{'fraction_non_alphanumeric': 0.05910144990355649, 'fraction_numerical': 0.011898311696755598, 'mean_word_length': 3.5576847096009483, 'pattern_counts': {'":': 0, '<': 37, '<?xml version=': 0, '>': 35, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 29, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Toeplitz operators on non-reflexive Fock spaces

We generalize several results on Toeplitz operators over Fock spaces to the case of non-reflexive Fock spaces. Among these results are the characterization of compactness and the Fredholm property of such operators, a well-known representation of the Toeplitz algebra, a characterization of the essential centre of the Toeplitz algebra. Further, we improve several results related to correspondence theory, e.g. we improve previous results on the correspondence of algebras and we give a correspondence theoretic version of the well-known Berger-Coburn estimates.


Introduction
The theory of Toeplitz operators on Fock spaces experienced significant attention and progress in recent years, cf. [2,3,4,7,9,10,15,16,24,25,26] for some works from the last few years. Many properties of Toeplitz operators and the algebras they generate have been investigated. One common feature among most of these works is that they deal only with Toeplitz operators on the reflexive Fock spaces. The aim of the present work is to present an approach which is suitable for both the reflexive and the non-reflexive cases. Simply speaking, we will apply tools from harmonic analysis to extend some results known from the reflexive cases to the non-reflexive world. Along the way, we will also obtain some results which are new even in the Hilbert space case.
Let us briefly discuss some of the obstacles one has to overcome when studying Toeplitz operators on non-reflexive Fock spaces. Details and definitions will be given later.
As is well-known [4], a bounded operator A on a reflexive Fock space is compact if and only if it satisfies two properties: First of all, it needs to be contained in the Banach algebra generated by all Toeplitz operators with bounded symbols (which we will refer to as the Toeplitz algebra). Secondly, its Berezin transform needs to be a function vanishing at infinity. When passing to non-reflexive Fock spaces, one faces the following two problems: Given a compact operator, it may happen that it is no longer contained in the Toeplitz algebra. Even worse, the Berezin transform may no longer be injective, i.e. there are non-trivial bounded linear operators with Berezin transform being constantly zero. These problems of course lead to other problems, e.g. difficulties in the characterization of Fredholm properties for operators from the Toeplitz algebra. In [10], it was shown for the reflexive cases that an operator from the Toeplitz algebra is Fredholm if and only if all of its limit operators are invertible. As a matter of fact, an operator from the Toeplitz algebra over a reflexive Fock space is Fredholm if and only if it is invertible in the Toeplitz algebra modulo the ideal of compact operators. Since, again, the compact operators are not necessarily contained in the Toeplitz operator when the Fock space is non-reflexive, this again causes obvious problems.
Let us now shift from describing obstacles towards describing the outcomes of this paper. To introduce some short-hand notation for this, we will denote by F p t the standard Fock spaces and by T F p t the (full) Toeplitz algebra over this Fock space. The precise definitions of these and other terms used will be given in Section 2 below. It will then be our first goal to present a suitable approach to the Correspondence Theory on non-reflexive Fock spaces. The basic concepts of Correspondence Theory originate from [21] and were already successfully used by the present author [9] to study Toeplitz operators in case of reflexive Fock spaces. This part of the paper, which will be carried out in Section 3, really forms the technical heart of the approach to non-reflexive Fock spaces, as most problems arrising from non-reflexivity will be explained and overcome in this section. In the next part, Section 4, we will present some applications of the Correspondence Theory. Some of them are already known from [9], we give some prominent examples: In [4], W. Bauer and J. Isralowitz proved the following compactness characterization for Toeplitz operators over Fock spaces: Theorem ( [4]). Let A ∈ L(F p t ), where 1 < p < ∞. Then, A ∈ K(F p t ) ⇐⇒ A ∈ T F p t and A ∈ C 0 (C n ).
In [24], J. Xia obtained (in principle) the following result: Theorem ( [24]]). It is We discussed in [9] how both of these results can be easily obtained from Correspondence Theory in the case of reflexive Fock spaces. Here, we will obtain (versions of) these results even over non-reflexive Fock spaces, adding some necessary modifications due to the non-reflexivity. Further, we will discuss, among other results in this direction, a generalization of following well-known fact: Theorem ( [5]). It is We will end that section by presenting a correspondence theoretic version of the Berger-Coburn estimates [6], generalizing results such as the compactness version of the Berger-Coburn estimates [2]. In the rather short Section 5, we add a small contribution to the "algebra question" of Correspondence Theory, initiated by the author in [9]. Finally, in Section 6, we derive the Fredholm characterization for operators from the Toeplitz algebra. Initially, this result has been proven for the reflexive Fock spaces: Theorem ( [10]). Let 1 < p < ∞ and A ∈ T F p t . Then, A is Fredholm if and only if A x is invertible for every x ∈ ∂C n .
Here, ∂C n denotes an appropriate boundary of C n (the boundary coming from the maximal ideal space of bounded uniformly continuous functions on C n , considered as a compactification of C n ) and A x is the operator obtained by "shifting" A to this boundary point in an appropriate way. While some of the results from [10] can be carried out in the case p = 1 (which will also be important for our discussion), some key arguments in the proof of above theorem break down in this case. We will replace these arguments by an investigation of the Banach algebra of all limit operators, resulting in a proof for the above Fredholm characterization even on the non-reflexive Fock spaces.
Let us end this introduction by mentioning that some of the results presented in sections 3 and 4 have already been obtained in the author's PhD thesis. Nevertheless, we have the feeling that our results deserve proper publication. In addition, we decided to give a more streamlined presentation here.

