). 7.55. Lemma. If 0 < α < 1, 0 < β < 1, α + β > 1, and if m, n are integers with m = n, then +−α + +−β ++ + + + +k − m + 1 + +k − n + 1 + ≤ cm − n1−α−β (7.55) + + + 2 2+ k∈Z
where c is some constant independent of m and n. Proof. Without loss of generality assume m < n. Then the function +−α + +−β + + 1 ++ ++ 1 ++ + f (x) := +x − m + + +x − n + + 2 2 is monotonically increasing on the intervals (−∞, m+1/2) and (d, n+1/2), and it is monotonically decreasing on the intervals (m + 1/2, d) and (n + 1/2, ∞), where d = [(n + 1/2)α + (m + 1/2)β]/(α + β). Hence, the sum in (7.55) is not larger than +−α + +−β ( ∞ + +−α + +−β + + + + 1 ++ ++ 1 ++ 1 ++ 1 ++ + + + c1 +m−n+ + + +m − n + + + +x − m + 2 + +x − n + 2 + dx. 2 2 −∞ The substitution y = (x − m + 1/2)/(m − n) in the integral gives the assertion. 7.56. Lemma. (a) If, for all k and j in Z+ , −γ + + 1 +k − j + bjk  ≤ c j + + 2 or
−γ + + 1 +k − j + bjk  ≤ c k + + 2
+γ−1 1 ++ 2+
(7.56)
+γ−1 1 ++ , 2+
(7.57)
where 0 < γ < 1/2 and c is some constant that does not depend on k and j, 2 then (bjk )∞ j,k=0 deﬁnes a bounded operator on . (b) If bjk = 0 for j < k and bjk  ≤ c(j + 1)−α (j − k + 1)−β (k + 1)−γ
for
j ≥ k,
where α + β + γ ≥ 1, α + β > 1/2, β < 1, and c is some constant independent 2 of k and j, then (bjk )∞ j,k=0 generates a bounded operator on .
7.6 Operators from algL(H 2 ) T (P C)
359
∞ 2 Proof. Let B = (bjk )∞ j,k=0 and x = {xk }k=0 ∈ .
(a) Suppose (7.56) is fulﬁlled. Then +∞ +2 ∞ + + + + bjk xk + Bx22 = + + + j=0 k=0 −2γ ∞ 1 ≤ c2 j+ 2 j=0 ∞ 2 xk (k + 1)δ 1 × k − j + 1/2(1−γ)/2 k − j + 1/2(1−γ)/2 (k + 1)δ k=0 −2γ ∞ 1 2 ≤c j+ 2 j=0 ∞ ∞ xk 2 (k + 1)2δ 1 × . k − j + 1/21−γ k − j + 1/21−γ (k + 1)2δ k=0
k=0
If δ ∈ (γ/2, (1 − γ)/2), then, by Lemma 7.55, −2γ γ−2δ ∞ ∞ 2 2δ x  (k + 1) 1 1 k 2 Bx2 ≤ c1 j+ j+ 2 k − j + 1/21−γ 2 j=0 k=0 +γ−1 −γ−2δ + ∞ ∞ + + 1 +j − k + 1 + = c1 xk 2 (k + 1)2δ , j+ + 2 2+ j=0 k=0
whence, again by Lemma 7.55, Bx22 ≤ c2
∞
xk 2 (k + 1)2δ (k + 1)−2δ = c2 x22 .
k=0
Passage to the adjoint yields the assertion for the case that (7.57) is satisﬁed. (b) We have Bx22 ≤ c2
∞
⎛ (j + 1)−2α ⎝
j=0
≤c
2
∞
−2α
(j + 1)
j=0
×
j
∞
⎞2 (j − k + 1)−β (k + 1)−γ xk ⎠
j=0 j
−2(β+ε)
(j − k + 1)
k=0
(j − k + 1) (k + 1)
k=0
Let ε > −1/2, δ > −1/2. Then
2ε
2δ
.
−2(γ+δ)
(k + 1)
xk 
2
360
7 Finite Section Method j
(j − k + 1)2ε (k + 1)2δ ≤ c1 (j + 1)2δ+2ε+1
k=0
and so Bx22 ≤ c2 = c2
j ∞ (j + 1)−2α+2δ+2ε+1 (j − k + 1)−2(β+ε) (k + 1)−2(γ+δ) xk 2 j=0
k=0
∞
∞
(k + 1)−2(γ+δ) xk 2
k=0
(j + 1)−2α+2δ+2ε+1 (j − k + 1)−2(β+ε) .
j=k
If 0 < 2α − 2δ − 2ε − 1 < 1, 0 < 2(β + ε) < 1, 2α + 2β − 2δ − 1 > 1, then, by Lemma 7.55, Bx22 ≤ c3 = c3
∞
(k + 1)−2(γ+δ) xk 2 (k + 1)−2(α+β−δ−1)
k=0 ∞
xk 2 (k + 1)−2(α+β+γ−1) ≤ c3 x22 .
k=0
It is not diﬃcult (but tedious) to see that there exist ε and δ with the properties required above, which completes the proof. (γ)
7.57. Lemma. Let µn then
be given as in 6.19. If 0 < Re γ ≤ δ < 1 and j ≥ k,
(γ)
(γ)
µj − µk  ≤ c(j − k)δ (k + 1)Re γ−δ , + + + 1 1 ++ + + (−γ) − (−γ) + ≤ c(j − k)δ (k + 1)Re γ−δ , +µ µk + j
(7.58) (7.59)
where c does not depend on k and j. (γ)
Proof. We have µn = Γ (γ + n + 1)/(Γ (γ + 1)Γ (n + 1)). Hence, using the formula ( ∞ xα−1 dx Γ (α)Γ (β) (Re α > 0, Re β > 0) = α+β (1 + x) Γ (α + β) 0 and Lemma 6.21 we get, for j > k, + + + + ( + 1 + 1 1 ++ ++ Γ (1 + γ) ∞ xγ−1 1 + − dx++ + (γ) − (γ) + = + γ k+1 j+1 +µ Γ (γ) (1 + x) (1 + x) (1 + x) 0 µj + k + + + + ( j j + ∞ + + Γ (γ + 1)Γ (s) + xγ dx + + + + = c1 ≤ c1 + + Γ (γ + s + 1) + (1 + x)γ+s+1 + 0 s=k+1
≤ c2
j s=k+1
s=k+1
j + µ(γ) + + s + s−1−σ ≤ c4 (k −σ − j −σ ), + ≤ c3 + s s=k+1
7.6 Operators from algL(H 2 ) T (P C)
361
where σ := Re γ, and thus + + + + 1 1 + + (γ) (γ) (γ) (γ) µj − µk  ≤ µj  µk  + (γ) − (γ) + ≤ c5 (j σ − k σ ). +µ µj + k
(7.60)
Since j σ − k σ ≤ (j − k)σ and j σ − k σ ≤ σ(j − k)kσ−1 (0 < σ < 1), we obtain 1−δ
δ−σ
j σ − k σ = (j − k) 1−σ (j σ − k σ ) 1−σ ≤ c6 (j − k)σ k σ−δ .
(7.61)
Now (7.58) results from (7.60) and (7.61). The proof of (7.59) is analogous. 7.58. Lemma. Let γ, δ ∈ C \ {−1, −2, . . .}. Then there exists a number c = 0 such that the operator I − cMγ−1 Mγ+δ Mδ−1 is compact on 2 . Proof. The nth diagonal entry dn of the operator is (γ+δ)
1−c
µn
(γ) (δ)
µn µn
=1−c
Thus, if we let c = 1/
n n (γ + δ + j)j γδ =1−c 1− . (γ + j)(δ + j) (γ + j)(δ + j) j=1 j=1 j=1 1 −
5∞
which implies compactness.
γδ , then dn → 0 as n → ∞, (γ + j)(δ + j)
In what follows the functions ξα , ηα , ϕα , etc., are always assumed to have the (possible) discontinuity at the same point τ ∈ T, i.e., ξα = ξα,τ , ηα = ηα,τ , ϕα = ϕα,τ , etc. 7.59. Proposition. If γ ∈ C and δ ∈ C, then Mγ T (ξγ )Mδ T (ξδ ) = Mδ T (ξδ )Mγ T (ξγ ), T (ηγ )Mγ T (ηδ )Mδ = T (ηδ )Mδ T (ηγ )Mγ .
(7.62) (7.63)
Proof. It suﬃces to verify (7.62), since (7.63) results from (7.62) by transponation. Fix δ ∈ C \ Z. We prove that (7.62) holds for all γ ∈ C \ Z satisfying Re γ > Re δ. Since the entries of the matrices in (7.62) are analytic functions of γ and δ, it then follows that (7.62) is true for all γ, δ ∈ C. After computing the j, j + n entry of both sides of (7.62) one sees that the following identity must be veriﬁed: −1 n γ δ+j+k δ δ+j (−1)k (−1)n−k k j+k n−k j k=0 −1 n γ+j δ γ+j+k γ k n−k = (−1) (−1) . (7.64) j k j+k n−k k=0
Formula (6.17) with a = δ + j + 1, b = −γ, c = j + 1 gives
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7 Finite Section Method
F (δ + j + 1, −γ; j + 1; x) = (1 − x)γ−δ F (−δ, γ + j + 1; j + 1; x)
(7.65)
(note that Re (γ − δ) > 0). On multiplying (7.65) by (1 − x)δ and computing the coeﬃcient of xn on both sides of the resulting equality one gets (7.64). 7.60. Proposition. Let γ, δ ∈ C and suppose −
1 1 < Re γ < , 2 2
−
1 1 < Re δ < , 2 2
−
1 1 < Re (γ + δ) < . 2 2
Then each of the following operators is bounded on 2 : A = Mγ+δ T (ξγ+δ )T (ξ−γ )Mγ−1 T (ξ−δ )Mδ−1 , −1 A−1 = Mδ T (ξδ )Mγ T (ξγ )T (ξ−γ−δ )Mγ+δ ,
B = Mγ−1 T (η−γ )Mδ−1 T (η−δ )T (ηγ+δ )Mγ+δ , −1 B −1 = Mγ+δ T (η−γ−δ )T (ηδ )Mδ T (ηγ )Mγ .
Proof. We only prove the boundedness of A. That the remaining three operators are bounded can be shown analogously. If Re δ = 0, we have A = [Mγ+δ T (ξδ )Mγ−1 ][T (ξ−δ )Mδ−1 ] and each bracket is a bounded operator by virtue of Corollary 6.22 and Proposition 6.44. Since, by Proposition 7.59, T (ξ−γ )Mγ−1 T (ξ−δ )Mδ−1 = T (ξ−δ )Mδ−1 T (ξ−γ )Mγ−1 ,
(7.66)
the case Re γ = 0 can be reduced to the case Re δ = 0. Now suppose Re (γ + δ) = 0. In this case it suﬃces to show that the operator C = T (ξ−γ )Mγ−1 T (ξ−δ )Mδ−1 is bounded. Let Re δ > 0. We have C = C1 C2 + C3 , where C1 = T (ξ−γ )Mγ−1 − Mγ−1 T (ξ−γ ),
C2 = T (ξ−δ )Mδ−1 ,
C3 = Mδ−1 T (ξ−γ−δ )Mγ−1 . Corollary 6.22 and Propositions 6.44, 6.45 immediately imply that C2 and C3 −γ (γ) (γ) are bounded. The j, k entry cjk of C1 is (1/µk − 1/µj ). Let α j−k satisfy Re γ < α < Re γ + 1/2. Then, by Lemmas 6.21 and 7.57, cjk  ≤ c(j − k + 1)−1−Re γ (j − k + 1)α (k + 1)Re γ−α and so Lemma 7.65(b) gives the boundedness of C1 . The case Re δ < 0 can be reduced to the case Re δ > 0 by taking into account (7.66). Thus, we may assume that Re γ = 0, Re δ = 0, Re (γ + δ) = 0. First suppose Re δ > 0. Write A = (A1 A2 + A3 + cI)A4 , where
7.6 Operators from algL(H 2 ) T (P C)
363
A1 = (Mγ+δ T (ξδ ) − T (ξδ )Mγ+δ )M−γ , −1 −1 A2 = cM−γ Mγ+δ T (ξ−δ ), −1 A3 = Mγ+δ T (ξδ )[Mγ−1 Mδ−1 Mγ+δ − cI]Mγ+δ T (ξ−δ ),
A4 = T (ξδ )Mδ T (ξ−δ )Mδ−1 . Here c is chosen so that Mγ−1 Mδ−1 Mγ+δ − cI is in C∞ (2 ) (Lemma 7.58). We ﬁrst show that A1 is bounded. Let ajk (j ≥ k) denote the jk entry of the operator A1 : 3 4 δ δ (γ+δ) (γ+δ) (−γ) ajk = µj − µk µk . j−k j−k Suppose Re (γ + δ) > 0 and choose α so that Re δ + Re γ < α < Re δ + 1/2. Then, by Lemmas 6.21 and 7.57, 1 α 1 −1−Re δ (k + 1)−Re γ j − k + (k + 1)Re (γ+δ)−α ajk  ≤ c j − k + 2 2 1 −1−Re δ+α = c j−k+ (k + 1)Re δ−α 2 and so Lemma 7.56(b) gives the boundedness of A1 . If Re (γ + δ) < 0, choose α so that Re γ + Re δ < α < Re γ + 1/2. Then, again by Lemmas 6.21 and 7.57, + + + + 1 1 δ + + (−γ) (γ+δ) (γ+δ) µk µj ajk  = + µk − (γ+δ) + (γ+δ) + j−k + µk µj 1 −1−Re δ+α ≤ c(j + 1)Re (γ+δ) j − k + (k + 1)−Re γ−α 2 and Lemma 7.56(b) again implies that A1 is in L(2 ). Lemma 6.21 shows that the jk entry ajk (j ≥ k) of A2 admits the estimate + ++ + 1 1 −1+Re δ 1 −δ + + (j + 1)−Re δ . ajk  = +c (−γ) (γ+δ) +≤c j−k+ j−k + + µ 2 µ j
j
Hence, by Lemma 7.56(a), A2 ∈ L(2 ). The operator A4 can be written as [T (ξδ )Mδ − Mδ T (ξδ )]T (ξ−δ )Mδ−1 + I and therefore its boundedness can be proved as for the operators C1 and C2 considered above. We are left with A3 . Let D = diag (dn )∞ n=0 denote the operator Mγ−1 Mδ−1 Mγ+δ − cI. Since dn = 1 −
∞ 1− j=n
γδ (γ + j)(δ + j)
−1 (Lemma 7.58),
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7 Finite Section Method
it is easily seen that dn = O(1/n) as n → ∞. Because A3 = c−1 Mγ+δ T (ξδ )DM−γ A2 , it suﬃces to show that Mγ+δ T (ξδ )DM−γ is bounded. We shall prove that this operator is even HilbertSchmidt. Indeed, for the jk entry ajk (j ≥ k) we have + + + (γ+δ) δ (−γ) + ajk = ++µj dk µk ++ j−k ≤ c(j + 1)Re γ+Re δ (j − k + 1)−1−Re δ (k + 1)−1−Re γ ≤ c(j + 1)Re γ−1 (k + 1)−1−Re γ
∞ and thus j,k=0 ajk 2 < ∞. This settles the proof for the case Re δ > 0. If Re δ < 0, write A = (A1 A2 + A3 + cI)A4 with A1 = cMγ+δ T (ξδ )M−γ , −1 −1 −1 A2 = M−γ (Mγ+δ T (ξ−δ ) − T (ξ−δ )Mγ+δ ), −1 A3 = Mγ+δ T (ξδ )[Mγ−1 Mδ−1 Mγ+δ − cI]Mγ+δ T (ξ−δ ),
A4 = T (ξδ )Mδ T (ξ−δ )Mδ−1 and then proceed in analogy to the case Re δ > 0. 7.61. Lemma. If α, γ, δ ∈ C and Re α < 1/2, then there is a c ∈ C \ {0} such that Mα T (ξα )(I − cMδ−1 Mδ+γ Mγ−1 )T (ξ−α )Mα−1 is HilbertSchmidt on 2 . Proof. The arguments we have used in the preceding proof to show that A3 is HilbertSchmidt also apply in the case at hand. 7.62. Proposition. Let β1 , . . . , βm ∈ C, suppose Re βk  <
1 , 2
k + + 1 + + Re βj + < + 2 j=1
∀ k ∈ {1, . . . , m},
(7.67)
and put Bm := T −1 (ϕβ1 ) . . . T −1 (ϕβm ). Then Cm := Mβ1 T (ξβ1 ) . . . Mβm T (ξβm )T (η−βm )M−βm . . . T (η−β1 )M−β1 is a bounded operator on 2 and there is a nonzero constant dm ∈ C such that Bm − dm Cm ∈ C∞ (2 ).
(7.68)
7.6 Operators from algL(H 2 ) T (P C)
365
Proof. Theorem 6.20 (with δ = −α and γ = α) gives T −1 (ϕα ) = Γα,−α Mα T (ξα )T (η−α )M−α .
(7.69)
On letting α = β1 we obtain all assertions for m = 1. If α := β1 + . . . + βm , 1 2 3 Zm Zm , where then Cm = Zm 1 Zm = Mβ1 T (ξβ1 ) . . . Mβm T (ξβm )T (ξ−α )Mα−1 , −1 2 Zm = Mα T (ξα )T (η−α )M−α = Γα,−α T −1 (ϕα ) ∈ L(2 ), 3 = Mα−1 T (ηα )T (η−βm )M−βm . . . T (η−β1 )M−β1 . Zm
Therefore, using Proposition 7.60, the boundedness of Cm can be proved by induction. We now prove (7.68) by induction. Suppose (7.68) is true for m − 1. Then −1 2 (ϕβm )Cm−1 = T −1 (ϕβm )(d−1 d−1 m−1 Bm − T m−1 Bm−1 − Cm−1 ) ∈ C∞ ( )
and it remains to show that T −1 (ϕβm )Cm−1 −c1 Cm ∈ C∞ (2 ) for some number c1 ∈ C \ {0}. For k = 1, . . . , m − 1, factorize Cm−1 as Cm−1 = Xk Yk , where 5k Xk = j=1 Mβj T (ξβj ). In the following ci always denotes a constant in C\{0}. Using (7.69) we get T −1 (ϕβm )Cm−1 = c2 Mβm T (ξβm )T (η−βm )M−βm Mβ1 T (ξβ1 )Y1 −1 = c2 [Mβm T (ξβm )T (η−βm )M−βm ][I − c3 M−β Mβ1 −βm Mβ−1 ]Cm−1 m 1 +c3 c2 Mβm T (ξβm )T (η−βm )Mβ1 −βm T (ξβ1 )Y1 . (7.70) The operator in the ﬁrst brackets is bounded by (7.69) and the operator in the second brackets is compact for some c3 due to Lemma 7.58. Since Cm−1 is known to be bounded, we are left with the second term on the right of (7.70). We have Mβm T (ξβm )T (η−βm )Mβ1 −βm T (ξβ1 )Y1 = c4 Mβm T (ξβm )Mβ1 T (ξβ1 )T (η−βm )M−βm Y1
(by (6.16))
= c4 Mβ1 T (ξβ1 )Mβm T (ξβm )T (η−βm )M−βm Y1 (by (7.62)) = c4 X1 Mβm T (ξβm )T (η−βm )M−βm Y1 = S1 + S2 , where −1 S1 := c4 X1 Mβm T (ξβm )T (η−βm )M−βm (I − c5 M−β Mβ2 −βm Mβ−1 )Y1 , m 2 S2 := c4 c5 X1 Mβm T (ξβm )T (ηβm )Mβ2 −βm T (ξβ2 )Y2 . −1 The operator c−1 4 c5 S2 equals
c6 X1 Mβm T (ξβm )Mβ2 T (ξβ2 )T (η−βm )M−βm Y2
(by (6.16))
= c6 X1 Mβ2 T (ξβ2 )Mβm T (ξβm )T (η−βm )M−βm Y2 (by (7.62)) −1 = c6 X2 Mβm T (ξβm )T (η−βm )M−βm (I − c7 M−β Mβ3 −βm Mβ−1 )Y2 m 3 +c6 c7 X2 Mβm T (ξβm )T (η−βm )Mβ3 −βm T (ξβ3 )Y3 .
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On repeating these computations we obtain that S2 =
m−1 k=3
Rk + R, where
Rk = c2k+7 Xk−1 Mβm T (ξβm )T (η−βm )M−βm −1 ×(I − c2k+8 M−β Mβk −βm Mβ−1 )Yk−1 , m k R = c2m+9 Xm−2 Mβm T (ξβm )T (η−βm )Mβm−1 −βm T (ξβm−1 )Ym−1 . Put α = β1 + . . . + βk . Then Rk (k = 3, . . . , m − 1) is a constant multiple of [Xk−1 T (ξ−α )Mα−1 ][Mα T (ξα )Mβm T (ξβm )T (η−βm )M−βm T (ξ−α )Mα−1 ] −1 ×[Mα T (ξα )(I − c2k+8 M−β Mβk −βm Mβ−1 )T (ξ−α )Mα−1 ][Mα T (ξα )Yk−1 ]. m m 1 and thus it is bounded, the boundThe operator in the ﬁrst brackets is Zk−1 edness of the operator in the fourth brackets can be proved similarly as the boundedness of Cm by representing it in the form Z 1 Z 2 Z 3 , and the operator in the third brackets becomes compact for a suitable c2k+8 by virtue of Lemma 7.61. The operator in the second brackets equals
Mβm T (ξβm )Mα T (ξα )T (η−βm )M−βm T (ξ−α )Mα−1 (by (7.62)) = c2m+10 Mβm T (ξβm )T (η−βm )Mα−βm T (ξα )T (ξ−α )Mα−1 (by (6.16)) −1 = c2m+10 [Mβm T (ξβm )T (η−βm )M−βm ][M−β Mα−βm Mα−1 ] m and hence it is bounded in view of (7.68) and Corollary 6.22. The same reasoning can be used to show that S1 is compact. We are left with R. Because c−1 2m+9 R is equal to c2m+11 Xm−2 Mβm T (ξβm )Mβm −1 T (ξβm−1 )T (η−βm )M−βm Ym−1
(by (6.16))
= c2m+11 Xm−2 Mβm−1 T (ξβm−1 )Mβm T (ξβm )T (η−βm )M−βm Ym−1 (by (7.62)) = c2m+11 Cm (recall that Cm−1 = Xm−1 Ym−1 ), it follows that T −1 (ϕβm )Cm−1 − c2m+11 Cm is compact, as desired.
βm be complex numbers and 7.63. Theorem (Roch/Verbitsky). Let β1 , . . . ,5 m suppose Re βj  < 1/2 for j = 1, . . . , m. Put A = j=1 T (ϕβj ). Then A ∈ Π{2 ; Pn } ⇐⇒ T
m + 1 + + + 2 ϕ 7 Re βj + < . βj ∈ GL( ) ⇐⇒ + 2 j=1 j=1
m
Proof. The second “⇐⇒” is an immediate consequence of Proposition 6.24. 5m must be invertible by virtue of If A ∈ Π{Pn }, then W(A) = T ϕ 7 β j j=1 Theorem 7.11 and Corollary 7.14. + m + Now suppose that Re βj  < 1/2 for all j and + j=1 Re βj + < 1/2. A little thought shows that there exists a permutation βj1 , . . . , βjm of the numbers β1 , . . . , βm such that Re βj1 + . . . + Re βjk  < 1/2 for k = 1, . . . , m. Because T (ϕα )T (ϕβ ) and T (ϕβ )T (ϕα ) only diﬀer by a compact operator we
7.6 Operators from algL(H 2 ) T (P C)
367
may in view of Corollary 7.17 a priori assume that (7.67) is satisﬁed. Hence, by Proposition 7.62, A−1 = dCm + K, where d ∈ C \ {0} and K ∈ C∞ (2 ). The operators Qn and Mα commute, and it is easy to verify that Qn T (ξα ) = Qn T (ξα )Qn ,
T (ηα )Qn = Qn T (ηα )Qn .
(7.71)
1 2 3 1 2 3 Q n Zm Q n Zm Qn , where Zm , Zm , Zm are as in Therefore, Qn Cm Qn = Qn Zm 1 the proof of Proposition 7.62. From Proposition 7.60 we infer that Zm and 3 2 i −1 Zm are in GL( ), and using (7.71) we see that Qn (Zm ) Qn is the inverse −1 i 2 Qn for all n ≥ 0 (i = 1, 3). Since Qn Zm Qn = Γα,−α Qn T −1 (ϕα )Qn , of Qn Zm 2 Qn is Corollary 7.16 and Theorem 7.37 (or Theorem 7.32) imply that Qn Zm invertible for all suﬃciently large n and that the norms of the inverses are uniformly bounded. Thus, Qn Cm Qn is invertible for all n large enough and the norms of the inverses are uniformly bounded. From the representation
Qn A−1 Qn = dQn Cm Qn (I + d−1 (Qn Cm Qn )−1 Qn KQn ) and Corollary 7.16 we ﬁnally deduce that A ∈ Π{Pn }. 7.64. Lemma. Let {Mτ2 }τ ∈T be the collection of localizing classes in SJπ (H 2 ) deﬁned in 7.53.
r 5s
r 5s (a) If A = j=1 k=1 T (ajk ), B = j=1 k=1 T (bjk ), where ajk and bjk are in L∞ , and ajk Xτ = bjk Xτ for all j, k and some τ ∈ T, then {Pn APn }πJ and {Pn BPn }πJ are Mτ2 equivalent from the left and the right. (b) If A ∈ algL(H 2 ) T (P C) and f ∈ C, then {Pn APn }πJ {Pn T (f )Pn }πJ = {Pn T (f )Pn }πJ {Pn APn }πJ . Proof. (a) We ﬁrst show that {Pn APn T (f )Pn }πJ = {Pn AT (f )Pn }πJ
∀ f ∈ C.
(7.72)
Since Pn AT (f )Pn − Pn APn T (f )Pn = Pn AQn T (f )Pn = Wn (Wn AV n+1 )H(f )Wn (by (7.11)) and Wn AV n+1 converges strongly to K(A) (by (7.24)) and H(f ) is compact, it follows that Pn AQn T (f )Pn = Wn KWn +Cn with K ∈ C∞ (H 2 ) and Cn → 0 as n → ∞. This proves (7.72). Further, since (A1 A2 − B1 B2 )T (f ) − (A1 − B1 )T (f )A2 − B1 (A2 − B2 )T (f ) is compact whenever A1 , A2 , B1 , B2 ∈ alg T (L∞ ) and f ∈ C, and because T (a) − T (b) T (f ) = T (a − b)f + K with K ∈ C∞ (H 2 ) if a, b ∈ L∞ and f ∈ QC, we deduce from (7.72) and the (trivial) fact that
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7 Finite Section Method
{Pn APn }πJ ≤
inf
K∈C∞ (H 2 )
A + K
that {Pn APn }πJ and {Pn BPn }πJ are Mτ2 equivalent from the left (recall the proof of Theorem 2.96). Their right Mτ2 equivalence can be shown analogously. (b) Since AT (f ) − T (f )A is compact, this follows from (7.72). 7.65. Lemma. Let R be a commutative ring with identity element e and let ajk (j = 1, . . . , r; k = 1, . . . , s) be given elements of R.
Then 5 there exists a matrix G ∈ RN ×N (N = r(s + 1) + 1) such that det G = rj=1 sk=1 ajk and all entries of G belong to the set {0, e, a11 , a12 , . . . , ars }. Proof. Without loss of generality assume s is odd. Deﬁne the matrices G0 , G1 , . . . , Gr by ⎤ ⎡ e 0 0 ... 0 ⎢ 0 0 0 ... 0 ⎥ ⎥ ⎢ ⎢ ... ... ... ... ... ⎥ ⎥ ⎢ Gn−1 ⎢ ... ... ... ... ... ⎥ ⎥ ⎢ ⎢ ... ... ... ... ... ⎥ ⎥ ⎢ 0 0 0 ... 0 ⎥ G0 = [0]. Gn = ⎢ ⎥, ⎢ ⎢ 0 0 . . . 0 e an1 0 . . . 0 ⎥ ⎥ ⎢ ⎢ 0 0 . . . 0 0 e an2 . . . 0 ⎥ ⎥ ⎢ ⎢ ... ... ... ... ... ... ... ... ... ⎥ ⎥ ⎢ ⎣ 0 0 . . . 0 0 0 0 . . . anr ⎦ e 0 ... 0 0 0 0 ... e It is easy to see that the dimension of Gn is n(s + 1) + 1 and that det Hn = 1, where Hn is the matrix resulting from Gn by cancelling the ﬁrst row and the ﬁrst column. Hence, by Laplace’s theorem, det Gn = det Gn−1 + (−1)(n+1)(s+1) an1 . . . ans det Hn−1 = det Gn−1 + an1 . . . ans , and thus G = Gr has the desired property.
7.66.
Lemma. 5s Let ajk ∈ P C0 (j = 1, . . . , r; k = 1, . . . , s) and suppose r A = j=1 k=1 T (ajk ) is Fredholm on H 2 . Then there exist bi ∈ P C0 (i = 5m 1, . . . , m) such that A − i=1 T (bi ) is compact on H 2 . Proof. Since alg T π (P C) is commutative, the previous lemma can be applied to deduce that there is a g ∈ (P C0 )N ×N such that Aπ = det T π (g). The2 and thus g ∈ GL∞ orem 1.14(c) implies that T (g) is Fredholm on HN N ×N . Therefore, g = ϕbψ, where ϕ and ψ are in GCN ×N and b ∈ (P C0 )N ×N ∩ GL∞ N ×N is an uppertriangular matrix function (recall 5.48(b)). It follows that T π (g) = T π (ϕ)T π (b)T π (ψ), whence det T π (g) = det T π (ϕ) det T π (ψ) det T π (b).
