Lifting the solutions of a Toeplitz type equation; the semigroup case X = TsXT*s

Note di Matematica 25, n. 1, 2005/2006, 231–257. Lifting the solutions of a Toeplitz type equation; the semigroup case X=TsXT ∗ s Sergio Bermudo, Carmen H. Mancera, Pedro J. Pa´ul i Departamento de Matemtica Aplicada II, Universidad de Sevilla [email protected], [email protected], [email protected] Abstract. A Toeplitz operator with respect to a contractive representation {Ts}of an abelian semigroup Σ in a Hilbert space His an operator X∈ B(H) such that X=TsXT ∗ s for all s∈Σ. We show that if {Ts}has a minimal isometric dilation {Us} ⊂ B(K), then Toeplitz operators can be obtained in a unique way as compressions of operators Y∈ B(K), called Toeplitz symbols, such that Y=UsY U∗ s. This approach to lifting the Toeplitz equation X=TsXT ∗ sis shown to be unitarily equivalent to the one proposed by Muhly in 1972. We use our approach to extend to this case a number of theorems about classical Toeplitz operators and, finally, we show that some classes of well-known operators —like Wiener-Hopf operators, Toeplitz operators in H2(Td) or Toeplitz operators in the sense of Murphy— fall within the class studied in this paper. The main objective of this paper is to provide a general framework that, we hope, will be useful in order to extend to wider classes of operators some of the more recent and deep advances in the theory of Toeplitz operators. Keywords: Toeplitz operators, Wiener-Hopf operators, operator semigroups, minimal isometric dilation. MSC 2000 classification: Primary: 47B35. Secondary: 47A10, 47A20, 47A53, 47A62. To the memory of Klaus Floret, a fine mathematician and a good man 1 Introduction Let {Ts}be a semigroup of contractions having a minimal isometric dilation {Us}. Can one lift simultaneously a collection of Toeplitz type equations X=TsXT∗ sto the corresponding collection Y=UsY U∗ sand recover Xas the compression of Y? It is widely known that the answer is affirmative for the semigroup {Bn:n= 0,1,2,...}, where Bis the co-isometric unilateral backward shift in H2(T), and that the solutions of the simultaneous equations X=BnXB∗nare the classical Toeplitz operators. iThis work has been partially supported by Consejer´ıa de Educaci´on y Ciencia de la Junta de Andaluc´ıa and by la Direcci´on General de Investigaci´on del Ministerio de Ciencia y Tecnolog´ıa, project number BFM2001-3735 232 S. Bermudo, C. H. Mancera, P. J. Pa´ul Every function φ∈L∞(T) defines a Toeplitz operator Tφ:H2(T)→H2(T) by means of Tφf:= P(φ·f) where Pis the orthogonal projection from L2(T) onto H2(T). In the language of dilation theory, Tφis the compression of the multiplication operator Mφinduced by φ; this function φis called the symbol of Tφand it is unique. Toeplitz operators satisfy a simple characteristic relation given by Brown and Halmos in 1963 [6]; it tells us that a bounded linear map X:H2(T)→H2(T) is a Toeplitz operator if, and only if, X=BXB∗, where Bis the backward shift in H2(T). The development of the theory of Toeplitz operators in the last decades can be found in [2], [5], [7], [18] or [19]. It is no wonder that a number of possible generalizations of the notion of a Toeplitz operator have been proposed in the literature. We shall concentrate on generalizations made by exploiting the properties of the characteristic relation. Namely, take a contraction Tdefined on a Hilbert space Hand study the properties of the operators X:H → H, called generalized Toeplitz operators, that satisfy X=TXT∗. The purpose of this line of research —started by Douglas and Pearcy [13], Douglas [14] and [15], Rosenblum [41] and Sz.-Nagy and Foia¸s [46] and [47], and continued by Muhly [26], Pt´ak and Vrbov´a [39], Pt´ak [38], two of the present authors [24] and [25], and K´erchy [22] and [23]— is two-fold: on one hand, to obtain new operators that share some of the properties that make the class of Toeplitz operators important and, on the other hand, to find out which of these interesting properties of the classical case depend only on the characteristic relation. Hyponormal operators and operators with one dimensional self-commutators are examples of generalized Toeplitz operators (see, respectively, [46] and [47], and [10]). To study up to what point the properties of classical Toeplitz operators are valid within this approach, an essential problem is to find out what operators play the role of symbols. In the classical case, the symbols are multiplication operators induced by functions from L∞(T) or, equivalently, operators Ysuch that Y=UY U∗where Uis the unitary bilateral backward shift in L2(T). From the point of view of dilation theory, Uis both the minimal isometric dilation and the minimal unitary dilation of B, and U∗is the minimal unitary extension of the unilateral forward shift S=B∗, hence the key step is lifting the solutions of BXB∗=Xto solutions of UY U∗=Y, and these families of solutions are oneto-one related by the fact that each Xis the compression of a Yto H2(T), that is X=PY |H2(T). This suggests that in the generalized setting the symbols should be solutions of a suitable lifting of the equation X=T XT ∗involving the minimal isometric-or-unitary dilation-or-extension of T. If Tis a co-isometry, then the approaches taken by Douglas, Sz.