Preliminaries
On C n we consider the family of probability measures µ t given by where |·| is the Euclidean norm on C n ∼ = R 2n , dz the standard Lebesgue measure and t > 0. Given any 1 ≤ p < ∞, we define The Fock space F p t is given by where Hol(C n ) denotes the entire functions on C n . This space is always endowed with its natural L p -norm, i.e.
For p = ∞, we set Here, two different Fock spaces come into play: First of all, there is Further, we also consider For each value of p, F p These Fock spaces are well-studied objects, the basic references being [17,26], where also the following facts can be found.
For p = 2, we clearly are in the Hilbert space setting. We will usually write the F 2 t inner product by ·, · t or even ·, · . As is well-known, F 2 t is even a reproducing kernel Hilbert space, the reproducing kernels being given by Here, w · z = n j=1 w j z j is the standard sesquilinear product on C n . Indeed, the functions K t z are contained in any of the spaces F p t and f ∞ t , and they span a dense subspace of F p t for 1 ≤ p < ∞ and f ∞ t . The normalized reproducing kernels are now defined as It follows from elementary computations that we indeed have k t Under the F 2 t inner product, the Fock spaces satisfy the following duality relations: Note that these identifications are not isometric, but they hold with an equivalence of norms. Nevertheless, we will always identify dual spaces in this way. If A is a bounded linear operator on any of the Fock spaces, A * will denote the adjoint, acting on the dual space, with respect to the F 2 t duality. We will consider L(X), the bounded linear operators on X, for X any Banach space. By K(X) we will denote the compact operators, N (X) will be our notation for the ideal of nuclear operators.
For p = 2, we of course have the well-known orthogonal projection P t ∈ L(L 2 t ) mapping onto F 2 t by As is well-known, P t (interpreted as an integral operator) gives rise to a bounded projection on L p t mapping onto F p t for any 1 ≤ p ≤ ∞ with P t | F p t = Id. Hence, for any f ∈ L ∞ (C n ) and 1 ≤ p ≤ ∞ the Toeplitz operator T t f given by is well defined and bounded, satisfying T t f F p t →F p t ≤ P t f ∞ . Further, every such Toeplitz operator leaves f ∞ t invariant. Therefore, we can also consider T t f ∈ L(f ∞ t ). For z ∈ C n we define the Weyl operator W t z by in particular W t z ∈ L(F p t ) for any 1 ≤ p ≤ ∞ and W t z ∈ L(f ∞ t ). These operators satisfy the following well-known properties, which we fix as a lemma: z is continuous in strong operator topology over F p t . The same holds true over f ∞ t . Note that (1) and (2) above follow from elementary computations, while (3) holds for 1 ≤ p < ∞ by Scheffé's lemma, and the case of f ∞ t can be proven by some standard ε-δ argument.
We will also frequently encounter the Berezin transform: As is well-known, the Berezin transform Under suitable growth conditions (in particular for f ∈ L ∞ (C n )), we then obtain There will be many statements we will consider over any of the Fock spaces. For this reason, we will write F t to denote a generic Fock space with respect to the parameter t > 0, that is: If we do not state otherwise, F t ∈ {F p t : 1 ≤ p ≤ ∞} ∪ {f ∞ t } arbitrary. For two quantities A, B we will write A B and A B for A ≤ cB and A ≥ cB, respectively, with some inessential constant c > 0. By A ≃ B we will mean A B and A B.

The Correspondence Theorem
For any z ∈ C n and A ∈ L(F t ), we set Further, for f ∈ L ∞ (C n ) and z ∈ C n we let We say that subspaces D 0 ⊂ L ∞ (C n ) and D 1 ⊂ L(F t ) are α-invariant if α z (f ) ∈ D 0 or α z (A) ∈ D 1 for any f ∈ D 0 or A ∈ D 1 , respectively, and any z ∈ C n . In the following, we always write The "classical counterpart" to C 1 (F t ) is the space of bounded uniformly continuous functions on C n , which we will denote by BUC(C n ).
In this section, we are going to establish the following theorem: Theorem 3.1. Assume D 0 is an α-invariant and closed subspace of BUC(C n ). Further, let A ∈ C 1 (F t ). Then, the following holds true: Note that this theorem essentially goes back (in the Hilbert space case) to [21] and has been proven by the author in [9] for the reflexive cases 1 < p < ∞. There, we made critical use of the so-called convolution formalism of Quantum Harmonic Analysis (which also goes back to [21]). To some extend, these concepts can be carried over to the non-reflexive case: At least on f ∞ t most things work out and hence can be carried over to F 1 t and F ∞ t by duality. Nevertheless, this approach somehow lacks the naturalness it has for p ∈ (1, ∞), as technical burdens keep stacking up. Therefore, we chose to present a different entrance point here, which is one of the two well-known Berger-Coburn estimates. While initially only proven for the Hilbert space case [6], they have recently been generalized to any value of p ∈ [1, ∞] [3]. 6,3]). Let f : C n → C be measurable such that f K t z ∈ L 2 t for any z ∈ C n . Then, for every s ∈ (0, t/2) there exists a constant C, depending only on n, s and t such that: Proof. Clearly, f (s) ∈ BUC(C n ) for every s > 0 whenever f ∈ L ∞ (C n ) ( f (s) is simply the convolution of f with an appropriate Gaussian). Hence, By uniform continuity, the right-hand side of this estimate goes to zero when |z| → 0, proving the statement.
In particular, this proves that T Ft ⊂ C 1 (F t ). Here, we denote by T Ft the Banach subalgebra of L(F t ) generated by all Toeplitz operators with bounded symbols.
We can now define an L 1 (C n ) module structure on C 1 (F t ) as follows. For any A ∈ C 1 (F t ) and f ∈ L 1 (C n ), we set This integral always exists as a Bochner integral in C 1 (F t ), hence we have f * A ∈ C 1 (F t ). It follows from basic properties of the Bochner integral that we have It is also imminent that this module structure satisfies In what follows, we will set g s (z) = 1 (πs) n e − |z| 2 s .