7.6 Operators from algL(H 2 ) T (P C)
369
Let b1 . . . bN denote the diagonal entries of b and put b0 = det ϕ det ψ. We then have Aπ = det T π (g) = T π (b0 )T π (b1 ) . . . T π (bN ), 5N from which we infer that A − i=0 T (bi ) is compact. 7.67. Deﬁnition. Let A ∈ algL(H 2 ) T (P C) and let a ∈ C(T × [0, 1]) be the Gelfand transform of Aπ ∈ alg T π (P C) (recall Theorem 4.86). Fix τ = eiθ0 τ on the unit circle T. We deﬁne the function RA ∈ C by ⎧ θ0 − θ θ0 − θ ⎪ ⎪ a(τ, 1) + 1 − a(τ, 0), θ ∈ (θ0 − π, θ0 ), ⎨ π τ π RA (eiθ ) := θ − θ0 ⎪ ⎪ , θ ∈ (θ0 , θ0 + π). ⎩ a τ, π τ (eiθ ) moves from a(τ, 1) to a(τ, 0) Thus, if θ ranges from θ0 − π to θ0 , then RA τ (eiθ ) along a straight line segment, and if θ ranges from θ0 to θ0 + π, then RA joins a(τ, 0) and a(τ, 1) along the curve traced out by {a(τ, µ) : µ ∈ [0, 1]}. If τ does not vanish on T (Theorem 4.86 both A and W(A) are Fredholm, then RA τ is well deﬁned (see 2.41). and Corollary 7.14). In that case the integer ind RA
7.68. Theorem. Let ajk ∈ P C0 (j = 1, . . . , r; k = 1, . . . , s) and put A=
r s j=1 k=1
T (ajk ),
W(A) = T
r s
a. jk .
j=1 k=1
Then if K ∈ C∞ (H 2 ), A + K ∈ Π{H 2 ; Pn } ⇐⇒ A + K ∈ GL(H 2 ), W(A) ∈ GL(H 2 ), τ ind RA = 0 ∀ τ ∈ T. Proof. We ﬁrst prove the implication “⇐=”. By Theorem 7.11, it suﬃces to show that {Pn APn }πJ is invertible in SJπ . Lemma 7.66 ensures the exof an operator L0 in C∞ (H 2 ) such that istence 5mof b1 , . . . , bm in P C0 and iθ0 A = j=1 T (bj ) + L0 . Let τ = e ∈ T, deﬁne cj (j = 1, . . . , m) by θ − θ0 θ − θ0 bj (τ − 0), θ ∈ (θ0 , θ0 + 2π), cj (eiθ ) := 1 − bj (τ + 0) + 2π 2π 5m and put B := j=1 T (cj ). Note that cj ∈ P C0 is continuous on T \ {τ } and that cj (τ ± 0) = bj (τ ± 0). Lemma 7.64(a) implies that {Pn APn }πJ is Mτ2 equivalent to {Pn BPn }πJ from the left and the right. So, by Theorem 1.32(a), the assertion will follow as soon as we have shown that {Pn BPn }πJ is invertible. Because A is invertible, we have, by 4.68 or 5.45, m 0 1 (1 − µ)bj (τ − 0) + µbj (τ + 0) = 0 ∀ µ ∈ [0, 1], j=1
(7.73)
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7 Finite Section Method
and hence, by 5.45 and 5.48, B ∈ Φ(H 2 ) and Ind B = 0. Consequently, there is an L ∈ C∞ (H 2 ) such that B + L ∈ GL(H 2 ). The operator W(B) equals τ T (c)1 . . . c. m ) (Corollary 7.14). The assumption that ind RA be zero now implies that T (c)1 . . . c. m ) is invertible. Thus, B + L and W(B) are invertible. We shall show that actually B +L ∈ Π{Pn }, which gives the invertibility of {Pn BPn }πJ by Theorem 7.11. −1/2 < Re βj ≤ 1/2 Each cj can be written 5mas cj = ϕβj ,τ dj , where 5m and dj ∈ C. Put B0 := j=1 T (ϕβj ,τ ) and d := j=1 dj . Then B diﬀers from B0 T (d) only by a compact operator, and since B is Fredholm of index zero, we deduce that Re βj  < 1/2 for all j and that T (d) is invertible. The invertibility of ) W(B) = T (ϕ β1 ,τ . . . ϕ βm ,τ )T (d) + compact operator + m + ) is invertible whenever implies that + j=1 Re βj + < 1/2 (note that T (d) T (d) is so). We now infer from Theorem 7.63 that B0 ∈ Π{Pn }, whence {Pn B0 Pn }πJ ∈ GSJπ and thus {Pn B0 T (d)Pn }πJ ∈ GSJπ . The proof of the implication “⇐=” is complete. Our next objective is to show the implication “=⇒”. In view of Theoτ = 0 for all τ ∈ T. Let bj , cj rem 7.11 we need only to prove that ind RA 5m and B be as above, put c := j=1 cj and ﬁx τ ∈ T. Because {Pn BPn }πJ is Mτ2 equivalent to {Pn APn }πJ ∈ GSJπ and, for t = τ , {Pn BPn }πJ is Mt2 equivalent to {c(t)Pn }πJ ∈ GSJπ , it follows that {Pn BPn }πJ is invertible. Since Ind B = 0 (by (7.73)), there is an L ∈ C∞ (H 2 ) such that B + L is invertible. By Theorem 2.38, W(B) = T () c) is at least onesided invertible. Assume T () c) is leftinvertible (in the case of rightinvertibility the proof is analogous). c) Thus, {Pn (B + L)Pn }πJ is invertible and both B + L and W(B + L) = T () are leftinvertible. The same argument as in the proof of Lemma 7.35 shows that then B + L ∈ Π{Pn } and so Theorem 7.11 gives the invertibility of τ τ = Ind T () c), we get ind RA = 0. T () c) = W(B + L). Because ind RA Remark. Let ajk , A, W(A) be as in the theorem and suppose the functions τ =0 ajk are continuous on T minus a single point τ . Then the equality ind RA is implied by the invertibility of both A + K and W(A), and hence A + K ∈ Π{Pn } ⇐⇒ A + K ∈ GL(H 2 ), W(A) ∈ GL(H 2 ) in this case. 7.69. alg D. For τ ∈ T, let χτ denote the characteristic function of the arc (τ, τ eiπ ) and abbreviate T (χτ ) to D. Let alg D refer to the smallest closed subalgebra of L(H 2 ) containing I and D. Since D = D∗ , the algebra alg D is a commutative singlygenerated C ∗ algebra with identity. Because the spectrum of D is the segment [0, 1] (see 2.36), we deduce from 1.19 and 1.26(a) that alg D is isometrically starisomorphic to C[0, 1] and that the inverse of the Gelfand map is given by
7.6 Operators from algL(H 2 ) T (P C)
Γ −1 : C[0, 1] → alg D,
371
f → f (D).
Consequently, if f ∈ C[0, 1], then sp f (D) = f ([0, 1]).
(7.74)
) We have χ .τ (t) := χτ (1/t) = χσ (t), where σ = −1/τ . Denote T (χ−1/τ ) by D. k If p(x) = α0 + α1 x + . . . + αk x is a polynomial, then, by Corollary 7.14, χτ ) W(p(D)) = T p(. χτ ) = T α0 + (α1 + . . . + αk ). ) = p(0)(I − D) ) + p(1)D, ) = p(0)I + p(1) − p(0) D and since W : LT (H 2 ) → L(H 2 ) is continuous (see (7.22)), it follows that ) + f (1)D ) W(f (D)) = f (0)(I − D) for every f ∈ C[0, 1]. In particular, if f ∈ C[0, 1] then sp W(f (D)) = (1 − λ)f (0) + λf (1) : λ ∈ [0, 1] .
(7.75)
(7.76)
Note that alg Dπ , the smallest closed subalgebra of L(H 2 )/C∞ (H 2 ) containing Dπ := D+C∞ (H 2 ) and the identity, is isometrically starisomorphic to C[0, 1], too (2.36, 1.19, 1.26(a)). Thus, if f ∈ C[0, 1] then f C[0,1] = f (D) = f (Dπ ) = f (D)π .
(7.77)
7.70. Proposition. Let A = f (T (χτ )) ∈ alg D. Then τ = 0. A ∈ Π{H 2 ; Pn } ⇐⇒ A ∈ GL(H 2 ), W(A) ∈ GL(H 2 ), ind RA
Proof. Let pk be polynomials in x which converge uniformly to f on [0, 1]. Then if f (D) ∈ Π{Pn }, we have pk (D) ∈ Π{Pn } for all suﬃciently large k (Corollary 7.4). Theorem 7.68 implies that ind Rpτ k (D) = 0, whence ind Rfτ (D) = 0. This gives the implication “=⇒”. To prove the reverse implication, suppose A and W(A) are invertible and τ = 0. We show that {Pn APn }πG is invertible in D/G. Proposition 7.3 ind RA then yields the assertion. τ = 0, Let p be a polynomial such that p(0) = f (0), p(1) = f (1), ind Rp(D) and ! # 1 f − pC[0,1] < min min f (x), min (1 − λ)f (0) + λf (1) . (7.78) 2 x∈[0,1] λ∈[0,1] From (7.75) we obtain that W(p(D)) = W(f (D)) and hence W(p(D)) is invertible. Denote 1/f by g. We have, by (7.77), A−1 = g(D) = gC[0,1] = 1/ min f (x) , x∈[0,1]
372
7 Finite Section Method
so, again by (7.77), A−1 −1 = min f (x) > f − pC[0,1] = A − p(D) x∈[0,1]
and it follows that p(D) is invertible. Theorem 7.68 now gives that p(D) is in Π{H 2 ; Pn }. Put a := {Pn APn }πG = {Pn f (D)Pn }πG ,
b := {Pn p(D)Pn }πG .
Thus, b is invertible in D/G. We claim that a − b < b−1 −1 . This will show that a is also invertible in D/G, as desired. By virtue of (7.3) and (7.77) we have a − b = f (D) − p(D) = f − pC[0,1] .
(7.79)
Since D = D∗ , it follows that bb∗ = b∗ b and, thus, that (b−1 )(b−1 )∗ = (b−1 )∗ (b−1 ). Consequently, b−1 is equal to the spectral radius (b−1 ) of b−1 . Because λ∈ / sp b ⇐⇒ b − λ ∈ G(D/G) ⇐⇒ p(D) − λ ∈ Π{H 2 ; Pn } (by 7.3) ⇐⇒ p(D) − λ ∈ GL(H 2 ), W(p(D)) − λ ∈ GL(H 2 ) (by 7.68) we obtain from (7.74) and (7.76) that sp b = p(x) : x ∈ [0, 1] ∪ (1 − λ)p(0) + λp(1) : λ ∈ [0, 1] . The invertibility of b along with the spectral mapping theorem implies that sp b−1 = (sp b)−1 , hence ! # 1 1 −1 , max (b ) = max max , x∈[0,1] p(x) λ∈[0,1] (1 − λ)p(0) + λp(1) and thus, because b−1 −1 = 1/(b−1 ), ! # −1 −1 b = min min p(x), min (1 − λ)p(0) + λp(1) . x∈[0,1]
(7.80)
λ∈[0,1]
Since p(0) = f (0) and p(1) = f (1), we deduce from (7.78) and (7.79) that a − b < min (1 − λ)p(0) + λp(1).
(7.81)
λ∈[0,1]
By (7.78), we have f (x) − p(x) ≤ f (x) − p(x) < (1/2) min f (x), whence x∈[0,1]
f (x)−(1/2) min f (x) < p(x), thus (1/2) min f (x) < min p(x), and x∈[0,1]
x∈[0,1]
x∈[0,1]
so (7.78) and (7.79) give a − b <
1 min f (x) < min p(x). 2 x∈[0,1] x∈[0,1]
From (7.80)–(7.82) we get a − b < b−1 −1 .
(7.82)
7.6 Operators from algL(H 2 ) T (P C)
373
7.71. Lemma. Let A be a Banach algebra with identity and let M be a bounded localizing class in A. Suppose a, b, bn , an are in A, an → a and bn → b as n → ∞, and an is M equivalent from the left (right ) to bn . Then a is M equivalent from the left (right ) to b. Proof. Let c := sup f . We have, for f ∈ M , f ∈M
(a − b)f ≤ (a − an )f + (an − bn )f + (bn − b)f ≤ c a − an + b − bn + (an − bn )f , there is an n0 such that c(a − an0 + b − bn0 ) < ε/2 and then one can ﬁnd an f ∈ M such that (an0 − bn0 )f < ε/2. 7.72. Theorem. If A ∈ algL(H 2 ) T (P C), then τ A ∈ Π{H 2 ; Pn } ⇐⇒ A ∈ GL(H 2 ), W(A) ∈ GL(H 2 ), ind RA = 0 ∀ τ ∈ T.
Proof. The implication “=⇒” follows from Theorems 7.11 and 7.68 along with an approximation argument (recall the ﬁrst part of the proof of Proposition 7.70). So we are left with the implication “⇐=”. Suppose A and W(A) τ = 0 for all τ ∈ T. We show that {Pn APn }πJ is are invertible and ind RA π invertible in SJ .
rk 5sk k k There are Ak := i=1 j=1 T (aij ) (aij ∈ P C0 ) such that A − Ak → 0 as k → ∞. Fix τ ∈ T and let χτ be as in 7.69. Put (akij )τ
:=
akij (τ
− 0)(1 − χτ ) +
akij (τ
+ 0)χτ ,
Aτk
:=
rk sk
T [(akij )τ ].
i=1 j=1
Now write Ak − Al as
5
T (bij ) and apply Theorem 4.86 to get " " " " (Aτk )π − (Aτl )π = " T π (bτij )" + + + + = max [(1 − µ)bτij (t − 0) + µbτij (t + 0)]+ + (t,µ)∈T×[0,1] + + + + = max + [(1 − µ)bij (τ − 0) + µbij (τ + 0)]+ µ∈[0,1] " " " " T π (bij )" = Aπk − Aπl ≤ Ak − Al . ≤"
Hence, (Aτk )π − (Aτl )π → 0 as
k, l → ∞.
(7.83)
The operators Aτk are polynomials in D. Indeed, if we let pτk (x) := akij (τ − 0)(1 − x) + akij (τ + 0)x , then Aτk = pτk (D). So (7.83) together with (7.77) gives that pτk −pτl C[0,1] → 0 as k, l → ∞. Thus, there is an fτ ∈ C[0, 1] such that pτk − fτ C[0,1] → 0 as k → ∞.
374
7 Finite Section Method
From Theorem 4.86 we infer that fτ (x) = (Γ Aπ )(τ, x), and since A and τ = 0, we obtain from (7.74), (7.76) that fτ (D) W(A) are invertible and ind RA and W(fτ (D)) are invertible and that ind Rfττ (D) = 0. Hence fτ (D) ∈ Π{Pn } by Proposition 7.70 and thus {Pn fτ (D)Pn }πJ ∈ GSJπ due to Theorem 7.11. Let Mτ2 be the (obviously bounded) localizing class introduced in 7.53. By Lemma 7.64(a), {Pn Ak Pn }πJ is Mτ2 equivalent to {Pn pτk (D)Pn }πJ from the left and the right. Since {Pn (A − Ak )Pn }πJ ≤ A − Ak , " " " Pn fτ (D) − pτk (D) Pn π " ≤ fτ (D) − pτk (D) = fτ − pτk , J we deduce from Lemma 7.71 that {Pn APn }πJ and {Pn fτ (D)Pn }πJ are Mτ2 equivalent from the left and the right. So Lemma 7.64(b) and Theorem 1.32(a) complete the proof. Remark. S. Roch [422] extended Theorem 7.72 to the matrix case and the space p .
7.7 FisherHartwig Symbols: H 2 () Theory 5m 7.73. Theorem. Let a = c j=1 ωαj ,tj , where c ∈ L∞ , t1 , . . . , tm are pairwise distinct points on T, ωαj ,tj is deﬁned as in 5.60, and Re αj  < min{1/p, 1/q} (1 < p < ∞, 1/p+1/q = 1). Let τ1 , . . . , τr be any pairwise distinct points on T such that the sets {t1 , . . . , tm } and {τ1 , . . . , τr } are disjoint, and let λ1 , . . . , λr be real numbers with −1/p < λj < 1/q for all j. Put λ (t) =
r
t − τj λj ,
j=1 r
ϕ−λ (t) =
j=1
α (t) =
m
t − tj Re αj ,
(7.84)
j=1
ϕ−λj ,τj (t),
ϕα (t) =
m
ϕαj ,tj (t)
(t ∈ T).
j=1
p Then T (ϕ−λ ϕα ) and T (ϕ−λ ϕ−1 α ) are in GL(H ) and we have
T (a) ∈ Π{H p (λ α ), H p (λ −1 α ); Pn } −1 −1 p ⇐⇒ T (ϕ−λ ϕα )T (ϕ−λ c)T (ϕ−λ ϕ−1 α ) ∈ Π{H ; Pn }. p results from Proof. The invertibility of T (ϕ−λ ϕα ) and T (ϕ−λ ϕ−1 α ) on H Corollary 5.33, Lemma 5.36, and Corollary 2.40. The proof of the implication (i) =⇒ (ii) of Proposition 7.3 shows that an operator A ∈ L(X, Y ) is invertible whenever A ∈ Π{X, Y ; Pn }. Thus, by Corollary 7.16, (7.85) T (a) ∈ Π{H p (λ α ), H p (λ −1 α ); Pn }
if and only if T (a) : H p (λ α ) → H p (λ −1 α ) is invertible, if
7.7 FisherHartwig Symbols: H 2 () Theory
375
p Qn T −1 (a)Qn : Qn H p (λ −1 α ) → Qn H (λ α )
is invertible for all suﬃciently large n, and if the norms of the inverses are uniformly bounded. Put ξα =
m j=1
ξαj ,tj ,
ηα =
m
ηαj ,tj ,
j=1
ξλ =
r
ξλj ,τj ,
ηλ =
j=1
r
ηλj ,τj .
j=1
Then a = ξα ηα c, ϕα = ξα−1 ηα , ϕ−λ = ξλ ηλ−1 . Hence, Qn T −1 (a)Qn = Qn T −1 (ξα ηα c)Qn = Qn T (ηα−1 )T −1 (c)T (ξα−1 )Qn −1 = Qn T (ξα−1 ϕ−1 (c)T (ϕα ηα−1 )Qn α )T −1 = Qn T (ξα−1 )Qn T (ϕ−1 (c)T (ϕα )Qn T (ηα−1 )Qn α )T
(recall (7.71)). From Lemma 5.61 and the fact that Qn is uniformly bounded on the spaces H p (λ α ), H p (λ ), H p (λ −1 α ) we obtain that the operators Qn T (ξα−1 )Qn : Qn H p (λ ) → Qn H p (λ −1 α ), Qn T (ηα−1 )Qn : Qn H p (λ α ) → Qn H p (λ ) are bounded and invertible for all n ≥ 0 and that their norms as well as the norms of their inverses are uniformly bounded. Thus, (7.85) holds if and only p if T −1 (ϕα )T (c)T −1 (ϕ−1 α ) ∈ GL(H (λ )), if the operators −1 Qn T (ϕ−1 (c)T (ϕα )Qn : Qn H p (λ ) → Qn H p (λ ) α )T
are invertible for all suﬃciently large n, and if the norms of their inverses are uniformly bounded. Since, again by Lemma 5.61 and the uniform boundedness of Qn on H p (λ ) and H p , the operators Qn T (ξλ )Qn : Qn H p (λ ) → Qn H p ,
Qn T (ηλ−1 )Qn : Qn H p → Qn H p (λ )
are bounded and invertible for all n ≥ 0 and their norms as well as the norms of their inverses are uniformly bounded, we conclude that (7.85) is true if and only if −1 p p T (ηλ )T −1 (ϕα )T (c)T −1 (ϕ−1 α )T (ξλ ) : H → H is invertible, if the operators −1 Qn T (ξλ )Qn T (ϕ−1 (c)T (ϕα )Qn T (ηλ−1 )Qn α )T −1 −1 = Qn T (ξλ )T (ϕα )T (c)T (ϕα )T (ηλ−1 )Qn
are invertible on Qn H p for all suﬃciently large n, and if the norms of their inverses are uniformly bounded. But −1 −1 (ϕα ηλ−1 )T (c)T −1 (ξλ ϕ−1 T (ηλ )T −1 (ϕα )T (c)T −1 (ϕ−1 α )T (ξλ ) = T α ) −1 −1 −1 −1 = T (ϕα ϕ−λ ξλ )T (c)T (ϕ−λ ηλ ϕα )
= T −1 (ϕα ϕ−λ )T (ξλ )T (c)T (ηλ−1 )T −1 (ϕ−λ ϕ−1 α ) −1 −1 −1 −1 = T (ϕα ϕ−λ )T (ξλ cηλ )T (ϕ−λ ϕα ) = T −1 (ϕα ϕ−λ )T (ϕ−λ c)T −1 (ϕ−λ ϕ−1 α ).
376
7 Finite Section Method
So Corollary 7.16 completes the proof. 5r 7.74. Corollary. Let (t) = j=1 t − τj λj be a Khvedelidze weight on Lp 5r (−1/p < λj < 1/q) and put ψ := j=1 ϕ−λj ,τj . Then T (ψ) ∈ GL(H p ), and if a ∈ L∞ , then T (a) ∈ Π{H p (); Pn } ⇐⇒ T −1 (ψ)T (aψ)T −1 (ψ) ∈ Π{H p ; Pn }. Proof. This is the preceding theorem with αj = 0. 5r 7.75. Corollary. Let (t) = j=1 t − τj λj be a Khvedelidze weight on L2 (−1/2 < λj < 1/2) and let a ∈ P C. Then T (a) ∈ Π{H 2 (); Pn } ⇐⇒ T (a) ∈ GL(H 2 (γ )) 5r where γ (t) := j=1 t − τj γλj .
∀ γ ∈ [−1, 1],
Proof. Assume T (a) ∈ Π{H 2 (); Pn }. We denote by Rn the projections Rn : L2 () → L2 (),
∞ k=−∞
fk χk →
n
f k χk .
k=−n
It is clear that T (a) is in Π{H 2 (); Pn } if and only if P aP + Q belongs to Π{L2 (); Rn } (recall 2.13 and notice that we abbreviate M (a) to a). Let Cn denote the (antilinear) operator on the space H 2 () deﬁned by (Cn f )(t) = tn (Pn f )(t). We have Cn f ≤ M f for all f ∈ H 2 (), where M is some constant independent of n. Clearly, Cn2 = Pn and Tn (a) = Cn Tn (a)Cn . Hence, if Tn (a) is invertible, then Tn (a) is also invertible, and because Tn−1 (a)Pn = Cn Tn−1 (a)Cn , we deduce that T (a) is in Π{H 2 (); Pn } and thus that P aP + Q belongs to Π{L2 (); Rn }. Since the operator Rn (P aP + Q)Rn ∈ L(L2 ()) is the adjoint of the operator Rn (P aP + Q)Rn ∈ L(L2 (−1 )), we obtain that both " −1 " sup " Rn (P aP + Q)Rn Rn " 2 n≥n0
and
L(L ())
" −1 " Rn "L(L2 (−1 )) sup " Rn (P aP + Q)Rn
n≥n0
are ﬁnite. So the SteinWeiss interpolation theorem implies that " −1 " sup " Rn (P aP + Q)Rn Rn "L(L2 (γ )) < ∞ ∀ γ ∈ [−1, 1], n≥n0
from which one easily sees that P aP + Q ∈ Π{L2 (γ ); Rn }, whence T (a) is in GL(H 2 (γ )). We now prove the implication “⇐=”. By virtue of Corollary 7.74, it suﬃces to show that5 A := T −1 (ψ)T (aψ)T −1 (ψ) (∈ alg T (P C)) belongs to Π{H 2 ; Pn }, where ψ := ϕ−λj ,τj . Let ξλ and ηλ be as in the proof of Theorem 7.73. Then
7.7 FisherHartwig Symbols: H 2 () Theory
T (aψ) = T (ξλ )T (a)T (ηλ−1 ),
377
T (aψ −1 ) = T (ξλ−1 )T (a)T (ηλ )
and therefore, by Lemma 5.61, the invertibility of T (a) on H 2 () and H 2 (−1 ) implies the invertibility of T (aψ) and T (aψ −1 ) on H 2 , from which we deduce that A and W(A) = T () aψ)−1 ) are invertible on H 2 . Thus the assertion will τ = 0 for all τ ∈ T. follow from Theorem 7.72 once we have shown that ind RA Let b ∈ C(T × [0, 1]) denote the Gelfand transform of Aπ ∈ alg T π (P C). If τ ∈ / {τ1 , . . . , τr }, then b(τ, µ) = [(1 − µ)a(τ − 0) + µa(τ + 0)]/ψ(τ ), τ see that ind RA = 0. So let τ = τj . and since T (aψ −1 ) is invertible on H 2 , we5 Abbreviate ϕ−λj ,τj to ϕj and put δ := 1/ k =j ϕ−λk ,τk (τj ). We have
b(τj , µ) = δ
(1 − µ)a(τj − 0)ϕj (τj − 0) + µa(τj + 0)ϕj (τj + 0) . [(1 − µ)ϕj (τj − 0) + µϕj (τj + 0)]2
(7.86)
Because T (a) is invertible on H 2 (λ ) for all γ ∈ [−1, 1], 5.47 and 5.48 imply the existence of a function aj ∈ P C0 such that aj (τj ± 0) = a(τj ± 0), aj is continuous on T \ {τj }, and T (aj ) is invertible on H 2 (γ ) for all γ ∈ [−1, 1]. τ τ Now (7.86) shows that RAj = δRAjj , where Aj = T −1 (ϕj )T (aj ϕj )T −1 (ϕj ). 2 The operators Aj and T (aj ϕ−1 j ) are invertible on H , and hence the curves Γ Aπj and Γ T π (aj ϕ−1 j ) have index zero (see 5.48). But these two curves coinτ cide for t = τj , and consequently, the curve RAjj must also have index zero. 5m 7.76. Corollary. Let a = b j=1 ωαj ,tj ϕβj ,tj , where t1 , . . . , tm are pairwise distinct points on T, Re αj  < 1/2 and Re βj  < 1/2 for all j, and suppose b ∈ C, b(t) = 0 on T and ind b = 0. Then
where (t) :=
T (a) ∈ Π{H 2 (), H 2 (−1 ); Pn },
5m j=1
t − tj Re αj .
5m Proof. Apply Theorem 7.73 with c = b j=1 ϕβj ,tj and {τ1 , . . . , τr } = ∅. The operators m m −1 −1 −1 ϕα = ϕαj ,tj , ϕβ = ϕβj ,tj A = T (ϕα )T (bϕβ )T (ϕα ) j=1
j=1
and W(A) = T ()b. ϕβ ) are invertible on H 2 . Hence, by virtue of Theorem 7.72, τ = 0 for all τ ∈ T. If τ ∈ / {t1 , . . . , tm }, then it remains to prove that ind RA τ = 0. Let τ = tj . Then clearly ind RA 0 1 (Γ Aπ )(τ, µ) = b(τ ) ϕβk ,tk (τ ) × (1 − µ)ϕβj ,tj (tj − 0) + µϕβj ,tj (tj + 0) k =j
0 1−1 × (1 − µ)ϕαj ,tj (tj − 0) + µϕαj ,tj (tj + 0) 0 1−1 −1 × (1 − µ)ϕ−1 αj ,tj (tj − 0) + µϕαj ,tj (tj + 0)
378
7 Finite Section Method t
τ and thus, RA is a constant multiple of RAjj , where
Aj = T −1 (ϕαj ,tj )T (ϕβj ,tj )T −1 (ϕ−1 αj ,tj ). ϕβj ,tj ) are invertible on H 2 , it is clear that Because Aj and W(Aj ) = T ( tj τ ind RAj = 0 (also see the remark after Theorem 7.68), whence ind RA = 0.
7.8 FisherHartwig Symbols: pµ Theory We now consider the ﬁnite section method for T (ξδ ηγ b) on weighted p spaces for the case that b is “regular,” Re (γ + δ) ≥ 0,
Re γ > −1,
Re δ > −1
(recall what was said before 6.41). More precisely, we shall construct pairs of spaces pr and ps such that (i) the equation T (ξδ ηγ b)x = y has a unique solution x ∈ ps for each y ∈ pr ; (ii) there exists a constant c such that xps ≤ cypr for all y ∈ pr ; (iii) Tn (ξδ ηγ b) is invertible for all suﬃciently large n, and for each y ∈ pr , Tn−1 (ξδ ηγ b)Pn y − xps → 0 as
n → ∞.
It is the nature of the matter to distinguish three cases (recall Theorems 6.48 and 6.49). (a) Re γ ≥ 0 and Re δ ≥ 0. (b) Re δ ≥ 0 and −1 < Re γ < 0; putting β = −γ and ν = γ + δ we have ξδ ηγ = ξν ϕ−β with Re ν ≥ 0 and 0 < Re β < 1. (c) Re γ ≥ 0 and −1 < Re δ < 0; then ξδ ηγ = ϕβ ην with β = −δ, ν = γ + δ and thus, 0 < Re β < 1, Re ν ≥ 0. Throughout the following let 1 < p < ∞, 1/p + 1/q = 1. 7.77. Proposition. Let γ, δ ∈ C \ {−1, −2, . . .} and suppose Re (γ + δ) > 1. Then Tn (ξδ ηγ ) is invertible for all n ≥ 0. Proof. Multiply equality (6.16) from the left and from the right by Pn and take into account that Pn T (ηγ ) = Pn T (ηγ )Pn ,
T (ξδ )Pn = Pn T (ξδ )Pn .
(7.87)
What results is Tn (ηγ )Pn Mγ+δ Pn Tn (ξδ ) = Γγ,δ Pn Mδ Pn Tn (ξδ ηγ )Pn Mγ Pn , and this gives the assertion at once.
(7.88)
7.8 FisherHartwig Symbols: pµ Theory
379
If Re γ ≥ 0 and Re δ ≥ 0, then T (ξδ ηγ ) is bounded and invertible as operator from Dµp (γ) onto Rµp (δ) for every µ such that −1/p < µ < 1/q, and its inverse T −1 (ξδ ηγ ) is in L(pµ+Re δ , pµ−Re γ ) (Theorem 6.48). 7.78. Proposition. Let Re γ ≥ 0, Re δ ≥ 0, −1/p < µ < 1/q. Then Tn−1 (ξδ ηγ )Pn y − T −1 (ξδ ηγ )ypµ−Re γ → 0
(n → ∞)
(7.89)
for each y ∈ pµ+Re δ , and if λ > 0 and µ + λ < 1/q, then as n → ∞, Tn−1 (ξδ ηγ )Pn − T −1 (ξδ ηγ )L(pµ+λ+Re δ ,pµ−Re γ ) = O
1 . nλ
(7.90)
Proof. From (7.87), (7.88) we obtain that −1 T −1 (ξδ ηγ )Pn = Γγ,δ Mγ T (ξ−δ )Mγ+δ Pn T (η−γ )Mδ
and from Theorem 6.20 we know that −1 T −1 (ξδ ηγ ) = Γγ,δ Mγ T (ξ−δ )Mγ+δ T (η−γ )Mδ .
Hence, Tn−1 (ξδ ηγ )Pn y − T −1 (ξδ ηγ )yp,µ−Re γ −1 = Γγ,δ  Mγ T (ξ−δ )Mγ+δ Qn T (η−γ )Mδ yp,µ−Re γ −1 ≤ cT (ξ−δ )Mγ+δ Qn T (η−γ )Mδ yp,µ
(by Corollary 6.22)
−1 cMγ+δ Qn T (η−γ )Mδ yp,µ+Re δ
≤ ≤ cQn T (η−γ )Mδ yp,µ−Re γ
(by Proposition 6.45) (by Corollary 6.22).
Here and throughout the following c denotes a constant independent of n but not necessarily the same at each occurrence. Again by Corollary 6.22 and Proposition 6.45, T (η−γ )Mδ yp,µ−Re γ ≤ cMδ yp,µ ≤ cyp,µ+Re δ , hence T (η−γ )Mδ y ∈ pµ−Re γ , and since Qn converges strongly to zero on pµ−Re γ , we get (7.89). To obtain (7.90) note that Qn T (η−γ )Mδ yp,µ−Re γ ≤ cn−γ Qn T (η−γ )Mδ yp,µ−Re γ+λ ≤ cn−γ Mδ yp,µ+λ (we use that µ + λ < 1/q) ≤ cn−λ yp,µ+λ+Re γ . Now consider T (ξν ϕ−β ) = T (ξν )T (ϕ−β ) with Re ν ≥ 0, 0 < Re β < 1. First let ν = 0. Then Proposition 6.24 shows that T (ϕ−β ) ∈ GL(pµ ) whenever Re β − 1/p < µ < 1/q.