-Nagy and Foia¸s, and Pt´ak and Vrbov´a are formally the same, but there is a slight difference in their points of view. They proved that if Tis a co-isometry then every solution Lifting the solutions of a Toeplitz type equation 233 Xof X=TXT∗is the compression of a solution Yof Y=UY U∗where, for Sz.-Nagy, Foia¸s, Pt´ak and Vrbov´a, Uis the minimal isometric dilation of T[47, Thm. 2], [39, Thm. 2.11], [38, Thm. 2.5], whereas for Douglas, U∗is the minimal unitary extension of T∗[14, Thm. 2]. These different points of view produce, however, different approaches for the case when Tis a general contraction, namely (DM) Douglas proposed the following approach. Let A:= √limnTnT∗nbe the asymptotic modulus of T∗and denote by Mthe closure of the range of A. Then there is an isometry Vdefined on Mand such that V A =AT ∗ and every solution Xof the equation X=T XT ∗can be represented in the form X=AX0Awhere X0is a solution of the equation X0=V∗X0V. Now, since V∗is a co-isometry, each of those X0can be represented as the compression of a solution Zof the equation Z=WZW ∗where W∗ is the minimal unitary extension of V. This approach was later used by Muhly [26] to show how to lift simultaneously a family of equations of the form X=TsXT∗ swhere {Ts:s∈Σ}is a contractive representation of an abelian semigroup Σ on a Hilbert space H. (SF) Sz.-Nagy and Foia¸s proposed the following approach. Let U∈ B(K) be the minimal isometric dilation of T, let Rbe the residual subspace of U, that is, the largest reducing subspace where Uis unitary, and denote by U|R the unitary part of the Wold decomposition of U. Then every solution Xof the equation X=TXT∗can be obtained from a solution Yof the equation Y= (U|R)Y(U|R)∗, by means of the equality X=P(H)Y P(R)|H, where P(H) and P(R) denote the respective orthogonal projections from K. (PV) Pt´ak and Vrbov´a proposed the following approach. With the same notation as in (SF), every solution Xof the equation X=T XT ∗is uniquely given as the compression X=P(H)Y|H of a solution Yof the equation Y=UY U∗. Moreover, these operators Yare essentially defined in R, in the sense that they satisfy the equalities Y=P(R)Y=Y P(R), and they commute with Uand U∗. As it is described in the last sentence, for the case of an arbitrary contraction T, the approaches (SF) and (PV) are essentially the same, although we prefer the latter because the relation between Xand Yis simpler in (PV) due to the fact that His a subspace of Kbut not necessarily a subspace of R. Within this approach a number of theorems have been extended from the classical to the generalized case in [24] and [25], complementing the results obtained in [14], [15], [46], [47], [39] and [38]. 234 S. Bermudo, C. H. Mancera, P. J. Pa´ul However, the approach (DM) is not, at least a priori, the same as the others if Tis not a co-isometry, and the starting purpose of this paper was to clarify the situation. Then we realized, inspired essentially by the techniques used in [38], that a (SF)-(PV) approach to lifting a family of equations X=TsXT ∗ s, where {Ts}is a semigroup of contractions, could be developed; this is done in Section 2 and we want to thank Prof. R. Douglas for calling our attention to Muhly’s paper [26] and for suggesting us to extend the (SF)-(PV) approach to the case of semigroups. In Section 3 we prove that our approach is unitarily equivalent to the approach (DM) used by Muhly. In Section 4 we give extensions to the semigroup case of several theorems about classical Toeplitz operators. Finally, in Section 5 we show that some classes of well-known operators —like WienerHopf operators, Toeplitz operators in H2(Td) or Toeplitz operators in the sense of Murphy— fall within the class studied in this paper. The main objective of this paper is, thus, to provide a general framework that, we hope, will be useful in order to extend to wider classes of operators some of the more recent and deep advances in the theory of Toeplitz operators. We thank the referee for his helpful suggestions. Notations. Our terminology and notations will be mostly standard, e.g., given two Hilbert spaces H1and H2, we shall denote by B(H1,H2) the set of all operators (bounded linear mappings) from H1into H2or simply B(H1) if H1= H2. The closure X(H1) of the range X(H) of an operator X∈ B(H1,H2) will be denoted by ran(X) and its kernel by ker(X). The spectrum, right-spectrum and left-spectrum of Xwill be denoted, respectively, by σ(X), σr(X) and σl(X). If His a closed subspace of K, the orthogonal projection from Konto Hwill be denoted by P(H). In general, we refer the reader to the excellent books by B¨ottcher and Silbermann [7], Douglas [16], Halmos [20] and [21], Nikolski [35], Sz.-Nagy and Foia¸s [45] and Young [50]. 