Then, elementary estimates show: For any
We will also use the following important identity, which holds for any A ∈ C 1 (F t ): This identity is well-known for p = 2 (see e.g. [5]) and can, for arbitrary F t , be verified in the same way, i.e. by comparing the Berezin transforms of the two operators: Note that the Berezin transform over L(F ∞ t ) is no longer injective, therefore it is not immediately clear that the above reasoning implies g t * A = T t A for this case. Indeed, the Berezin transform is injective on C 1 (F ∞ t ). We will defer showing this for a moment.
Recall that, by Wiener's approximation theorem, for any N ∈ N there are finitely many constants c N j ∈ C, z N j ∈ C n such that Proof of the Correspondence Theorem 3.1 for F t = F ∞ t . Let N ∈ N and the constants c N j , z N j as above. Then, we have: We therefore obtain that A can be approximated by Toeplitz operators with symbols in D 0 , whenever D 0 is an α-invariant subspace of BUC(C n ) containing A. In particular, A ∈ D 0 implies A ∈ T Ft lin (D 0 ). On the other hand, if A ∈ T Ft lin (D 0 ), then A ∈ D 0 follows easily by translation invariance.
Let us come back to the problem we mentioned earlier, namely the noninjectivity of the Berezin transform on L(F ∞ t ). This problem can be resolved by observing the following facts: (c) The restriction of the Berezin transform to Proof. Let us consider Then, for any A ∈ A we have where B is a pre-adjoint in L(F 1 t ), and since the Berezin transform is injective the Berezin transform is injective on A. As before, we obtain now g t * A = T t A for A ∈ A. Now, one can imitate the proof of the Correspondence Theorem, i.e. show that any A ∈ A can be approximated by linear combinations of translates of T t A . Since every Toeplitz operator (with bounded symbol) on F ∞ t leaves f ∞ t invariant, we obtain that every operator from A leaves f ∞ t invariant. From this, we can now prove (d): First, we recall that we already know that z → W t z f is continuous in hence we only need to show the reverse inclusion.
From this, we then obtain Then, the statement from automatically follows from the reasoning above.
So let A ∈ C 1 (F ∞ t ) and pick any f ∈ f ∞ t . As explained above, the rank 1 Remark 3.5. Indeed, once one sees the characterization of f ∞ t in (d), it is not at all surprising that this holds true. Nevertheless, it seems that this was not observed before. Obtaining a direct proof of (d) should be a nice exercise.
Proof of the Correspondence Theorem 3.1 for F t = F ∞ t . Follows now as in the other cases, having the injectivity of the Berezin transform on C 1 (F ∞ t ) at hand.
be an α-invariant, closed subspace. Then, for some unique α-invariant and closed subspace D 0 of BUC(C n ).
Proof. Set Since D 1 is α-invariant, the same is true for D 0 . The proof of the Correspondence Theorem shows where the right-hand side was defined as a Bochner integral with values in D 1 , hence is again contained in D 1 . This shows equality of D 1 and T Ft lin (D 0 ). The Correspondence Theorem now easily gives uniqueness of D 0 .

