Fredholm Toeplitz operators with VMO symbols and the duality of generalized Fock spaces with small exponents

Abstract We characterize Fredholmness of Toeplitz operators acting on generalized Fock spaces of the n-dimensional complex space for symbols in the space of vanishing mean oscillation VMO. Our results extend the recent characterizations for Toeplitz operators on standard weighted Fock spaces to the setting of generalized weight functions and also allow for unbounded symbols in VMO for the first time. Another novelty is the treatment of small exponents 0 < p < 1, which to our knowledge has not been seen previously in the study of the Fredholm properties of Toeplitz operators on any function spaces. We accomplish this by describing the dual of the generalized Fock spaces with small exponents.


Introduction
The Fock space (also known as the Segal-Bargmann space) consists of all holomorphic functions in the n-dimensional complex Euclidean space C n square-integrable with respect to the Gaussian measure exp(−|z| 2 ) dv, where dv is the Lebesgue measure on C n . It plays an important role in many areas of analysis and its applications, especially in quantum mechanics. Its study is genuinely different from other function spaces, such as the Hardy space of the unit circle or the Bergman space of the unit ball, and features unique phenomena that require a distinct set of tools and techniques. Some of these applications arise from the theory of operators, such as Toeplitz and Hankel operators, and there is currently considerable interest in the study of these operators on the Fock space and its generalizations, which illustrates c 2019 The Royal Society of Edinburgh. This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/ 4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited. the fruitful interplay between geometry, function theory of several complex variables and functional analysis; see [18].
Indeed, relatively recently, Schuster and Varolin [15] began the study of Toeplitz operators T f with symbol f on generalized Fock spaces F p ϕ of C n with respect to the measure exp(−pϕ) dv, where the weight function ϕ is of C 2 with dd c ϕ comparable to the Euclidean Kähler form. When n = 1, this simply means that m Δϕ M for some positive constants m and M . Their results characterize boundedness and compactness of T μ in terms of the so-called Carleson and vanishing Carleson conditions for symbols that are positive measures. This was followed by the work [6] of Lv and one of the present authors, in which similar characterizations are given for bounded and compact Toeplitz operators between possibly different Fock spaces F p ϕ and F q ϕ , covering the full range of parameters 0 < p, g < ∞ and also symbols of bounded mean oscillation when p = q. For further results and recent developments in the study of operators and operator algebras on these function spaces, we refer to the papers [8,9] and the references therein.
The goal of the present paper is to study the Fredholm properties of Toeplitz operators T f on generalized Fock spaces in terms of the Berezin transform of f at infinity for symbols of vanishing mean oscillation. Fredholm theory of Toeplitz operators originated in the works of Coburn, Douglas, Gohberg and Krupnik in the late 1960s, and it is closely connected with (singular) integral operators, Hankel operators and the theory of function spaces, such as Hardy, Bergman and Fock spaces.
In the late 1980s, Berger and Coburn [3], who were the first to study Fredholm Toeplitz operators on the Fock space F 2 , showed that for f ∈ L ∞ ∩ V MO, the Toeplitz operator is Fredholm if and only if there are positive numbers R and such that |f (z)| whenever |z| > R. Their approach was based on C * -algebra techniques and Hilbert space methods, which they used to obtain isomorphic representations as quotients of function algebras for the image in the Calkin algebra generated by Toeplitz operators with symbols in L ∞ ∩ V MO. A few years later, Stroethoff [16] obtained the same characterization using methods more suited to operators on Banach spaces. Indeed, very recently, the Fredholm theory was extended to the setting of standard weighted Fock spaces F p α for 1 < p < ∞ in [2] (in the one-dimensional setting) using similar ideas and elementary methods and in [4] using newly developed limit operator techniques. It is worth noting that analogous characterizations are well known in Hardy spaces of the unit circle for symbols in the Douglas algebra C + H ∞ and in Bergman spaces of the unit ball for symbols in the algebra L ∞ ∩ V MO.
In the present paper, we prove that the Toeplitz operator wheref is the Berezin transform of the symbol f . Our result covers Berger and Coburn's characterization and its recent generalizations to standard weighted Fock spaces F p α . Another important feature of our analysis is the discovery of the second inequality in (1.1), which allows us to treat Toeplitz operators with all symbols in V MO, in contrast to the previous results, which are all restricted to bounded symbols in V MO. Finally, we emphasize that our results remain valid for any parameter 0 < p < ∞, which to our knowledge is a novelty in the study of the Fredholm properties of Toeplitz operators on any function space. For this reason, we also provide an account of the theory of Fredholm operators on quasi-Banach spaces in § 4 and show that the dual of F p ϕ with 0 < p < 1 can be identified with F ∞ ϕ ; the dual spaces of the other generalized Fock spaces F p ϕ with 1 p ∞ were described in [15] while the treatment of the standard weighted Fock spaces can be found in [10,17]. This will likely stimulate new interest and research activity in the study of operators on these and other small exponent function spaces.