2 Toeplitz operators with respect to a semigroup Operator representations of semigroups. Let Σ be an abelian (additive) semigroup with identity e; we regard Σ as a directed set by saying that r≤s if s=r+qfor some q∈Σ. A family of contractions (respectively isometries, co-isometries, unitary operators) {Ts:s∈Σ} ⊂ B(H) is said to be a contractive (respectively isometric, co-isometric, unitary)representation of the semigroup Σ in the Hilbert space Hif Teis the identity operator id(H) on Hand Tr+s= TrTsfor all r, s ∈Σ. An isometric representation {Us:s∈Σ}of Σ in Kis said to be an isometric dilation of a contractive representation {Ts:s∈Σ}in Hif His a subspace of Kand Ts=P(H)Us|H for all s∈Σ, and is said to be minimal if Kis the Lifting the solutions of a Toeplitz type equation 235 smallest subspace containing Hand Us-invariant for all s∈Σ or, equivalently, if K=Ws∈ΣUsH. For instance, given any contraction T∈ B(H), the family {Tn:= Tn:n= 0,1,2,...}is a contractive representation of the (additive) semigroup Z+of non-negative integers in H, and if U∈ B(K) is the minimal isometric dilation of T, then {Un:= Un:n= 0,1,2,...}is a minimal isometric dilation of {Tn:n∈Z+}; in this case, all the minimal isometric dilations are the same up to unitary equivalence. However, it is not always the case that a contractive representation {Ts:s∈Σ}of an arbitrary semigroup Σ has a minimal isometric dilation, and it may also happen that two minimal isometric dilations of the same representation are not isomorphic. Although we refer the reader to [45, I.6– I.9], where this topic is discused in detail, let us mention the following examples of contractive representations {Ts:s∈Σ}of a semigroup Σ that have a minimal isometric dilation: (a) Σ = Z+,(b) Σ = Z+2,(c) Σ is the additive semigroup R+of all non-negative real numbers and {Ts:s≥0}is continuous (that is, lims→0+Ts=T0in the strong operator topology), in this case {Ts:s≥0}is usually called a continuous one-parameter semigroup of contractions, (d) {Ts: s∈Σ}is a co-isometric representation (in which case the elements of a minimal isometric dilation {Us}are, in fact, unitary; namely, {U∗ s}is the minimal unitary extension of the isometric representation {T∗ s}), and (e) {Ts:s∈Σ}is doubly commuting (that is, TsT∗ r=T∗ rTsfor all r, s ∈Σ). For more recent advances on the existence of minimal isometric or unitay dilations of a semigroup and related problems about lifting operator equalities and inequalities, we refer the interested reader to [3,4,8,9,17,34,48], and [49]. In what follows, Σ stands for a fixed abelian semigroup with identity eand {Ts}for a contractive representation of Σ in a Hilbert space Hhaving a minimal isometric dilation {Us} ⊂ B(K). 1 Lemma. For every s∈Σthe following assertions hold true: (1) P(H)Usk=TsP(H)kfor all k∈ K. (2) UsH⊥⊂ H⊥. (3) U∗ sH ⊂ H. (4) U∗ s|H =T∗ s. Proof. Since K=Wr∈ΣUrH, to prove that (1) holds it will be enough to check that P(H)UsUrh=TsP(H)Urhfor all r∈Σ and h∈ H. But P(H)UsUrh=P(H)Us+rh=Ts+rh=TsTrh=TsP(H)Urh, 236 S. Bermudo, C. H. Mancera, P. J. Pa´ul as desired. Now, a straightforward computation shows that the four assertions are, indeed, equivalent (part (2) is essentially proved in [43] as a matter of fact). QED Toeplitz operators and Toeplitz symbols. A bounded operator X: H → H is said to be a Toeplitz operator with respect to {Ts}if X=TsXT∗ sfor all s∈Σ. In this equation one could consider X:H1→ H2, a {T1s}representation of Σ in H1on the right, and a {T2s}representation of the same semigroup in H2on the left. Muhly [26] showed how to reduce this apparently more general situation to the case when both representations are the same, namely, Xis a solution of X=T2sXT∗ 1sif, and only if, 0X 0 0 is a solution of 0X 0 0 =T2s0 0T1s0X 0 0 T2s0 0T1s∗. However, this way of reducing the equation X=T2sXT ∗ 1sto the equation X= TsXT∗ sis not suitable to study some properties, like invertibility. For the sake of avoiding a cumbersome notation, we shall stick to the equations X=TsXT ∗ s but the reader can check that all of the results contained in this paper are true, with the obvious changes, for equations X=T2sXT ∗ 1s(and we shall make explicit use of this fact in the proof of assertion (2) of Theorem 3 below). When Tis a contraction and {Tn}is the corresponding representation of Z+, we recover the notion of generalized Toeplitz operator described in the Introduction. An operator Y:K → K is said to be a Toeplitz symbol with respect to {Ts} if Y=UsY U∗ sfor all s∈Σ, that is, if Yis a Toeplitz operator with respect to the minimal isometric dilation {Us}of {Ts}. The key point here is that if Yis a Toeplitz symbol with respect to {Ts}, then the compression X=P(H)Y|H is a Toeplitz operator with respect to {Ts}because, by using Lemma 1 twice, for all s∈Σ and h∈ H we have TsXT∗ sh=TsXU∗ sh=TsP(H)Y U∗ sh=P(H)UsY U∗ sh=P(H)Y h =Xh. Our first main result is that there is a one-to-one and isometric relation between Toeplitz operators and Toeplitz symbols. 