Applications of the Correspondence Theorem
Here, we give some applications of the Correspondence Theorem. In what follows, we will denote by T Ft the full Toeplitz algebra, i.e. the Banach algebra generated by all Toeplitz operators with L ∞ (C n ) symbols over the respective Fock space F t .
Theorem 4.1. The following holds true: Proof. Follows from the Correspondence Theorem with D 0 = BUC(C n ).
was first obtained by J. Xia in [24]. In [9], we obtained this result in the reflexive cases.
The following result extends the characterization of compact operators on Fock spaces, which was first obtained in [4] for the reflexive cases, to the nonreflexive cases. This initial result is contained in part (1) of the theorem.
. Then, we have: Then, we have: Before we discuss the proof, let us briefly mention that some additional assumptions in the cases p = 1, ∞+ are indeed necessary: On F 1 t , there are clearly compact operators not contained in T F 1 t . As an example, One readily checks that this operator is not contained in Proof of Theorem 4.4. Recall that the statement for the reflexive cases have been proven in [9] using the Correspondence Theorem. In principle, the proofs here work analogously. Let us, for completeness, quickly go through the details to verify that nothing strange is happening in the non-reflexive cases. First, we discuss why any f ∈ C 0 (C n ) gives rise to a compact Toeplitz operator T t f even in the non-reflexive cases, which seems to be not that well-known in the literature. Let us denote . By approximation, we obtain compactness for Showing that a compact operator A ∈ C 1 (F t ) has Berezin transform vanishing at infinity follows from standard methods: On F p We want to add a nice observation which we borrow from [19], where it was presented for the Hilbert space case. Recall that a function f ∈ L ∞ (C n ) is said to be slowly oscillating if for every ε > 0 there exists K > 0 and δ > 0 such that Obviously, bounded uniformly continuous functions are slowly oscillating. Even more so, a slowly oscillating function is continuous if and only if it is uniformly continuous.
Recall Pitt's improvement of the classical Wiener Tauberian theorem: ). Let f ∈ L ∞ (R n ) be slowly oscillating and φ ∈ L 1 (R n ) a regular function (i.e. its Fourier transform vanishes nowhere).
Since the Berezin transform f (t) is simply the convolution by a Gaussian, which is a regular function (in the sense that its Fourier transform vanishes nowhere), we obtain the following characterization of compactness for Toeplitz operators: Corollary 4.6. Let f ∈ L ∞ (C n ) be slowly oscillating. Then, the following are equivalent over any F t : 3) T t f is compact. In the following, we want to consider the quotient of the Toeplitz algebra modulo the compact operators. Since the compact operators are not entirely contained in T Ft for F t = F 1 t , F ∞ t , we will abbreviate Suppressing F t in this notation will not cause any confusion.
The following result is well-known in the Hilbert space case [5] and has recently extended to the reflexive cases [12].We present it now also for the nonreflexive cases.
Theorem 4.7. The essential center of T Ft , i.e.
is given by Here, VO ∂ (C n ) is the space of functions of vanishing oscillation at infinity: It is not difficult to see that VO ∂ (C n ) is an α-invariant closed subspace of BUC(C n ).
Proof of Theorem 4.7. It is easy to show that essCen(T Ft ) is closed and αinvariant. Hence, let D 0 denote the unique α-invariant and closed subspace of BUC(C n ) in the sense of Corollary 3.6. Since the result is already proven in the Hilbert space case, we know by the compactness characterization that for Hence, using the compactness characterization we have just proven for any choice of F t , this gives that For the other inclusion, we can literally argue as in the Hilbert space case [5], which we briefly do for completeness:

This in turn implies
But such fuctions f are well-known to be contained in VO ∂ (C n ) [5].
We also have the following consequence, which is again well-known in the Hilbert space case.
Corollary 4.8. It is: Proof. As in the previous proof, one sees that for any A ∈ T Ft lin (VO ∂ (C n )) that Therefore, which finishes the proof.
Recall that the Calkin algebra is the quotient algebra L(X)/K(X). It is a Banach algebra when endowed with its canonical quotient norm. This quotient norm is indeed nothing else but the essential norm of the representatives: Let us, for simplicity, abbreviate K = K(X) in the following. Recall again that Hence, we may consider the quotient T Ft /K = C 1 (F t )/K as a subalgebra of the Calkin algebra. Further, since the shifts leave K invariant, they descend to a group action in the Calkin algebra: In this case, we obtain the following: the following holds true: Proof. For A + K ∈ T Ft /K, it is clear that the shifts act continuously on them with respect to the norm (keep in mind that A + K ≤ A op ). Clearly, the set of all A+ K on which the shifts act norm continuously forms a closed subalgebra of L/K. Denote this closed subalgebra of L(F t )/K by C Ft 1,ess . We can continue as in the proof of the Correspondence Theorem: Define an L 1 module action on C Ft 1,ess by considered as a Bochner integral in C Ft 1,ess . It is then again not hard to verify that as A + K ∈ A + C 0 (C n ). Now, continue as in the proof of the Correspondence Theorem to show equality.
Remark 4.10. Indeed, from applying the proof of the Correspondence Theorem as sketched above, one does not only get the equality T Ft /K = C Ft 1,ess , but also a statement analogous to the Correspondence Theorem in the sense of a correspondence between closed, translation-invariant subspaces of T Ft /K and BUC(C n )/C 0 (C n ).
The statement of the following corollary is contained in [12] for 1 < p < ∞. Hence, the statement is only new over f ∞ t .
The assumption clearly implies A−α z (A) ∈ K. Hence, α z (A+K) = A+K in L(F t )/K. The previous theorem yields A + K ∈ C Ft 1,ess = T Ft /K. Therefore, there are B ∈ T Ft and K ∈ K with A = B + K, proving A ∈ T Ft .
Here is another corollary, which will be more important to us in the following: and A ∈ T Ft be Fredholm. Then, for any Fredholm regularizer B ∈ L(F t ), i.e.
Proof. Being a Fredholm regularizer is just the same as saying that B + K is an inverse to A + K in the Calkin algebra. Now, since the Calkin algebra is a Banach algebra with unit, inversion is continuous here. Hence, we obtain that z → α z (B + K) is continuous in the Calkin algebra whenever z → α z (A + K) is continuous in the Calkin algebra. Theorem 4.9 implies that B + K ∈ T Ft /K whenever A ∈ T Ft and B + K is an inverse to A + K. As in the proof of the previous corollary, this implies B ∈ T Ft .
While the previous corollary does not carry over to the cases F t = F 1 t , F ∞ t , its essence still prevails: The Fredholm theory of operators from T Ft is entirely contained in T Ft : Theorem 4.13. For any A ∈ T Ft we have: Proof. By the previous results, we only need to deal with the cases Since a bounded linear operator on a Banach space is Fredholm iff its adjoint is Fredholm, we can conclude that A| f ∞ t is Fredholm. Therefore, there exist B ∈ T f ∞ t and K 1 , Clearly, we have B * * , K * * 1 , K * * 2 ∈ T F ∞ t , K * * 1 and K * * 2 are still compact and AB * * = Id +K * * 1 , B * * A = Id +K * * 2 .
This settles the case of F t = F ∞ t . For p = 1 and A ∈ T F 1 t , we can apply the same construction to A * ∈ T F ∞ t to obtain where B is a Fredholm regularizer of (A * )| f ∞ t . Let us end this section by discussing correspondence versions of the Berger-Coburn estimates. Recall that the Toeplitz operator T t f with (possibly unbounded) symbol f is defined as As we already noted above (cf. Theorem 3.2), one of the two Berger-Coburn estimates has been proven to hold true over any Fock space by W. Bauer and the present author in [3]. The other Berger-Coburn estimate, also obtained initially in [6] for the case p = 2, can also be carried over to the general case: Theorem 4.14. Let f : C n → C such that f K t z ∈ L 2 t for any z ∈ C n . Then, there exist constants C > 0 (depending on s, t, p, n) such that for any s ∈ ( t 2 , 2t): Let us very briefly sketch the original proof for the Hilbert space setting and then discuss why this proof still works for arbitrary F t .
First of all, there is nothing to prove for s ≥ t. For s < t, Berger and Coburn obtained the inequalities by considering the operators Here, e t α are the elements of the standard orthonormal basis of F 2 t , i.e. e t α (z) = 1 t |α| α! z α , z ∈ C n , α ∈ N n 0 , using standard multiindex notation. Clearly, P k is of finite rank. Further, for s ∈ ( t 2 , t), the sum defining T   More generally, Now, using the standard estimate | Tr(AB)| ≤ A S 1 B op , where the norms denote the trace norm and operator norm, respectively, yields In principle, all of this carries over upon replacing the trace ideal by the ideal of nuclear operators. First of all, we still have the identity Tr(T (s) , which follows from the Dominated Convergence Theorem upon writing out the definition of the trace, provided that T (s) 0 is indeed a nuclear operator (note that the nuclear trace is well-defined, as the Fock spaces possess the approximation property -see Section 6.1 below for a short discussion of this). Those are indeed the two points which need clarification. For doing so, we first observe that the result on F ∞ t follows immediately from the result on are equivalent. Hence, we may exclude the case F t = F ∞ t in the following discussions for convenience.
It is therefore the first logical step to prove the following proposition. Proof. Using the tensor product structure of the standard basis, it suffices to consider the case n = 1. For q = ∞, basic calculus shows that For p = 1 we obtain Thus, we arrive at Stirling's approximation Clearly, the case p = ∞, q = 1 follows the same asymptotic behaviour. For p ∈ (1, ∞), the uniform bound is immediately obtained from a version of Littlewood's interpolation inequality for Fock spaces.
For completeness, we give the Fock space version of the interpolation inequality we just mentioned: Lemma 4.17. For f ∈ F 1 t and 1 < p < ∞ it is: We omit the proof of this inequality, as it is a standard consequence of Hölder's inequality.
Proof of Proposition 4.15. Recall that the nuclear norm of A ∈ N (F t ) is defined as As usually, we will identify (F p t ) ′ with F q t , which are identical up to passing to an equivalent norm. Recall that we excluded the case F t = F ∞ t from our discussion.
We clearly have converges absolutely in N (F t ). As the nuclear ideal is complete, the series therefore converges and T (s) 0 ∈ N (F t ).
Proposition 4.18. Let f : C n → C such that f K t z ∈ L 2 t for any z ∈ C n . Further, let s ∈ ( t 2 , t). If α −z (T t f ) is bounded on any of the spaces F p t or f ∞ t , then Proof. Without loss of generality, we may assume z = 0. Since the nuclear operators form an ideal, we have T (s) 0 T t f ∈ N (F t ) provided that the Toeplitz operator is bounded. We may compute its nuclear trace as follows: Note that for any polynomial q, |q| 2 can be dominated by a linear combination of functions |K t z | 2 . Since f K t z ∈ L 2 t , we also get f q ∈ L 2 t , hence T t f (q) = P t (f q), thus: It is an easy exercise to show that f ∈ L 2 t implies f ∈ L 1 s whenever s ∈ (t/2, t). Thus, we may apply the Dominated Convergence Theorem to obtain: Combining all the steps, we have now seen that the Berger-Coburn estimates carry over to any Fock space, i.e. for f with f K t z ∈ L 2 t we have seen that where the operator norm can be taken over any of the Fock spaces F p t , 1 ≤ p ≤ ∞, and f ∞ t . The Correspondence Theorem now immediately gives the following improvement of this result: Theorem 4.19. Let f : C n → C such that f K t z ∈ L 2 t for any z ∈ C n . Further, let D 0 ⊂ BUC(C n ) be a closed, α-invariant subspace.