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7 Finite Section Method
7.79. Lemma. Let A ∈ GL(pµ ) and suppose An := Pn APn Im Pn is invertible. Then, for y ∈ pµ , −1 −1 p p A−1 ypµ ≤ 1 + A−1 ypµ . n Pn y − A n Pn L(µ ) AL(µ ) Qn A Proof. −1 −1 y ≤ A−1 y + Qn A−1 y A−1 n Pn y − A n Pn y − Pn A −1 ≤ An Pn Pn y − Pn APn A−1 y + Qn A−1 y −1 ≤ A−1 y + Qn A−1 y. n Pn Pn AQn A
7.80. Proposition. If 0 < Re β < 1/q and Re β − 1/p < µ < 1/q, then Tn−1 (ϕ−β )Pn y − T −1 (ϕ−β )ypµ → 0
(n → ∞)
(7.91)
for each y ∈ pµ , and if λ > 0 and µ + λ < Re β + 1/q, then, as n → ∞, Tn−1 (ϕ−β )Pn − T −1 (ϕ−β )L(pµ+λ ,pµ ) = O
1 . nλ
(7.92)
Proof. That (7.91) holds is a consequence of Theorem 7.37. From Lemma 7.79, (7.91), and Proposition 6.23 we infer that Tn−1 (ϕ−β )Pn y − T −1 (ϕ−β )yp,µ ≤ cQn T −1 (ϕ−β )yp,µ ≤ cn−λ Qn T −1 (ϕ−β )yp,µ+λ for all n large enough. Using (6.16), Proposition 6.23, and Corollary 6.22 we obtain Qn T −1 (ϕ−β )yp,µ+λ = Qn Γ−β,β M−β T (ϕβ )Mβ yp,µ+λ ≤ cT (ϕβ )Mβ yp,µ+λ−Re β ≤ cMβ yp,µ+λ−Re β ≤ cyp,µ+λ , which gives (7.92).
Now let Re ν ≥ 0. Then, by virtue of Propositions 6.24 and 6.47, the operator T (ξν ϕ−β ) = T (ξν )T (ϕ−β ) is bounded and invertible from pµ onto Rµp (ν) and its inverse T −1 (ξν ϕ−β ) is in L(pµ+Re ν , pµ ). 7.81. Proposition. Let Re ν ≥ 0 and 0 < Re β < 1. If Re β < 1/q and Re β − 1/p < µ < 1/q, then Tn−1 (ξν ϕ−β )Pn y − T −1 (ξν ϕ−β )ypµ → 0
(n → ∞)
(7.93)
for each y ∈ pµ+Re ν , and if λ > 0 and µ + λ < Re β + 1/q, then, as n → ∞, Tn−1 (ξν ϕ−β )Pn − T −1 (ξν ϕ−β )L(pµ+λ+Re ν ,pµ ) = O
1 . nλ
(7.94)
7.8 FisherHartwig Symbols: pµ Theory
381
Proof. We have Pn − Pn T −1 (ξν ϕ−β )Pn T (ξν ϕ−β )Pn = Pn T −1 (ξν ϕ−β )Qn T (ξν ϕ−β )Pn −1 = Pn T −1 (ξν ϕ−β )Mν+β T (η−β )Qn T (ηβ )Mν+β Qn T (ξν ϕ−β )Pn −1 = Pn T −1 (ξν ϕ−β )Mν+β T (η−β )Qn T (ηβ )Mν+β T (ξν ϕ−β )Pn −1 −Pn T −1 (ξν ϕ−β )Mν+β T (η−β )Qn T (ηβ )Mν+β Pn T (ξν ϕ−β )Pn ,
and since, by (6.16) and (7.87), Qn T (ηβ )Mν+β T (ξν ϕ−β )Pn −1 −1 = cQn T (ηβ )Mν+β Mν+β T (η−β )Mν T (ξν+β )M−β Pn −1 = cMν Qn T (ξν+β )Pn M−β = 0,
we arrive at the formula Tn−1 (ξν ϕ−β )Pn − Pn T −1 (ξν ϕ−β )Pn −1 T (η−β )Qn T (ηβ )Mν+β Pn . = −Pn T −1 (ξν ϕ−β )Mν+β
(7.95)
−1 Here T −1 (ξν ϕ−β )Mν+β ∈ L(pµ−Re β , pµ ) and the term in braces equals
0 −1 10 10 1 −1 T (ϕ−β )M−β M−β T (ξ−β )Qn T (ηβ )Mβ Pn Mβ−1 Mν+β .
(7.96)
The operator in the ﬁrst brackets is in L(pµ , pµ−Re β ), the operator in the third brackets belongs to L(pµ+Re ν , pµ ), and the operator in the middle brackets is Pn M−β T (ξ−β )Qn T (ηβ )Mβ Pn + Qn M−β T (ξ−β )Qn T (ηβ )Mβ Pn = Pn T −1 (ϕ−β )Pn − Tn−1 (ϕ−β )Pn + Qn T −1 (ϕ−β )Pn = T −1 (ϕ−β )Pn − Tn−1 (ϕ−β )Pn and thus, by (7.91), it converges strongly to zero on pµ . We now deduce from (7.95) that Tn−1 (ξν ϕ−β )Pn y − Pn T −1 (ξν ϕ−β )Pn yp,µ → 0 (n → ∞) for each y ∈ pµ+Re ν , and since Tn−1 (ξν ϕ−β )Pn y − T −1 (ξν ϕ−β )yp,µ ≤ Tn−1 (ξν ϕ−β )Pn y − Pn T −1 (ξν ϕ−β )Pn yp,µ +Pn T −1 (ξν ϕ−β )Qn yp,µ + Qn T −1 (ξν ϕ−β )yp,µ ,
(7.97)
and Qn converges strongly to zero on pµ+Re ν and pµ , we obtain (7.93). Due to (7.92) the L(pµ+λ , pµ ) norm of the operator in the middle brackets of (7.96) is O(1/nλ ), and because the operator in the third brackets of (7.96) belongs to L(pµ+λ+Re ν , pµ+λ ), we get from (7.95) that
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7 Finite Section Method
Tn−1 (ξν ϕ−β )Pn − Pn T −1 (ξν ϕ−β )Pn L(pµ+λ+Re ν ,pµ ) = O
1 nλ
as n → ∞. Finally, since Qn T −1 (ξν ϕ−β )yp,µ ≤ cn−λ Qn T −1 (ξν ϕ−β )yp,µ+λ ≤ cn−λ T −1 (ξν ϕ−β )yp,µ+λ ≤ cn−λ yp,µ+λ+Re ν and T −1 (ξν ϕ−β )Qn yp,µ ≤ cQn yp,µ+Re ν ≤ cn−λ yp,µ+λ+Re ν , inequality (7.97) yields estimate (7.94). Out next objective is to extend the results hitherto obtained for the “pure singularity” ξδ ηγ to symbols which still involve a “regular part” b. As usual, this will be done by applying a perturbation argument (see, e.g., Pr¨ ossdorf, Silbermann [407]). The following theorem is just what is needed in our situation. 7.82. Theorem. Suppose (a) X, Y, Z, U are Banach spaces and Pn (n = 0, 1, 2, . . .) are projections deﬁned and bounded on each of the spaces X, Y, Z, U ; (b) Z ⊂ Y and X ⊂ U , the embeddings being continuous; (c) A ∈ L(X, Y ) is (boundedly) invertible; (d) the operators An := Pn APn ∈ L(Pn X, Pn Y ) are invertible for all −1 zU → 0 as n → ∞; suﬃciently large n and, for each z ∈ Z, A−1 n Pn z − A (e) T ∈ C∞ (U, Z); (f) A + T ∈ L(X, Y ) is (boundedly) invertible. Then (g) the operators Pn (A + T )Pn ∈ L(Pn X, Pn Y ) are invertible for all n large enough; (h) (Pn (A + T )Pn )−1 Pn z − (A + T )−1 zU → 0 (n → ∞) ∀ z ∈ Z; (k) there is a constant c independent of n and z such that (Pn (A + T )Pn )−1 Pn z − (A + T )−1 zU −1 −1 ≤ cA−1 zU + cA−1 T L(U ) zZ . n Pn z − A n Pn T − A Proof. Obviously, I + A−1 T ∈ GL(X): the inverse is (A + T )−1 A. We claim that I +A−1 T is also in GL(U ). Since A−1 T ∈ C∞ (U ), it follows that I +A−1 T is Fredholm on U and has index zero there. Thus we must show that it has a trivial kernel. Let (I + A−1 T )u = 0 for some u ∈ U . Then u = −A−1 T u,
7.8 FisherHartwig Symbols: pµ Theory
383
hence u ∈ X, hence Au = −T u, hence (A + T )u = 0, and this gives u = 0, as desired. −1 T From (d) and (e) we conclude that I + A−1 n Pn T converges to I + A uniformly on U . Therefore, by what has been proved in the preceding paragraph, I + A−1 n Pn T is in GL(U ) for all suﬃciently large n, say n ≥ n0 . Let Bn ∈ L(U ) denote the inverse: Bn + Bn A−1 n Pn T = I,
Bn + A−1 n Pn T Bn = I.
(7.98)
The second equality in (7.98) implies that Bn ∈ L(X). It also implies that Pn Bn Pn = Bn Pn . Thus, for y ∈ Y , −1 −1 Pn (A + T )Pn Bn A−1 n Pn y = An Bn An Pn y + Pn T Bn An Pn y −1 = An Bn A−1 n Pn y + (An − An Bn )An Pn y = An A−1 n Pn y = Pn y
and, for x ∈ X, −1 −1 Bn A−1 n Pn Pn (A + T )Pn x = Bn An Pn An x + Bn An Pn T Pn x = Bn Pn x + (I − Bn )Pn x = Pn x.
It results that Pn (A + T )Pn ∈ L(Pn X, Pn Y ) is invertible for all n ≥ n0 and that −1 −1 −1 An Pn ∈ L(Pn Y, Pn X) Bn A−1 n Pn = (I + An Pn T ) is the inverse. Now let z ∈ Z and n ≥ n0 . Since (I + A−1 T )−1 A−1 ∈ L(Y, X) is the inverse of A + T ∈ L(X, Y ), we get (Pn (A + T )Pn )−1 Pn z − (A + T )−1 Pn zU −1 ≤ ((I + A−1 − (I + A−1 T )−1 )A−1 n Pn T ) n Pn zU −1 −1 −1 +(I + A T ) (A−1 P z − A z) n U n −1 −1 −1 −1 ≤ (I + A−1 P T ) − (I + A T ) n L(U ) An Pn L(Z,U ) zZ n −1 +(I + A−1 T )−1 L(U ) A−1 zU . n Pn z − A −1 But (I + A−1 − (I + A−1 T )−1 L(U ) is not larger than n Pn T ) −1 T L(U ) (I + A−1 T )−1 2L(U ) A−1 n Pn T − A
−1 T 1 − (I + A−1 T )−1 L(U ) A−1 n Pn T − A L(U ) −1 and since A−1 T L(U ) → 0 as n → ∞, the proof is complete. n Pn T − A
7.83. Conventions. Here and throughout Sections 7.84–7.85 and 7.87–7.89 we shall assume that the “regular part” b is a function with (at least) absolutely convergent Fourier series which does not vanish on
T and has index zero. So 2.41(e) implies that b has a logarithm log b = n∈Z (log b)n χn in
384
7 Finite Section Method
W = F 1,1 0,0 . If we deﬁne G(b), b− , b+ as in Corollary 6.55, then b = G(b)b− b+ . For what follows in this chapter we may without loss of generality assume that G(b) = 1. Let ε0 > 0 denote a real number which can be chosen as small as desired but remains ﬁxed throughout the following. Given a real number x we deﬁne (x + 0) := ε0
if
x ≤ 0,
(x + 0) := x if
x > 0.
Also recall how (x)◦ was deﬁned in 6.52. If Re γ ≥ 0, Re δ ≥ 0, and T (b) ∈ GL(pµ ) (−1/p < µ < 1/q), then T (ξδ ηγ b) is an invertible operator in L(Dµp (γ), Rµp (δ)) and its inverse T −1 (ξδ ηγ b) belongs to L(pµ+Re δ , pµ−Re γ ) (Theorem 6.48). 7.84. Theorem. Let Re γ ≥ 0, Re δ ≥ 0, −1/p < µ < 1/q. (a) If q,p b ∈ F 1,1 (Re γ−µ)◦ ,(µ+Re δ)◦ ∩ F 1/p+(Re γ+0),1/q+(Re δ+0) ,
then Tn (ξδ ηγ b) is invertible for all suﬃciently large n and if y ∈ pµ+Re δ , then Tn−1 (ξδ ηγ b)Pn y − T −1 (ξδ ηγ b)ypµ−Re γ → 0 (n → ∞). (b) If λ > 0 and µ + λ < 1/q, and if 1,1 q,p b ∈ F (Re γ−µ)◦ ,(µ+λ+Re δ)◦ ∩ F 1/p+λ+(Re γ+0),1/q+λ+(Re δ+0) ,
then, as n → ∞, Tn−1 (ξδ ηγ b)Pn − T −1 (ξδ ηγ b)L(pµ+λ+Re δ ,pµ−Re γ ) = O
1 . nλ
Proof. (a) We apply Theorem 7.82 with X = Dµp (γ),
Y = Rµp (δ),
Z = pµ+Re δ ,
U = pµ−Re γ .
The projections Pn are clearly bounded on Z and U , and it is easy to prove that they are bounded on X and Y (see the remark following the proof). Proposition 6.47 shows that Z and X are continuously embedded in Y and U , respectively. Put A = T (b+ )T (ξδ ηγ )T (b− ),
A + T = T (b− )T (ξδ ηγ )T (b+ ).
Note that A + T = T (ξδ ηγ b). From Theorem 6.54 and Proposition 6.53 it can be deduced that T (b± ) ∈ GL(X) and T (b± ) ∈ GL(Y ). Consequently, both A and A + T are bounded and invertible from X to Y . Since −1 −1 (Pn APn )−1 Pn = T (b−1 − )Pn Tn (ξδ ηγ )Pn T (b+ )
(7.99)
7.8 FisherHartwig Symbols: pµ Theory
385
−1 (recall 7.77) and T (b−1 − ) ∈ L(U ), T (b+ ) ∈ L(Z), we infer from Proposition 7.78 that the hypothesis (d) of Theorem 7.82 is satisﬁed. Using Proposition 2.14 we get
T = T (b+ )H(ξδ ηγ )H(b− ) +H(b+ )H(ξ)δ η. γ )T (b− ) . +H(b+ )T (ξ)δ η. γ )H(b− ).
(7.100)
Lemma 6.51 implies that p p ). H(ξδ ηγ )H(b. − ), H(b+ )H(ξδ η γ ) ∈ C∞ (µ−Re γ , µ+Re δ ),
and Lemma 6.50 shows that H(b+ ) ∈ C∞ (pτ , pµ+Re δ ),
H(b− ) ∈ C∞ (pµ−Re γ , pτ ),
where τ := µ + 1/p − 1/q. The Toeplitz operators still occurring in (7.100) q are bounded on the corresponding spaces (in particular, T (ξ)δ η. γ ) ∈ L(τ ) by Proposition 6.44), and so it follows that T ∈ C∞ (U, Z). Thus, Theorem 7.82 can be applied and its conclusions (g) and (h) give the assertion. (b) Under the stronger restrictions imposed on the smoothness of b, all arguments of the proof of part (a) remain true with µ + Re δ replaced by µ + λ + Re δ. Hence, combining (7.99) and (7.90) we get 1 (Pn APn )−1 Pn − A−1 L(pµ+λ+Re δ ,pµ−Re γ ) = O λ n and, consequently, (Pn APn )−1 Pn T − A−1 T L(pµ−Re γ ) ≤ cn−λ T L(pµ−Re γ ,pµ+λ+Re δ ) = O
1 . nλ
Conclusion (k) of Theorem 7.82 ﬁnishes the proof. Remark. We emphasize that we do not assert the uniform boundedness of the projections Pn on X and on Y . It can be shown that, for µ > −1/p, sup Pn L(Rµp (α)) < ∞ ⇐⇒ Re α <
1 q
sup Pn L(Dµp (α)) < ∞ ⇐⇒ Re α <
1 . p
n
and, for µ < 1/q,
n
Now let ξδ ηγ = ξν ϕ−β , where Re ν ≥ 0 and 0 < Re β < 1. If T (b± ) is in GL(pµ ) (−1/p < µ < 1/q), then T (ξν ϕ−β b) = T (ξν )T (b− )T (ϕ−β )T (b+ ) is bounded and invertible from pµ onto Rµp (ν) and its inverse T −1 (ξν ϕ−β b) is in L(pµ+Re ν , pµ ).
386
7 Finite Section Method
7.85. Theorem. Let Re ν ≥ 0,
0 < Re β < 1,
Re β <
1 , q
Re β −
1 1 <µ< . p q
Let ε be a real number which can be chosen arbitrarily small. (a) If q,p b ∈ F 1,1 (−µ)◦ ,(µ+Re ν)◦ ∩ F 1/p+ε,1/q+(Re ν+0) ,
then Tn (ξν ϕ−β b) is invertible for all n large enough and if y ∈ pµ+Re ν , then Tn−1 (ξν ϕ−β b)Pn y − T −1 (ξν ϕ−β b)ypµ → 0 (n → ∞). (b) If λ > 0 and µ + λ < 1/q, and if 1,1 q,p b ∈ F (−µ) ◦ ,(µ+λ+Re ν)◦ ∩ F 1/p+λ+ε,1/q+λ+(Re ν+0) ,
then, as n → ∞, Tn−1 (ξν ϕ−β b)Pn − T −1 (ξν ϕ−β b)L(pµ+λ+Re ν ,pµ ) = O
1 . nλ
Proof. Similar arguments as in the proof of Theorem 7.84 apply. Remark 1. We would like to draw attention to the following. Unless b− ≡ 1, the ﬁrst term in (7.100) involves H(ξν ϕ−β ) and the largest weight exponent µ + λ + Re ν such that H(ξν ϕ−β ) maps into pµ+λ+Re ν is less than 1/q + Re ν. Therefore we now require µ + λ < 1/q, although Proposition 7.81 is valid for µ + λ < Re β + 1/q. Remark 2. The case where the symbol is ην ϕβ b with Re ν ≥ 0, 0 < Re β < 1 can be settled by “taking adjoints in all arguments” we have above applied to ξν ϕ−β b. So, for instance, if Re β < 1/q, Re β − 1/p < µ < 1/q, λ > 0, µ + λ < 1/q, and if b satisﬁes the smoothness conditions imposed upon b in Theorem 7.85(a) and (b), respectively, then, as n → ∞, Tn−1 (ην ϕβ b)Pn y − T −1 (ην ϕβ b)yq−µ−Re ν → 0 Tn−1 (ην ϕβ b)Pn − T −1 (ην ϕβ b)L(q−µ ,q−µ−λ−Re ν )
∀ y ∈ q−µ , 1 =O λ . n
7.86. Estimates for the inverses of ﬁnite Toeplitz matrices. Let b ∈ C, b(t) = 0 for t ∈ T and ind b = 0. Then A := T (b) ∈ Π{2 ; Pn } and thus, if we −1 −1 let An = Tn (b) and A−1 n = Tn (b)Pn , An L(2 ) = O(1) as n → ∞. The jk −1 and A are entries of A−1 n −1 (A−1 n )jk = (An ek , ej ),
Hence, since A ∈ Π{2 ; Pn }, we have
(A−1 )jk = (A−1 ek , ej ).
7.8 FisherHartwig Symbols: pµ Theory −1 −1 (A−1 )jk  ≤ A−1 ek 2 ej 2 = O(1) n )jk − (A n ek − A
387
(n → ∞).
If, in addition, b is suﬃciently smooth, say bn (n + 1)µ < ∞ for some µ > 0, n∈Z
then b ∈ F 1µ ⊂ C2,µ and so T (b) ∈ Π{2−µ ; Pn } (Theorem 7.25), which implies that −1 −1 (A−1 )jk  ≤ A−1 ek 2−µ ej 2µ n )jk − (A n ek − A
≤ cQn A−1 ek 2−µ ≤ cn−µ A−1 ek 2
(Lemma 7.79) 1 (n → ∞). =O µ n
The purpose of the next three sections is to establish estimates for Tn−1 (ξδ ηγ b)L(2 ) and [Tn−1 (ξδ ηγ b)]jk −[T −1 (ξδ ηγ b)]jk  as n → ∞. We denote Tn (ξδ ηγ b) resp. Tn (ξν ϕ−β b) by An and T (ξδ ηγ b) resp. T (ξν ϕ−β b) by A. 7.87. Theorem. (a) If Re γ ≥ 0, Re δ ≥ 0, and if 1,1 2,2 b ∈ F Re γ,Re δ ∩ F 1/2+(Re γ+0),1/2+(Re δ+0) ,
then Tn−1 (ξδ ηγ b)Pn L(2 ) = O(nRe γ+Re δ ) (n → ∞). (b) If Re ν ≥ 0, 0 < Re β < 1/2, and if, for example, 2,2 b ∈ F 1,1 0,Re β ∩ F 101/200,1/2+(Re ν+0) ,
then Tn−1 (ξν ϕ−β b)Pn L(2 ) = O(nRe ν ) (n → ∞). (c) If Re ν ≥ 0, 1/2 ≤ Re β < 1, and if, for example, 1,1 b ∈ F 101/200,Re ν+1 ,
then Tn−1 (ξν ϕ−β b)Pn L(2 ) = O(n2Re β−1+Re ν+ε ) (n → ∞), where ε can be chosen as small as desired. Proof. (a) Note that A−1 n Pn y2,0 is not larger than Re γ nRe γ A−1 Pn y2,Re δ ≤ cnRe γ+Re δ Pn y2,0 , n Pn y2,−Re γ ≤ cn
the ﬁrst “≤” resulting from Theorem 7.84(a) (p = q = 2, µ = 0). (b) Apply Theorem 7.85(a) with p = q = 2, µ = 0. (c) Deﬁne q and µ by 1/q = Re β + ε/2 and µ = 2Re β − 1 + ε. H¨ older’s −1 inequality gives A−1 n Pn y2,0 ≤ cAn Pn yp,µ and Theorem 7.85(a) shows that µ+Re ν Pn y2,0 . A−1 n Pn yp,µ ≤ cPn yp,µ+Re ν ≤ cn
388
7 Finite Section Method
In the following two theorems we think of A−1 as given by its matrix representation with respect to the standard basis {en }∞ n=0 . Note that −1 T −1 (ξδ ηγ b) = T (b−1 + )T (η−γ )T (ξ−δ )T (b− ), −1 where T (b−1 + ), T (η−γ ) are lower and T (ξ−δ ), T (b− ) are upper triangular ma−1 trices. Therefore, the jk entry [T (ξδ ηγ b)]jk can be computed through a ﬁnite number of operations (and involves no inﬁnite sums). Also recall that ξν ϕ−β is ξδ ηγ with δ = ν +β, γ = −β. The parts (a) of Theorems 7.84 and 7.85 imply that
[Tn−1 (ξδ ηγ b)]jk − [T −1 (ξδ ηγ b)]jk = o(1) (n → ∞). The parts (b) can be used to say more about the o(1). Also, in the following two theorems ε > 0 is a real number which can be chosen as small as desired and c denotes a constant which does not depend on k, j, n. 7.88. Theorem. Let Re γ ≥ 0, Re δ ≥ 0, and A = T (ξδ ηγ b). If p and q are any real numbers such that 1 < p < ∞, 1/p + 1/q = 1, and if q,p b ∈ F 1,1 1/p+Re γ,1/q+Re δ ∩ F 1+1/p+Re γ,1+1/q+Re δ , −1 )jk  is not larger than then (A−1 n )jk − (A
c(k + 1)1/q+Re δ−ε/2 (j + 1)1/p+Re γ−ε/2 n−1+ε . Proof. Apply Theorem 7.84(b) with µ = −1/p + ε/2 and λ = 1 − ε: −1 −1 (A−1 ek , ej ) ≤ A−1 ek p,µ−Re γ ej q,−µ+Re γ n Pn ek − A n Pn ek − A −1 −1 p p ≤ An Pn − A L(µ+λ+Re δ ,µ−Re γ ) ek p,µ+λ+Re δ ej q,−µ+Re γ .
7.89. Theorem. Put A = T (ξν ϕ−β b). (a) Let Re ν ≥ 0 and 0 < Re β < 1. If q is any real number such that Re β < 1/q < 1, if 1/p + 1/q = 1, and if 1,1 q,p b ∈ F 1/p,1/q+Re ν ∩ F 1+1/p−Re β,1+1/q+Re ν−Re β , −1 )jk  is not larger than then (A−1 n )jk − (A
c(k + 1)1/q+Re ν−ε/2 (j + 1)1/p−Re β−ε/2 n−1+Re β+ε . (b) Let Re ν = 0 and 0 < Re β < 1. If q is any real number such that q ≤ 2, Re β < 1/q < 1, if 1/p + 1/q = 1, and if b ∈ F 11/q+Re β , then −1 )jk  is not larger than (A−1 n )jk − (A c(k + 1)1/q+Re β−ε/2 (j + 1)1/p−Re β−ε/2 n−1+ε .
7.9 Invertibility Versus Finite Section Method
389
Proof. (a) Theorem 7.85(b) with µ = Re β − 1/p + ε/2, λ = 1 − Re β − ε. (b) Put µ = Re β − 1/p + ε/2. Lemma 7.79 gives −1 −1 p p A−1 yp,µ ≤ (1 + A−1 yp,µ n Pn y − A n Pn L(µ ) AL(µ ) )Qn A
≤ cQn A−1 yp,µ
(7.85(a), 6.44, 6.23).
Put λ = 1 − ε. Then again by Propositions 6.23 and 6.44, Qn A−1 yp,µ ≤ n−λ Qn A−1 yp,µ+λ −1 ≤ cn−λ T (b−1 (ϕ−β )T (ξ−ν )T (b−1 + )T − )yp,µ+λ ≤ cn−λ yp,µ+λ . Thus
1 nλ and now one can proceed as in the proof of Theorem 7.88. Tn−1 (ξν ϕ−β b)Pn − T −1 (ξν ϕ−β b)L(pµ+λ ,pµ ) = O
7.90. Toeplitz operators on p,+∞ . Recall the deﬁnitions and results . We say that A of 6.57. Let A be a linear and bounded operator on p,+∞ N ; P } if there is an n such that for each y ∈ p,+∞ the belongs to Π{p,+∞ n 0 N N equations Pn APn x = Pn y have a unique solution xn ∈ Im Pn for all n ≥ n0 and if xn converges in the topology of p,+∞ to a solution x ∈ p,+∞ of the N N p,+∞ denote the collection of all linear and bounded equation Ax = y. Let CN having the following properties: operators K on p,+∞ N ; (i) K is deﬁned on the whole space pN and maps pN into p,+∞ N (ii) K ∈ C∞ (pN , p,µ N ) for all µ ∈ Z+ . p,+∞ ∞ ∞ . Finally for a ∈ CN Example: if c ∈ CN ×N , then H(c) ∈ CN ×N , deﬁne ) a as usual by ) a = a(1/t) (t ∈ T). The following result was established by B¨ottcher [61] (for N = 1 see also Gorodetsky [244]). p,+∞ ∞ . Then T (a) + K ∈ Π{p,+∞ ; Pn } if and Let a ∈ CN ×N and K ∈ CN N p,+∞ only if T (a) + K is invertible on N and T () a) is invertible on p,−∞ . N
7.9 Invertibility Versus Finite Section Method We now assume that the underlying space is H 2 . In 7.28(c) we observed that there is a natural continuous algebraic homomorphism ∆π of alg T FJπ (A) onto alg T π (A). Moreover, we showed that ∆π is even an isomorphism if A is a closed algebra between C and C + H ∞ or if A = P C or A = P QC. 7.91. Theorem. There exist {An } ∈ alg T F (L∞ ) such that Aπ is invertible in alg T π (L∞ ), where A := s lim An , but {An }πJ is not invertible in n→∞
alg T FJπ (L∞ ). In particular, ∆π is not an isomorphism between alg T FJπ (L∞ ) and alg T π (L∞ ).
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Proof. If {An } ∈ alg T F (L∞ ) and if {An }πJ is invertible, then {Wn An Wn } also belongs to alg T F (L∞ ) and {Wn An Wn }πJ is invertible, too. Hence, since ∆π is an algebraic homomorphism, the invertibility of {An }πJ implies that W{An } (A) = s lim Wn An Wn is Fredholm. Therefore, in order to prove the n→∞
theorem it suﬃces to ﬁnd A ∈ Φ(H 2 ) and {An } ∈ alg T F (L∞ ) such that / Φ(H 2 ). An → A strongly and W{An } (A) ∈ ∞ is an inﬁnite Blaschke product, put A = I and Suppose b ∈ H An = Tn (b)Tn (b). Then {An } ∈ alg T F (L∞ ), An → A strongly, and W{An } (A) = T ()b)T ()b). Assume T ()b)T ()b) ∈ Φ(H 2 ). Then T ()b) ∈ Φ+ (H 2 ), and because T ()b)T ()b−1 ) = I, we have T ()b) ∈ Φ− (H 2 ) and thus T ()b) ∈ Φ(H 2 ). But Theorem 2.65 shows that T ()b) cannot be Fredholm, since the harmonic (= analytic) extension of an inﬁnite Blaschke product is not bounded away from zero near the unit circle T. The following theorem shows that there exist even a ∈ L∞ such that the previous theorem is true with An = Tn (a). 7.92. Theorem (Treil). There exists a function a ∈ L∞ such that T (a) is / Π{H 2 ; Pn }. invertible on H 2 but T (a) ∈ Proof. We shall construct three sequences {ak }, {nk }, {fk } (k = 0, 1, 2, . . .): . {ak } consists of unimodular functions ak = ei(ϕk +ψk ) ∈ L∞ (ϕk , ψk are real∞ valued functions in L and the tilde will always refer to the conjugation operator); {nk } is a sequence of positive integers satisfying nk < nk+1 ; {fk } is constituted by polynomials fk ∈ Im Pnk satisfying fk 2 = 1. The construction is required to provide ak , ϕk , ψk , nk , fk which fulﬁl the following conditions: (i) Tnk (ak )Pnk − Tnk (ak+1 )Pnk L(H 2 ) ≤ 1/2k ; (ii) Tnk (ak )fk 2 ≤ 1/2k ; (iii) ∃ α, β > 0 : ϕk ∞ ≤ α < π/2, ψk ∞ ≤ β < ∞; (iv) supp ϕk ∩ supp ψk = ∅; (v) ϕk − ϕk+1 2 < 1/2k , ψk − ψk+1 2 < 1/2k . Condition (v) implies that there are ϕ, ψ ∈ L2 such that ϕk − ϕ2 → 0, ψk − ψ2 → 0 as k → ∞. Taking into account (iii) we see that actually ) ϕ, ψ ∈ L∞ and ϕ∞ ≤ α, ψ∞ ≤ β. Hence, if we deﬁne a = ei(ϕ+ψ) , then . T (a) is invertible due to Theorem 2.23. Since ak = ei(ϕk +ψk ) converges in the ) .k − ψ ) 2 → 0 by the continuity of L2 norm to a = ei(ϕ+ψ) (also note that ψ 2 the conjugation operator on L ), we deduce from 1.1(d) that T (ak ) converges strongly to T (a) on H 2 . This observation combined with (i) and (ii) gives that
7.9 Invertibility Versus Finite Section Method
391
∞ " " " " Tnk (a)fk 2 = " Pnk T (aj+1 ) − T (aj ) fk + Pnk T (ak )fk "
≤
2
j=k ∞
" " "Pn T (aj+1 ) − T (aj ) fk " + Pn T (ak )fk 2 k k 2
j=k ∞ 1 1 1 ≤ + k < k−2 , 2j 2 2 j=k
from which it is easily seen that T (a) ∈ / Π{H 2 ; Pn }. . We now construct the sequences {ak } = {ei(ϕk +ψk ) }, {nk }, {fk }. Condition (iv) is needed to carry out the construction. Let b0 be the function deﬁned by (4.64). Since T (b0 ) is invertible, there exist c ∈ R and realvalued functions u, v ∈ L∞ such that b0 = ei(u+)v+c) ,
u∞ ≤ α <
π , 2
v∞ ≤ β < ∞.