2 Theorem. If X∈ B(H)is a Toeplitz operator with respect to {Ts}, then there exists a unique Toeplitz symbol Y∈ B(K)such that X=P(H)Y|H. This symbol is given by Y k = lim s∈ΣUsXP(H)U∗ skfor all k∈ K Lifting the solutions of a Toeplitz type equation 237 and it satisfies that kYk=kXk. Proof. Since K=Wr∈ΣUrHand {UsXP(H)U∗ s:s∈Σ}is uniformly bounded, to prove that {UsXP (H)U∗ sk:s∈Σ}is a convergent net for each k∈ K, it will be enough to check that {UsXP(H)U∗ s(Urh) : s∈Σ}converges for all r∈Σ and h∈ H. Fix r∈Σ and take s≥rso that s=q+rfor some q∈Σ. Now, define kq:=UsXP(H)U∗ sUrh=Uq+rXP(H)U∗ q+rUrh =Uq+rXP(H)U∗ qU∗ rUrh=Uq+rXP(H)U∗ qh=Uq+rXT∗ qh. Let us see that hkp+q, kqi=hkq, kqifor all p∈Σ. Indeed, hkp+q, kqi=Up+q+rXT∗ p+qh, Uq+rXT∗ qh=UpXT∗ p+qh, XT∗ qh =P(H)UpXT∗ p+qh, XT∗ qh=TpXT∗ pT∗ qh, XT∗ qh =XT∗ qh, XT∗ qh=Uq+rXT∗ qh, Uq+rXT∗ qh=hkq, kqi. Now, it follows that kkp+q−kqk2=kkp+qk2−kkqk2and this shows that {kkqk2: q∈Σ}is an increasing net of real numbers. Since this net is also bounded by khk2, it is convergent. The equality kkp+q−kqk2=kkp+qk2−kkqk2tells us now that {kq:q∈Σ}is a Cauchy net in K, hence it converges and this finishes the proof that Y k = lim s∈ΣUsXP(H)U∗ skfor all k∈ K defines a bounded operator in Ksuch that kYk ≤ kXk. Let us see now that Xis the compression of Yto H. Indeed, by Lemma 1, for all h∈ H and s∈Σ we have P(H)UsXP(H)U∗ sh=TsXT∗ sh=Xh, hence P(H)Y h = lims∈ΣP(H)UsXP(H)U∗ sh=Xh. It also follows kXk ≤ kYk so that, as a matter of fact, kXk=kYk. To prove that Yis a Toeplitz symbol, simply note that for each r∈Σ and k∈ K we have UrY U∗ rk= lim s∈ΣUrUsXP(H)U∗ sU∗ rk= lim s∈ΣUr+sXP(H)U∗ r+sk=Y k. Finally, to prove the uniqueness of the symbol, asume that Zis a Toeplitz symbol such that X=P(H)Z|H. Then, for all s∈Σ and k∈ K we have Zk =UsZU∗ sk=UsP(H) + P(H⊥)ZU∗ sk =UsP(H)ZP(H) + P(H⊥)U∗ sk+UsP(H⊥)ZU∗ sk =UsP(H)ZP(H)U∗ sk+UsP(H)ZP(H⊥)U∗ sk+UsP(H⊥)ZU∗ sk =UsXP(H)U∗ sk+UsP(H)ZP(H⊥)U∗ sk+UsP(H⊥)U∗ sZk. 238 S. Bermudo, C. H. Mancera, P. J. Pa´ul If we prove that lims∈ΣP(H⊥)U∗ s= 0 in K, this will imply that the last two summands of the last line of the chain of equalities displayed above are convergent to zero and, therefore, Z= lims∈ΣUsXP(H)U∗ s=Y. But, by using once again that K=Wr∈ΣUrH, in order to prove that lims∈ΣP(H⊥)U∗ s= 0 in K, it will be enough to check up on elements of the form Urhwith h∈ H. Indeed, for s≥r, we have P(H⊥)U∗ sUrh=P(H⊥)U∗ s−rh= 0 because, according to Lemma 1, the space His U∗ s-invariant for all s∈Σ. QED As we mentioned above, when we have a contractive representation {Tn} of Σ = Z+, the subspace R, the residual part of the Wold decomposition of the minimal isometric dilation Uof T, plays an essential role because, in that case, the Toeplitz symbols Yare essentially defined in Rin the sense that Y=P(R)Y=Y P(R). In the general semigroup case, as we are about to see (Theorem 2 below), we have the same situation: the symbol is essentially defined in the unitary part Rof the Wold decomposition of a semigroup of isometries introduced by Suciu [44] as follows. Since Usis an isometry, we have that UsU∗ s is a projection for each s∈Σ. Therefore, {UsU∗ s:s∈Σ}is a decreasing net of commuting projections that converges strongly to the orthogonal projection P(R) onto the subspace defined by R:= Ts∈ΣUsK. Moreover, Ris Us-reducing and Us|R is unitary for all s∈Σ [44, Thm. 1]. (The Wold decomposition of a semigroup of isometries has three parts called unitary, totally non-unitary or shift, and strange or evanescent; we refer the interested reader to [44] and [27]). Let us note at this point that if {Ts}is a co-isometric representation of Σ, then each Usis unitary, hence R=K. This shows that the importance of the role played by Ris hidden in the classical case because if Tis the backward shift Bin H2(T), then R=K=L2(T). In our case, each single Usmight not be the minimal isometric dilation of the corresponding Tsand Rmight not be the residual subspace of the Wold decomposition of Usbut, nevertheless, this subspace Rhas similar properties, so we shall call it the residual subspace of the minimal isometric dilation {Us}of {Ts}. We record now for later use some of the properties of the representations {Ts}and {Us}related to R(see, e.g., [38, Lemmas 2.3 and 2.4] and [45, II.3] for the case of a single contraction). 3 Lemma. Let R:= Tt∈ΣUtKbe the residual subspace of the minimal isometric dilation {Us}of {Ts}. Let Pbe the closure of P(R)H. Then the following assertions hold: (1) For each s∈Σ, the residual subspace Ris a Us-reducing subspace of K contained in the residual subspace of the Wold decomposition of Us. (2) Us|R is a unitary operator. (3) R=P ⊕(R∩H⊥), where H⊥is the orthocomplement of Hin K. Lifting the solutions of a Toeplitz type equation 239 (4) P(P)h=P(R)hfor every h∈ H. (5) P(H)P(P) = P(H)P(R). (6) For all h∈ H and s∈Σthe following chain of equalities holds U∗ sP(P)h=P(P)U∗ sh=P(P)T∗ sh=U∗ sP(R)h=P(R)U∗ sh=P(R)T∗ sh. (7) Pis U∗ s-invariant and U∗ s|P is an isometry. (8) For each s∈Σ, define the co-isometry Rs:= (U∗ s|P)∗∈ B(P). Then {Rs}is a co-isometric representation of Σin Pand {Us|R} is a minimal isometric (unitary, in fact) dilation of {Rs}. Proof. (1) Since R=Tt∈ΣUtK, it is clear that Ris Us-invariant. Now, given an element k= limtUtU∗ tk∈ R, we have U∗ sk= lim tU∗ sUtU∗ tk= lim rU∗ sUs+rU∗ s+rk= lim rUrU∗ rU∗ sk=P(R)U∗ sk, and it follows that Ris U∗ s-invariant. Finally, if Rsis the residual subspace of the isometry Usthen, as it is well-known, Rs=\ n∈N Un sK=\ n∈N UnsK ⊃ \ t∈Σ UtK=R. (2) follows from (1). (3) For every x∈ R and h∈ H we have hx, hi=hP(R)x, hi=hx, P(R)hi so it follows, by using that P(R)His dense in P, that hx, Pi = 0 if, and only if, x∈ H⊥. Noting that P(R∩H⊥)h= 0 for all h∈ H, we have that (4) follows from (3). (5) also follows from (3) because P(H)P(R) = P(H)P(P) + P(R∩H⊥)=P(H)P(P). (6) By Lemma 1, U∗ sh=T∗ sh∈ H. By using now assertions (1) and (4), we have U∗ sP(P)h=U∗ sP(R)h=P(R)U∗ sh=P(P)U∗ sh=P(P)T∗ sh. By using (5) we obtain the whole chain of equalities. (7) follows from (6) and (2). 