In what follows, we first introduce basic background material that is essential to the rest of the paper. In § 3, we discuss some more advanced preliminary results, such as characterizations of boundedness and compactness of Toeplitz and Hankel operators in terms of functions of bounded and vanishing oscillation and the Berezin transform. In § 4, we recall the basic Fredholm theory both in the Banach and quasi-Banach space settings, describe the dual of F p ϕ for 0 < p < 1 and then proceed to prove our main result (theorem 4.5). We conclude the paper with a list of open problems related to our work and other important questions about Toeplitz and Hankel operators on Fock spaces.

Notation and definitions
In this section, we define the generalized Fock spaces, Toeplitz and Hankel operators, the spaces of mean oscillation, and their close relatives.
Let C n be the n-dimensional complex Euclidean space. For z = (z 1 , . . . , z n ) and w = (w 1 , . . . , w n ) in C n , we write (z, w) = z 1 w 1 + · · · + z n w n and |z| = (z, z). We denote by B(z, r) the Euclidean ball in C n , that is, We write ω 0 = dd c |z| 2 for the Euclidean Kähler form on C n , where d c = √ −1 4 (∂ − ∂). Throughout the paper, we assume that ϕ ∈ C 2 (C n ) is real-valued and that is, there are two positive numbers M 1 , M 2 satisfying the two inequalities in the sense of currents; see [15] for more details. For 0 < p < ∞, the space L p ϕ consists of all Lebesgue measurable functions f on C n for which where v is the Lebesgue volume measure on C n . With the norm · p,ϕ , it is easy to check that L p ϕ is a Banach space when p 1 and a quasi-Banach space when 0 < p < 1 (see definition 4.1). We denote by L ∞ the usual space of all bounded Lebesgue measurable functions f on C n with the norm given by Finally, we denote by H(C n ) the family of all holomorphic functions on C n . We can now define the weighted Fock spaces.
Definition 2.1. For 0 < p < ∞, the Fock space with weight ϕ is defined by It is well known that each F p ϕ is a closed subspace of L p ϕ , and hence it is a Banach space when p 1 and a quasi-Banach space when 0 < p < 1.
Functions of bounded and vanishing mean oscillation, which we will define next, play an important role in our analysis, and can be used to characterize boundedness and compactness of large classes of Toeplitz and Hankel operators. The mean oscillation MO p r (f ) of a locally integrable function f on C n for r > 0 and 1 p < ∞ is defined by where the averaging function f r is given by To describe the structure of both BM O and V MO, we define the following related function spaces.
are all independent of r, and different values of r correspond to equivalent semi-norms on BM O p , BO and BA p . For this reason, we write ω(·) for ω 1 (·). We also note that Other useful and well-known properties of these spaces are seen in the following two canonical decompositions and which hold for 1 p < ∞ and are obtained by writing Let K(·, ·) be the reproducing kernel (also known as the Bergman kernel) of F 2 ϕ . The orthogonal projection P of L 2 ϕ onto F 2 ϕ can be represented as the integral operator for z ∈ C n . Using this expression, P can be extended to a bounded linear operator from L p ϕ to F p ϕ for 1 p < ∞, and in particular, we have for all f ∈ F p ϕ , which was verified for 1 p ∞ in [15], and the other case 0 < p < 1 follows from the fact that F p ϕ ⊂ F 1 ϕ . The normalized reproducing kernel of F 2 ϕ is denoted by k z , that is, We set As we have not found the following density result in the literature in the case 0 < p < 1, we include its proof for completeness. Recall first that a sequence (z k ) in C n is an r-lattice with r > 0 if the balls B(z k , r) cover C n and B(z k , r/2) are pairwise disjoint. It is not difficult to see that for any r > 0, an r-lattice exists in C n and there is an N ∈ N such that each z ∈ C n can belong to at most N balls in which is a contradiction. Suppose next that 0 < p < 1 and let f ∈ F p ϕ . By (2.5) and lemma 2.4 of [9], there exists a constant C > 0 such that and define for w ∈ C n . Then, as in the proof of theorem 4.4 of [7], it follows that S r,R f ∈ S and Definition 2.6. Let Γ be the family of all measurable functions f on C n satisfying fk z ∈ ∪ p 1 L p ϕ for each z ∈ C n . Given some f ∈ Γ, we define the Toeplitz operator T f and Hankel operator H f on F p ϕ as and respectively, where I is the identity operator on L p ϕ .