(7.101)
Let ϕ0 , ψ0 be any functions satisfying (iii) (with α, β given by (7.101)) and (iv). Put n0 = 0 and f0 = χ0 . Now suppose a0 , . . . , ak , n0 , . . . , nk , f0 , . . . , fk satisfying (i)–(v) are deﬁned; we shall deﬁne ak+1 , nk+1 , fk+1 so that (i)–(v) are again satisﬁed. By virtue of (iv), there is an open subarc I of T such that I ∩ supp ϕk = ∅ and I ∩ supp ψk = ∅. For N ∈ Z+ , let cN := χI (u ◦ χN ),
dN := χI (v ◦ χN ),
i.e., cN (t) = χI (t)u(tN ), dN (t) = χI (t)v(tN ) (t ∈ T), where u and v are given by (7.101). Then put ϕk+1 := ϕk + cN ,
ψk+1 := ψk + dN .
Since, by (7.101), cN 22 = It * is clear 2that ϕk+12, ψk+1 satisfy (iii) and (iv). 2 u ◦ χN  dm ≤ α I and analogously dN 2 ≤ β 2 I, it follows that (v) is I satisﬁed if only I is taken small enough. Moreover, by choosing I suﬃciently small one can guarantee that .
ak − ak+1 2 = ei(ϕk −ψk ) − ei(ϕk+1 +ψk+1 ) 2 becomes as small as desired, which implies that (i) is also fulﬁlled (note that all norms on a ﬁnitedimensional space are equivalent to each other). Thus, if we choose I small enough, then (i), (iii), (iv), (v) are satisﬁed. We now show that if N is chosen suﬃciently large, then there exist nk+1 and fk+1 such that (ii) is fulﬁlled (with k replaced by k + 1). Let I0 be any open subarc of T whose closure is contained in I and suppose 2π/I0  = M is an integer. We claim that, for eiθ ∈ I0 , the function ak+1 can be written as ak+1 (eiθ ) = b0 (eiN θ )gN (eiθ ), where b0 is as in (7.101), the functions θ → gN (eiθ ) are continuously diﬀerentiable, and
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7 Finite Section Method
+ + d + + sup + gN (eiθ )+ : eiθ ∈ I0 , N ∈ Z+ < ∞. dθ
(7.102)
Indeed, for t = eiθ ∈ I0 ⊂ I, .k (t) + (χI (v ◦ χN )))(t) ak+1 (t) = exp i u(tN ) + ψ .k (t) = exp i u(tN ) + v)(t) − (χT\I (v ◦ χN )))(t) + ψ .k (t) = b0 (tN ) exp i − c − (χT\I (v ◦ χN )))(t) + ψ and since the functions χT\I (v ◦ χN ) and ψk vanish identically on I one can verify straightforwardly (using e.g. (1.18)) that the function in braces is continuously diﬀerentiable and that, on {θ : eiθ ∈ I0 }, its derivative is bounded by a constant independent of N . This proves our claim. Now let fk+1 be any analytic polynomial satisfying fk+1 2 = 1,
fk+1 χT\I0 2 <
1 2k+2
(7.103)
(it is readily checked that such an fk+1 exists) and put nk+1 := max{deg fk+1 , nk + 1}. We have
+( + + + + Tnk+1 (ak+1 )fk+1 2 = sup + ak−1 fk+1 h dm++ H +( + T +( + + + + + + + + ≤ sup + ak+1 fk+1 h dm+ + sup + ak+1 fk+1 h dm++ , + + H H T\I0
(7.104)
I0
the supremum over H := {h ∈ Im Pnk+1 : h2 ≤ 1}. The ﬁrst term on the right of (7.104) is less than 1/2k+2 by virtue of (7.103). To estimate the second term, let N = mM (m ∈ Z+ ) and write (
(
(b ◦ χN )gN fk+1 h dm =
ak+1 fk+1 h dm = I0
I0
m ( j=1
(b ◦ χN )q dm Ij
(q := gN fk+1 h), where Ij (j = 1, . . . , m) are pairwise disjoint subarcs of T whose length is 2π/N and whose union is I0 . Taking into account (7.102) and using the fact that all norms on Im Pnk+1 are equivalent to each other we see that + # !+ + iθ +d iθ + + sup + (gN fk+1 h)(e )+ : e ∈ I, N ∈ Z+ , h ∈ H =: γ < ∞. dθ This implies that, for t ∈ Ij , the function q can be represented in the form q(t) = q(tj ) + p(t)(t − tj ), where tj ∈ Ij and p∞ ≤ γ. Hence,
7.9 Invertibility Versus Finite Section Method
393
+ + + + + + ( ( m + + + +m + + + + ≤ q(t (b ◦ χ )q dm ) (b ◦ χ ) dm + + 0 N j 0 N + + + Ij + j=1 + +j=1 Ij + + m +( + + + + + (b0 ◦ χN )p(χ1 − tj ) dm+ . (7.105) + Ij + j=1
* Since the 0th Fourier coeﬃcient of b0 is zero, we have I (b0 ◦ χN ) dm = 0 for every arc I of length 2π/N , and therefore the ﬁrst term on the right of (7.105) is zero. The second term on the right of (7.105) is not larger than 2 m ( m 2π b0 ∞ γ θχ1 − tj  dm ≤ b0 ∞ γγ 2 N j=1 Ij ≤
2b0 ∞ γγ π 2 = o(1) (m → ∞). M 2m
Consequently, if we choose N = mM large enough, then (7.104) is less than 1/2k+1 , as desired. Remark. (Treil [524]). A slight modiﬁcation of the previous proof (replace b0 by a function for which u and v in (7.101) are smooth and replace χI by a smooth function εI which vanishes outside I and is identically 1 in some open subarc of I) shows that there are even a ∈ C(T◦ ) such that T (a) ∈ GL(H 2 ) but T (a) ∈ / Π{H 2 ; Pn }. 7.93. Projection methods generated by inner functions. Let Λ be either of the index sets {l0 , l0 + 1, l0 + 2, . . .} (l0 ∈ Z+ ) or (r0 , ∞) (r0 ∈ R+ ). Given an inner function θ ∈ H ∞ put Kθ := H 2 θH 2 and let Pθ denote the orthogonal projection of H 2 onto Kθ . A (generalized) sequence of inner functions {θλ }λ∈Λ is said to be ordered if (i) θλ divides θµ (i.e., there exists an inner function ω ∈ H ∞ such that ωθλ = θµ ) whenever λ ∈ Λ, µ ∈ Λ, λ ≤ µ, (ii) Pθλ → I strongly on H 2 as λ → ∞. Let {θλ } be an ordered sequence of inner functions and let a ∈ L∞ . In accordance with 7.1, we write T (a) ∈ Π{H 2 ; Pθλ } if there is a λ0 ∈ Λ such that Tθλ (a) := Pθλ T (a)Kθλ is invertible for all λ > λ0 and if, for each g ∈ H 2 , (a)Pθλ g converges in the norm of H 2 to a solution f ∈ H 2 of the equation Tθ−1 λ T (a)f = g as λ → ∞. If Λ = Z+ and θn = χn (n ∈ Z+ ), then Kθn = Im Pn−1 and Pθn = Pn−1 , where Pn is as in 7.5. Thus, in that case we have T (a) ∈ Π{H 2 ; Pθn } if and only if T (a) ∈ Π{H 2 ; Pn }. For the case that Λ = R+ and θτ (τ ∈ R+ ) is the singular inner function t+1 θτ (t) := exp τ (t ∈ T) (7.106) t−1
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7 Finite Section Method
an interesting interpretation of what T (a) ∈ Π{H 2 ; Pθτ } means will be given in 9.41. The following two results were established by Treil [524]. The ﬁrst result shows that to each invertible Toeplitz operator a certain “individual” projection method is applicable, and the second result generalizes Theorem 7.92. (a) Let a ∈ L∞ and suppose T (a) ∈ GL(H 2 ). Then there exists an ordered sequence of inner functions {θn }n∈Z+ (which depends on a) such that T (a) is in Π{H 2 ; Pθn }. By the DouglasRudin theorem (see, e.g., Garnett [211, Chap. V, Theorem 2.1]), there exist functions hn ∈ H ∞ and an ordered sequence of inner functions {θn }n∈Z+ such that a−1 − θn hn ∞ → 0 as n → ∞. The sequence {θn } obtained in this way is a sequence such that T (a) ∈ Π{H 2 ; Pθn }. (b) Let {θλ }λ∈Λ be an ordered sequence of inner functions. Suppose there exists an open subarc I of T such that each of the functions θλ can be continued to a function analytic in some open subset of C containing I. Then there exists a function a ∈ L∞ with only one point of discontinuity such that T (a) / Π{H 2 ; Pθλ }. is invertible on H 2 but T (a) ∈
7.10 Pseudospectra 7.94. Preliminaries. The εpseudospectrum of an operator A ∈ L(X) is deﬁned as the set spε A = λ ∈ C : (A − λI)−1 ≥ 1/ε . Here we put (A − λI)−1 = ∞ if A − λI is not invertible. Thus, the spectrum sp A is always a subset of spε A. We want to know the behavior of spε Tn (a) as n → ∞. This amounts to studying the behavior of the norms (Tn (a) − λI)−1 = Tn−1 (a − λ) and so leads to questions on the algebras alg T F (AN ×N ). We therefore begin by citing some more results on such algebras and then move step by step to pseudospectra. Let 1 ≤ p < ∞ and deﬁne Cp as in Section 2.43. Furthermore, let p p p algp T F (CN ×N ) := algL(N ) T F (CN ×N )
be the smallest closed subalgebra of the algebra S(pN ) that contains the set p {{Tn (a)} : a ∈ CN ×N }. If p = 2, we deﬁne alg2 T F (P CN ×N ) := algL(2N ) T F (P CN ×N ) analogously. Recall that G is the set of all {Cn } ∈ S(pN ) for which Cn p → 0 as n → ∞, where ·p is the norm in L(pN ). We know from Proposition 7.27(b)
7.10 Pseudospectra
395
p that G is a closed twosided ideal of algp T F (CN ×N ) and alg2 T F (P CN ×N ). Recall also that {An } + G = lim sup An p n→∞
and that the strong limits A := s lim An , n→∞
W(A) := W{An } (A) := s lim Wn An Wn n→∞
p exist for every {An } in algp T F (CN ×N ) and alg2 T F (P CN ×N ) and belong p to algp T (CN ×N ) and alg2 T (P CN ×N ), respectively. Here alg2 T F (P CN ×N ) is the smallest closed subalgebra of L(pN ) containing all operators T (a) with p a ∈ CN ×N , and alg2 T F (P CN ×N ) is deﬁned similarly. Proofs of the following two theorems can be found in B¨ ottcher, Grudsky, ottcher [73] for P C (p = 2). Silbermann [87] for Cp (1 ≤ p < ∞) and in B¨
7.95. Theorem. Let 1 ≤ p < ∞. The mappings p p p algp T F (CN ×N )/G → algp T (CN ×N ) × algp T (CN ×N )
and alg2 T F (P CN ×N )/G → alg2 T (P CN ×N ) × alg2 T (P CN ×N ) deﬁned by {An } + G → (A, W(A)) are isometric Banach algebra homomorphisms. Thus, lim sup An p = max Ap , W(A)p n→∞
p for every {An } in algp T F (CN ×N ) (1 ≤ p < ∞) or alg2 T F (P CN ×N ) (p = 2). Moreover, we have lim sup An p = lim An p n→∞
n→∞
for these sequences. p We see in particular that if a ∈ CN ×N (1 ≤ p < ∞) or a ∈ P CN ×N (p = 2), K and L are compact, and Cn → 0, then lim Tn (a) + Pn KPn + Wn LWn + Cn p = max T (a) + Kp , T () a) + Lp . n→∞
p 7.96. Theorem. If {An } belongs to algp T F (CN ×N ) (1 ≤ p < ∞) or alg2 T F (P CN ×N ) (p = 2), then −1 (7.107) lim A−1 p , (W(A))−1 p . n p = max A n→∞
In accordance with the convention to put B −1 p = ∞ if B is not invertible, the righthand side of (7.107) is ∞ if A or W(A) is not invertible. This theorem contains Corollary 7.30 (with Cp + Hp∞ replaced by Cp ) and Theorem 7.54 (for p = 2). In fact, the theorem is a striking reﬁnement of part of the previous results: it replaces the equivalence
396
7 Finite Section Method p lim sup A−1 n p < ∞ ⇐⇒ A and W(A) are invertible on N n→∞
by equality (7.107). We emphasize the special case lim (Tn (a) + Pn KPn + Wn LWn + Cn )−1 p = max (T (a) + K)−1 p , (T () a) + L)−1 p , n→∞
where a, p, K, L, Cn are as in 7.95. In particular, a)p . lim Tn−1 (a)p = max T −1 (a)p , T −1 () n→∞
In the scalar case (N = 1) and for p = 2 we have T −1 (a)2 = T −1 () a)2 because T () a) is simply the transposed operator of T (a). In Grudsky, Kozak [253] and B¨ ottcher, Grudsky, Silbermann [87] (also see B¨ ottcher, Silbermann [116, Example 7.14]) one can ﬁnd examples of functions a ∈ W for which a)p . T −1 (a)p = T −1 () 7.97. Deﬁnitions. For ε > 0, the εpseudospectrum of an operator A ∈ L(X) on a Banach space X is deﬁned by spε,X A = λ ∈ C : (A − λI)−1 L(X) ≥ 1/ε . This deﬁnition has several modiﬁcations and generalizations. One of them is socalled structured pseudospectra (also called spectral value sets). In this context one is given two operators B, C ∈ L(X) and one deﬁnes −1 BL(X) ≥ 1/ε . spB,C ε,X A = λ ∈ C : C(A − λI) Clearly, spε,X A is just spI,I ε,X A. The following theorem provides us with alternative characterizations of structured pseudospectra. 7.98. Theorem. If X is a Hilbert space or if X is a Banach space and at least one of the operators B and C is compact, then / spB,C sp (A + BKC), (7.108) ε,X A = K≤ε
the union over all K ∈ L(X) with the given norm constraint. Proof. To prove that the righthand side of (7.108) is contained in the lefthand side, it suﬃces to show that if A is invertible, CA−1 B < 1/ε, and K ≤ ε, then A+BKC is invertible. So let A be invertible, CA−1 B < 1/ε, and K ≤ ε. Then CA−1 BK < 1 and hence I + CA−1 BK is invertible. Since I + M N is invertible if and only if so is I + N M (in which case (I + M N )−1 = I − M (I + N M )−1 N ), we conclude that I + A−1 BKC and thus also A + BKC = A(I + A−1 BKC) are invertible, as desired.
7.10 Pseudospectra
397
In order to prove that the lefthand side of (7.108) is a subset of the righthand side, suppose A is invertible and CA−1 B = 1/δ > 0. We show that there is a K ∈ L(X) such that K = δ and A + BKC is not invertible. The operator A + BKC = A(I + A−1 BKC) is invertible if and only if so is I + A−1 BKC, which, by the I + M N versus I + N M trick employed above, is equivalent to the invertibility of I + KCA−1 B. Assume ﬁrst that X is a Hilbert space. Abbreviate CA−1 B to S and put K = −δ 2 S ∗ . Then K = δ and I + KCA−1 B = I − δ 2 S ∗ S. The spectral radius of the positive semideﬁnite operator S ∗ S coincides with its norm, that is, with S ∗ S = S2 = 1/δ 2 . It follows that 1/δ 2 ∈ sp S ∗ S and hence that 0 ∈ sp (I − δ 2 S ∗ S). If X is a Banach space and B or C is compact, then CA−1 B is compact. Consequently, there is a u ∈ X such that u = 1 and CA−1 Bu = 1/δ. By the HahnBanach theorem, there exists a functional ϕ ∈ X ∗ such that ϕ = 1 and ϕ(CA−1 Bu) = CA−1 Bu = 1/δ. Let K ∈ L(X) be the rankone operator deﬁned by Kx = −δϕ(x)u. Clearly, K ≤ δ. We have BKCA−1 Bu = B(−δϕ(CA−1 Bu)u) = −δϕ(CA−1 Bu)Bu = −δ(1/δ)Bu = −Bu
(7.109)
Put y = A−1 Bu. If y = 0, then CA−1 Bu = Cy = 0, which contradicts the inequality CA−1 Bu = 1/δ > 0. Thus, y = 0. From (7.109) we infer that BKCy = −Bu = −Ay, whence (A+BKC)y = 0. This implies that A+BKC is not invertible. Open problem. Is Theorem 7.98 true for Banach spaces without the extra assumption that one of the operators B and C be compact? 7.99. Norm of the resolvent. The modulus of a nonconstant analytic function cannot be locally constant. This is no longer true for operatorvalued analytic functions. Indeed, the function λ0 f : C → L(C2 ), λ → 01 is analytic but f (λ)2 = max{λ, 1} is constant for λ ≤ 1. However, if f (λ) is not an arbitrary operatorvalued analytic function but the resolvent of an operator, f (λ) = (A − λI)−1 , things are diﬀerent. In this case f (λ) cannot be locally constant in two important situations. (a) Andrzej Daniluk proved in 1994 that if X is a Hilbert space and A is in L(X), then (A − λI)−1 cannot be locally constant. (b) In B¨ ottcher, Grudsky, Silbermann [87] it is shown that if X is LpN (Ω, dµ) with 1 < p < ∞ and A ∈ L(X), then (A − λI)−1 is nowhere locally constant. Full proofs of these two results are also in B¨ ottcher, Grudsky [86].
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7 Finite Section Method
Open problem. Is the norm (A − λI)−1 nowhere locally constant for an operator A ∈ L(X) on an arbitrary Banach space X? 7.100. Limiting sets. Let {En }∞ n=1 be a sequence of sets En ⊂ C. The (uniform) limiting set lim inf En is deﬁned as the set of all λ ∈ C for which there are λ1 ∈ E1 , λ2 ∈ E2 , . . . such that λn → λ as n → ∞, and the (partial ) limiting set lim sup En is the set of all λ ∈ C for which there exist natural numbers n1 < n2 < . . . and λnk ∈ Enk such that λnk → λ as k → ∞. If all En and E are compact, then the two equalities lim inf En = lim sup En = E are equivalent to the convergence of En to E in the Hausdorﬀ metric, which means that d(En , E) → 0 with d(A, B) := max max dist (a, B), max dist (b, A) . a∈A
b∈B
This is a result of Hausdorﬀ [268, Section 2.8]. A full proof is also in Hagen, Roch, Silbermann [263, Sections 3.1.1 and 3.1.2]. p 7.101. Theorem. If {An } is in algp T F (CN ×N ) (1 < p < ∞) or in alg2 T F (P CN ×N ) (p = 2), then
lim inf spε,pN An = lim sup spε,pN An = spε,pN A ∪ spε,pN W(A). Here spε,pN An refers to the εpseudospectrum of An as an operator on Im Pn with the p norm. Proof. We abbreviate spε,pN to spε . We ﬁrst prove that spε A ⊂ lim inf spε An .
(7.110)
If λ ∈ sp A, then (An − λI)−1 p → ∞ by Proposition 7.3. Consequently, (An − λI)−1 p ≥ 1/ε for all n ≥ n0 , which implies that λ ∈ spε An for all n ≥ n0 . Thus, λ is in the righthand side of (7.110). Now suppose that λ ∈ spε A \ sp A. Then (A − λI)−1 p ≥ 1/ε. Let U ⊂ C be any open neighborhood of λ. From 7.99 we deduce that there is a point µ ∈ U such that (A − µI)−1 p > 1/ε (since otherwise (A − zI)−1 p would be constant in U ). Hence, we can ﬁnd a natural number k0 such that (A − µI)−1 p ≥
1 ε − 1/k
∀ k ≥ k0 .
As U was arbitrary, we can therefore ﬁnd µ1 , µ2 , . . . such that µk ∈ spε−1/k A and µk → λ. Since −1 Bp (7.111) B −1 p = inf y =0 yp
7.10 Pseudospectra
399
for every invertible operator B and (A − µk I)−1 p ≥ 1/(ε − 1/k), it follows that 1 inf (A − µk I)yp ≤ ε − . k yp =1 Thus, there are yk ∈ pN such that yk p = 1 and (A−µk I)yk p < ε−1/(2k). We have (An − µk I)Pn yk p → (A − µk I)yk p ,
Pn yk p → yk p = 1
as n → ∞. Consequently, 1 (An − µk I)Pn yk p <ε− Pn yk p 3k for all n > n0 (k). Again by (7.111), 1 −1 1 (An − µk I)−1 p > ε − > 3k ε and thus µk ∈ spε An for all n > n0 (k). This implies that λ = lim µk belongs to the righthand side of (7.110). Repeating the above reasoning with Wn An Wn and W(A) in place of An and A we get spε W(A) ⊂ lim inf spε Wn An Wn , and since Wn is an isometry on Im Pn , we have spε Wn An Wn = spε An . At this point we have proved that spε A ∪ spε W(A) ⊂ lim inf spε An , and we are left with the inclusion lim sup spε An ⊂ spε A ∪ spε W(A).
(7.112)
Let λ ∈ / spε A ∪ spε W(A). Then 1 1 , (W(A) − λI)−1 p < , ε ε and from Theorem 7.96 we deduce that 1 1 ∀ n ≥ n0 (An − λI)−1 p < − δ < ε ε (A − λI)−1 p <
(7.113)
with some δ > 0. If µ − λ is suﬃciently small, then An − µI is invertible together with An − λI and (An − µI)−1 p ≤
(An − λI)−1 p . 1 − µ − λ (An − λI)−1 p
(7.114)
Let µ − λ < εδ(1/ε − δ)−1 . Then (7.113) and (7.114) give (An − µI)−1 p <
1 1/ε − δ = . 1 − εδ(1/ε − δ)−1 (1/ε − δ) ε
Consequently, µ ∈ / spε An for all n ≥ n0 . This proves that λ is not in the lefthand side of (7.112).
400
7 Finite Section Method
p 7.102. Corollary. Let a be in CN ×N (1 < p < ∞) or in P CN ×N (p = 2), let K and L be compact operators, and let Cn p → 0 as n → ∞. Then lim inf spε,pN Tn (a) + Pn KPn + Wn LWn + Cn = lim sup spε,pN Tn (a) + Pn KPn + Wn LWn + Cn a) + L . = spε,pN T (a) + K ∪ spε,pN T ()
Proof. This is a special case of Theorem 7.101.
7.103. Corollary. If a ∈ P C (p = 2 and N = 1) then lim inf spε,2 Tn (a) = lim sup spε,2 Tn (a) = spε,2 T (a). Proof. Since T () a) is the transposed operator of T (a), the two pseudospectra spε,2 T (a) and spε,2 T () a) coincide.
7.11 Notes and Comments 7.1–7.6. Since the papers by Baxter [50] and Reich [417] the ﬁnite section method has been the subject of numerous investigations by many authors. The development culminated with Gohberg and Feldman’s book [220] (a preliminary edition of which appeared in 1967), in which a ﬁrst systematic and comprehensive theory of projection methods for convolution equations was given and which is a basic reference on this topic till now. At the beginning of the seventies, under the impression of I. B. Simonenko’s local Fredholm theory of Toeplitz operators, V. B. Dybin brought forth the idea that local methods ought to be applicable to projection methods too, and A. V. Kozak [317], [318], [319], a student of his, was the ﬁrst to carry out this program. He algebraized and essentialized as in 7.2, generalized Simonenko’s local principle to the case of arbitrary Banach algebras, and then developed a selfcontained theory of the ﬁnite section method for (one and higherdimensional) operators with continuous symbols. Here we conﬁne ourselves to the ﬁnite section method for Toeplitz operators. For other projection methods and their applications to other classes of operators (singular integral or pseudodiﬀerential operators) we refer to Gohberg, Feldman [220], Pr¨ ossdorf, Silbermann [407], [408], Verbitsky [540], Silbermann [479], Roch, Silbermann [426], [427], Mikhlin, Pr¨ ossdorf [361], Hagen, Roch, Silbermann [262], [263], Rabinovich, Roch, Silbermann [411]. Recent developments which are close to the algebraization and localization strategy of the present book include Roch, Santos, Silbermann [424], Santos, Silbermann [450], and Santos [449]. The problem of the connection between the MoorePenrose inverse T + (a) of a normally solvable Toeplitz operator T (a) and the MoorePenrose inverses Tn+ (a) of the ﬁnite sections as n → ∞ is studied in Heinig, Hellinger [272]
7.11 Notes and Comments
401
and Silbermann [486] (also see B¨ ottcher, Silbermann [116] and Hagen, Roch, Silbermann [263]). The ﬁnite section method for Toeplitz operators with operatorvalued symbols is studied in Gohberg, Kaashoek [224] and B¨ ottcher, Silbermann [115]. 7.7. Widom [569]. 7.8–7.11. The history is as follows. Since all eﬀort made to extend Kozak’s local theory to the case of discontinuous symbols failed (see 8.61 for the why), the further development paused many years. At the end of the seventies we turned our attention to Toeplitz determinants, and in 1980 one of the authors (B¨ottcher [59]) found a separation technique for treating Toeplitz determinants with discontinuous symbols (we were not aware of the fact that Basor [20] had worked out exactly the same method some time before). We then observed that this separation technique could also be applied to the ﬁnite section method, and in the note B¨ottcher, Silbermann [104] we solved a series of problems on the applicability of the ﬁnite section method to Toeplitz operators with discontinuous symbols which had been open for a long time. In particular we showed that the results of Verbitsky and Krupnik [543] pertaining to the case of only one discontinuity (and being the main achievement of the development in the middle of seventies) extend to the case of a ﬁnite number of discontinuities. The deciding discovery was ﬁnally made by Silbermann [478], who established the results of 7.8–7.11. Of course, Widom’s formula 7.7 played a crucial role for recognizing how to “essentialize” in the right way. Note that Theorem 7.11 admits not only a “separation” of singularities, but even a “localization,” i.e., it turns out to be just the right tool for considering symbols with inﬁnitely many discontinuities. Moreover, this theorem works equally good in both the scalar and matrix case, which cannot be said at all about previous methods. 7.12–7.14. These results were established by Roch and Silbermann [426]. In the Hilbert space case similar results were also obtained by Barr´ıa and Halmos [17]. We remark that the paper Roch, Silbermann [426] appeared as Preprint PMATH–22/83, Akad. Wiss. DDR, Inst. Math., Berlin 1983, and that we did not know of the paper by Barr´ıa and Halmos [17] at that time. 7.15–7.16. It is a delicate problem to say who made such observations for the ﬁrst time. The essence of 7.15 and 7.16, except for formula (7.27), is already contained in Devinatz, Shinbrot [155]. We learned formula (7.27) from A. V. Kozak (private communication). 7.18. This is perhaps well known to specialists, but we know of no explicit reference. 7.19. This is also well known. The proof of the index equality in (f) is from B¨ottcher, Silbermann [106]. 7.20. Baxter [50] and Reich [417] showed that T (a) ∈ Π{Pn ; 1 } if a ∈ GW p and ind a = 0. Theorem 7.20 for a ∈ CN ×N (resp. a ∈ CN ×N ) and K = 0
402
7 Finite Section Method
is Gohberg and Feldman’s [220]. Results on the ﬁnite section method for operators with C + H ∞ symbol were obtained by several authors. The earliest reference we know is Devinatz, Shinbrot [155], who proved Theorem 7.20(a) for p = 2 and K = 0 (and considered even arbitrary operatorvalued symbols). Independently the same result was established by Ambartsumyan [7] and Widom [569]. Note that each of these authors had his own proof. Theorem 7.20(a) as it is stated here appeared in Silbermann [478] for the ﬁrst time. It should be remarked that all previous results on compact perturbations of Toeplitz operators made use of 7.17 and so gave only the implication T (a), T (a) + K, T (a) invertible =⇒ T (a) + K ∈ Π{Pn }, while the implication T (a) + K, T (a) invertible =⇒ T (a) + K ∈ Π{Pn } was ﬁrst shown in Silbermann [478]. Part (b) of Theorem 7.20 is published here for the ﬁrst time. A result of the type of part (c) is already contained in Widom [569]. For two more proofs of Theorem 7.20 see also B¨ ottcher, Silbermann [106, 3.7 and 3.10]. 7.21. A similar formula is in Widom [569], the identity under consideration was ﬁrst explicitly stated and exploited in B¨ ottcher [61] and B¨ ottcher, Silbermann [106]. 7.22. This kind of argument appeared in Silbermann [474] for the ﬁrst time. 7.23. Verbitsky, Krupnik [543]. 7.24–7.25. Verbitsky and Krupnik [543] proved Theorem 7.25 for N = 1 using diﬀerent methods. Their method is not applicable in the matrix case. The proof given here as well as Proposition 7.24 are new. 7.26–7.33. Proposition 7.27 and the results of 7.29 and 7.33 (for B = C) are from B¨ ottcher, Silbermann [105]; all other results are due to Silbermann [482], [483]. That the ﬁnite section method is applicable in the space H 2 to invertible Toeplitz operators whose symbol is locally sectorial over C had already been shown in Gohberg [218] (see also Gohberg, Feldman [220]). 7.35–7.36. Lemma 7.35 and Proposition 7.36(b) are implicit in Roch, Silbermann [426, Part II]. Proposition 7.36(a) is well known. 7.37. Verbitsky, Krupnik [543]. Seybold [464] extended this result to the spaces p (w) with Muckenhoupt weights of the form wk = (k + e)δ+ϑ sin(log log(k+e)) . Notice that these weights are nontrivial in the sense that their socalled indices of powerlikeness are not equal to each other (see B¨ottcher, Seybold [101]), in contrast to power Khvedelidze weights (k + 1)µ , for which the indices of powerlikeness coincide and are equal to µ.