246 S. Bermudo, C. H. Mancera, P. J. Pa´ul 9 Corollary. For the case of a single contraction, the approaches (DM) and (PV) described above are unitarily equivalent. 4 Properties of Toeplitz operators Within the framework of Toeplitz operators with respect to a single contraction, we gave in [24] and [25] appropriate extensions, as well as clarifying examples, of a number of results about classical Toeplitz operators; namely, Wintner’s theorem of invertibility of analytic Toeplitz operators, Widom and Devinatz’s invertibility criteria for Toeplitz operators with unitary symbols, Hartman and Wintner’s theorem about Toeplitz operators having Fredholm symbols, Hartman and Wintner’s estimate of the norm of a compactly perturbed Toeplitz operator, the non-existence of compact classical Toeplitz operators due to Brown and Halmos, and some spectral properties that complemented the work done by Sz.-Nagy and Foia¸s. The tools that we used in [24] and [25] (the existence and uniqueness of symbols, the residual subspace, the associated Toeplitz operator Y0, etc.) work similarly in the semigroup case, so the proofs of our results there can be carried out almost word by word to the present situation. To prevent from making this paper unnecessarily long, we shall only state and prove, under the assumption that {Ts}is a co-isometric representation of Σ, the extensions of Wintner’s theorem of invertibility of analytic Toeplitz operator, the spectral inclusions between the left and right spectra of a Toeplitz symbol and the corresponding spectra of its associated Toeplitz operator, and Hartman and Wintner’s theorem about Toeplitz operators having Fredholm symbols. The hypothesis that {Ts}is a co-isometric representation of Σ in H, implies that every Us=Wsis a unitary operator on K=R=N, that H=M=P, that X=X0=Y0, that Z=Y, and that Xis an analytic Toeplitz operator if, and only if, XT∗ s=T∗ sX. These consequences help us to avoid technicalities —otherwise necessary in the most general case as the examples in [24] and [25] show— in the arguments. Nevertheless, the results that we offer give a flavor of the situation and can be still applied to some examples, as we shall see in Section 5. Let us remark at this point that these results will sound undoubtly familiar to the reader of the papers we quote here; please bear in mind that our main purpose here is to show that they fit into a common framework. 10 Theorem. Let Xbe an analytic Toeplitz operator with respect to the coisometric representation {Ts}. Then Xis invertible if, and only if, its symbol Yis invertible and Y−1is also an analytic Toeplitz symbol, in which case the Toeplitz operator associated to Y−1is X−1. Lifting the solutions of a Toeplitz type equation 247 Proof. Assume that Xis invertible. Since Xis analytic, it commutes with each T∗ s. According to Douglas’s characterization [15, Thm. 2] quoted after Lemma 3 above to prove that X−1is also an analytic Toeplitz operator it will be enough to prove that X−1commutes with each T∗ s. Indeed, T∗ sX−1=X−1XT∗ sX−1=X−1T∗ sXX−1=X−1T∗ s. Let Y0be the analytic symbol of X−1. We have to prove now that Y Y0=Y0Y= id(K). Since K=Ws∈ΣUsH, we only need to check up elements of the form Urh with r∈Σ and h∈ H. But, by using Theorem 1, we have Y Y0Urh= lim s∈ΣY UsX−1P(H)U∗ sUrh = lim s>r Y UsX−1P(H)U∗ sUrh= lim s>r Y UsX−1P(H)U∗ s−rh. Now, by using that Ycommutes with each Us(by Theorem 2), that His U∗ s−rinvariant (by Lemma 1) and that X=Y|H due to the analyticity, we have Y Y0Urh= lim s>r Y UsX−1P(H)U∗ s−rh= lim s>r UsY X−1U∗ s−rh = lim s>r UsXX−1U∗ s−rh= lim s>r UsU∗ s−rh=Urh. The equality Y0Y Urh=Urhfollows analogously. Conversely, if Yis invertible and Y−1is also an analytic symbol with Toeplitz operator X0, then His Y-invariant and Y−1-invariant so that XX0=Y|HY−1|H =Y Y −1|H = id(H). Analogously, one can prove that X0X= id(H). Therefore Xis invertible. QED 11 Corollary. Let {Ts}be a co-isometric representation of a semigroup Σ and let {Us}be its minimal unitary dilation. Let Xbe a unitary operator that commutes with every Ts. Then Xis an analytic Toeplitz operator with respect to {Ts}that can be uniquely extended to a unitary operator Y∈ B(K), its analytic symbol, that commutes with every Us. Proof. By taking adjoints in the equality TsX=XTsand using that Xis unitary, it follows that X∗T∗ s=T∗ sX∗and that XT∗ s=T∗ sXfor every s∈ Σ. Therefore, Xand X∗are both analytic Toeplitz operators with respect to {Ts}. Since Xis invertible and its inverse is X∗, it follows from Theorem 4 that its symbol Yis invertible and that its inverse is the symbol Y∗of X∗. Hence Yis unitary, extends Xbecause it is analytic, and Y Us=UsYby Theorem 2. QED 248 S. Bermudo, C. H. Mancera, P. J. Pa´ul When every Tsis a co-isometry, it is clear that λid(H) is an analytic Toeplitz operator with respect to {Ts}for each complex number λand it follows from the theorem that for analytic Toeplitz operators we have σ(Y)⊂σ(X). This is true for arbitrary Toeplitz operators with respect to a co-isometric representation, as was proved by Muhly [26, Thm. III]. We can go a bit further. 