Observe that, since BM O 1 is contained in Γ, it follows from proposition 2.5 that both T f and H f are well-defined on F p ϕ for 0 < p < ∞ when f ∈ BM O 1 . We finish the section with the definition of the Berezin transform, which is another useful tool for studying the properties of Toeplitz and Hankel operators, and indeed, we will use it heavily in our present work. For a function f in Γ, the Berezin transformf of f is defined bỹ

Preliminaries
Because the Bergman kernel of the generalized Fock space F p ϕ has no explicit expression, we need to rely on the following estimates instead. The first inequality is due to Christ in the case n = 1 and to Delin when n 2; for further details, references and an alternate proof, see [15].
(i) There exist positive numbers C and θ such that The meaning of the notation above is explained in (2.2).
We can use the preceding lemma to obtain the following connection between functions of vanishing oscillation and their Berezin transform.
Proof. Given ε > 0, by lemma 3.1, we have some R > 0 such that, for all z ∈ C n , which gives (3.4).
Next we show that f is continuous on C n . Indeed, for ε > 0, we can choose R as in (3.6), and then by lemma 3.1 and Lebesgue's dominated theorem, it follows that Therefore, Recall that an operator T : X → Y between two normed spaces is said to be compact if for every bounded sequence (f n ) in X, the sequence (T f n ) contains a convergent subsequence. When studying Toeplitz and Hankel operators, the equivalent formulation of compactness as the property that the image of the unit ball of X under T is relatively compact in Y is useful. Another useful simple observation is that T f and H f are compact if f has compact support.
As a consequence of the previous lemma and a general description of compact Toeplitz operators in [9], we obtain the following characterization for symbols of vanishing oscillation. ( Proof. We deduce the two assertions from theorem 3.2 of [9], which states that the for all r > 0. Suppose that f ∈ V O. To prove the first assertion, we only need to verify (3.7) under the assumption that lim z→∞ f (z) = 0. To see this, let ε > 0 and use lemma 3.2 to obtain an R > 0 such that for |ξ| > R. It follows that |f (ξ)| < 2ε whenever |ξ| > R, and therefore, for w ∈ B(z, r), by lemma 3.1, we get when |z| is sufficiently large. From this the estimate (3.7) follows.
To prove the second assertion, we set g = f − f . Then, by lemma 3.2, we have g ∈ L ∞ and lim z→∞ g(z) = 0. Thus, the condition in (3.7) implies that T g is compact on F p ϕ for all 0 < p < ∞, which completes the proof.
We need one more preliminary result on Toeplitz operators.
Proof. For 0 < p < s < ∞, there is a constant C such that f s,ϕ f p,ϕ for all f ∈ H(C n ); see [15]. We may therefore assume that 0 < p 1.
Let ε > 0. Since f ∈ V O, lemma 3.1 implies that there is an R > 1 such that for all z ∈ C n . Furthermore, for the fixed ε and R, we have some ρ > 0 so that whenever |z| > ρ. Then for |z j | > ρ, by lemma 2.4 of [9], we get The constants C above are independent of ε. This, together with lemma 3.5 and the obvious inequality which gives the limit in (3.8).
Relative to the Bergman projection, we define an integral operator P + by setting for z ∈ C n .
Lemma 3.5. The operator P + defined above has the following properties.
(i) For 1 p ∞, P + is bounded on L p ϕ .
(ii) For 0 < p < 1, P + is bounded from F p ϕ to L p ϕ .
Proof. We start with the first assertion. For f ∈ L 1 ϕ , by lemma 3.1, we have By interpolation, we obtain the desired conclusion. For the second assertion, let f ∈ H(C n ). Applying lemma 2.4 of [9] with Ω = C n and h = fK(·, z), we have which completes the proof.
As is well known, Hankel operators play an important role in the study Toeplitz operators and their spectral properties, and in particular we make use of compact Hankel operators to prove our main result on Fredholmness of Toeplitz operators. To show that Hankel operators with symbols of vanishing oscillation are compact (see theorem 3.8 below, which is also of independent interest), we need the following auxiliary result.