2)
If T t f ∈ T Ft lin (D 0 ), then f (s) ∈ D 0 for every t 2 < s < 2t. Proof. 1) By the Berger-Coburn estimate, T t f is a bounded operator. As in the proof of Corollary 3.3, f (s) ∈ BUC(C n ) implies even T t f ∈ C 1 (F t ). Now, apply the Correspondence Theorem.
2) Recall that g s denotes the standard Gaussian g s (z) = 1 (πt) n e − |z| 2 s . By comparing Berezin transforms, it is not difficult to see that g s * T t f = T t f (s) , where the star denotes the L 1 (C n ) module structure of C 1 (F t ). Since T Ft lin (D 0 ) is, by definition, closed and α-invariant, we get T t f (s) ∈ T Ft lin (D 0 ). Note that one can prove the following: A function g ∈ BUC(C n ) is contained in D 0 iff g (t) is contained in D 0 for some t > 0. This can be shown by mimicking the prove of the Correspondence Theorem, i.e. use that convolution by g s is an approximate identity on BUC(C n ) and then, for suitable constants c j ∈ C, z j ∈ C n , show that ∈ D 0 . Since f (s) is in BUC(C n ) by the Berger-Coburn estimates, the previous comment imply that f (s) ∈ D 0 . Remark 4.20. We want to emphazise that, after this theorem has been presented in the author's thesis, Wu and Zheng independently proved the first part of the theorem for p = 2 in their recent preprint [22]. Their methods differ significantly from our proof. For D 0 = C 0 (C n ) and p = 2, this result is already contained in [2].
The Berger-Coburn estimates naturally lead to the Berger-Coburn conjecture, which asks if T t f is bounded iff f (t/2) is bounded (with f under the assumptions of the Berger-Coburn estimtates). So far, this conjecture is only resolved for certain classes of symbols, most notably non-negative symbols and symbols of bounded mean oscillation. For such symbols, we have the following well-known fact: For this class of symbols, we now get the following correspondence statement: Theorem 4.22. Let f : C n → C such that f K t z ∈ L 2 t for any z ∈ C n and let D 0 be a closed, α-invariant subspace of BUC(C n ). Further, assume that either f ≥ 0 or f ∈ BMO(C n ). Then, the following are equivalent: 3. f (s) ∈ D 0 for any s > 0.
Proof. As we have already mentioned, the property of f (s) being bounded is independent of s > 0 for such symbols. Therefore the property of f (s) being in BUC(C n ) is independent of s > 0 (simply write f (s) = g ε * f (s−ε) ). Now,

Correspondence of algebras
In [9], we started the investigation of the following question: Let D 0 be a closed, α-invariant subspace of BUC(C n ). Is there a relation between D 0 being an algebra and T Ft lin (D 0 ) being an algebra?
As the author already noted in [9,Corollary 3.9], the property of T Ft lin (D 0 ) being an algebra is independent of the particular choice of F t (there, it was stated for F t = F p t with 1 < p < ∞, but this is true for every possible F t by the same proof). Our investigations led to [9,Theorem 3.13], which we shall not formulate here. Instead, we want to note that parts of the proof of said theorem needed some additional structure on D 0 , namely that it is self-adjoint and βinvariant (β is the operator βf (z) = f (−z)). The assumption of β-invariance can be overcome within the frame of our initial proof, but our proof strictly hinges on the self-adjointness of D 0 . Nevertheless, in their recent preprint [23], S. Wu and X. Zhao could give a better statement and proof in this particular direction, not using self-adjointness. Combining now all the available results, we arrive at the following theorem: 9,23]). Let D 0 be a closed, α-invariant subspace of BUC(C n ). Then, the following are equivalent: (1) D 0 is an algebra; (2) T Ft lin (D 0 ) is an algebra for every F t and every t > 0; (3) T Ft lin (D 0 ) is an algebra for some F t and every t > 0. If D 0 satisfies (1)-(3) above and I 0 is a closed and α-invariant subspace of D 0 , then the following are equivalent: (1*) I 0 is an ideal of D 0 ; (2*) T Ft lin (I 0 ) is a (left-or right-)ideal of T Ft lin (D 0 ) for every F t and every t > 0; (3*) T F:t lin (I 0 ) is a (left-or right-)ideal of T Ft lin (D 0 ) for some F t and every t > 0. We will add another contribution to this particular question. For this, we introduce the notation δ λ f (z) = f (λz) for any λ > 0. It is easy to see that δ λ leaves BUC(C n ) invariant for every λ > 0. For typographic reasons, we will reserve δ for the action on symbols and write C λ f (z) = f (λz) for the operator on Fock space elements. Then, [26,Lemma 2.6] shows that the linear operator C λ ∈ L(F t , F t/λ 2 ) is an isometric isomorphism.
The following statement is certainly well-known. Nevertheless, it seems that it is not contained in the standard reference [26], hence we provide the simple proof.
2) Let A ∈ L(F t ) and λ > 0. Then, Here, we write the Berezin transform A for typographic reasons as B(A).

Proof. 1) The equality follows by direct computations: It is
We use the substitution λw = u, dw = 1 λ 2 du: .
2) Recall that C 1/λ AC λ ∈ L(F tλ 2 ). Thus, Adding δ-invariance of D 0 , we can now fully characterize the property of T Ft lin (D 0 ) being an algebra. We say that a subset of BUC(C n ) is δ-invariant if it is invariant under δ λ for every λ > 0. (1) D 0 is a Banach algebra; (2) T Ft lin (D 0 ) is a Banach algebra for every t > 0; (3) T Using Lemma 5.2 we get: Since we assumed that T Ft 0 lin (D 0 ) is an algebra, we obtain Since D 0 is δ-invariant, we get which is now of course the same as By the Correspondence Theorem, for every λ > 0. Hence, T Fs lin (D 0 ) is a Banach algebra for every s > 0. Using literally the same reasoning, one proves that (3*) implies (2*) and then concludes the proof.