12 Theorem. Let Xbe a Toeplitz operator with respect to a co-isometric representation {Ts}and let Ybe the symbol of X. Then σl(Y)⊂σl(X)and σr(Y)⊂σr(X). Moreover, if Xis analytic then σl(Y) = σl(X). Proof. Since X−λid(H) is a Toeplitz operator with symbol Y−λid(K), to prove that σl(Y)⊂σl(X) it will be enough to prove that if Xis left-invertible then so is Y. If Xis left-invertible, then there exists ε > 0 such that kXhk ≥ εkhk for each h∈ H. Now, for each r∈Σ and h∈ H we have kY Urhk=kUrY hk=kY hk ≥ kP(H)Y hk=kXhk ≥ εkhk=εkUrhk. Now take a finite linear combination k=Ps∈ΦUshsand consider r=Ps∈Φs. Using again that His U∗ s-invariant for all s∈Σ, we have that h=Ps∈ΦU∗ r−shs is in H. Hence, by the inequality we have just established, it follows that kY kk=Y(X s∈Φ Ushs)=Y Ur(X s∈Φ U∗ r−shs)=kY Urhk ≥ εkUrhk=εkkk. Since K=Ws∈ΣUsH, we have kY kk ≥ εkkkfor all k∈ K so that Yis leftinvertible. To prove that σr(Y)⊂σr(X), simply note that X∗is a Toeplitz operator with Toeplitz symbol Y∗and take complex conjugates in the inclusion σl(Y∗)⊂ σl(X∗). Now assume that Xis analytic and that Xis not left-invertible. Then there exists a sequence of unit vectors (hn)⊂ H such that limnkXhnk= 0. This implies, by using that X=Y|H, that limnkY hnk= 0, so that Yis not leftinvertible. This shows that if Xis analytic then the inclusion σl(X)⊂σl(Y) also holds. QED It is well-known that for φ∈L∞the spectrum of Mφis the essential range of φ. On the other hand, Wintner’s Theorem (see, e.g., [16, 7.21], [21, Prob. 247] or [35, p. 320]) says that the spectrum of a classical analytic Toeplitz operator Tφequals ˜ φ(D) where ˜ φis the analytic extension of φto the open unit disc D. Lifting the solutions of a Toeplitz type equation 249 So even under the most favourable (but non-trivial) conditions there is no hope of obtaining that σ(Y) = σ(X) or, for that matter, σr(Y) = σr(X). Now, denote by F(H) the set of all Fredholm operators. Atkinson’s theorem [16, 5.17] tells us that F(H) can be written as F(H) = F+(H)∩F−(H) where F+(H) = {A∈ B(H) : AHis closed and ker Ais finite-dimensional} F−(H) = {A∈ B(H) : AHis closed and ker A∗is finite-dimensional}. 13 Theorem. Let Xbe a Toeplitz operator with respect to a co-isometric representation {Ts}and let Ybe the symbol of X. Then the following hold: (1) If X∈ F+(H)then Y∈ F+(K). (2) If X∈ F−(H)then Y∈ F−(K). (3) If X∈ F(H)then Y∈ F(K). Proof. Since an operator is in F+if, and only if, its adjoint is in F−, we only need to prove (1). Now, if Kis finite-dimensional then it is obvious that Y∈ F+(K) and there is nothing to prove. So we may, and do, assume that Kis infinitedimensional. We shall use the following general characterization for operators in F+(H) [7, 1.11(g)]: Let Hbe a Hilbert space and take A∈ B(H). If A∈ F+(H)and P0is the orthogonal projection from Honto ker(A)then there exists δ > 0such that kAxk+kP0xk ≥ δkxkfor all x∈ H. Conversely, if there is a finite number of compact operators K1, K2,...,Kn∈ B(H)and δ > 0such that kAxk+ Pn j=1 kKjxk ≥ δkxkfor all x∈ H, then A∈ F+(H). So assume X∈ F+(H) and let P0be the orthogonal projection from Honto ker(X). Then, according to the characterization written above, there exists δ > 0 such that kXhk+kP0hk ≥ δkhkfor all h∈ H. Therefore, kP(H)Y P(H)kk+kP0P(H)kk ≥ δkP(H)kkfor all k∈ K. Add δP(H⊥)kto both sides of this inequality to obtain kP(H)Y P(H)kk+δP(H⊥)k+kP0P(H)kk ≥ δkkkfor all k∈ K. In particular, fix k∈ K such that k6= 0, since U∗ sis unitary for all s∈Σ, we have kP(H)Y P(H)U∗ skk+δP(H⊥)U∗ sk+kP0P(H)U∗ skk ≥ δkU∗ skk=δkkk. 250 S. Bermudo, C. H. Mancera, P. J. Pa´ul Now use that each Usis an isometry to write kUsP(H)Y P(H)U∗ skk+δP(H⊥)U∗ sk+kP0P(H)U∗ skk ≥ δkkk.(i) We shall analyze the behaviour of each one of the three summands in the left hand side of (i). For the first one we know, by Theorem 1 and the equality XP(H) = P(H)Y P(H), that lims∈ΣUsP(H)Y P(H)U∗ sk=Y k. Therefore, there exists s0∈Σ, depending on k, such that if s≥s0then kUsP(H)Y P(H)U∗ skk−kY kk ≤ kUsP(H)Y P (H)U∗ sk−Y kk<δ 4kkk.(ii) Concerning the second summand, use that K=Ws∈ΣUsHto find h1, h2, . . . , hn∈ Hand r1, r2,...,rn∈Σ such that kk−Pn i=1 Urihik ≤ 1/4kkk. Then, by using that His U∗ s-invariant, for all s≥s1:= r1+r2+···+rn, we have P(H⊥)U∗ sk≤P(H⊥)U∗ sk− n X i=1 Urihi+P(H⊥)U∗ s n X i=1 Urihi ≤1/4kkk+P(H⊥) n X i=1 U∗ s−rihi= 1/4kkk. (iii) Now, for the third summand note that given x∈ K all the terms of the net C(x) := {P0P(H)U∗ s:s∈Σ} belong to the finite-dimensional subspace ker(X). Since it is clear that C(x) is bounded, it follows that C(x) is relatively compact and Tikhonov’s theorem ensures that the product set Qx∈K C(x) is relatively compact for the product topology. Therefore, the net {(P0P(H)U∗ sx)x∈K :s∈Σ} has an adherent point (Cx)x∈K in the product space Qx∈K C(x). This gives us a function C:K → ker(X) and a standard proof shows that Cis a bounded linear mapping with finite rank, hence Cis a compact operator. Note that C does not depend on our previously fixed k∈ K. Going back to this k∈ K, there exists an cofinal set Σk⊂Σ such that for all s∈Σkthe following holds kP0P(H)U∗ skk−kCkk<δ 4kkk.