7.11 Notes and Comments
403
7.39–7.44. Proposition 7.41 was suggested by B¨ ottcher [64] and a result like (7.48) was ﬁrst proved in B¨ ottcher, Roch, Silbermann [100]; note that the proof makes essential use of Lemma 7.35 and Proposition 7.36. Theorem 7.42 was established in B¨ ottcher, Silbermann [104] for the case of ﬁnitely many jumps and N = 1 and in Silbermann [478] in the form presented here. The ottcher, Roch, Silbermann H p analogues of 7.43 and 7.44 were stated in B¨ [100]. 7.45. This result was ﬁrst proved by B¨ ottcher (the proof is published in Roch, Silbermann [426, Part I]). For the case of a single jump the result is Verbitsky and Krupnik’s [543]. 7.48. Verbitsky [537]. 7.50–7.54. B¨ottcher, Roch, Silbermann [100]. 7.55–7.58. Lemma 7.55 and Lemma 7.56(a) are due to Duduchava [169], Lemma 7.56(b) was established by Roch [419], Lemma 7.57 is Verbitsky’s [539]. 7.59–7.63. All these results were obtained by Roch [419]. Some of the arguments go back to Verbitsky [539], who proved Theorem 7.63 for the case Re β1 = . . . = Re βm . A key observation for proving Theorem 7.63 is of course formula (6.16). 7.64–7.66. Lemma 7.64 is a result of B¨ottcher, Silbermann [110]. Lemmas 7.65 and 7.66 are taken from Gohberg, Krupnik [229] (where Lemma 7.65 is attributed to V. L. Pinski); also see Krupnik [329]. 7.67–7.72. These things are the result of a hard birth. Roch [419], [420] developed all of the machinery needed to prove Theorems 7.68 and 7.72 (in particular, the approach of 7.69 and 7.70, which is far from being trivial, is due to him), but he did unfortunately not notice that these theorems are only τ = 0 for all τ ∈ T (incidentrue with the additional requirement that ind RA tally neither did the authors). Only A. Rathsfeld observed that this condition must be added to ensure the validity of Theorems 7.68 and 7.72 (private communication). Thus, if you will add this condition to the corresponding results of Roch [419], [420] and Roch, Silbermann [425], then all results stated there become true. The results of Roch, Silbermann [426, Part II] pertaining to alg T (P QC) still require a correction. τ = 0} We have not checked the details, but the sets {λ ∈ C : ind RA−λI τ certainly coincide with the local spectra of {Pn APn }J (recall Theorem 7.50). 7.73–7.76. Theorem 7.73 and Corollary 7.76 were established in B¨ ottcher, Silbermann [108], [110]. Corollary 7.74 is Verbitsky’s [540], who also proved the implication “=⇒” of Corollary 7.75 and established the reverse implication for symbols satisfying a(τj − 0) = a(τj + 0) for all j using diﬀerent methods. 7.77–7.89. These results are from B¨ottcher, Silbermann [111]. Notice that Theorem 7.82 is nontrivial and should be distinguished from other theorems
404
7 Finite Section Method
of this type; e.g., if such a result had been known earlier, one had been able to avoid a series complications which were to overcome in Pr¨ossdorf, Silbermann [407]. The paper Vladimirov, Volovich [548] may serve as a motivation for studying estimates of the matrices Tn−1 (ξδ ηγ b). The evolution of the entries of Tn−1 (a) has been of interest for about a century. Suppose a is of the special form a(t) = t − t0 2α b(t) (t ∈ T) where α > 0 is a real number and b is a suﬃciently smooth function with values in (0, ∞). Then Tn (a) is positive deﬁnite and hence invertible for all n ≥ 0. There is a continuous function Gα : [0, 1]2 → (0, ∞) such that 0 1 lim n1−2α Tn−1 (a) nx,ny =
n→∞
1 Gα (x, y) b(t0 )
(7.115)
uniformly in (x, y) ∈ [0, 1]2 ; here z denotes the integral part of z. A weakened version of this result was in principle already established by Widom [560], [561], showed that the integral operator on L2 (0, 1) with the ker0 who 1 1−2α −1 Tn (a) nx,ny converges uniformly to the integral operator with nel n the kernel Gα (x, y)/b(t0 ). In the form cited here, (7.115) is due to Rambour, Seghier [412], [413], [414]. See also B¨ ottcher [77] and B¨ ottcher, Grudsky [86]. We remark that if α is a natural number, then Gα (x, y) is Green’s kernel for the boundary value problem (−1)α u(2α) (x) = v(x) on [0, 1] and u(k) (0) = u(k) (1) = 0 for k = 0, 1, . . . , α − 1. Diﬀerent questions on the behavior of the entries of the inverses of Toeplitz ¨ matrices are considered in Strohmer [509] and Deift, Ostensson [150]. 7.91–7.93. We do not know who was the ﬁrst to raise the question on whether the ﬁnite section method is applicable to every Toeplitz operator T (a) in ottcher, Silbermann [106, p. 76] and GL(H 2 ), but it is an old question. In B¨ Silbermann [480] we formulated this question and conjectured that the answer be yes. We even conjectured that the algebras alg T π (L∞ ) and alg T FJπ (L∞ ) are isomorphic to each other. However, we soon realized that the two algebras are not isomorphic (Theorem 7.91), which then made us become thoroughly convinced in that the answer to the ﬁrst question is also negative. By the way, this conviction had also been held by the colleagues in Kishinev for a long time. We then tried to ﬁnd a symbol in P3 C which generates an invertible Toeplitz operator to which the ﬁnite section method is not applicable, but our endeavor was not (and has not yet been) crowned with success. The state of aﬀairs was fortunately altered by Treil’s [524] result 7.92, which is undoubtedly one of the most signiﬁcant achievements in this ﬁeld. Notice that Treil’s theorem 7.92 is a “positive” result in the following sense: if it would have turned out that the ﬁnite section method were applicable to every invertible Toeplitz operator, then a major part of all earlier work were for nothing; Treil’s result aposteriorily justiﬁes this work. The material of 7.92 and 7.93 is from Treil [525]. The following result is shown in B¨ ottcher, Grudsky [80].
7.11 Notes and Comments
405
Suppose b ∈ C, T (b) ∈ GL(H 2 ), and the zeroth Fourier coeﬃcient b0 of b is zero. If a ∈ L∞ is given by t+1 a(t) = b exp (t ∈ T), t−1 / Π{H 2 ; Pn }. then T (a) ∈ GL(H 2 ) but T (a) ∈ Notice that a and b are related by the formula x−i a (7.116) = b(eix ) (x ∈ R). x+i √ √ The function b(t) = ( 3−t)2 ( 3−1/t)−2 satisﬁes all hypotheses and delivers the symbol 2 √ x−i 3 − eix (x ∈ R) a = √ x+i 3 − e−ix satisfying T (a) ∈ GL(H 2 ) but T (a) ∈ / Π{H 2 ; Pn }. Furthermore, the symbol x−i a = eif (x) (x ∈ R) x+i deﬁned by f (x) = π −
8 π
cos 3x cos 5x + + . . . cos x + 32 52
can also be represented in the form (7.116) with b ∈ C, T (b) ∈ GL(H 2 ), b0 = 0 (simply let b be the function given by (4.62)). This symbol is almost periodic and reveals that we need not look for symbols a with T (a) ∈ GL(H 2 ) but T (a) ∈ / Π{H 2 ; Pn } in the entire abyss of L∞ . 7.94–7.103. Trefethen and Embree’s book [523] is an excellent source on all aspects of pseudospectra. The philosophy is that some basic phenomena for an operator can be understood by considering the norm (A−λI)−1 . If A is a normal Hilbert space operator, then (A−λI)−1 = 1/dist (λ, sp A), and hence knowledge of the spectrum alone is suﬃcient to understand (A − λI)−1 . However, for nonnormal Hilbert space operators, or more generally, for Banach space operators, the information contained in the spectrum does not provide a precise description of (A − λI)−1 . In that case pseudospectra do a perfect job, since they encode all information about (A − λI)−1 on the one hand and do this in a visual manner (as subsets of the plane) on the other. The Toeplitz operators Tn (a) and T (a) are normal in rare cases only, and hence pseudospectra are expected to tell us more on Toeplitz operators than spectra. The pioneering works on pseudospectra of Toeplitz matrices are due to Henry Landau [331], [332], [333] and Reichel and Trefethen [418]. They essentially established Corollary 7.103 for smooth symbols and for p = 2 by using diﬀerent methods. Corollary 7.103 for P CN ×N (p = 2) was ﬁrst proved
406
7 Finite Section Method
p in B¨ ottcher [73] (by C ∗ algebra methods), and for CN ×N (1 < p < ∞) it is from Grudsky, Kozak [253] and B¨ ottcher, Grudsky, Silbermann [87]. Theorems 7.95, 7.96, 7.101 are from B¨ ottcher [73], Roch, Silbermann [430] in the ottcher, Grudsky, Silbermann [87] 2 case and from Grudsky, Kozak [253], B¨ for p spaces. See also Beam, Warming [51]. For B = C = I, Theorem 7.98 is in principle already in Trefethen [521], [522]. In the general case, this theorem is due to Gallestey, Hinrichsen, Pritchard [207], [208], Hinrichsen, Kelb [278], Hinrichsen, Pritchard [279]. Our proof is based on ideas of Gallestey, Hinrichsen, Pritchard [208] and follows B¨ottcher, Grudsky [82]. The question whether the norm of the resolvent may be locally constant was posed by one of us (A.B.) during a Banach semester in Warsaw in 1994. A few months later, Andrzej Daniluk of Cracow sent us a proof of the result of 7.99(a). In B¨ ottcher, Grudsky, Silbermann [87] we proved 7.99(b) for X = LpN (Ω, dµ). The fact that (A − λI)−1 is nowhere locally constant is equivalent to saying that the pseudospectra spε,X A do not jump as ε varies continuously. Thus, 7.99(a) tells us that the usual pseudospectra of Hilbert space operators cannot jump. In contrast to this, structured pseudospectra of Hilbert space operators can jump. An example is given in B¨ ottcher, Grudsky [82]. There it is shown that if U = L(χ1 ) is the forward shift on 2 (Z) and P1 is the projection that leaves x0 and x1 unchanged and replaces xj with zero for 1 ,P1 j ∈ / {0, 1}, then the structured pseudospectrum spP ε,2 (Z) U does jump. The > √ precise result is as follows. Put ε0 = (3 − 5)/2. There is a continuous and strictly monotonically increasing function h : [ε0 , ∞) → [0, ∞) such that h(ε0 ) = 0, h(∞) = ∞, and ⎧ for 0 < ε < ε0 , ⎨ {λ ∈ C : λ = 1} 1 ,P1 {λ ∈ C : 1 ≤ λ ≤ 1 + h(ε)} for ε0 ≤ ε < 1, spP U = 2 ε, (Z) ⎩ {λ ∈ C : 0 ≤ λ ≤ 1 + h(ε)} for 1 ≤ ε. 1 ,P1 Thus, spP ε,2 (Z) U jumps at ε = 1. Results on pseudospectra of Toeplitz operators (= inﬁnite Toeplitz matrices) T (a) can be found in B¨ ottcher, Grudsky [86, Section 7.4], B¨ ottcher, Grudsky, Silbermann [87], and B¨ ottcher, Silbermann [116, Section 3.6]. There in particular the following is shown. If a ∈ L∞ , then
spL(2 ) T (a) + ε D ⊂ spε,2 T (a) ⊂ conv R(a) + ε D, and each of the two inclusions may be proper. Here D is the closed unit disk. If a ∈ L∞ and spL(2 ) T (a) is convex, then spε,2 T (a) = conv R(a) + ε D. Chapter 14 of B¨ ottcher, Grudsky [86] contains results on structured pseudospectra of ﬁnite and inﬁnite Toeplitz matrices which were obtained in joint work with Mark Embree, Viatcheslav Sokolov, and Marko Lindner.
7.11 Notes and Comments
407
Another issue concerns the speed of convergence of the pseudospectra spε,2 Tn (a) to spε,2 T (a). If a ∈ L∞ is rational, a ∈ R, and ind (a−λ) = 0, then Tn−1 (a − λ)2 increases exponentially fast (see Reichel, Trefethen [418] and B¨ottcher, Grudsky [81]). Consequently, the inequality Tn−1 (a − λ)2 ≥ 1/ε is already satisﬁed for moderately sized n and it follows that the convergence of ottcher, Embree, Trefethen [78] it spε,2 Tn (a) to spε,2 T (a) is very fast. In B¨ was observed that if a ∈ P C, then Tn−1 (a−λ)2 may grow only polynomially. This implies that the convergence of spε,2 Tn (a) to spε,2 T (a) may be spectacularly slow (and almost invisible on the computer’s screen). In B¨ ottcher, Grudsky [83] it was shown that such a slow convergence is generic even within the class of continuous symbols. We remark that the proofs of [78] and [83] make essential use of formulas for Toeplitz determinants, which have FisherHartwig symbols in [78] and symbols with nonvanishing index in [83]. The numerical range (= Hausdorﬀ range = ﬁeld of values) of Toeplitz matrices behaves as nicely as the pseudospectrum. Roch [423] showed that if A is a bounded linear operator on 2 , then the numerical range of the ﬁnite sections Pn APn Im Pn converges to the closure of the numerical range of A in the Hausdorﬀ metric. For a ∈ L∞ (T), the numerical range of T (a) was determined by Klein [313]. Theorem 1 of [313] says that if the Toeplitz operator has a nonconstant symbol and is normal, so that the spectrum is a closed interval [γ, δ] ⊂ C, then the numerical range is the corresponding open interval (γ, δ). Theorem 2 of [313] states that if the Toeplitz operator is not normal, then its numerical range is the interior of the convex hull of its spectrum. Polynomial numerical hulls are another alternative to spectra. They were independently invented by Nevanlinna [364] and Greenbaum [248]. For polynomial numerical hulls of Toeplitz matrices we refer to Faber, Greenbaum, and Marshall [197], Burke, Greenbaum [128], and Chapter 8 of B¨ ottcher, Grudsky [86]. Theorem 7.96 shows in particular that if a ∈ P C (scalar case) and T (a) is invertible on 2 , then Tn−1 (a)2 converges to T −1 (a)2 . This implies that the condition number κ(Tn (a)) := Tn (a)2 Tn−1 (a)2 converges to κ(T (a)) := T (a)2 T −1 (a)2 , which is a ﬁnite number. If T (a) is not invertible, then κ(Tn (a)) grows to inﬁnity. Results on the speed of the growth can be found in B¨ ottcher, Grudsky [81], [86]. When working with structured matrices it is frequently reasonable to replace the usual condition numbers by socalled structured condition numbers. See D.J. Higham and N.J. Higham [276], N.J. Higham [277], and Rump [445] for more on this topic. Special attention to the Toeplitz structure is paid in B¨ottcher, Grudsky [85], [86, Chapter 13], Graillath [247], Rump [445]. In [85] it is shown that structured condition numbers of Toeplitz matrices with rational symbols are rarely better (that is, essentially smaller) than the usual condition numbers.
8 Toeplitz Operators over the QuarterPlane
8.1 Function Classes on the Torus 8.1. p (Z2 ) and p (Z2++ ). Let 1 ≤ p < ∞ and let Ω be a subset of Zk (k = 1, 2 . . .). Deﬁne p (Ω) := x = {xj }j∈Ω : xpp := xj p < ∞ . j∈Ω
If Ω1 and Ω2 are subsets of Z and if x = {xj } ∈ p (Ω1 ) and y = {yk } ∈ p (Ω2 ) then x ⊗ y ∈ p (Ω1 × Ω2 ) is deﬁned by (x ⊗ y)jk = xj yk (j ∈ Ω1 , k ∈ Ω2 ). Note that obviously x ⊗ yp = xp yp . Given subsets A ⊂ p (Ω1 ) and B ⊂ p (Ω2 ), let A " B refer to the subset of p (Ω1 × Ω2 ) consisting of all ﬁnite sums i xi ⊗ yi (xi ∈ p (Ω1 ), yi ∈ p (Ω2 )) and let A ⊗ B denote the closure of A " B in the norm of p (Ω1 × Ω2 ) (if both Ω1 and Ω2 are ﬁnite sets, then A " B = A ⊗ B). It can be easily veriﬁed that p (Z2 ) = p (Z) ⊗ p (Z),
p (Z2++ ) = p ⊗ p ,
where Z2++ := Z+ × Z+ and p := p (Z+ ). Also note that p (Z2 ) = 0 (Z) ⊗ 0 (Z), p (Z2++ ) = 0 (Z+ )⊗0 (Z+ ), where 0 refers to the sequences with ﬁnite support. Let pN (Ω) denote the space of all columnvectors x = (x1 , . . . , xN ) whose components are in p (Ω) and deﬁne a norm in pN (Ω) by (x1 , . . . , xN ) pp := x1 pp + . . . + xN pp . Note that one can also think of pN (Ω) as the space of CN valued sequences
p 1/p (see 2.92). over Ω with ﬁnite norm j∈Ω xj CN Again let Ω1 , Ω2 ⊂ Z. For x = (x1 , . . . , xN ) ∈ pN (Ω1 ) and y ∈ p (Ω2 ) deﬁne x ⊗ y ∈ pN (Ω1 × Ω2 ) as (x1 ⊗ y, . . . , xN ⊗ y). It is readily seen that x ⊗ ypN (Ω1 ×Ω2 ) = xpN (Ω1 ) yp (Ω2 ) .
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8 Toeplitz Operators over the QuarterPlane
Given subsets A ⊂ pN (Ω1 ) and B ⊂ p (Ω2 ), we let A " B stand
for the subset of pN (Ω1 × Ω2 ) the elements of which are the ﬁnite sums i xi ⊗ yi (xi ∈ pN (Ω1 ), yi ∈ p (Ω2 )) and we denote the closure of A " B in pN (Ω1 × Ω2 ) by A ⊗ B. In particular, pN (Z2 ) = pN (Z) ⊗ p (Z),
pN (Z2++ ) = pN ⊗ p .
In analogy to the preceding paragraph one can deﬁne x ⊗ y for x ∈ p (Ω1 ) and y ∈ pN (Ω2 ), A " B and A ⊗ B for A ⊂ p (Ω1 ) and B ⊂ pN (Ω2 ). We have, for example, pN (Z2++ ) = pN ⊗ p = p ⊗ pN = (p ⊗ p )N . 8.2. Lp (T2 ) and H p (T2 ). Let T2 denote the torus T × T. We deﬁne Lp (T2 ) (1 ≤ p < ∞) as the Banach space of all (classes of) measurable functions f on T2 for which ( 2π ( 2π 1 p f p := f (eiθ , eiψ )p dθ dψ < ∞. (2π)2 0 0 The m, n Fourier coeﬃcient of f ∈ L1 (T2 ) is given by ( 2π ( 2π 1 fm,n := f (eiθ , eiψ )e−imθ e−inψ dθ dψ, (2π)2 0 0 and H p (T2 ) is deﬁned as the (obviously closed) subspace of Lp (T2 ) consisting of all f ∈ Lp (T2 ) for which fm,n = 0 unless m ≥ 0 and n ≥ 0. For f, g ∈ Lp deﬁne f ⊗ g ∈ Lp (T2 ) by (f ⊗ g)(s, t) := f (s)g(t) (s, t ∈ T). p Clearly, f ⊗ gp = f p gp . If A and
B are subsets of L , we let A " B denote the collection of all ﬁnite sums i fi ⊗ gi (fi ∈ A, gi ∈ B), and A ⊗ B will denote the closure of A " B in Lp (T2 ). The n, n partial sum sn,n f of the Fourier series of a function f ∈ Lp (T2 ) is sn,n f = fjk χj ⊗ χk , j≤n k≤n
and it is well known that f − sn,n f p → 0 as n → ∞ (1 ≤ p < ∞). This implies that Lp (T2 ) = P ⊗ P and H p (T2 ) = PA ⊗ PA , and thus that 2 ) = H p ⊗ H p. Lp (T2 ) = Lp ⊗ Lp and H p (T
Notice that the mapping fij χi ⊗χj → {fij } is an isometric isomorphism of L2 (T2 ) onto 2 (Z2 ) as well as of H 2 (T2 ) onto 2 (Z2++ ). p (T2 ) are the spaces of columnvectors f = (f 1 , . . . , f N )
LpN (T2 ) and HN with components in Lp (T2 ) and H p (T2 ), respectively. The norm in these spaces is given by (f
1
, . . . , f N ) pp
:=
( ( N T
T
k=1
p/2 f (s, t) k
2
dms dmt .
8.1 Function Classes on the Torus
411
For f = (f 1 , . . . , f N ) ∈ LpN and g ∈ Lp , deﬁne f ⊗ g = (f 1 ⊗ g, . . . , f N ⊗ g) ,
g ⊗ f = (g ⊗ f 1 , . . . , g ⊗ f N ) .
It can be easily veriﬁed that f ⊗ gp = g ⊗ f p = f p gp . Given sets p " B (resp. B " A) denote the collection of all A ⊂ LpN and
B ⊂ L , let A
ﬁnite sums i fi ⊗ gi (resp i gi ⊗ fi ) with fi ∈ A, gi ∈ B, and let A ⊗ B (resp. B ⊗ A) denote the closure of A " B (resp. B " A) in LpN (T2 ). Thus, we have LpN (T2 ) = LpN ⊗ Lp = Lp ⊗ LpN ,
p p p HN (T2 ) = HN ⊗ H p = H p ⊗ HN .
p 8.3. Tensor products of operators on H p and p . Let Y be LpN or HN , let Z be Lp or H p , and let A ∈ L(Y ) and B ∈ L(Z). For a ﬁnite sum h = i fi ⊗ gi (fi in PN resp. (PA )N , gi in P resp. PA ) deﬁne (A ⊗ B) fi ⊗ gi := Afi ⊗ Bgi . (8.1) i
i
We claim that (A ⊗ B)hY ⊗Z ≤ AL(Y ) BL(Z) hY ⊗Z .
(8.2)
To see this notice ﬁrst that (( " "p " " (A ⊗ I)hp = (Afi )(s)gi (t)" dms dmt " CN
( ( " i "p " " = gi (t)fi (s)" dms dmt "A CN
( ( "i "p " " p ≤ A fi (s)gi (t)" dms dmt = Ap hp , " i
CN
observe that, similarly, (I ⊗ B)h ≤ B h, and thus (A ⊗ B)h = (A ⊗ I)(I ⊗ B)h ≤ A B h, as desired. From (8.2) we deduce that A ⊗ B extends to a bounded operator on Y ⊗ Z, which will be denoted by A ⊗ B, too. If we choose f ∈ Y, g ∈ Z so that f = g = 1 and Af ≥ (1 − ε)A, Bg ≥ (1 − ε)B, then (A ⊗ B)(f ⊗ g) = Af ⊗ Bg = Af Bg ≥ (1 − ε)2 A B. Consequently, A ⊗ BL(Y ⊗Z) = AL(Y ) BL(Z) . pN (Z)
pN (Z+ ),
(8.3)
In case Y is or Z is p (Z) or p (Z+ ), A ∈ L(Y ) and B ∈ L(Z), we deﬁne (A ⊗ B)h for ﬁnite sums h = i fi ⊗ gi (fi in 0N (Z) resp. 0N (Z+ ), gi in 0 (Z) resp. 0 (Z+ )) again by (8.1). A similar reasoning as above shows
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8 Toeplitz Operators over the QuarterPlane
that (8.2) holds and that, therefore, A ⊗ B extends to a bounded operator Y ⊗ Z, which satisﬁes (8.3). Analogously one can deﬁne A ⊗ B for the case that A acts on scalarvalued and B acts on vectorvalued functions or sequences. Examples. (a) If Ω ⊂ Z2 , then pN (Ω) may be viewed as a subspace of pN (Z2 ). Let PΩ denote the projection of pN (Z2 ) onto pN (Ω) parallel to pN (Z2 \ Ω). It is clear that PΩ has norm 1. Let P be the canonical projection of pN (Z) onto pN (Z+ ). Then for Ω = Z+ × Z, Ω = Z × Z+ , Ω = Z2++ = Z+ × Z+ , the projection PΩ equals P ⊗ I, I ⊗ P , and P++ := P ⊗ P , respectively. Here P ∈ L(pN (Z)) and I ∈ L(p (Z)) if pN (Z+ × Z) is identiﬁed with pN (Z+ ) ⊗ p (Z), while P ∈ L(p (Z)) and I ∈ L(pN (Z)) if pN (Z+ × Z) is identiﬁed with p (Z+ ) ⊗ pN (Z). p (b) Let P : LpN → HN (1 < p < ∞) denote the Riesz projection (see 1.42). From what was said above we infer that P++ := P ⊗ P is bounded on LpN (T2 ) p (T2 ) = P++ LpN (T2 ). (1 < p < ∞) and has norm P 2L(Lp ) . Clearly, HN
m p ∞ or ci ∈ MN (c) Let b = i=1 ci ⊗ di , where ci ∈ L∞ N ×N and di ∈ L ×N p and di ∈ M . Then the operators M (ci ) ⊗ M (di ), M (ci ) ⊗ T (di ), i
i
T (ci ) ⊗ M (di ),
i
T (ci ) ⊗ T (di )
i
can be identiﬁed with the operators (I ⊗ I)b(I ⊗ I), (I ⊗ P )b(I ⊗ P )Im (I ⊗ P ), (P ⊗ I)b(P ⊗ I)Im (P ⊗ I), (P ⊗ P )b(P ⊗ P )Im (P ⊗ P ), respectively. Here b refers to the operator of multiplication by (or convolution with) b, which will be deﬁned precisely in the next section. (d) If αi ∈ C, Ai ∈ L(Y ), and I denotes the identity operator on Z, then m i=1
Ai ⊗ αi I =
m
αi Ai ⊗ I.
i=1
8.4. Multiplication operators. Let L∞ (T2 ) denote the C ∗ algebra of all (classes of) measurable and essentially bounded functions on T2 . If a is in L∞ (T2 ), then the operator M2 (a) : Lp (T2 ) → Lp (T2 ),
ϕ → aϕ
(8.4)
is obviously bounded. It is called the multiplication operator on Lp (T2 ) with symbol a. The arguments of the proof of Proposition 2.2 can be used to show the following. If A ∈ L(Lp (T2 )) (1 < p < ∞) and if there are complex
8.1 Function Classes on the Torus
413
numbers amn (m, n ∈ Z) such that A has the matrix representation (ai−k,j−l ) with respect to the basis {χm ⊗ χn }m,n∈Z in Lp (T2 ), then there exists an a ∈ L∞ (T2 ) such that A = M2 (a). In that case {amn } is the Fourier coeﬃcient sequence of a and M2 (a)L(Lp (T2 )) = aL∞ (T2 ) . Denote the set of sequences {γmn }m,n∈Z with ﬁnite support by 0 (Z2 ). Let a ∈ L1 (T2 ) have Fourier coeﬃcients sequence {amn }m,n∈Z . For x in 0 (Z2 ) deﬁne ai−k,j−l xkl (i, j ∈ Z). (a ∗ x)ij = k,l∈Z
Let M p (T2 ) (1 ≤ p < ∞) denote the collection of all a ∈ L1 (T2 ) with the following property: if x ∈ 0 (Z2 ), then a ∗ x ∈ p (Z2 ) and # ! a ∗ xp : x ∈ 0 (Z2 ), x = 0 < ∞. sup xp If a ∈ M p (T2 ), then the mapping 0 (Z2 ) → p (Z2 ), x → a ∗ x extends to a bounded operator M2 (a) : p (Z2 ) → p (Z2 ),
x → a ∗ x,
(8.5)
which is referred to as the multiplication operator on p (Z2 ) with symbol a. From what was said above we know that M 2 (T2 ) = L∞ (T2 ) and it is easy to show that M 1 (T2 ) coincides with W (T2 ), the algebra of all functions on T2 with absolutely convergent Fourier series (see 2.5). It can be proved as in the onedimensional case (see 2.5) that, for 1 < r < p < 2, 1/r + 1/s = 1, 1/p + 1/q = 1, W (T2 ) ⊂ M r (T2 ) = M s (T2 ) ⊂ M p (T2 ) = M q (T2 ) ⊂ L∞ (T2 ), the embeddings being continuous, and that M p (T2 ) is a Banach algebra under the norm a := M2 (a)L(p (Z2 )) . p 2 2 For a ∈ L∞ N ×N (T ) resp. a ∈ MN ×N (T ) the multiplication operators p p 2 2 M2 (a) on LN (T ) resp. N (Z ) are deﬁned in the natural manner. We introp 2 2 duce norms on L∞ N ×N (T ) resp. MN ×N (T ) by setting 2 = M2 (a)L(L2 (T2 )) , aL∞ N ×N (T ) N
aMNp ×N (T2 ) = M2 (a)L(pN (Z2 )) .
From 1.29(a) we infer that 2 = a∞ := ess sup a(s, t)L(C ) . aL∞ N N ×N (T )
(s,t)∈T2
8.5. Tensor products of subalgebras of M p . If a and b are in M p , then a ⊗ b is in M p (T2 ). This and the equality a ⊗ bM p (T2 ) = aM p bM p follow p from 8.3. Let A and B be two
closed subalgebras of M . Deﬁne A " B as the collection of all ﬁnite sums i ai ⊗ bi (ai ∈ A, bi ∈ B) and let A ⊗ B denote the closure of A " B in M p (T2 ). It is clear that A ⊗ B is a closed subalgebra of
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8 Toeplitz Operators over the QuarterPlane
M p (T2 ). The collection of all functions a ⊗ χ0 (a ∈ A) is a closed subalgebra of A ⊗ B and it will be identiﬁed with A. Hence, if ω ∈ M (A ⊗ B) is a multiplicative linear functional on A ⊗ B, then ωA and ωB belong to M (A) and M (B), respectively. 8.6. Proposition. Let A and B be closed subalgebras of M p having the following property: for each open subset of the maximal ideal space there exists a nonzero element of the algebra whose Gelfand transform is supported in this subset. Then the mapping ϕ : M (A ⊗ B) → M (a) × M (B),
ω → (ωA, ωB)
is a homeomorphism of M (A ⊗ B) onto M (a) × M (b), where the latter space is provided with the product topology. Proof. To show that ϕ is onetoone, suppose ω1 A = ω2 A = α and ω1 B = ω2 B = β. Then ω1 ai ⊗ b i = ai (α)bi (β) = ω2 ai ⊗ b i i
i
i
for every ﬁnite sum i ai ⊗bi = i (ai ⊗χ0 )(χ0 ⊗bi ), and since these sums are dense in A⊗B, it follows that ω1 = ω2 . It is easily seen that ϕ is continuous and because M (A ⊗ B) is compact, ϕ(M (A ⊗ B)) is a closed subset of the product M (A)×M (B). Assume ϕ(M (A⊗B)) is not equal to M (A)×M (B). Then there are open sets U ⊂ M (A) and V ⊂ M (B) such that (U ×V )∩ϕ(M (A⊗B)) = ∅. Choose nonzero a ∈ A and b ∈ B so that supp a ⊂ U and supp b ⊂ V . Then ω(a ⊗ b) = 0 for all ω ∈ M (A ⊗ B), so the spectrum of a ⊗ b in A ⊗ B is {0}, and thus the spectrum of a ⊗ b in L∞ (T2 ) also equals {0}. Since the spectrum in L∞ (T2 ) is the essential range, it follows that either a = 0 or b = 0 , which is a contradiction. Thus ϕ is onto. 8.7. Subalgebras of M p (T2 ). (a) The previous proposition applies to A = B = Cp (1 ≤ p < ∞). Note that Cp ⊗ Cp coincides with Cp (T2 ), the closure in M p (T2 ) of P(T2 ) = P " P. So the fact that M (Cp ⊗ Cp ) = T2 could also be proved using the reasoning of the proof of Proposition 2.46(a). (b) From Proposition 6.28 we see that if U is any open subset of M (P Cp ), there is a nonzero a ∈ P K such that the support of the Gelfand transform of a is entirely contained in U . So the preceding proposition gives that M (P Cp ⊗ P Cp ) = (T × {0, 1}) × (T × {0, 1}). We claim that for each a ∈ P Cp ⊗ P Cp the four limits a(σ ± 0, τ ± 0) =
lim
θ → θ0 ± 0 ψ → ψ0 ± 0
a(eiθ , eiψ )
(8.6)
8.1 Function Classes on the Torus
415
exist and are ﬁnite for each (σ, τ ) = (eiθ0 , eiψ0 ) ∈ T2 . Choose an ∈ P Cp " P Cp so that a − an M p (T2 ) → 0. It is clear that the limits ln := an (σ − 0, τ − 0) exist and that ln − lm  ≤ an − am ∞ . Hence {ln }n∈Z+ is a Cauchy sequence and, consequently, there is an l ∈ C such that ln − l → 0 as n → ∞. We have a(eiθ , eiψ ) − l ≤ a(eiθ , eiψ ) − an (eiθ , eiψ ) + an (eiθ , eiψ ) − ln  + ln − l. (8.7) Given any ε > 0, there is an n0 such that the ﬁrst and the third terms on the right of (8.7) are smaller than ε/3 for n = n0 and then one can ﬁnd a δ = δ(ε, n0 ) such that the second term is smaller than ε/3 whenever θ ∈ (θ0 − δ, θ0 ), ψ ∈ (ψ0 − δ, ψ0 ). This implies that the limit a(σ − 0, τ − 0) exists and equals l. The proof is the same for the remaining three limits in (8.6). Now it is obvious that, for (σ, τ ) ∈ T2 , the Gelfand transform of a function a ∈ P Cp ⊗ P Cp is given by a(σ, 0; τ, 0) = a(σ − 0, τ − 0), a(σ, 1; τ, 0) = a(σ + 0, τ − 0),
a(σ, 0; τ, 1) = a(σ − 0, τ + 0), a(σ, 1; τ, 1) = a(σ + 0, τ + 0).