On the Fredholm property
As is well-known by now, the Fredholm property of operators from T Ft can be characterized in terms of limit operators, at least for F p t with 1 < p < ∞ [10]. Unfortunately, the method of proof from [10] has an important drawback: It does not carry over to the non-reflexive cases. We will now discuss another approach to the problem, which goes through a better understanding of the limit structures, i.e. the algebra of limit operators.
We want to start by briefly recalling the definition of limit functions and limit operators. We note that theory of limit operators has a long history on sequence spaces, see e.g. [18]. The moralic ancestors our the limit operators we use were first considered in [20] for the Bergman space over the complex ball and in [4] over (reflexive) Fock spaces. Since the limit operators are really at the heart of the Fredholm theory, we describe them in some detail.
By M(BUC) we denote the maximal ideal space of the unital C * algebra BUC(C n ). As is well-known, maximal ideal spaces of unital C * subalgebras of C b (C n ) (the bounded continuous functions), which separate the points of C n and contain C 0 (C n ), are in 1:1 correspondence with compactifications of C n . This can be achieved by mapping each z ∈ C n to the functional of point evaluation δ z . Then, {δ z : z ∈ C n } forms a dense subspace of M(BUC).
In particular, any f ∈ BUC(C n ) can be considered as a function Γ(f ) ∈ C(M(BUC)) (Γ being the Gelfand transform). In an abuse of notation, we will not distinguish between f and Γ(f ) (identifying f (z) with Γ(f )(δ z )) and will also write f (x) := Γ(f )(x) for x ∈ M(BUC). For any x ∈ M(BUC) and z ∈ C n we write Let (z γ ) γ be a net in C n converging to x ∈ M(BUC). Then, since M(BUC) is considered with the w * topology, we have pointwise and independently of the precise choice of the net (z γ ) γ . An easy application of the Arzelà-Ascoli Theorem shows that this convergence is indeed uniform on compact subsets of C n and further f x ∈ BUC(C n ) for any x ∈ M(BUC).
Given a net (z γ ) γ in C n converging to x ∈ M(BUC), using the fact that in strong operator topology, at least for F t = F ∞ t : If q a holomorphic polynomial on C n , α zγ (T t f )q → T t fx q is readily established. Then, use that polynomials are dense and α zγ (T t f ) is uniformly bounded in operator norm. From this, it follows that for every A ∈ C 1 (F t ) and every x ∈ M(BUC), there exist A x such that α zγ (A) → A x in strong operator topology when z γ → x: Note that the group action of C n induces an operation on M(BUC). Since the symol α is reserved for the action on functions and operators, we denote this new action by τ and define it by letting τ z (x)(f ) := x(α z (f )). Then, τ z (x) is clearly again a multiplicative functional, i.e. τ z (x) ∈ M(BUC). Further, the group action βf (w) = f (−w) induces the action ν on M(BUC) in the same way, i.e. by letting νx(f ) = x(βf ). These group actions on M(BUC) are well-behaved in the sense that α z (f x ) = f τ−z(x) and α z (A x ) = A τ−z(x) .
In [10], we obtained the following characterization of the Fredholm property: Theorem 6.1 ([10, Theorem 28]). Let F t = F p t with 1 < p < ∞ and t > 0. Then, A ∈ T Ft is Fredholm if and only if for every x ∈ M(BUC) \ C n , the limit operator A x is invertible.
As already mentioned, we aim to generalize this result to any of the spaces F t by different means. In the following, we will usually abbreviate M := M(BUC) and ∂C n := M \ C n .

Some auxiliary facts
Recall that for F t = F ∞ t , an operator A ∈ L(F t ) is called sufficiently localized (cf. [25,15]) if there are constants C > 0, β > 2n: It is not hard to see, for F t = F ∞ t and A ∈ L(F t ), A is given by the integral expression Over F ∞ t , we say that A ∈ L(F ∞ t ) is sufficiently localized if it satisfies both (6.1) and (6.2) (where the second condition is not automatically true, as the Berezin transform is not injective). As is well-known, Toeplitz operators with bounded symbols are sufficiently localized. Therefore, the Banach algebra A Ft sl generated by sufficiently localized operators contains T Ft . It was first proven in [24] for In [12], the result was extended to F p t for 1 < p < ∞. Indeed, the proof given in [12] does not at all depend on the reflexivity and can verbatim be used in the case p = 1. Further, if A is sufficiently localized over some F t , it is not hard to show that the integral operator (6.2) defines is a bounded operator on each F t . This is a consequence of Young's inequality, cf. [12]. Since its formal adjoint, the integral operator is again sufficiently localized and therefore bounded on each F t , a standard duality argument generalizes the following result from [12] to each F t : Let us recall a few facts on certain approximation properties. A Banach space X is said to have the approximation property (AP) if for each compact subset K ⊂ X and each ε > 0 there exists an operator T ∈ L(X) of finite rank with This implies in particular that each compact operator can be approximated by finite rank operators in uniform topology.
X has the λ-bounded approximation property (λ-BAP), where λ ≥ 1, if there is a net T γ of finite rank operators on X, T γ ≤ λ for every γ, with T γ → Id in SOT. Finally, X has the λ-bounded compact approximation property (λ-BCAP) if there is a net K γ of compact operators on X, K γ ≤ λ, with K λ → Id in SOT.
Proof. We will show that F t has ((1 + ε)-BCAP) for every ε > 0. Together with (AP) and the previous lemma, this implies the desired result.
For s < 1 set K s f (z) = f (sz). Then, for 1 ≤ p < ∞: Another simple estimate shows that K s ≤ 1 over F ∞ t (hence also over f ∞ t ). We will show that K s is compact. For this, we will show that K s ∈ T Ft .
It is then an easy exercise to compute K s (z) = e − |z| 2 t + s|z| 2 t ∈ C 0 (C n ), which shows the compactness.
Membership of K s in T Ft will be established by showing that K s is sufficiently localized -then the statement is a consequence of Theorem 6.2. We compute: Finally, note that K s → Id in SOT as s ↑ 1. This is well-known, as e.g. shown in [26, Proposition 2.9] for F p t , 1 ≤ p < ∞. Over f ∞ t , this is an easy exercise. Therefore, (K s ) 1>s≥s0 is a net as needed in the definition of ((1 + ε)-BCAP) for s 0 ∈ (0, 1) appropriate.