(iv) Now, since Σkis cofinal, we may take s∈Σksuch that s≥s0and s≥s1. With this splug (ii), (iii) and (iv) in (i) to obtain kY kk+kCkk>kUsP(H)Y P(H)U∗ skk+δP(H⊥)U∗ sk +kP0P(H)U∗ skk− 3δ 4kkk ≥ δ 4kkk. (v) Lifting the solutions of a Toeplitz type equation 251 Since kwas arbitrary in K, the characterization quoted at the beginning of the proof tells us that Y∈ F+(K). QED 5 Examples The purpose of this final section is to show that a number of widely studied classes of operators can be defined as the families of all Toeplitz operators with respect to suitable semigroups. Toeplitz operators in the sense of Murphy. In the series of papers [29–32] and [33], Murphy has extended the notion, and many of the properties, of classical Toeplitz operator to Hardy spaces generated by function algebras (a similar extension was introduced by Cowen and Douglas [11]); we shall describe his framework as explained in [32]. Let Ω be a function algebra on a compact Hausdorff space Ghaving a unique representing measure µfor a character of Ω. Then a great deal of the theory of Hardy spaces on Textends to this setting. Let H2(µ) be the closure of Ω in L2(µ). The functions in H2(µ) are called analytic; in particular, the analytic and unimodular functions are called µ-inner, as in the classical case. Every function φ∈L∞(µ) defines a Toeplitz operator Tφ:H2(µ)→H2(µ) by Tφf:= P(φ·f) where Pis the orthogonal projection from L2(µ) onto H2(µ). Let Σ be the semigroup of all µ-inner functions, then Murphy proved [32, Thm. 2.1] that X:H2(µ)→H2(µ) is a Toeplitz operator if, and only if, X=T∗ φXTφfor every µ-inner φ. Whence, in our language, X:H2(µ)→H2(µ) is a Toeplitz operator in the sense of Murphy if, and only if, Xis a Toeplitz operator with respect to the semigroup {T∗ φ:φ∈Σ}. Our results in Section 4 yield some of the results obtained by Murphy for individual Toeplitz operators. He also produced a number of interesting contributions to the study of the Toeplitz algebra generated by Toeplitz operators; we refer the interested reader to his papers. Toeplitz operators on H2(Td).Let L2(Td) and H2(Td) be the corresponding Lebesgue and Hardy spaces of functions of dvariables f:Td→C. As in the one dimensional case, every function φ∈L∞(Td) defines a Toeplitz operator on H2(Td) by Tφf:= P(φ·f), where Pis the orthoprojection from L2(Td) onto H2(Td). The operator Tφis said to be analytic if the symbol φ∈H∞(Td). This class of operators has been studied by many authors; we refer the reader to the book by B¨ottcher and Silbermann [7, Ch. 8] and references therein (see also [1]). Given ~n = (n1, n2,...,nd)∈Zd +consider the shift S~n defined on H2(Td) by (S~nf)(ζ1, ζ2,...,ζd) := ζn1 1ζn2 2···ζnd df(ζ1, ζ2,...,ζd). If we now define T~n := S∗ ~n, then it is clear that {T~n :~n ∈Zd +}is a co-isometric representation of Zd +in H2(Td) that has a minimal isometric dilation {U~n :~n ∈ 252 S. Bermudo, C. H. Mancera, P. J. Pa´ul Zd +}defined on L2(Td) by (U~nf)(ζ1, ζ2,...,ζd) := ζ−n1 1ζ−n2 2···ζ−nd df(ζ1, ζ2,...,ζd). (Recall that an arbitrary contractive representation of Zd +does not necessarily have a minimal isometric dilation if d > 2 [45, I.6.3].) Following the lines of the proof given by Brown and Halmos for the one dimensional case, it is easy to see, and surely well-known, that X∈ B(H2(Td)) is a Toeplitz operator if, and only if, it is a Toeplitz operator with respect to {T~n : ~n ∈Zd +}and that analytic Toeplitz operators Tφwith φ∈H∞(Td) correspond to analytic Toeplitz operators with respect to {T~n :~n ∈Zd +}. Theorems 4 and 6 above do not provide new information, only alternative proofs, about analytic Toeplitz operators in H2(Td). Nevertheless, one of the consequences of Theorem 5, the fact that if φ∈H∞(Td), then σl(φ) = σl(Tφ), is probably well-known at least, as we mentioned above, for Toeplitz operators in the sense of Murphy [31, 5.6] so, in particular, for the one dimensional case; however, we have been unable to locate a reference for the d-dimensional case. Recall that if φ∈H∞(Td) then σ(φ) = σl(φ) hence, in fact, the part covered by the spectrum of φwithin the spectrum of its Toeplitz operator Tφ(which is greater in general) is exactly the left spectrum of Tφ. Toeplitz operators with respect to a continuous semigroup. Let {Ts:s≥0}be a continuous one-parameter semigroup of contractions and let A:= lim s→0+ Ts−id(H) sand T:= A+ id(H)A−id(H)−1 be, respectively, the generator and the co-generator of the semigroup. As it is well-known, Ais a closed linear mapping densely defined in Hbut generally unbounded, and Tis a contraction in Hthat determines {Ts:s≥0}uniquely (see [45, III. 8] for details). As a matter of fact, the following hold T= lim s→0+φs(Ts) where φs(ζ) = ζ−1 + s ζ−1−sfor ζ∈D,and Ts=es(T) where es(ζ) = exps(ζ+ 1)/(ζ−1)for ζ∈D. It is well-known that {Ts:s≥0}is, respectively, isometric, co-isometric