(c) If A is a C ∗ subalgebra of L∞ = M 2 (e.g., A = C, P C, QC, P QC, CE , QCE , L∞ ), then A satisﬁes the hypothesis of Proposition 8.6. Thus, A ⊗ A is a C ∗ subalgebra of L∞ (T2 ) which
is isometrically starisomorphic to the C ∗ algebra C(M (A) × M (A)). If i ci ⊗ di ∈ A ⊗ A and x, y ∈ M (A), then ci ⊗ di (x, y) = ci (x)di (y) (x, y ∈ M (A)). i
i ∞
(d) We want to show that L ⊗ L∞ does not coincide with L∞ (T2 ). Deﬁne χ ∈ L∞ (T2 ) by χ(s, t) = χU (st−1 ) (s, t ∈ T), where χU ∈ P C is the characteristic function of the upper half circle. Assume χ ∈ L∞ ⊗ L∞ . Then
(n) (n) there are bn := i ci ⊗di ∈ L∞ "L∞ such that χ−bn ∞ → 0 as n → ∞.
(n) (n) Put an := i ci di . Then an ∈ L∞ , and because obviously an − am L∞ ≤ bn − bm L∞ (T2 ) , there exists a function a ∈ L∞ with a − an ∞ → 0 as n → ∞. For (s, t) ∈ T2 we have χ(s, t) − a(t) ≤ χ(s, t) − bn (s, t) + bn (s, t) − an (t) + an (t) − a(t). (8.8) There is an n0 such that the ﬁrst and the third terms on the right of (8.8) are smaller than 1/8 for n = n0 and almost all (s, t) ∈ T2 and t ∈ T, respectively. From writing ci (teih ) − ci (t) di (t) bn0 (teih , t) − an0 (t) = i
we get
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8 Toeplitz Operators over the QuarterPlane
( χ(te , t) − a(t) dm ≤ 2π ih
T
1 1 + 8 8
*
+
i
( di ∞
T
ci (teih ) − ci (t) dm,
c(teih , t) − c(t) dm = 0 for every c ∈ L1 , it follows that there * is a δ > 0 such that ω(h) := T χ(teih , t) − a(t) dm < π for all h ∈ (−δ, δ). But from the deﬁnition of χ we obtain that ( ω(0 + 0) := lim ω(h) = a(t) dm, h→0+0 (T ω(0 − 0) := lim ω(h) = 1 − a(t) dm, and since lim
h→0 T
h→0−0
*
*
T
*
and because T a(t) dm + T 1 − a(t) dm ≥ T dm = 2π it is impossible that both ω(0 + 0) and ω(0 − 0) are smaller than π. This contradiction shows that χ is not in L∞ ⊗ L∞ . (e) Let A and B be closed subalgebras of M p . For functions a ∈ AN ×N p 2 and b ∈ BN ×N deﬁne a ⊗ b ∈ MN Let ×N (T ) by (a ⊗ b)(s, t) = a(s)b(t).
AN ×N " BN ×N denote the collection of all ﬁnite sums of the form i ai ⊗ bi (ai ∈ AN ×N , bi ∈ BN ×N ) and denote the closure of AN ×N " BN ×N in p 2 MN ×N (T ) by AN ×N ⊗ BN ×N .
The collection of all ﬁnite sums i ai ⊗ bi , where ai is in AN ×N and bi in BN ×N is of the form bi = diag (ci , . . . , ci ) with ci ∈ B (of course, one can also think of bi as a scalarvalued function) will be denoted by AN ×N " B. p 2 We let AN ×N ⊗ B refer to the closure of AN ×N " B in MN ×N (T ). Similarly A " BN ×N and A ⊗ BN ×N are deﬁned. It is not diﬃcult to verify that AN ×N " BN ×N = AN ×N " B = A " BN ×N = (A " B)N ×N . The same is therefore true with " replaced by ⊗. We merely
introduced AN ×N " B for the following reason: in the sequel, when writing i ai ⊗ bi ∈ AN ×N " B we shall always assume that bi refers to a diagonal matrix function whose entries on the diagonal are equal to each other (or, equivalently, bi is a scalarvalued function). ∞ (f) Finally, let H ∞ (T2 ) = H++ (T2 ) denote the space of all functions a in 2 L (T ) whose Fourier coeﬃcients amn are zero if m < 0 or n < 0. The spaces ∞ ∞ ∞ ∞ (T2 ), H−+ (T2 ), H+− (T2 ) are deﬁned analogously. Note that H±± (T2 ) H−− ∞ 2 are closed
subalgebras of L (T ). a given by If a = m,n≥0 amn χm ⊗ χn is in H ∞ (T2 ), then the function m n a z w is holomorphic and bounded in D × D. Con a(z, w) = mn m,n≥0 versely, if b is a holomorphic and bounded function in D × D, then there is an a. a ∈ H ∞ (T2 ) such that b = ∞
8.8. Tensor products of operators
on L∞ . Abbreviate L∞ N ×N to Y and f ⊗ g ∈ Y " Y deﬁne (A ⊗ B)h as let A, B ∈ L(Y ). For a ﬁnite sum h = i i i
Af ⊗ Bg . Then i i i
8.2 Elementary Properties of QuarterPlane Operators
417
(A ⊗ B)hY ⊗Y ≤ AL(Y ) BL(Y ) hY ⊗Y . Indeed, if we let X = M (L∞ ), " " " " fi ⊗ Bgi " (I ⊗ B)hY ⊗Y = " i
Y ⊗Y
" " " " = max " fi (x)(Bgi )(y)" x,y∈X
i
L(CN )
(see 8.7(c))
" " " " fi (x)gi (y)" = max "B x,y∈X
≤ BL(Y )
L(CN )
i
" " " " max " fi (x)gi (y)"
x,y∈X
L(CN )
i
= BL(Y ) hY ⊗Y
(again by 8.7(c)),
which implies the asserted inequality as in 8.3. Hence, A ⊗ B extends to a bounded operator on Y ⊗ Y , which will be denoted by A ⊗ B, too. Finally, the argument given in 8.3 also shows that A ⊗ BL(Y ⊗Y ) = AL(Y ) BL(Y ) . ∞ ∞ ∞ For A ∈ L(L∞ N ×N ) and B ∈ L(L ), one can deﬁne A⊗B ∈ L(LN ×N ⊗L ) in a similar fashion. It can be shown as above that ∞ = AL(L∞ BL(L∞ ) . A ⊗ BL(L∞ N ×N ⊗L ) N ×N )
(8.9)
Example. Suppose {Kλ }λ∈Λ is an approximate identity. For λ ∈ Λ, deﬁne kλ ∈ L(L∞ N ×N ) by kλ (ajk ) := (kλ ajk ), where kλ ajk is as in 3.14. Thus, if λ and µ are in Λ, then the operator given by ∞ kλ ⊗ kµ : L∞ fi ⊗ gi → kλ fi ⊗ kµ gi , N ×N " LN ×N → CN ×N " CN ×N , i
i
∞ extends to an operator in L(L∞ N ×N ⊗ LN ×N , CN ×N ⊗ CN ×N ), for which the equality kλ ⊗ kµ = kλ kµ holds.
8.2 Elementary Properties of QuarterPlane Operators 2 8.9. Toeplitz operators over the quarterplane. For a ∈ L∞ N ×N (T ) the p 2 Toeplitz operator T2 (a) with symbol a on HN (T ) (1 < p < ∞) is the (obviously bounded) operator acting by the rule p p (T2 ) → HN (T2 ), T2 (a) : HN
ϕ → P++ (aϕ),
(8.10)
p 2 and for a ∈ MN ×N (T ) (1 ≤ p < ∞) the Toeplitz operator T2 (a) with symbol p 2 a on N (Z++ ) is the (clearly bounded) operator given by
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8 Toeplitz Operators over the QuarterPlane
T2 (a) : pN (Z2++ ) → pN (Z2++ ),
ϕ → P++ (a ∗ ϕ).
(8.11)
Sometimes these operators are also called twodimensional Toeplitz operators. If p = 2, then the operators deﬁned by (8.10) and (8.11) are unitarily equivalent through the isomorphism 2 (T2 ) → 2N (Z2++ ), ϕij χi ⊗ χj → {ϕij }. HN
p p 2 2 Clearly, if ϕ = ϕij χ
i ⊗ χj ∈ HN (T ) resp. ϕ = {ϕij } ∈ N (Z++ ) (ϕij ∈ CN ), then T2 (a)ϕ = ψij χi ⊗ χj resp. T2 (a)ϕ = {ψij }, where ψij = ai−k,j−l ϕkl (i, j ∈ Z+ ) k,l∈Z+
and amn ∈ CN ×N is the m, n Fourier coeﬃcient of a. p ∞ ∞ p Suppose
a = i ci ⊗ di belongs to LN ×N " L resp. MN ×N " M . Then T2 (a) = i T (ci ) ⊗ T (di ) (recall Example (c) in 8.3, but also 8.7(d)). p 2 Every function a ∈ MN ×N (T ) can be written as a(s, t) =
aij si tj =
i,j
bi (t) :=
si bi (t) =
i j
aij t ,
cj (s) :=
j
tj cj (s),
j
aij s
i
(s, t ∈ T).
i
For i ∈ Z+ , let Hi and Ki denote the subspaces Hi := pN ({i} × Z+ ) and Ki := pN (Z+ × {i}) of pN (Z2++ ). Then ·
·
·
·
·
·
pN (Z2++ ) = H0 + H1 + H2 + . . . = K0 + K1 + K2 + . . .
(8.12)
and a simple computation shows that T2 (a) has the matrix representation ∞ (T (bj−k ))∞ j,k=0 and (T (cj−k ))j,k=0 with respect to the ﬁrst and second decomposition in (8.12), respectively. Thus, a Toeplitz operator over the quarterplane can be interpreted as a onedimensional Toeplitz operator with operator entries that are themselves onedimensional Toeplitz operators. The following proposition is the twodimensional analogue of formula (2.20), but it does not even nearly play the same role in the twodimensional theory as formula (2.20) does in the onedimensional situation. p 2 ∞ 2 8.10. Proposition. (a) Suppose a−− , a++ ∈ MN ×N (T ) ∩ HN ×N (T ) and p b ∈ MN ×N (T2 ). Then
T2 (a−− ba++ ) = T2 (a−− )T2 (b)T2 (a++ ). ∞ (T2 )]N ×N , then (b) If a±∓ , b±∓ ∈ [M p (T2 ) ∩ H±∓
T2 (a+− b+− ) = T2 (a+− )T2 (b+− ),
T2 (a−+ b−+ ) = T2 (a−+ )T2 (b−+ ).
8.2 Elementary Properties of QuarterPlane Operators
419
Proof. (a) Abbreviate M2 (f ) to f . Because P++ a−− ba++ P++ equals P++ a−− P++ bP++ a++ P++ + P++ a−− (I − P++ )ba++ P++ +P++ a−− P++ b(I − P++ )a++ P++ +P++ a−− (I − P++ )b(I − P++ )a++ P++ , and since a−− and a++ leave Im (I −P++ ) and Im P++ , respectively, invariant, and since P++ (I − P++ ) = (I − P++ )P++ = 0, we get the asserted formula. (b) The proof goes similarly. We now state three theorems about twodimensional Toeplitz operators whose proofs can be carried out using the same arguments as in the proofs of their onedimensional analogues. 2 8.11. Theorem. (a) If 1 < p < ∞ and a ∈ L∞ N ×N (T ), then
c1 a∞ ≤ T2 (a)Φ(HNp (T2 )) ≤ T2 (a)L(HNp (T2 )) ≤ c2 a∞ ,
(8.13)
with certain positive constants c1 and c2 independent of a. p 2 (b) If 1 ≤ p < ∞ and a ∈ MN ×N (T ), then
T2 (a)Φ(pN (Z2++ )) = T2 (a)L(pN (Z2++ )) = aMNp ×N (T2 ) .
(8.14)
Proof. (a) The second and the third inequalities are obvious. To see that c1 a∞ ≤ T2 (a)ess proceed as in the proofs of Propositions 4.1(a) and p p (T2 )), then, because V n ⊗ V n → 0 weakly on HN ⊗ Hp 4.4(d): if K ∈ C∞ (HN (−n) (−n) n n and (V ⊗V )T2 (a)(V ⊗ V ) = T2 (a), we have T2 (a) ≤ lim inf (V (−n) ⊗ V (−n) )(T2 (a) + K)(V n ⊗ V n ) n→∞ ≤ sup P++ U −n ⊗ U −n T2 (a) + K V n ⊗ V n n
= P++ T2 (a) + K, whence T2 (a) ≤ P++ T2 (a)ess , and since T2 (a) ≥ cP++ M2 (a)P++ = c(U −n ⊗ U −n )P++ (U n ⊗ U n )M2 (a)(U −n ⊗ U −n )P++ (U n ⊗ U n ) ≥ cM2 (a) ≥ c1 a∞ (note that (U −n ⊗ U −n )P++ (U n ⊗ U n ) → I strongly on LpN ⊗ Lp ), we get the inequality c1 a∞ ≤ T2 (a)ess . (b) This follows from the arguments of part (a) along with the equality P++ = 1.
420
8 Toeplitz Operators over the QuarterPlane
2 8.12. Toeplitz operators over the halfplane. For a ∈ L∞ N ×N (T ) resp. p 2 a ∈ MN ×N (T ) the (obviously bounded) operators p p ⊗ Lp → HN ⊗ Lp , T+· (a) : HN
ϕ → (P ⊗ I)(M2 (a)ϕ),
p HN
ϕ → (I ⊗ P )(M2 (a)ϕ),
T·+ (a) : Lp ⊗ T+· (a) : T·+ (a) :
→ Lp ⊗
pN (Z+ × Z) pN (Z × Z+ )
→ →
p HN ,
pN (Z+ × Z), pN (Z × Z+ ),
x → (P ⊗ I)(M2 (a)x), x → (I ⊗ P )(M2 (a)x)
are called Toeplitz operators over the half plane. If we set Hi := pN ({i} × Z), Ki := pN (Z+ × {i}), then the operator T+· (a) ∞ has the matrix representations (M (bj−k ))∞ j,k=0 , (T (cj−k ))j,k=−∞ with respect to the decompositions ·
·
·
·
·
·
pN (Z+ × Z) = H0 + H1 + . . . = . . . + K−1 + K0 + K1 + . . . , where b and c are as in 8.9. 2 8.13. Theorem. (a) If 1 < p < ∞ and a ∈ L∞ N ×N (T ), then
M2 (a) ∈ Φ± (LpN (T2 ))
⇐⇒ M2 (a) ∈ GL(LpN (T2 )) 2 ⇐⇒ a ∈ GL∞ N ×N (T ),
p p T2 (a) ∈ Φ± (HN ⊗ H p ) =⇒ T+· (a) ∈ GL(HN ⊗ H p) 2 =⇒ a ∈ GL∞ N ×N (T ), p 2 T+· (a) ∈ Φ± (HN ⊗ H p ) =⇒ a ∈ GL∞ N ×N (T ). p 2 (b) If 1 ≤ p < ∞ and a ∈ MN ×N (T ), then
M2 (a) ∈ Φ(pN (Z2 ))
⇐⇒ M2 (a) ∈ GL(pN (Z2 )) p 2 ⇐⇒ a ∈ GMN ×N (T ),
T2 (a) ∈ Φ(pN (Z2++ ))
=⇒ T+· (a) ∈ GL(pN (Z+ × Z)) p 2 =⇒ a ∈ GMN ×N (T ),
p 2 T+· (a) ∈ Φ(pN (Z+ × Z)) =⇒ a ∈ GMN ×N (T ).
Proof. The assertions for the multiplication operators can be proved similarly as their scalarvalued onedimensional analogues (2.28 and 2.29). The proof of Theorem 2.30 with an appropriate replacements of U ±n by I ⊗ U ±n and U ±n ⊗ I gives the implication concerning Toeplitz operators. 2 2 8.14. Theorem. Suppose a ∈ L∞ N ×N (T ) is sectorial on T , i.e., there are c, d ∈ GCN ×N and ε > 0 such that Re (ca(s, t)dz, z) ≥ εz2 for all z ∈ CN 2 and almost all (s, t) ∈ T2 . Then T2 (a) ∈ GL(HN (T2 )).
8.3 Continuous Symbols
421
Proof. See the proof of Corollary 3.62. 8.15. Corollary. If a ∈ L∞ (T2 ), then R(a) ⊂ spΦ(H 2 (T2 )) T2 (a) ⊂ spL(H 2 (T2 )) T2 (a) ⊂ conv R(a), where R(a) = spL∞ (T2 ) (a) is the essential range of a. Proof. Combine Theorems 8.13 and 8.14.
8.3 Continuous Symbols 8.16. Deﬁnitions. (a) Let x ∈ X = M (L∞ ) and let Γx denote the operator in L(L∞ ) which assigns the constant function a(x) to a function a ∈ L∞ . From 8.8 we know that the operator Γx ⊗ I is well deﬁned and bounded ∞ ∞ 1 ⊗ L∞ ⊗ L∞ on L
N ×N . For a ∈ L N ×N , deﬁne
ax := (Γx ⊗ I)a. Thus, if ∞ ∞ 1 a = i ci ⊗ di ∈ L " LN ×N , then ax (t) = i ci (x)di (t) (t ∈ T). Moreover, we have 2 ≤ aL∞ ∀ a ∈ L∞ ⊗ L∞ (8.15) a1x L∞ N ×N . N ×N N ×N (T ) 1 If A is a C ∗ subalgebra of L∞ and a ∈ A ⊗ L∞ N ×N , then ax is the same ∞ function in LN ×N for all x in the ﬁber Xα (α ∈ M (A)). This function will ∞ be denoted by a1α . For example, if a = i ci ⊗ di ∈ P C " LN ×N , then 1 2 aτ ±0 (t) = i ci (τ ± 0)di (t) (t ∈ T). Finally, deﬁne ax as (I ⊗ Γx )a, and for all 2 2 a ∈ L∞ N ×N ⊗ A and α ∈ M (A) deﬁne aα as ax , where x is any point in Xα .
(b) Now let a = i ci ⊗ di ∈ P " P. If τ ∈ T, then a1τ ∈ P and
a1τ ∞ = max a(τ, t) ≤ a∞ . t∈T
We also have ( · W being the Wiener norm) + + + ++ + + + a1τ W = ci (τ )(di )n + = (ci )m (di )n τ m + + + n
n
m
i + i+ + ++ + + + m = τ (ci )m (di )n + ≤ (ci )m (di )n + + + n
m
" " " " =" ci ⊗ d i " i
n,m
i W ⊗W
i
= aW ⊗W .
Hence, I ⊗ M (a1τ )L(p ⊗p ) ≤ M2 (a)L(p ⊗p ) for p = 1, 2, and the RieszThorin interpolation theorem combined with passage to adjoints extends this p to all values p ∈ [1, ∞). So it is clear that a1τ ∈ CN ×N is well deﬁned for p p a ∈ C ⊗ CN ×N and that a1τ MNp ×N ≤ caMNp ×N (T2 )
p 2 ∀ a ∈ CN ×N (T ),
422
8 Toeplitz Operators over the QuarterPlane
with some c independent of a. Of course, the matrix function a2τ given by a2τ (t) = a(t, τ ) has similar properties as a1τ .
p (c) Assume a = i ci ⊗ di ∈ P K " MN ×N (recall 6.25). We claim that there is a constant c depending only on p and N such that a1α MNp ×N ≤ caMNp ×N (T2 )
∀ α = (τ, j) ∈ T × {0, 1} = M (P Cp ), (8.16)
where a1(τ,0) (t) := i ci (τ − 0)di (t) and a1(τ,1) (t) := i ci (τ + 0)di (t). To prove this claim we may conﬁne ourselves to the case N = 1. Put bi = ci − ci (τ − 0)χ0 . Then there exists an ε > 0 such that bi (t) = 0 whenever arg τ − ε < arg t < arg τ . Choose u ∈ P K so that u(t) = 1 for t satisfying arg τ − ε/2 < arg t < arg τ and u(t) = 0 otherwise. So bi (t)u(t) = 0 for all t ∈ T. We have 1 = u∞ ≤ uM p ≤ cp u∞ + V1 (u) ≤ cp (1 + 2) = 3cp and hence, " " " " ci (τ − 0)di " "
Mp
" " " " ≤ uM p " ci (τ − 0)di "
" "i " " = "u ⊗ ci (τ − 0)di " " " i " " =" ci u ⊗ d i "
M p (T2 )
i
" " " " ≤ 3cp " ci ⊗ d i " i
Mp
M p (T2 )
i " " " " =" ci (τ − 0)u ⊗ di "
" i " " " =" ci ⊗ di (u ⊗ χ0 )"
M p (T2 )
i
M p (T2 )
M p (T2 )
.
As the same arguments applies to τ + 0 in place of τ − 0, we get our claim. p 1 Thus, for α ∈ M (P Cp ) and a ∈ P Cp ⊗ MN ×N the matrix function aα is well p deﬁned and (8.16) holds. A similar statement is valid for a ∈ MN ×N ⊗ P Cp and a2α . (d) For a ∈ GC(T2 ) deﬁne the mapping i1 by i1 : T → Z, τ → ind a2τ . From 2.41(b) we know that i1 is continuous, and therefore i1 must be constant. Let ind1 a denote this constant value and deﬁne ind2 a analogously. It can be shown that the abstract index group of C(T2 ) (see Douglas [162, 2.10]) is isomorphic to Z2 . A function a ∈ GC(T2 ) belongs to the connected component of GC(T2 ) containing the identity if and only if ind1 a = ind2 a = 0. Also notice the following fact: if a ∈ GC(T2 ) and f1 , f2 are continuous mappings of T into T, then ind a(f1 (t), f2 (t)) = (ind1 a)(ind f1 ) + (ind2 a)(ind f2 ).
(8.17)
8.17. Deﬁnitions. Suppose {Y, Z} ⊂ {H p , Lp } (1 < p < ∞) or {Y, Z} ⊂ {p (Z+ ), p (Z)} (1 ≤ p < ∞). Given subset A ⊂ L(YN ) and B ⊂ L(Z) deﬁne
8.3 Continuous Symbols
423
A " B as the collection of all ﬁnite sums i Ai ⊗ Bi (Ai ∈ A, Bi ∈ B) and let A ⊗ B denote the closure of A " B in L(YN ⊗ Z). It is easy to see that C∞ (YN ) ⊗ L(Z), L(YN ) ⊗ C∞ (Z), and C∞ (YN ) ⊗ C∞ (Z) are closed twosided ideals of L(YN ) ⊗ L(Z). It is clear that C∞ (YN ) ⊗ C∞ (Z) is a subset of C∞ (YN ⊗ Z). Since the (ﬁniterank) projections Pn deﬁned in 7.5 converge strongly to I on both YN and Z and since each ﬁniterank operator on YN ⊗ Z is readily seen to be in C∞ (YN ) ⊗ C∞ (Z), it results that C∞ (YN ) ⊗ C∞ (Z) is actually equal to C∞ (YN ⊗ Z). Put Lπ1 (YN ⊗ Z) := L(YN ) ⊗ L(Z)/C∞ (YN ) ⊗ L(Z), Lπ2 (YN ⊗ Z) := L(YN ) ⊗ L(Z)/L(YN ) ⊗ C∞ (Z), Lπ12 (YN ⊗ Z) := L(YN ) ⊗ L(Z)/C∞ (YN ) ⊗ C∞ (Z), and for A ∈ L(YN ) ⊗ L(Z) let Aπ1 , Aπ2 , Aπ12 denote the coset in the corresponding quotient algebra containing A. Notice that the above deﬁnitions can also be made for the case that Z is provided with the subscript N , i.e., that A ⊂ L(Y ) and B ⊂ L(ZN ). 8.18. Lemma. Let A ∈ L(YN ) ⊗ L(Z) and suppose Aπ1 ∈ GLπ1 (YN ⊗ Z) and Aπ2 ∈ GLπ2 (YN ⊗ Z). Then Aπ12 is in GLπ12 (YN ⊗ Z) and if B1π and C2π are the inverses of Aπ1 and Aπ2 , respectively, then (B + C − BAC)π12 is the inverse of Aπ12 . Proof. If BA = I + K1 and CA + I + K2 with K1 ∈ C∞ (YN ) ⊗ L(Z) and K2 ∈ L(YN ) ⊗ C∞ (Z), then (B + C − BAC)A = I − K1 K2 , and a little thought shows that K1 K2 ∈ C∞ (YN ) ⊗ C∞ (Z). Before turning to Toeplitz operators over the quarterplane we state two propositions on Toeplitz operators over the halfplane. The ﬁrst provides a suﬃcient condition for the invertibility of a halfplane operator for a large class of symbols, and the second shows that this condition is even necessary for the operator to be Fredholm in case the symbol is continuous. 8.19. Proposition. Let A be a C ∗ subalgebra of L∞ containing the constants resp. A ∈ {Cp , P Cp } and let H p (1 < p < ∞) resp. p (1 ≤ p < ∞ for A = Cp and 1 < p < ∞ for A = P Cp ) be the underlying space. Let b ∈ (L∞ ⊗ A)N ×N resp. b ∈ (M p ⊗ A)N ×N . If T (b2α ) is invertible for each α ∈ M (A), then T+· (b) is invertible in p L(HN ) ⊗ L(Lp )
resp.
L(pN (Z+ )) ⊗ L(p (Z))
p and, consequently, invertible in L(HN ⊗ Lp ) resp. L(pN (Z+ ) ⊗ p (Z)).
Proof. We only consider the H p case; apart from some technical details (such as in the proof of Theorem 2.69), the p case can be treated similarly. For α ∈ M (A), let Nα denote the collection of all ϕ ∈ A ∼ = C(M (A)) such that 0 ≤ ϕ ≤ 1 and ϕ is identically 1 in some neighborhood of α (depending on
424
8 Toeplitz Operators over the QuarterPlane
ϕ). Put Mα = {I ⊗ M (ϕ) : ϕ ∈ Nα }, where I refers to the identity operator p . It can be easily veriﬁed that {Mα }α∈M (A) is a covering system of on HN p p that T+· (b) commutes with every localizing classes in L(HN
) ⊗ L(L ) and operator in Mα . If b = i ci ⊗ di ∈ L∞ N ×N " A and ϕ ∈ Nα , then α
[T+· (b) − T (b2α ) ⊗ I] [I ⊗ M (ϕ)] = T (ci ) ⊗ M (di ) − T (ci ) ⊗ di (α)I [I ⊗ M (ϕ)] i
=
i
1 0 T (ci ) ⊗ M di − di (α)χ0 .
i
This implies that T+· (b) and T (b2α ) ⊗ I are Mα equivalent from the left. Lemma 7.70 shows that this is also true for b ∈ L∞ N ×N ⊗ A. The right equivalence can be proved analogously. To complete the proof is remains to apply Theorem 1.32(a). 8.20. Proposition. Let B be a C ∗ algebra between C and QC resp. B = Cp and let H p (1 < p < ∞) resp. p (1 < p < ∞) be the underlying space. If b ∈ (B ⊗ B)N ×N , then the following are equivalent. (i) T+· (b) is Fredholm. (ii) T+· (b) is invertible. p (iii) T+· (b) is invertible in L(HN ) ⊗ L(Lp ) resp. L(pN (Z+ )) ⊗ L(p (Z)).
(iv) T (b2β ) is invertible for each β ∈ M (B). Proof. Again we only consider the H p case. The implication (iv) =⇒ (iii) results from the preceding proposition and the implications (iii) =⇒ (ii) =⇒ (i) are trivial. So suppose (i) holds. From Theorem 8.13 we obtain that b p ). It remains to show that is in G(B ⊗ B)N ×N , and hence T (b2β ) ∈ Φ(HN Ker T (b2β ) = Ker T ∗ (b2β ) = {0}. Deﬁne Nβ (β ∈ M (B)) as in the previous proof and put p Mπβ = I ⊗ M (ϕ) + C∞ (HN ⊗ Lp ) : ϕ ∈ Nβ . Then {Mπβ }β∈M (B) is a covering system of localizing classes in the algebra p p p ⊗ Lp )/C∞ (HN ⊗ Lp ), T+· (b) + C∞ (HN ⊗ Lp ) commutes with Mπβ and L(HN β p ⊗ Lp ). So is Mπβ equivalent from the left and the right to T (b2β ) ⊗ I + C∞ (HN p p ⊗Lp ), Theorem 1.32(a) implies that there are Ai ∈ L(HN ⊗Lp ), Ki ∈ C∞ (HN and ϕi ∈ B (i = 1, 2) such that A1 T (b2β ) ⊗ I I ⊗ M (ϕ1 ) = I ⊗ M (ϕ1 ) + K1 , (8.18) 2 I ⊗ M (ϕ2 ) T (bβ ) ⊗ I A2 = I ⊗ M (ϕ2 ) + K2 . (8.19)
8.3 Continuous Symbols
425
We show that (8.18) implies that Ker T (b2β ) = {0}. Taking the adjoint of (8.19) we then obtain analogously that Ker T ∗ (b2β ) = {0}. Assume T (b2β )f = 0. Then (8.18) gives that f ⊗ M (ϕ1 )U n χ0 = −K1 (I ⊗ U n )(f ⊗ χ0 ) ∀ n ∈ Z+ p and since I ⊗ U n converges weakly to zero on HN ⊗ Lp and K1 is compact, it n follows that f ⊗ M (ϕ1 )U χ0 → 0 as n → ∞. But
f ⊗ M (ϕ1 )U n χ0 = f M (ϕ1 )U n χ0 = f ϕ1 , hence f ϕ1 = 0 and thus f = 0, as desired.