Proof. Write B(F p ϕ ) for the unit ball of F p ϕ . Observe that B(F p ϕ ) is a normal family. Therefore, to show that H f (B(F p ϕ )) is relatively compact in L p ϕ , it suffices to prove that lim j→∞ H f (g j ) p,ϕ = 0 (3.9) for any bounded sequence {g j } ∞ j=1 in F p ϕ converging to 0 uniformly on any compact subset of C n . Also, without loss of generality, we may assume that the support of f is contained in B(0, σ). For such f , we have Hence, for 1 p < ∞, and so, by lemma 3.5, we have (3.10) For 0 < p 1, using lemma 2.4 of [9] with Ω = B(0, σ) and h(ξ) = g(ξ)K(ξ, z), we get According to lemma 3.1, Since the constants C above in (3.10) and (3.11) are independent of {g j }, the limit in (3.9) follows for those {g j } which converge to 0 uniformly on any compact subset of C n . The proof is complete.
The simultaneous compactness of the Hankel operators H f and Hf from the setting of the standard weighted Fock spaces F p α was recently described in [12] as follows. We can now give sufficient conditions for boundedness and compactness of Hankel operators, which are needed for the study of the Fredholm properties of Toeplitz operators.
Dividing the both sides above by |z|, we obtain an R > r + 2 such that Then For R − 1 < |z| < 2R + 1 and w ∈ B(z, 1), using (3.13), we get This completes the proof of the inequality in (3.12). Therefore, by the first assertion, Since h has compact support, H h is compact from F p ϕ to L p ϕ , and hence H f is also compact from F p ϕ to L p ϕ . The proof is completed.

Fredholm theory
In this section, we prove that (1.1) is a necessary and sufficient condition for Toeplitz operators with symbols in V MO to be Fredholm and discuss some of the consequences of this result. We also show that the dual of F p ϕ can be identified with F ∞ ϕ when 1 < p < 1, which allows us to deal with Fredholmness of Toeplitz operators on Fock spaces with small exponents. We start by recalling some of the basic theory of Fredholm operators that are needed in our proof. For operators on Banach spaces, all is well known and can be easily found in the literature, while for operators acting on more general vector spaces, the theory is significantly less developed. Indeed, in the context of Toeplitz operators on Hardy, Bergman or Fock spaces, to our knowledge, their Fredholm theory has been previously studied only in the setting of Banach spaces.
A linear mapping T on a topological vector space X is said to be Fredholm if dim ker T < ∞ and dim X/T (X) < ∞.

When X is a Banach space, it is well known that T is Fredholm if and only if T + K(X) is invertible in the Calkin algebra B(X)/K(X), where B(X) and K(X)
stand for the spaces of bounded and compact operators, respectively. From this, it follows that an operator on a Banach space is Fredholm if and only if there are bounded operators A and B on X such that AT = I + K 1 and T B = I + K 2 for some compact operators K 1 and K 2 acting on X. Because two Toeplitz operators often commute modulo compact operators, the previous characterization for their Fredholmness is almost tailor-made for large classes of symbols. These characterizations of Fredholm operators are not true in general if X is not a Banach space. However, an adequate theory can still be developed for quasi-Banach spaces under some additional conditions, which is important in certain PDE problems; see, e.g. [11]. A pair (X, · ) is said to be a quasi-Banach space if · satisfies all the properties of a norm except for the triangle inequality and if there is a constant C > 0 such that for all x, y ∈ X. Observe that all generalized Fock spaces F p ϕ are quasi-Banach spaces. We now define an additional property for quasi-Banach spaces as in [13].
Definition 4.1. A quasi-Banach space X is said to be dual rich if for all nonzero vectors x ∈ X, there is a continuous linear functional x * such that x * (x) = 1.
As an example, we mention that every Banach space is dual rich, and so are p with 0 < p < 1, while none of the L p (C n , dv) spaces with 0 < p < 1 is dual rich; see [13].
In order to see that generalized Fock spaces are all dual rich, we describe the dual of F p ϕ when 0 < p < 1. For this reason, define where L ∞ ϕ consists of all Lebesgue measurable functions f on C n for which For g ∈ F ∞ ϕ , we define a linean functional F g : F p ϕ → C by If f ∈ F p ϕ , then there is a constant C > 0 such that where the last inequality follows from the inclusion that is, the mapping is bijective.