or unitary if, and only if, so is the co-generator T. On the other hand, we also know that if Uis the minimal isometric dilation of T, then Uis the co-generator of a continuous semigroup {Us:s≥0}which is the minimal isometric dilation of {Ts:s≥0}; moreover, in this case, the reducing subspace of {Us:s≥0} equals the reducing susbpace of U. By using these properties and the fact that the symbols commute with the dilations on the residual subspace, it is easy to prove the following result. Lifting the solutions of a Toeplitz type equation 253 14 Theorem. An operator X∈ B(H)is a Toeplitz operator with respect to a continuous one-parameter semigroup of contractions {Ts:s≥0}if, and only if, it is a generalized Toeplitz operator with respect to the co-generator Tof the semigroup. Wiener-Hopf operators. Wiener-Hopf operators fall within the class of Toeplitz operators with respect to a continuous semigroup described above. To see this, let us consider the translation co-isometries Tsdefined on L2(R+) for each s≥0 by (Tsf)(x) = χR+(x)f(x+s) for f∈L2(R+). Then {Ts:s≥0}is a continuous co-isometric semigroup and its minimal isometric, in fact unitary, dilation is the semigroup {Us:s≥0}consisting of the translation operators Usdefined on L2(R) by (Usf)(x) = f(x+s). It is also known [45, III. 9] that the co-generator of {Ts:s≥0}is the co-isometry T defined on L2(R+) by (Tf)(x) = f(x)−2exZ∞ x f(ξ)e−ξdξ for f∈L2(R+). According to our Theorem 2, the Toeplitz symbols with respect to {Ts:s≥0} are the operators Y∈ B(L2(R)) that commute with the translations Usfor all s∈Rand, as it is well-known [7, 9.2], these are precisely the operators of the form Y=F−1MφFwhere Fis the Fourier transform on L2(R) and Mφis the operator of multiplication by a function φ∈L∞(R). Consequently, X∈ B(L2(R+)) is a Toeplitz operator with respect to {Ts:s≥0}if, and only if, it is of the form X=χR+F−1MφF|L2(R+). The operators of this form are called Wiener-Hopf integral operators and have been widely studied in the literature (we refer the reader to [7, Ch. 9], see also [11] and [28] for the study of algebras generated by Wiener-Hopf operators). It was proved by Rosenblum and Devinatz that Wiener-Hopf operators are unitarily equivalent to classical Toeplitz operators on H2(T) (see [18] for a different approach), and we now see that both classes are particular cases of a more abstract situation. By using Theorem 7 above we obtain that X∈ B(L2(R+)) is a Wiener-Hopf operator if, and only if, X=S∗XS where Sis the adjoint of the co-generator Tor, in other words, Sthe isometry defined by (Sf)(x) = f(x)−2e−xZx 0 f(ξ)eξdξ for f∈L2(R+). (This isometry Sis a disguised form of the Laguerre shift Rdefined on L2(R+) by (Rf)(x) = f(x)−2e−x/2Rx 0f(ξ)eξ/2dξ which can also be used to define Wiener-Hopf operator as the solutions of X=R∗XR; see [42, Ch. 3]). 254 S. Bermudo, C. H. Mancera, P. J. Pa´ul When the function φin the symbol Y=F−1MφFis in H∞(R) the WienerHopf operator X=χR+Y|L2(R+) is said to be analytic; this corresponds to the analyticity of the classical Toeplitz operator which is unitarily equivalent to X. Let us see that this is also consistent with our definition of analytic Toeplitz operator with respect to {Ts:s≥0}. Indeed, since the semigroup {Ts:s≥0} is co-isometric, a symbol Y=F−1MφFis analytic (in the sense introduced in Section 2 above) if Y L2(R+)⊂L2(R+) or, equivalently, F−1MφFL2(R+)⊂L2(R+). But, since by the Paley-Wiener representation theorem Fmaps L2(R+) unitarily onto H2(R), this is the same as saying that Mφmaps H2(R) into itself and this happens if, and only if, φ∈H∞(R) as desired. Bearing in mind that the multiplication operator Mφby a function φ∈ L∞(R) is Fredholm if, and only if, it is invertible, our results in Section 4 yield the well-known characterizations of invertible analytic Wiener-Hopf operators and Fredholm Wiener-Hopf operators [7, Ch. 9]. Again, one of the consequences of Theorem 5 is the fact that if φ∈H∞(R), then the left spectrum of the analytic Wiener-Hopf operator defined by φcoincides with the spectrum of corresponding symbol that, as in the classical case, is the essential range of φ. Let us finally mention that, along a series of papers, Devinatz, Pellegrini (also for unbounded operators), Reeder and Shinbrot considered invertibility properties of so-called generalized Wiener-Hopf operators, defined as the class of operators X∈ B(H) which are compressions X=P(H)Y|H of operators Y defined on a superspace Kof H(see, e.g., [12], [36], [37] [40], and references therein). The class of generalized Wiener-Hopf operators is obviously larger than the class of Toeplitz operators with respect to semigroups considered in this paper and share some of the properties in a weaker sense (no Toeplitz type equations or commutativity involved). Acknowledgements. This research has been partially supported by la Consejer´ıa de Educaci´on y Ciencia de la Junta de Andaluc´ıa and by la Direcci´on General de Investigaci´on del Ministerio de Ciencia y Tecnolog´ıa, project number BFM2001-3735. References [1] J. A. Ball, W. S. Li, D. Timotin, T. T. 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