Remark 1. See also Corollary 8.80. Remark 2. If b ∈ WN ×N (T2 ) and T+· (b) is invertible on 1N (Z+ × Z), then T (a2τ ) is in GL(1N (Z+ )) for each τ ∈ T. To see this localize as in the proof of Proposition 8.19 to get the above equalities (8.18), (8.19) with K1 = K2 = 0. 8.21. Theorem. (a) Let B be a C ∗ algebra between C and QC, let a be in (B ⊗ B)N ×N , and let 1 < p < ∞. Then p p T2 (a) ∈ Φ(HN (T2 )) ⇐⇒ T (a1β ), T (a2β ) ∈ GL(HN )
∀ β ∈ M (B).
(b) Let a ∈ (Cp ⊗ Cp )N ×N and 1 ≤ p < ∞. Then T2 (a) ∈ Φ(pN (Z2++ )) ⇐⇒ T (a1τ ), T (a2τ ) ∈ GL(pN )
∀ τ ∈ T.
(c) Under the hypotheses of (a) or (b), if T2 (a) is Fredholm then T+· (a) and T·+ (a) are invertible and −1 −1 −1 −1 P++ T+· (a) + T·+ (a) − T+· (a)P++ M2 (a)P++ T·+ (a) P++ is a regularizer of T2 (a). Proof. (a), (c) Theorem 8.13 and Proposition 8.20 give the implication “=⇒”. Conversely, suppose T (a1β ) and T (a2β ) are invertible for all β ∈ M (B). From Proposition 8.20 (Proposition 8.19) we deduce that T+· (a) is invertible and p ) ⊗ L(Lp ). We have that its inverse is in L(HN −1 (P ⊗ P )T+· (a)(P ⊗ P )T2 (a) = (P ⊗ P )[(P ⊗ I)a(P ⊗ I)]−1 (P ⊗ I)a(P ⊗ I)(P ⊗ P )
−(P ⊗ P )[(P ⊗ I)a(P ⊗ I)]−1 (P ⊗ Q)a(P ⊗ P ). (8.20)
The ﬁrst term on the right equals P ⊗ P . If a = i ci ⊗ di ∈ QCN ×N " QC, then p P ci P ⊗ Qdi P ∈ L(HN ) " C∞ (Lp ), (P ⊗ Q)a(P ⊗ P ) = i
426
8 Toeplitz Operators over the QuarterPlane
and since QCN ×N ⊗ QC is the closure of QCN ×N " QC, we conclude that p ) ⊗ C∞ (Lp ) for every a ∈ QCN ×N ⊗ QC. It follows (P ⊗ Q)a(P ⊗ P ) is in L(HN p ) ⊗ C∞ (Lp ). A that the second term on the right of (8.20) belongs to L(HN similar reasoning for T·+ (a) in place of T+· (a) and Lemma 8.18 yield the implication “⇐=” and part (c). (b), (c) Proceed as in the H p case and take into account Remark 2 of 8.20. 8.22. Corollary. Let a ∈ C(T2 ) resp. a ∈ Cp (T2 ). Then T2 (a) is Fredholm on H p (T2 ) (1 < p < ∞) resp. p (Z2++ ) (1 ≤ p < ∞) if and only if a(s, t) = 0
∀ (s, t) ∈ T2 ,
ind1 a = ind2 a = 0.
(8.21)
If T2 (a) is Fredholm, then Ind T2 (a) = 0. Proof. It remains to prove that Ind T2 (a) = 0 if (8.21) is fulﬁlled. Since the functions satisfying (8.21) are just the functions belonging to the connected component of GC(T2 ) containing the identity and since the index is constant on each connected component of GC(T2 ) (Theorem 8.11(a)), we obtain the equality Ind T2 (a) = 0 in the H p case. To see that the same is true in the p case note that there is a b ∈ P(T2 ) such that Ind T2 (a) = Ind T2 (b) and b ∈ GC(T2 ), ind1 b = ind2 b = 0. Remark. Sazonov [458] showed that if a ∈ C(T2 ) and T2 (a) is normally solvable on H 2 (T2 ), then either a(s, t) = 0 for all (s, t) ∈ T2 or a vanishes identically. He also proved that if a ∈ GC(T2 ), then there exists an operator K ∈ C∞ (H 2 (T2 )) such that T2 (a) + K is normally solvable on H 2 (T2 ). Open problem. Let a ∈ QC ⊗ QC and suppose T2 (a) ∈ Φ(H 2 (T2 )). Is Ind T2 (a) equal to zero? 8.23. Factorable symbols. (a) Assume a = a−− a−+ a+− a++ , where a±± 2 ∞ 2 and a−1 ±± are in C(T ) ∩ H±± (T ) (here and in the following “±±” means one of the four pairs “+−”, “++”, “−+”, “−−”, that is, the second sign does not depend on the ﬁrst; equality of pairs is understood as usual). In that case T+· (a) and T·+ (a) are invertible on the corresponding H p spaces. The inverses are −1 −1 −1 −1 T+· (a) = (P ⊗ I)a−1 ++ a+− (P ⊗ I)a−+ a−− (P ⊗ I), −1 −1 −1 −1 T·+ (a) = (I ⊗ P )a−1 ++ a−+ (I ⊗ P )a+− a−− (I ⊗ P ).
This can be readily veriﬁed by a direct calculation (take into account that, for example, (Q ⊗ I)a+· (P ⊗ I) = 0 if the Fourier coeﬃcient sequence of a+· is supported in the right halfplane Z+ × Z). (b) Thus, under the hypothesis of (a), Theorem 8.21(c) provides a regularizer of T2 (a) which can be explicitly computed. Another (somewhat simpler) regularizer was discovered by Strang [508]:
8.3 Continuous Symbols
427
−1 −1 R = P++ T+· (a) + T·+ (a) − M2 (a−1 ) P++ . Let us prove this. Write P+− = P ⊗ Q, P−+ = Q ⊗ P , P−− = Q ⊗ Q. Then −1 −1 P++ T+· (a)P++ aP++ = P++ − P++ T+· (a)P+− aP++ −1 −1 −1 = P++ − P++ a−1 ++ a+− (P++ + P+− )a−+ a−− P+− aP++ , −1 −1 P++ T·+ (a)P++ aP++ = P++ − P++ T·+ (a)P−+ aP++ −1 −1 −1 = P++ − P++ a−1 ++ a−+ (P++ + P−+ )a+− a−− P−+ aP++ ,
−P++ a−1 P++ aP++ = −P++ + P++ a−1 P+− aP++ + P++ a−1 P−+ aP++ + P++ a−1 P−− aP++ , adding these equalities we arrive at RT2 (a) = P++ + P++ a−1 P−− aP++ −1 −1 −1 +P++ a−1 ++ a+− (P−+ + P−− )a−+ a−− P+− aP++ −1 −1 −1 +P++ a−1 ++ a−+ (P+− + P−− )a+− a−− P−+ aP++ ,
and since P++ ϕP−− , P+− ϕP−+ , P−+ ϕP+− , P−− ϕP++ are compact whenever ϕ ∈ C ⊗ C, it follows that RT2 (a) − P++ is compact, as desired. (c) The results of (a) and (b) remain valid for Toeplitz operators on p 2 ∞ 2 spaces if one requires that a±± and a−1 ±± are in Cp (T ) ∩ H±± (T ). (d) If a ∈ W (T2 ), a(s, t) = 0 for all (s, t) ∈ T2 , and ind1 a = ind2 a = 0, then a admits a factorization a = a−− a−+ a+− a++ with a±± and a−1 ±± in ∞ (T2 ). This follows from the fact that under these assumptions W (T2 ) ∩ H±± a is in the connected component of GW (T2 ) containing the identity, so that a = exp b with some b ∈ W (T2 ) (see 1.16(a)), and therefore a±± = exp(P±± b) are the factors of the wanted factorization. (e) If a ∈ (C ⊗ C)N ×N is suﬃciently smooth (e.g., if the second partial derivatives satisfy a H¨ older condition) and if T+· (a) and T·+ (a) are invertible on H 2 , then a = a−· a+· = a·− a·+ , −1 where the functions a±· , a−1 ±· and a·± , a·± are smooth and belong to the spaces ∞ 2 ∞ 2 [H±· (T )]N ×N and [H·± (T )]N ×N , respectively. Under these conditions T2 (a) 2 (T2 )) and the index of T2 (a) equals is in Φ(HN
−1 1 −1 −1 tr [S1 , a−· ]a−1 −· [S2 , a·− ]a·− − tr a+· [S1 , a+· ]a·+ [S2 , a·+ ] . 4 This formula is due to Duduchava [172]. Here S1 = S ⊗ I, S2 := I ⊗ S (S := 2P −I being the Cauchy singular integral operator), [A, B] := AB−BA, and if 2 (T2 )) is an integral operator with suﬃciently smooth matrixkernel K ∈ C1 (HN
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8 Toeplitz Operators over the QuarterPlane
k(s1 , s2 ; t1 , t2 ) = (kij (s1 , s2 ; t1 , t2 ))N i,j=1 (the operators in the parentheses of the above expression are of this kind), then tr K =
N (( j=1
T2
kjj (t1 , t2 ; t1 , t2 ) dt1 dt2 .
sn −tm (f) If a(s, t) = (n, m ∈ Z+ ), then T2 (a) ∈ Φ(H22 (T2 )) but t−m s−n Ind T2 (a) = −nm. This can be veriﬁed with the help of the formula in (e). Another proof of this fact is in Douglas [160, pp. 45–46]. Also see Coburn, Douglas, Singer [145].
8.4 The Invertibility Problem Corollary 8.22 provides us with an eﬀective criterion for deciding whether a given Toeplitz operator over the quarterplane is Fredholm. However, it is far from easy to decide whether such an operator is invertible. In the onedimensional scalar case this question can be answered by computing the index (Corollary 2.40). A halfplane operator is invertible if and only if it is Fredholm (Proposition 8.20), and thus it has always index zero. We also know that the index of a quarterplane operator with scalarvalued continuous symbol is always zero, but we shall see that there exist such operators which are Fredholm but not invertible. The purpose of what follows is to list some classes of symbols for which the invertibility problem for the corresponding Toeplitz operators is solved (also see 8.45(e)). ∞ (T2 )]N ×N and 1 < p < ∞. Then the 8.24. Theorem. (a) Let a ∈ [H±± following are equivalent. p (i) T2 (a) ∈ Φ(HN (T2 )). p (ii) T2 (a) ∈ GL(HN (T2 )). ∞ (iii) a ∈ G[H±± (T2 )]N ×N . ∞ (iv) det a ∈ GH±± (T2 ).
(v)
inf (z,w)∈D×D
(det a)(z ±1 , w±1 ) > 0.
p ∞ 2 (T2 )]N ×N ∩ MN (b) Let a ∈ [H±± ×N (T ) and 1 ≤ p < ∞. Then the conditions p (T2 ) by pN (Z2++ ) in (i), (ii) (i)–(v) are equivalent provided one replaces HN p 2 and adds “and a ∈ GMN ×N (T )” in (iii)–(v).
Proof. (a) The equivalences (iii) ⇐⇒ (iv) ⇐⇒ (v) can be easily veriﬁed (recall 8.7(f)). The implications (iii) =⇒ (ii) =⇒ (i) are trivial. So assume (i) holds, ∞ (T2 )]N ×N (the other cases and for the sake of deﬁniteness let a = a+− ∈ [H+−
8.4 The Invertibility Problem
429
∞ 2 can be treated similarly). By Theorem 8.13, b = a−1 +− ∈ LN ×N (T ) and also p q p ∗ q T+· (a+− ) ∈ GL(HN ⊗ L ) and T·+ (a+− ) ∈ GL(L ⊗ HN ). Hence, there exist ϕ+· ∈ (H p ⊗ Lp )N ×N and ψ·+ ∈ (Lq ⊗ H q )N ×N such that
a+− ϕ+· = (P ⊗ I)(a+· ϕ+· ) = IN ×N ,
a∗+− ψ·+ = (I ⊗ P )(a∗+− ψ·+ ) = IN ×N ,
∗ p p q q therefore b = a−1 +− = ϕ+· = ψ·+ ∈ (H ⊗ L )N ×N ∩ (L ⊗ H )N ×N , which ∞ 2 implies that b ∈ [H+− (T )]N ×N .
Remark. Thus, if a is either of the form a = a−− a−+ a++ or of the form ∞ (T2 )]N ×N , then T2 (a) is invertible on a = a−− a+− a++ with a±± ∈ G[H±± p −1 −1 −1 2 HN (T ) and the inverse is T2 (a) = T2 (a−1 ++ )T2 (a∓± )T2 (a−− ). 8.25. Toeplitz operators with analytic symbols on 2,∞ (Z2++ ). Let 2,∞ (Z2++ ) be the linear space of all sequences x = {xjk }∞ 2,∞ ++ := j,k=0 for which x2m :=
∞
xjk 2 (j + 1)2m (k + 1)2m < ∞ ∀ m ∈ Z+ .
j,k=0 2,∞ On deﬁning a metric in analogy to (6.27) we make ++ into a Fr´echet space. ∞ 2 If a ∈ C ∞ (T2 ), then T2 (a) is obviously bounded on 2,∞ ++ . Put C±± (T ) := ∞ (T2 ). The following results are due to Gorodetsky [243]. C ∞ (T2 ) ∩ H±± ∞ (T2 ), then (a) If a++ ∈ C++ 2,∞ 2 2 T (a++ ) ∈ Φ(2,∞ ++ ) ⇐⇒ T (a++ ) ∈ GL(++ ) ⇐⇒ T (a++ ) ∈ GL( (Z++ )). ∞ ∞ (T2 ) and λ ∈ T, deﬁne a− (b) For a function a+− ∈ C+− λ ∈ C (T) by := a+− (λ, t) (t ∈ T). Then the following are equivalent.
a− λ (t)
(i) T (a+− ) ∈ Φ(2,∞ ++ ). (ii) T (a+− ) ∈ GL(2,∞ ++ ). (iii) a+− (z, ∞) = 0 for all z ≤ 1 and the operator T (a− λ ) is invertible on 2,∞ (Z+ ) for all λ ∈ T. (iv) a+− (z, ∞) = 0 for all z ≤ 1, and for each λ ∈ T, the function a− λ has (z) = 0 for z > 1. at most ﬁnitely many zeros of integral order on T and a− λ Examples: If a(s, t) = 1 − sn t−m (n, m ∈ Z+ ) and b(s, t) = 2 − (1 + s)t−1 , 2,∞ . then T2 (a) and T2 (b) are invertible on ++ ∞ ∞ (T2 ) and λ ∈ T, deﬁne a− (c) For a function a−− ∈ C−− λ ∈ C (T) by = a−− (t, λt) (t ∈ T). Then the following are equivalent.
a− λ (t)
(i) T2 (a−− ) ∈ Φ(2,∞ ++ ). 2,∞ (ii) T2 (a−− ) ∈ GL(++ ).
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8 Toeplitz Operators over the QuarterPlane
j k (iii) a−− (∞, ∞) = 0, and if j,k≥0 bjk z w is the series expansion of the −1 , w−1 ) in a neighborhood of (0, 0), then function a−1 −− (z bjk  ≤ C(j + 1)m (k + 1)m
∀ j, k ≥ 0,
where C > 0 and m > 0 are certain constants independent of j and k. 2,∞ (Z+ )) for all λ ∈ T. (iv) T (a− λ ) ∈ GL(
(v) For each λ ∈ T the function a− λ has at most ﬁnitely many zeros of integral (z) = 0 for z > 1. order on T and a− λ Examples: 1 − s−n t−m (n, m ∈ Z+ ) and 2 − (1 + s−1 )t−1 generate invertible 2,∞ . Toeplitz operators on ++ 8.26. Lemma. If A is invertible on Y and Fredholm of index zero on Z and if Z ⊂ Y or Z ∗ ⊂ Y ∗ , then A is invertible on Z. Proof. If Z ⊂ Y (resp. Z ∗ ⊂ Y ∗ ), then the kernel of A in Z (resp. the kernel of A∗ in Z ∗ ) is contained in the kernel of A in Y (resp. the kernel of A∗ in Y ∗ ). The next theorem shows that Toeplitz operators over the quarterplane with, in a sense, almost analytic symbols are invertible if and only if they are Fredholm. 8.27. Theorem (Malyshev/Douglas). Let b++ be in C(T2 )∩H ∞ (T2 ) resp. Cp (T2 ) ∩ H ∞ (T2 ), let (α, β) ∈ D × D, and let a(s, t) = (s − α)−1 (t − β)−1 b++ (s, t)
(s, t ∈ T).
Then T2 (a) is invertible on H p (T2 ) (1 < p < ∞) resp. p (Z2++ ) (1 ≤ p < ∞) if and only if (8.21) holds. This result was established by Malyshev [344] for α = β = 0 and p = 1, was then generalized to the case (α, β) ∈ D × D and p = 2 by Douglas [160], and was explicitly stated in the present form by Duduchava [174]. A proof for p = 2 is in Douglas [160, pp. 47–48]. Lemma 8.26 and Corollary 8.22 extend this result to other values of p. 8.28. Theorem (Osher). Let a, b, c ∈ W and g(s, t) = a(s)b(t) + c(t)
(s, t ∈ T).
Then T2 (g) is invertible on H p (T2 ) (1 < p < ∞) resp. p (Z2+ ) (1 ≤ p < ∞) if and only if (8.21) is satisﬁed. Proof. In view of Corollary 8.22 and Lemma 8.26 it suﬃces to consider the case p = 2. So assume T2 (g) ∈ Φ(2 (Z2++ )). Note that the operator T2 (g) = T (a) ⊗ T (b) + I ⊗ T (c) has the matrix representation A = (bj−k T (a) + cj−k I)∞ j,k=0 with respect to the second decomposition in (8.12).
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431
We claim that for each µ ∈ sp T (a) the function µb + c does not vanish on T and has index zero. For µ ∈ a(T) this is immediate from (8.21). Let µ0 ∈ sp T (a) \ a(T) and suppose µ0 b(t0 ) + c(t0 ) = 0 for some t0 ∈ T. If b(t0 ) = 0, then c(t0 ) = 0, and this is impossible. Hence b(t0 ) = 0 and thus, µ0 = −c(t0 )/b(t0 ). It follows that inds (a(s) − µ0 ) = inds (a(s) + c(t0 )/b(t0 )) = inds (a(s)b(t0 ) + c(t0 )) = 0 (again by (8.21)), which is impossible for µ0 ∈ sp T (a). This proves our claim. Thus, for each µ ∈ sp T (a) we have a (uniquely determined) factorization µb(t) + c(t) = d− (µ, t)d+ (µ, t)
(t ∈ T),
d− (µ, ∞) = 1,
∞ and d±1 (µ, ·) ∈ W ∩H ∞ . Hence T (µb+c) ∈ GL(1 ), where d±1 − (µ, ·) ∈ W ∩H + −1 and since d+ (µ, t) = [T −1 (µb + c)χ0 ](t), it follows that, for each ﬁxed t ∈ T, the four mappings sp T (a) → C, µ → d±1 ± (µ, t) are continuous and that
max µ∈sp T (a)
d±1 ± (µ, ·)W ≤ M < ∞.
Consequently, by 1.19 and 1.26(a), b(t)T (a) + c(t)I = d− (T (a), t)d+ (T (a), t), d−1 ± (T (a), t)d± (T (a), t) = I
(t ∈ T).
If we write d± (µ, t) =
n d± n (µ)t ,
d−1 ± (µ, t) =
n∈Z±
n e± n (µ)t ,
n∈Z±
then A = D− D+ and D± E± = E± D± = I, where ∞ ∞ D± := d± , E± := e± j−k T (a) j−k T (a) j,k=0
.
j,k=0
± Since n d± n (T (a)) = n dn C(sp T (a)) ≤ M and the same is also true for ± ± en in place of dn , each of the matrices D± , E± represents a bounded operator, from which we infer that T2 (g) is invertible. p 2 8.29. Homogeneous symbols. A function a ∈ MN ×N (T ) is said to be p 2 homogeneous if there exists a b ∈ MN ×N (T ) such that a(s, t) = b(st−1 ) for 8.7(d) is homogeneous. (s, t) ∈ T2 . Note that the function χ constructed in
−1 n ) be homogeneous. If b(t) = Let a(s, t) = b(st n∈Z bn t , then a(s, t) =
n −n 2 (s, t ∈ T). For n ∈ Z+ , put Ωn = {(j, k) ∈ Z++ : j + k = n}, n∈Z bn s t ·
·
·
and let En = pN (Ωn ). Then pN (Z2++ ) = E0 + E1 + E2 + . . . and it is easily seen that En is an invariant subspace of T2 (a). If x ∈ pN (Ωn ) and y = T2 (a)x, then
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8 Toeplitz Operators over the QuarterPlane
yi,n−i =
ai−j,n−i−k xjk =
(j,k)∈Ωn
n
bi−j xj,n−j .
j=0
Consequently, T2 (a) has the diagonal matrix representation diag (T0 (b), T2 (b), T2 (b), . . .) ·
·
·
with respect to the decomposition pN (Z2++ ) = E0 + E1 + E2 + . . . (here Tn (b) is deﬁned as in 7.5). A ﬁrst consequence of this representation is that the functions a given by p p 2 a(s, t) = b(st−1 ) (s, t ∈ T) is in MN ×N (T ) whenever b ∈ MN ×N (T). The following theorem provides some much more interesting consequences of this representation. p 8.30. Theorem (Douglas/Howe). Let a(s, t) = b(st−1 ), where b ∈ MN ×N (1 ≤ p < ∞). Then
T2 (a) ∈ Φ(pN (Z2++ )) ⇐⇒ T (b) ∈ Π{pN ; Pn } and T2 (a) ∈ GL(pN (Z2++ )) ⇐⇒ T (b) ∈ Π{pN ; Pn } and det Tn (b) = 0 ∀ n ∈ Z+ . If T2 (a) is Fredholm, then Ind T2 (a) = 0. There exist homogeneous functions a ∈ P(T2 ) such that T2 (a) is Fredholm but not invertible on p (Z2++ ). p invertible for all n ≥ n0 , then Proof. n } and Tn (b) is If T (b) ∈ Π{N ; P−1 (b), . . . is clearly a regularizer of T2 (a) diag P0 , P1 , . . . , Pn0 −1 , Tn0 (b), Tn−1 0 +1 (recall Proposition 7.3). Now suppose T2 (a) is Fredholm. Then there exists an n0 ∈ Z+ such that Tn (b) is invertible for all n ≥ n0 , since otherwise the kernel of T2 (a) would have inﬁnite dimension. Assume there is a sequence {nk } ⊂ Z+ such (b)Pnk → ∞ as k → ∞. Without loss of generality assume that Tn−1 k (b)P ≥ k 2 (take a subsequence if necessary). Then one can ﬁnd Tn−1 n k k that yk = Tnk (b)xk , xk = 1, yk ≤ 1/k2 . Put xk , y k ∈
nIm Pnk such
∞ wn = k=0 yk , w = k=0 yk . It is clear that wn ∈ Im T2 (a), wn → w, but w ∈ / Im T2 (a). The conclusion is that T2 (a) is not normally solvable and this is a contradiction. Thus, sup Tn−1 (b)Pn =: M < ∞. We have n≥n0
Pn x ≤ Tn−1 (b)Pn Tn (b)Pn x for all x ∈ pN and n ≥ n0 . Passage to the limit n → ∞ gives x ≤ M T (b)x for all x ∈ pN . Since Tn−1 (b)Pn L(pN ) = Tn−1 (b∗ )Pn L(XN ) , where X = q for 1 < p < ∞ and X = c0 for p = 1, it follows analogously that zX ≤ M T (b)zX for all z ∈ X. This proves that T (b) is invertible, and now Proposition 7.3 yields that T (b) is in Π{pN ; Pn }.
8.4 The Invertibility Problem
433
The implication concerning invertibility and the fact that Ind T2 (a) = 0 are now obvious. Finally, let 3 3 2 b(t) = 16t2 − 36t + 27t−1 = 16t−1 t + . t− 4 2 Then b(t) = 0 for t ∈ T and ind b = 0, so that T (b) ∈ Π{p ; Pn }. However, T0 (b) = 0. 8.31. Toeplitz operators with kernels supported in a halfplane. Let γ and δ be real numbers and suppose (γ, δ) = (0, 0). Put Πγ,δ := (x, y) ∈ R2 : γx + δy ≥ 0 . Given a function a ∈ W (T2 ) deﬁne supp a as the support of the Fourier
coeﬃcients sequence, i.e., if a = ajk χj ⊗ χk , then supp a := (j, k) ∈ Z2 : ajk = 0 . Finally, let
Wγ,δ := a ∈ W (T2 ) : supp a ⊂ Πγ,δ .
Note that Wγ,δ is a closed subalgebra of W (T2 ). Theorem 8.35 will provide an invertibility criterion for Toeplitz operators over the quarterplane with symbols in Wγ,δ . To prove
this theorem we need some lemmas. We shall always assume that a = γj+δk≥0 ajk χj ⊗ χk and
that b0 is deﬁned by b0 = γj+δk=0 ajk χj ⊗ χk . In particular, if the ascent of the line γx + δy = 0 is irrational, then b0 (s, t) = a00 for all (s, t) ∈ T2 . 8.32. Lemma. Let γ and δ be positive integers and let a ∈ Wγ,δ . If 1 ≤ p < ∞ and T2 (a) ∈ GL(p (Z2++ )), then Ker T2 (b0 ) = {0}. Proof. Without loss of generality assume the largest common divisor of γ and δ equals 1. Put Ωn := (j, k) ∈ Z2++ : γj + δk = n , En = p (Ωn ). Of course, it may happen that Ωn = ∅. In that case we let En = {0}. We ·
·
then have p (Z2++ ) = E0 + E1 + . . . and it is easily seen that the matrix representation of T2 (a) with respect to this decomposition is lower triangular: ⎛ ⎞ B00 ⎜ ⎟ ⎜ B10 B11 ⎟ ⎜ ⎟ (8.22) ⎜ ⎟. ⎜ B20 B21 B22 ⎟ ⎝ ⎠ .. .. .. . . . . . . The corresponding representation of T2 (b0 ) is of diagonal form and results from (8.22) by putting B10 = B20 = B21 = . . . = 0. This observation gives the assertion straightforwardly.
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8 Toeplitz Operators over the QuarterPlane
8.33. Lemma. Let a ∈ W (T2 ) and suppose there are µ, λ ∈ (0, 1) such that aµ,λ := ajk µj λk χj ⊗ χk is also in W (T2 ). If 1 ≤ p < ∞ and Ker T2 (aµ,λ ) = {0} in p (Z2++ ), then Ker T2 (a) = {0} in p (Z2++ ). Proof. Deﬁne p (µ, λ) as the Banach space of all sequences x = {xjk }∞ j,k=0 for which µjp λkp xjk p < ∞ (1 ≤ p < ∞), xpp := j,k≥0
x∞ := sup µj λk xjk  : (j, k) ∈ Z2++ < ∞ (p = ∞) and let Λ be the isometric isomorphism Λ : p (µ, λ) → p (Z2++ ),
{xjk } → {µj λk xjk }.
It is easy to verify that T2 (aµ,λ ) = ΛT2 (a)Λ−1 . Now assume T2 (a)x = 0 for some nonzero x ∈ p (Z2++ ). Since p (Z2++ ) is embedded in p (µ, λ), we have 0 = y := Λx ∈ p (Z2++ ) and thus, T2 (aµ,λ )y = ΛT2 (a)Λ−1 Λx = ΛT2 (a)x = 0, which is impossible if Ker T2 (aµ,λ ) = {0} in p (Z2++ ).
8.34. Lemma. Let γ and δ be nonzero real numbers and let a ∈ Wγ,δ . Suppose a does not vanish on T2 and ind1 a = ind2 a = 0. Then (a) b0 does not vanish on T2 and ind1 b0 = ind2 b0 = 0; (b) a ∈ GWγ,δ . Proof. Without loss of generality assume γ > 0 and δ > 0. (a) First divisor is 1.
suppose γ and δ are integers whose largest common
∞ Put bn = γj+δk=n ajk χj ⊗ χk and, for µ ∈ clos D, deﬁne gµ := n=0 µn bn . If µ = 1, then gµ (s, t) = a(µγ s, µδ t). Hence, if we think of s and t as being ﬁxed and of g as a function of µ, we have gµ (s, t) = 0 for µ ∈ T and indµ gµ (s, t) = γ ind1 a + δ ind2 a = 0 (see (8.16)). This implies that gµ (s, t) = 0 for all (s, t) ∈ T2 and µ ∈ clos D. Therefore, a = g1 and b0 = g0 belong to the same connected component of GC(T2 ), which gives the assertion immediately. Now suppose the ascent of the straight line γx+δy = 0 is irrational. There is an N ∈ Z+ such that the N, N th partial sum sN N a of the Fourier series of a is in GC(T2 ) and satisﬁes ind1 sN N a = ind2 sN N a = 0. Application of the above homotopy argument to sN N a shows that a00 = 0.