Proof. We already verified that (g) ∈ (F p ϕ ) * and (g) C g ∞,ϕ for all g ∈ F ∞ ϕ . Suppose next that F is a bounded linear functional on F p ϕ . Define a function g on C n by g(w) = F (K w ). We claim that and for all f ∈ F p ϕ . By lemma 3.1, and so g ∈ L ∞ ϕ . Next we show that g is holomorphic. For z, w ∈ C n , write w = (w 1 , w ), where w ∈ C n−1 and Δw 1 ∈ C with |Δw 1 | < 1 2 . Then, by Cauchy's estimate (see theorem I.1.6 of [14]), we get where the constant C is independent of w and z. Furthermore, the estimate for the Bergman kernel in (3.1) tells us that Because of this and we can apply the Lebesgue Dominated Theorem to get This implies, for F ∈ (F p ϕ ) * , Similarly, ∂g/∂w j (w) exists for 2 j n. Therefore, g ∈ H(C n ) and (4.1) is proved. It remains to prove (4.2). Let f ∈ F p ϕ . As in (2.7), for an r-lattice {w where R > 0 is sufficiently large. The right-hand side of (4.3) is only a finite sum of holomorphic functions, so S r,R (f ) ∈ H(C n ). We claim that To see this, write ∇ w K(w, z) = (∂K/∂w 1 , ∂K/∂w 2 , . . . , ∂K/∂w n ). Applying Cauchy's estimates again, we obtain the inequality sup w∈B(0,R) Then, for w ∈ B(w (r) This implies, when r → 0, and hence (4.4) follows. Furthermore, by (4.1), Notice that g(·) = F (K(·, w)), applying the Lebesgue Dominated Theorem again to get (4.5) Therefore, by (2.6), (2.8), (4.4) and (4.5), we have which is (4.2), and the theorem is proved. Corollary 4.3. If 0 < p < 1, then the generalized Fock space F p ϕ is a dual rich quasi-Banach space.
The following result is needed in the next section when we characterize Fredholm operators on F p ϕ for 0 < p < 1. Remark 4.7. We give an example that shows that (4.12) fails iff is replaced by f in it. Define f : C → C by On the other hand, f (C \ B(0, R)) = C for all R > 0, and hence σ ess (T f ) cannot coincide with the right side of (4.12) iff is replaced by f in it.
In order to state one more consequence of our main result, we recall the definition of block Toeplitz operators. Let N ∈ N and 0 < p < ∞. Suppose that a = (a jk ) 1 j,k N with each a jk ∈ BM O 1 . The block Toeplitz operator T a is defined on F p ϕ,N by As in [2], we can use the standard Banach algebra techniques to obtain the following characterization for Fredholmness of block Toeplitz operators. Compared with theorem 22 of [2], we observe that in addition to more general weights, we can also deal with more general symbols in the following theorem. Proof. If f, g are in V MO 1 withf,g ∈ L ∞ , then, T f and T g are both bounded and for some compact operator K by theorem 3.7. Therefore, as in lemma 21 of [2], it follows that det T a − T det a is compact, and hence T a is Fredholm if and only if det T a is Fredholm, which is equivalent to T a being Fredholm by what we already proved. Thus, (4.13) is both necessary and sufficient for T a to be Fredholm by theorem 4.5.

Open problems
An immediate question related to our main result is what can be said about the Fredholm properties of Toeplitz operators on (more general) doubling Fock spaces F p φ . The Hilbert space case F p φ (C) was recently studied in [1]. The general case 0 < p < ∞ and n ∈ N is much more difficult because the techniques of the present paper are no longer sufficient; however, there are a number of methods and inequalities for doubling Fock spaces that may prove fruitful and we hope to deal with the general case in a future publication.
It is also worth noting that here we have described Fredholmness of Toeplitz operators but are currently unable to say anything about their Fredholm index. In fact, to our knowledge, the index formula is only known for Toeplitz operators on Fock spaces with standard weights α > 0 and with weights of the form |z| β where β > 0 (see [1]). In both of these cases, the space has a nice basis and an explicit formula for the reproducing kernel, which allows for a reduction to an index computation of a simpler operator. New ideas are required to deal with Fock spaces that lack these nice properties.
Another major challenge is the case of matrix-valued symbols. In theorem 4.8, we have merely reduced the study to the scalar-valued case; more general cases, where one needs to deal with block Toeplitz operators directly, are considerably more difficult, but the results known for Hardy and Bergman spaces may offer some clues as to what could be expected.
Finally, in our present paper, we have used and further developed the theory of Hankel operators. Their theory is also important in its own right and there are natural further questions about their compactness, for example, which arise from our work. Another aspect of Hankel operators, which is completely different, and perhaps surprising, from many other function spaces, such as Hardy and Bergman spaces, is the property that if H f is compact, then Hf is also compact. This was recently proved for standard weighted Fock spaces F p α in [5] and it would be interesting to know whether this property remains true for Hankel operators acting on other weighted Fock spaces.
second author was also supported by Engineering and Physical Sciences Research Council (EPSRC) grants EP/M024784/1 and EP/T008636/1.