8.4 The Invertibility Problem
435
(b) Since the maximal ideal space of W (T2 ) can be identiﬁed with T2 , it follows that d := a−1 ∈ W (T2 ). Again let us ﬁrst consider the case that γ and δ are integers. Deﬁne gµ as above and recall that
we have proved that gµ (s, t) = 0 for all (s, t) ∈ T2 and µ ∈ clos D. Put dn = γj+δk=n djk χj ⊗ χk . If µ = 1, then d(µγ s, µδ t) =
1 1 = . a(µγ s, µδ t) gµ (s, t)
2 Consequently, for ﬁxed (s, t) ∈ T
, d(µγ s, µδ t) is an analytic function of µ in γ δ D. Thus, because d(µ s, µ t) = n∈Z µn dn (s, t), we conclude that dn = 0 for n < 0, i.e., that d ∈ Wγ,δ . We are left with the case that the ascent of the straight line γx + δy = 0 is irrational. Let sN N a be as in the proof of part (a) and put rN a := a − sN N a. Then ∞ r a n N a−1 = (sN N a)−1 (−1)n . s N Na n=0
From what was proved in part (a) we know that (sN N a)−1 ∈ Wγ,δ , and since obviously rN a ∈ Wγ,δ , it results that d = a−1 ∈ Wγ,δ . 8.35. Theorem. Let (γ, δ) ∈ R2 \ {(0, 0)} and a ∈ Wγ,δ . Put S = (x, y) ∈ R2 : x ≥ 0, y ≥ 0, x2 + y 2 > 0 and suppose 1 ≤ p < ∞. (a) If the intersection of the straight line γx + δy = 0 and S is not empty, then T2 (a) ∈ GL(p (Z2++ )) ⇐⇒ T2 (a) ∈ Φ(p (Z2++ )). (b) If the intersection of the straight line γx + δy = 0 and S is empty, then T2 (a) ∈ GL(p (Z2++ )) ⇐⇒ T2 (a) ∈ Φ(p (Z2++ )) and T2 (b0 ) ∈ GL(p (Z2++ )). Proof. (a) Suppose T2 (a) is Fredholm. Then, by Corollary 8.22, a ∈ GW (T2 ) and ind1 a = ind2 a = 0. If γ = 1 and δ = 0, then the function a admits a factorization a = a+− a++ as in 8.23(d), and so Proposition 8.10 implies −1 that T2 (a−1 ++ )T2 (a+− ) is the inverse of T2 (a). The case γ = 0 and δ = 1 can be settled analogously. Thus, let γ < 0 and δ > 0. Lemma 8.34 shows that a ∈ GWγ,δ and that b0 ∈ GW (T2 ), ind1 b0 = ind2 b0 = 0. We want to show that a is even in the connected component of GWγ,δ containing the identity. First suppose the straight line γx + δy = 0 has rational ascent. A similar reasoning as in the proof of Lemma 8.34(a) shows that a and b0 are in the same connected component of GWγ,δ . Without loss of generality assume the largest common divisor of γ and δ is 1. Then
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8 Toeplitz Operators over the QuarterPlane
b0 (s, t) =
ajk sj tk =
σ(t) :=
a−δn,γn s−δn tγn .
n∈Z
γj+δk=0
Put
a−δn,γn tn
(t ∈ T).
(8.23)
n∈Z
There are α, β ∈ Z such that αγ + βδ = 1. Hence, a−δn,γn (tαγ+βδ )n = b0 (t−β , tα ). σ(t) = n∈Z
It follows that σ does not vanish on T and that ind σ = 0 (by (8.16)). Consequently, there is a homotopy σµ of σ1 = σ to σ0 = 1 through GC(T2 ). Let τµ (s, t) := σµ (s−δ tγ ). Then τµ ∈ Wγ,δ ∩ GC(T2 ), ind1 τµ = ind2 τµ = 0, and hence τµ ∈ GWγ,δ (Lemma 8.34(b)). Since τ0 = 1 and τ1 (s, t) = a−δn,γn s−δn tγn = b0 (s, t), n∈Z
we conclude that b0 and thus also a is in the connected component of Wγ,δ containing the identity. If the ascent of γx + δy = 0 is irrational, then choose a number N so large that a and sN N a are in the same connected component of GWγ,δ and that ind1 sN N a = ind2 sN N a = 0, and then proceed as in the case where γ and δ are integers to show that sN N a and a00 are in the same connected component of GWγ,δ . Thus, in either case a belongs to the connected component of GWγ,δ containing the identity. Therefore a = exp f with f ∈ Wγ,δ (1.16(a)) and since ∞ (T2 ), we deduce from Propof = f−− + f−+ + f++ ; with f±± ∈ W (T2 ) ∩ H±± −1 −1 −1 sition 8.10 that T2 (a++ )T2 (a−+ )T2 (a−− ) (a±± = exp f±± ) is the inverse of T2 (a). The case γ > 0 and δ < 0 is analogous. (b) Without loss of generality assume γ > 0 and δ > 0; otherwise consider adjoints. Suppose T2 (a) is Fredholm and T2 (b0 ) is invertible. For ∈ (0, 1), put µ = γ and λ = δ . We have (recall Lemma 8.33) ajk µj λk sj tk aµ,λ (s, t) := γj+δk≥0
= b0 (s, t) +
ajk γj+δk sj tk = b0 (s, t) + b (s, t),
γj+δk>0
where b W (T2 ) → 0 as → 0. Hence, since T2 (b0 ) is invertible, the operators T2 (aµ,λ ) are invertible for all suﬃciently small , so Lemma 8.33 gives that Ker T2 (a) = {0}, which implies that T2 (a) is invertible (Corollary 8.22). Conversely, suppose now that the operator T2 (a) is invertible. If the straight line δx + γy = 0 has irrational ascent, then b0 (s, t) = a00 = 0
8.4 The Invertibility Problem
437
(Lemma 8.34(a)) and thus T2 (b0 ) = a00 I is invertible. So let γ and δ be integers. From Lemma 8.32 we deduce that Ker T2 (b0 ) = {0} and Lemma 8.34(a) along with Corollary 8.22 implies that T2 (b0 ) is Fredholm with index zero. Hence, T2 (b0 ) is invertible. Remark. If the straight line γx + δy = 0 has irrational ascent, then the requirement in (b) that T2 (b0 ) be invertible is redundant, since T2 (b0 ) = a00 I = 0 (Lemma 8.34(a)). In case the ascent of γx + δy = 0 is rational, the invertibility of T2 (b0 ) may be replaced by the condition that det Tn (σ) = 0 for all n ∈ Z+ , where σ is given by (8.23). This follows from the representation (8.22), Lemma 8.34(a), Corollary 8.22, and Theorem 8.30. 8.36. Convolutions over angular sectors in Z2 . A subset W0 of R2 is called an angular sector in R2 with vertex (0, 0) if W0 is of the form W0 = (λx, λy) ∈ R2 : λ ∈ [0, ∞), (x, y) ∈ Ψ , where Ψ is a closed connected subset of T = {(x, y) ∈ R2 : x2 + y 2 = 1} containing at least two points. An angular sector in Z2 with vertex (0, 0) is a set of the form K = W0 ∩ Z2 , where W0 is angular sector in R2 with vertex (0, 0). Given an angular sector K in Z2 with vertex (0, 0) and a function a ∈ Cp (T2 ) (1 ≤ p < ∞) deﬁne the operator TK (a) on p (K) by TK (a) : p (K) → p (K), {xij }(i,j)∈K → ai−k,j−l xkl . (k,l)∈K
(i,j)∈K
The following result is due to Simonenko [495]. For TK (a) to be in Φ(p (K)) it is necessary and suﬃcient that (a) a ∈ GC(T2 ) if W0 = R2 ; (b) a ∈ GC(T2 ) and (ind1 a, ind2 a) ∈ ∂W0 ∩ Z2 (∂W0 being the boundary of W0 ) if W0 is a halfplane; (c) a ∈ GC(T2 ) and (ind1 a, ind2 a) = (0, 0) in the remaining cases. Note that a ∈ Cp (T2 ) ∩ GC(T2 ) implies that a ∈ GCp (T2 ) (8.7(a)). In the cases (a) and (b) Fredholmness yields invertibility and in the case (c) Fredholmness yields that
the index is zero. Now suppose a = ajk χj ⊗ χk ∈ Wγ,δ , where (γ, δ) ∈ R2 \ {(0, 0)}, and
2 put b0 = γj+δk=0 ajk χj ⊗ χk . Also assume that W0 is neither R nor a halfplane. Put W0 = W0 \ {(0, 0)} if the opening of W0 is less than π and W0 = clos (R2 \ W0 ) \ {(0, 0)} if the opening of W0 is larger than π. The following invertibility criterion was proved in B¨ ottcher [62]. (d) If the intersection of the straight line γx+δy = 0 and W0 is not empty, then
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8 Toeplitz Operators over the QuarterPlane
TK (a) ∈ GL(p (K)) ⇐⇒ TK (a) ∈ Φ(p (K)). (e) If the intersection of the straight line γx + δy = 0 and W0 is empty, then TK (a) ∈ GL(p (K)) ⇐⇒ TK (a) ∈ Φ(p (K)) and TK (b0 ) ∈ GL(p (K)). If the ascent of γx + δy = 0 is irrational, then the condition that TK (b0 ) be in GL(p (K)) is redundant. In case γ and δ are nonzero integers, it can be replaced by the requirement that det Tn () = 0 for all n belonging to a certain countable set N depending on γ, δ, W0 . Here = σ (resp. = 1/σ) if the opening of W0 is smaller (resp. larger) than π, and σ is given by (8.23). Examples: (i) N = Z+ if W0 = {2x − y ≥ 0, −x + y ≥ 0} and γ = δ = 1; (ii) N = {2k + [k/2] : k ∈ Z+ } if W0 = {x + 3y ≥ 0, 2x + y ≥ 0} and γ = δ = 1; (iii) N = {k + [2k/3] : k ∈ Z+ } if W0 = {x + 3y ≥ 0, 2x + y ≥ 0} and γ = 1, δ = 2. In (ii) and (iii), [x] denotes the largest integer n satisfying n ≤ x.
8.5 Bilocal Fredholm Theory The possibility of constructing a regularizer for Fredholm Toeplitz operators over the quarterplane with (quasi) continuous symbols enabled us to avoid local techniques, except for Theorem 8.20, where we localized over the maximal ideal space M (B) of a C ∗ algebra B between C and QC. Local techniques are a natural and powerful tool for the treatment of operators with discontinuous symbols. Duduchava [172] was the ﬁrst to show how local methods can be used to establish a Fredholm criterion for operators with symbols from the algebra P C ⊗ P C on 2 (Z2++ ). He localized over the maximal ideal space M (P C) = T × {0, 1}; this reduced the problem to the study of “local representatives” of the form T (a) ⊗ T (b) + I ⊗ T (c) with a, b, c ∈ P C, and he succeeded to overcome the diﬃculties arising when investigating such “rather complicated” local representatives. In our paper [105], we pointed out that the things can be substantially simpliﬁed by localizing over the maximal ideal space M (alg T π (P C)) = T × [0, 1]; then the local representatives take the extremely simple form T (a) ⊗ I, where a ∈ P C. In what follows we shall show that localization over M (alg T π (A)) leads to a fairly simple Fredholm theory of the operators T2 (a) with a ∈ A ⊗ A, provided A satisﬁes some additional conditions. Note that M (alg T π (A)) = M (A) in case A is a C ∗ algebra between C and QC. Moreover, since the techniques
8.5 Bilocal Fredholm Theory
439
employed in the following do not cause essential complications when passing from the “pure” Toeplitz operators T2 (a) (a ∈ A ⊗ A) to operators from alg T (A) ⊗ alg T (A) we shall without delay turn to the study of operators belonging to alg T (A) ⊗ alg T (A). Throughout the following let 1 < p < ∞ and 1/p + 1/q = 1. 8.37. Deﬁnitions. (a) Let A be a C ∗ subalgebra of L∞ containing C and put AπN ×N := algL(HNp )/C∞ (HNp ) T π (AN ×N ).
AN ×N := algL(HNp ) T (AN ×N ),
Abbreviate A1×1 and Aπ1×1 to A and Aπ , respectively. Suppose Aπ is comπ mutative and the Shilov boundary of the maximal ideal space NA p := M (A ) A π A coincides with the whole space Np . Let Γp : A → C(Np ) denote the Gelfand map. For example, one can take A = C, QC, P C, P QC if 1 < p < ∞ and A = CE , QCE if p = 2. Notice that the proof of ∂S M (alg Tp (P QC)) = ottcher, Spitkovsky M (alg Tp (P QC)) for p = 2 is based on the results from B¨ [120] and the arguments of the proof of Theorem 5.46.
1 (b) Given B = i Ci ⊗ Di ∈ A " AN ×N and ν ∈ NA p deﬁne Bp,ν ∈ AN ×N by 1 Bp,ν := (Γp Ciπ )(ν)Di . (8.24) If B = T2 (b) = where
For B =
i
i 1 T (ci ) ⊗ T (di ) with ci ∈ A, d ∈ AN ×N , then Bp,ν = T (b1p,ν ), Γp T π (ci ) (ν)di . (8.25) b1p,ν := i
Ci ⊗ Di ∈ AN ×N " A and b = i ci ⊗ di ∈ AN ×N " A we deﬁne 2 Bp,ν := Ci (Γp Diπ )(ν)I, b2p,ν := ci Γp Tiπ (di ) (ν)I.
i
i
i
If A is a C ∗ algebra between C and QC, then NA p is naturally homeomorphic to M (A) (Theorems 4.79 and 5.31) and so in this case the function deﬁned by (8.25) is just the function deﬁned in 8.16(a). (c) Put p BN ×N := algL(pN ) T (P CN ×N ),
p BπN ×N := algL(pN )/C∞ (pN ) T π (P CN ×N ),
let B = B1×1 , Bπ := Bπ1×1 , denote the maximal ideal space of Bπ by Np , and π let Γ
p : B → C(Np ) be the Gelfand map. For B = i Ci ⊗ Di ∈ B " BN ×N , p 1 1 , ν ∈ N , deﬁne B b = i ci ⊗ di ∈ P Cp " P CN p p,ν and bp,ν by (8.24) and ×N 2 2 (8.25). Bp,ν and bp,ν are deﬁned similarly. 8.38. Proposition. Let A resp. B be as in the previous section and let H p resp. p be the underlying space (1 < p < ∞).
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8 Toeplitz Operators over the QuarterPlane
(a) If E1 , . . . , Em are in A resp. B and if there is a ν in NA p resp. Np such π l r that (Γp Ei )(ν) = 0 for i = 1, . . . , m, then there exist Un , Un (n = 1, 2, . . .) in A resp. B such that Unl = Unr = 1 and m
Unl Ei → 0
and
i=1
m
Ei Unr → 0
as
n → ∞.
i=1
p resp. pN , (b) If B is in AN ×N resp. BN ×N and if B is not invertible on HN then there exist Un (n = 1, 2, . . .) in AN ×N resp. BN ×N such that Un = 1 and Un B → 0 or BUn → 0 as n → ∞.
Proof. We only consider the H p case, as the proof for the p case is analogous. (a) By virtue of 1.20(c) there are Vj ∈ A such that Vjπ = 1 and → 0 as j → ∞ for all i. Hence, there exist Kij ∈ C∞ (H p ) and Cij ∈ A such that
Vjπ Eiπ
Vj Ei = Kij + Cij ,
Cij → 0 (j → ∞, ∀ i).
Let Pk denote the projections deﬁned in 7.5 and put Qk := I − Pk , M := sup Qk L(H p ) . We have Qk Vj Ei = Qk Kij +Qk Cij . Given any n ∈ N there is a k
j0 such that Qk Cij0 < 1/(2n) for all k and i, and then one can ﬁnd a k0 such that Qk0 Kij0 < 1/(2n) for all i (1.3(d)). It follows that Qk0 Vj0 Ei < 1/n for all i. Because Pk0 Vj0 is a ﬁniterank operator, we have Qk0 Vj0 = Vj0 − Pk0 Vj0 ≥ Vjπ0 = 1. Thus, if we let Unl = Qk0 Vj0 /Qk0 Vj0 (notice that k0 and j0 depend on n), then Unl = 1 and Unl Ei < 1/(nQk0 Vj0 ) ≤ 1/n. The existence of Unr can be shown analogously. (b) Let B = (Bij )N i,j=1 with Bij ∈ A. First suppose B is not Fredholm. Then B π is not invertible in AπN ×N and so (det B)π cannot be invertible in π Aπ . Consequently, there is a ν ∈ NA p such that (Γp (det B) )(ν) = 0. This implies that there exists an ω0 = (ω0ij ) ∈ CN ×N \ {0} satisfying N π ω0 (Γp B π )(ν) := ω0 (Γp Bij )(ν) i,j=1 = 0. From part (a) we infer that there are Vn ∈ A such that " " π "Vn bij − (Γp Bij )(ν)I " → 0 as n → ∞ for all i, j and Vn = 1 for all n. If we put Un = ω0 Vn , then Un AN ×N ≥ δ max ω0ij Vn A = δ max ω0ij  =: ε > 0. i,j
i,j
8.5 Bilocal Fredholm Theory
441
Since ω0 Vn = Vn ω0 , we get " " Un B ≤ ω0 "Vn B − (Γp B π )(ν)I " + Vn ω0 (Γp B π )(ν)I " " ≤ ω0 ∆ max "Vn B − (Γp B π )(ν)I " = o(1) (n → ∞). i,j
Thus, Un = Un /Un has the desired properties. p . Let Y Now suppose B is Fredholm but Ker B ∗ = {0}, i.e., Im B = HN p p be any ﬁnitedimensional subspace of HN such that HN decomposes into the ·
p onto Y parallel direct sum Im B + Y and let PY denote the projection of HN to Im B. Then PY ≥ 1 and PY B = 0. Hence, if we let Un := PY /PY for all n, then Un has the required properties. Finally suppose B is Fredholm but Ker B = {0}. Choose any closed sub·
p p space Z of HN such that HN = Z + Ker B and let PZ be the projection of p HN onto Ker B parallel to Z. Again PZ ≥ 1, but now we have BPZ = 0. So Un := PZ /PZ is as we wanted.
8.39. Lemma. (a) Let A be as in 8.37(a), let B ∈ A " AN ×N and ν ∈ NA p. Then 1 L(HNp ) ≤ BΦ(HNp (T2 )) . Bp,ν (b) If B ∈ B " BN ×N and ν ∈ Np , then 1 Bp,ν L(pN ) ≤ BΦ(pN (Z2++ )) .
m Proof. (a) Let B = i=1 Ci ⊗ Di with Ci ∈ A, Di ∈ AN ×N and let L be p 2 any compact
soperator on HN (T ).p Let ε > 0 pbe arbitrarily given. There exist operators j=1 Kj ⊗ Mj ∈ C∞ (H ) " C∞ (HN ) and R ∈ A such that L=
s
Kj ⊗ Mj + R,
R <
j=1
ε . 3
Put Ei := Ci − (Γp Ciπ )(ν)I. Then (Γp Eiπ )(ν) = 0 for i = 1, . . . , m, and since Kjπ = 0, we obtain that (Γp Kjπ )(ν) = 0 for j = 1, . . . , s. Hence, by Proposition 8.38(a), there are Un ∈ A such that Un = 1, We have
Un Ei → 0 (n → ∞, ∀ i),
Un Kj → 0 (n → ∞, ∀j).
" " " " 1 1 Un Bp,ν = Un ⊗ Bp,ν = "Un ⊗ (Γp Ciπ )(ν)Di " " " " i " " " " " π =" Un (Γp Ci )(ν)I ⊗ Di " = " Un (Ci − Ei ) ⊗ Di " i
≤ Un B + L + (Un ⊗ I)L +
i
i
Un Ei Di .
(8.26)
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8 Toeplitz Operators over the QuarterPlane
s Since (Un ⊗ I)L ≤ j=1 Un Kj Mj + R, there is an n0 such that the second term in (8.26) is smaller than 2ε/3 for all n ≥ n0 . Clearly, if n0 is large enough, then the third term in (8.26) is smaller than ε/3 for all n ≥ n0 . Thus, (8.26) gives that 1 1 = Un Bp,ν ≤ Un B + L + ε = B + L + ε, Bp,ν p (T2 )) can be chosen arbitrarily, it follows that and as ε > 0 and L ∈ C∞ (HN 1 Bp,ν ≤ Bess .
(b) The proof is the same. i 8.40. Corollary. The mappings B → Bp,ν (i = 1, 2) deﬁned in 8.37 can be naturally extended to mappings of (A ⊗ A)N ×N onto AN ×N resp. (B ⊗ B)N ×N onto BN ×N .
Proof. Immediate from the preceding lemma.
Remark. From the results of 8.7(b) we deduce that for b ∈ (P C ⊗ P C)N ×N , p C 2 1 1 ν = (τ, λ) ∈ NP p , and HN (T ) as underlying space the function bp,ν = bp;τ,λ is given by b1p;τ,λ (t) = 1 − σp (λ) b(τ − 0, t) + σp (λ)b(τ + 0, t), and that for b ∈ (P Cp ⊗ P Cp )N ×N , ν = (τ, λ) ∈ Np , and pN (Z2++ ) as underlying space we have b1p;τ,λ (t) = 1 − σq (λ) b(τ − 0, t) + σq (λ)b(τ + 0, t), with σr as in 5.12 (also recall Remark 6.33). Our next objective is to prove Theorem 8.43, which provides necessary and suﬃcient conditions for an operator in (A ⊗ A)N ×N resp. (B ⊗ B)N ×N to be Fredholm. The following proposition settles the “necessity portion” and Section 8.42 prepares the proof of the “suﬃciency part.” 8.41. Proposition. (a) Let A be as in 8.37(a) and let B ∈ (A ⊗ A)N ×N . If p p p 1 2 (T2 )), then Bp,ν ∈ GL(HN ) and Bp,ν ∈ GL(HN ) for all ν ∈ NA B ∈ Φ(HN p. (b) Let B be as in 8.37(c) and let B ∈ (B ⊗ B)N ×N . If B ∈ Φ(pN (Z2++ )), 1 2 ∈ GL(pN ) and Bp,ν ∈ GL(pN ) for all ν ∈ Np . then Bp,ν Proof. (a) Assume, contrary to what we want, there is a ν0 ∈ NA p such that p 1 is not invertible on H . There exists an ε > 0 with the property that Bp,ν N 0
m p (T2
)) whenever B − B < ε. Choose i=1 Ci ⊗ Di ∈ A " AN ×N B ∈ Φ(HN so that B − i Ci ⊗ Di < ε/2. Corollary 8.40 and Lemma 8.39(a) imply that " " " ε " " " 1 " " 1 − (Γp Ciπ )(ν0 ) ⊗ Di " = "Bp,ν − (Γp Ciπ )(ν0 )Di " < "I ⊗ Bp,ν 0 0 2 i i
8.5 Bilocal Fredholm Theory
443
(also see example (d) in 8.3). Hence " " " " 1 π C − − (Γ C )(ν )I ⊗ D "B − I ⊗ Bp,ν i p i 0 i " < ε. 0 i
m 1 Put Ei = Ci − (Γp Ciπ )(ν0 )I and B := I ⊗ Bp,ν + i=1 Ei ⊗ Di . Then B 0 p 1 is in Φ(HN (T2 )). From Proposition 8.38(b) we deduce that Bp,ν is a left or 0 right topological divisor of zero. For the sake of deﬁniteness, assume there 1 → 0 as n → ∞. Let are Un in AN ×N such that Un = 1 and Un Bp,ν 0 p p 2 2 (T )) and K ∈ C (H (T )) satisfy B F = I + K, and choose F ∈ L(H ∞ N N
s p p L ⊗ M in C (H ) " C (H ) so that K = j j ∞ ∞ N j Lj ⊗ Mj + R with j=1 R < 1/3. Since (Γp Eiπ )(ν0 ) = 0 and Lπi = 0, Proposition 8.38(a) shows that there are Unl ∈ A such that Unl = 1 and Unl Ei → 0 (n → ∞, ∀ j),
Unl Lj → 0 (n → ∞, ∀ i).
It follows that 1 (Unl ⊗ Un )B F ≤ Un Bp,ν F + 0
m
Unl Ei Di F
i=1
is smaller than 1/3 for all suﬃciently large n, while " " " " (Unl ⊗ Un )B F = "(Unl ⊗ Un ) I + Lj ⊗ Mj + R " ≥ 1−
j
Unl Lj Mj − R > 1 −
j
1 1 1 − = 3 3 3
if only n is large enough. This contradiction completes the proof. (b) The proof for the p case is analogous.
8.42. Bilocalization. Let A be as in 8.37(a). In accordance with 8.17 put Aπ1 := A ⊗ AN ×N /C∞ (H p ) ⊗ AN ×N . Let Uπ1 denote the closure in Aπ1 of the set of all elements of the form (B ⊗I)π1 , p . It is clear that Uπ1 is where B ∈ A and I is the identity operator on HN π contained in the center of A1 . Denote the maximal ideal space of Uπ1 by MA p. The mapping γ1 : Aπ → Uπ1 , C π → (C ⊗ I)π1 is well deﬁned (C π = Dπ =⇒ (C ⊗I)π1 = (D⊗I)π1 ), it is obviously an algebraic homomorphism, and since (C ⊗ I)π1 = inf C ⊗ I + K : K ∈ C∞ (H p ) ⊗ AN ×N ≤ inf C ⊗ I + L ⊗ I : L ∈ C∞ (H p ) = inf C + L : L ∈ C∞ (H p ) = C π ,
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8 Toeplitz Operators over the QuarterPlane
it follows that γ1 is continuous. Hence, if ϕ is a multiplicative linear functional on Uπ1 , then ϕ ◦ γ1 is a multiplicative linear functional on Aπ , and because C π ∈ Ker (ϕ ◦ γ1 ) if and only if (C ⊗ I)π1 ∈ Ker ϕ, we deduce that the set γ1−1 (µ) ⊂ Aπ is a maximal ideal of Aπ whenever µ ⊂ Uπ1 is a maximal ideal 1 π of Uπ1 . For µ ∈ MA p , let Jµ denote the smallest closed twosided ideal of A1 containing µ. From what was said above we infer that J1µ = closidAπ1 (C ⊗ I)π1 : Γp C π γ1−1 (µ) = 0 . (8.27) In a completely analogous fashion we deﬁne Aπ2 := AN ×N ⊗ A/AN ×N ⊗ C∞ (H p ), then Uπ2 and γ2 : Aπ → Uπ2 , and for µ in the maximal ideal space of Uπ2 the closed twosided ideal of Aπ2 generated by µ can be shown to be of the form J2µ = closidAπ2 (I ⊗ D)π2 : Γp Dπ γ2−1 (µ) = 0 . (8.28) Also in accordance with 8.17 we let p Aπ12 = (A ⊗ A)N ×N /C∞ (HN (T2 )).
Bilocalization is nothing else than the following. To show that an operator p B ∈ (A ⊗ A)N ×N is Fredholm on HN (T2 )) it suﬃces to show that B1π is in π π π π is in GAπ12 ), GA1 and that B2 is in GA2 (Lemma 8.18 then implies that B12 π π and in order to show that Bi is in GAi it is enough to show that Biπ + Jiµ is invertible in Aπi /Jiµ for each µ ∈ MA p (Theorem 1.35(a)). Finally, in view of π equalities (8.27) and (8.28) we need not know anything about MA p = M (Ui ), A π we have merely to know what Np = M (A ) is. Notice that all the above deﬁnitions make sense if A is replaced by the algebra B introduced in 8.37(c) and at the same time H p is replaced by p . 8.43. Theorem. (a) Let A be as in 8.37(a) and B ∈ (A ⊗ A)N ×N . Then the following are equivalent: p (i) B ∈ Φ(HN (T2 )). π (ii) B12 ∈ GAπ12 . p p 1 2 (iii) Bp,ν ∈ GL(HN ), Bp,ν ∈ GL(HN ) ∀ ν ∈ NA p.
(b) Let B be as in 8.37(c) and B ∈ (B ⊗ B)N ×N . Then the following are equivalent: (i) B ∈ Φ(pN (Z2++ )). π (ii) B12 ∈ GBπ12 . 1 2 (iii) Bp,ν ∈ GL(pN ), Bp,ν ∈ GL(pN ) ∀ ν ∈ Np .
8.5 Bilocal Fredholm Theory
445
Proof. Proposition 8.41 gives the implications (i) =⇒ (iii). The implications (ii) =⇒ (i) are trivial. So suppose (iii) is satisﬁed, and for the sake of deﬁniteπ ∈ GAπ12 . ness let us consider the H p case. We shall prove that B12 −1 A A Let µ ∈ Mp and put ν = γ1 (µ). Recall that ν ∈ Np . We claim that
If B =
1 )π1 ∈ J1µ . (B − I ⊗ Bp,ν
i
(8.29)
Ci ⊗ Di ∈ A " AN ×N , then 1 B − I ⊗ Bp,ν = Ei ⊗ Di = (Ei ⊗ I)(I ⊗ Di ), i
i
where Ei := Ci − (Γp Ciπ )(ν)I, and since (Γp Eiπ )(ν) = 0, we get (8.29). If B ∈ A ⊗ AN ×N , choose Bn ∈ A " AN ×N so that B − Bn → 0 as n → ∞. 1 − (Bn )1p,ν → 0 as n → ∞, and because Corollary 8.40 implies that Bp,ν (8.29) holds for Bn in place of B and J1µ is closed, it results that (8.29) is true for every B ∈ A ⊗ AN ×N . p 1 is in Φ(HN ). The commutativity of Aπ along with TheDue to (iii), Bp,ν 1 ∈ Φ(H p ). If there would exist a ν0 ∈ NA orem 1.14(c) show that det Bp,ν p 1 π such that (Γp (det Bp,ν ) )(ν0 ) = 0, then combining our assumption that A 1 π ∂S NA p = Np with 1.20(c), it would follow that (det Bp,ν ) is a topological π p p divisor of zero in A and thus in L(H )/C∞ (H ), which is clearly impos1 1 is Fredholm. Thus (det Bp,ν )π ∈ GAπ and therefore sible in case det Bp,ν 1 π π 1 and because (Bp,ν ) ∈ GAN ×N . So there is a regularizer R ∈ AN ×N of Bp,ν p 1 −1 1 belongs (Bp,ν ) − R ∈ C∞ (HN ) ⊂ AN ×N , it follows that the inverse of Bp,ν to AN ×N . Since 1 1 )−1 (I ⊗ Bp,ν ) = I ⊗ I, I ⊗ (Bp,ν we conclude from (8.29) that 1 B1π + J1µ = (I ⊗ Bp,ν )π1 + J1µ ∈ G(Aπ1 /J1µ ).
Recalling 8.42 we see that the proof is complete.
8.44. Toeplitz operators with P C ⊗ P C symbols. (a) Suppose that a ∈ (P C ⊗ P C)N ×N . Then the preceding theorem and the remark in 8.40 p (T2 )) if and only if give that T2 (a) is in Φ(HN 0 1 p T 1 − σp (λ) a(τ − 0, ·) + σp (λ)a(τ + 0, ·) ∈ GL(HN ), 1 0 p T 1 − σp (λ) a(·, τ − 0) + σp (λ)a(·, τ + 0) ∈ GL(HN ) for all (τ, λ) ∈ T × [0, 1]. Here σp (λ) is as in 5.12. (b) Analogously, if a ∈ (P Cp ⊗ P Cp )N ×N , then T2 (a) ∈ Φ(pN (Z2++ )) if and only if 0 1 T 1 − σq (λ) a(τ − 0, ·) + σq (λ)a(τ + 0, ·) ∈ GL(pN ), 1 0 T 1 − σq (λ) a(·, τ − 0) + σq (λ)a(·, τ + 0) ∈ GL(pN ) for all (τ, λ) ∈ T × [0, 1]. Here 1/p + 1/q = 1.
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8 Toeplitz Operators over the QuarterPlane
(c) After interpreting the function ap : T × [0, 1] → C (see 5.37) resp. ap : T × [0, 1] → C (see 6.31) for a ∈ P C resp. a ∈ P Cp as a function given on T (see 2.79) a similar reasoning as in the proof of Corollary 8.22 can be used to show that the index of a scalar Fredholm Toeplitz operator on H p (T2 ) resp. p (Z2++ ) with symbol from P C ⊗ P C resp. P Cp ⊗ P Cp is always zero. 8.45. Locally sectorial symbols. (a) Let A and B be C ∗ algebras between C and L∞ . A function a ∈ L∞ ⊗ L∞ ∼ = C(X × X) is said to be locally sectorial over A ⊗ B if it is sectorial