The algebra of limit operators
Definition 6.5. A compatible family of limit operators is a map γ : ∂C n → T Ft , satisfying the following properties: 1) x → γ(x)1, 1 is continuous.
2) For every x ∈ ∂C n and z ∈ C n it is 3) It is sup x∈∂C n γ(x) < ∞.

4) The family of operator-valued functions
is uniformly equicontinuous with respect to the operator norm.
Here, we say that a family of operator-valued functions (f i ) i∈I , where f i : C n → L(F t ), is uniformly equicontinuous, if for each ε > 0 there exist δ > 0 such that for z, w ∈ C n with |w| < δ it is Of course, limit operators give a compatible family of limit operators: Lemma 6.6. Let A ∈ T Ft . Then, γ(x) = A x , x ∈ ∂C n , is a compatible family of limit operators.
Proof. Continuity of the map x → A x 1, 1 follows from continuity in strong operator topology, which is obtained directly from the construction of the limit operators (for F t = F ∞ t , note that A x 1, 1 = (A| f ∞ t ) x 1, 1 ). Also, we have (using that the Weyl operators are isometries), If (w γ ) is a net in C n converging to x ∈ ∂C n , we have where in the last step we possibly need to pass to a subnet. Finally, uniform equicontinuity follows from the estimate By limT Ft we will denote the set of all compatible families of limit operators. Endowed with the pointwise sum, this obviously turns into a complex vector space. Indeed, it even turns into an algebra upon endowed with the pointwise product, but this is not clear a priori. What is clear is that the pointwise adjoint and pre-adjoint of a compatible family of limit operators is again a compatible family of limit operators. We endow limT Ft with the norm The following fact is now immediate: Lemma 6.7. The range of the map is a normed algebra. The map is homomorphism of normed algebras onto its range, which also is a contraction with kernel T K.

Reconstructing operators from limit operators
Given a function f ∈ BUC(C n ), it is not hard to see that any other g ∈ BUC(C n ) with g| ∂C n = f | ∂C n is of the form g = f + h for some h ∈ C 0 (C n ). Let us reformulate this simple fact and fix it as a lemma: Proof. ∂C n is a closed subspace of M, which is compact and Hausdorff, hence normal. Thus, Tietze's extension theorem allows us to extend f ∈ C(∂C n ) to f 0 ∈ C(M). But any function in C(M) is in BUC(C n ). The uniqueness modulo C 0 (C n ) has already been mentioned above.
Note that, in the notation of the lemma, f contains all the information about the limit functions of f 0 : (f 0 ) x (w) is simply βf (τ w (x)). Hence, the above lemma says that we can reconstruct functions from BUC(C n ) from their limit functions, at least modulo C 0 (C n ). The same is indeed true for operators from T Ft , nevertheless the proof needs more explanation.
The principal result in the discussion of compatible families of limit operators is the following: Theorem 6.11. Let γ be a compatible family of limit operators. Then, there exists A ∈ T Ft such that γ(x) = A x for every x ∈ ∂C n . A is unique modulo T K.
The proof will make use of the following lemma: is uniformly equicontinuous with respect to the operator norm, i.e. for each ε > 0 there exist δ > 0 such that sup A∈S α z (A) − A < ε for |z| < δ. For each N ∈ N, let c N j ∈ C, z N j ∈ C n be finitely many constants such that Then, the convergence Proof. As in the proof of the Correspondence Theorem, we have The second term clearly converges uniformly to 0, as sup A∈S A < ∞. For the first term, observe that Now, for ε > 0 let us choose δ > 0 such that for every A ∈ S. Pick N large enough such that B(0,δ) c g t/N (w) dw < ε. Then, This proves uniform convergence of the approximation scheme.
Pick a function f 0 ∈ BUC(C n ) according to Lemma 6.10 such that f 0 | ∂C n = f . Let the constants c N j and z N j be as in the previous lemma. Note that the limit operators of j c N j α z N j (T t f0 ) are given by  where the limit functions are (f 0 ) x (w) = f (ν(τ w (x)) = f (τ −w (ν(x))).
Proof. In Lemma 6.7 we already established that the map is an injective homomorphism of normed algebras onto its range. By Theorem 6.11, the map is onto limT Ft . By Proposition 6.8, it is continuous with continuous inverse. Since T Ft /T K is complete, we also obtain that limT Ft is complete.

Fredholmness through invertibility of limit operators
Before coming to the Fredholm property, observe the following: Proposition 6.14. Let A ∈ T Ft with A x invertible for each x ∈ ∂C n . Then, sup x∈∂C n A −1 x < ∞.
Proof. For 1 < p < ∞, the statement is contained in [10]. Again note that, while not every proof of that paper carries over to p = 1, all the ingredients needed for this particular statement can be proven verbatim. Hence, the same proof yields the proposition over F 1 t . Now, using the fact that A x is invertible iff A * x is, and further A −1 x ≃ (A −1 x ) * = (A * x ) −1 , the statement follows also over f ∞ t and F ∞ t .
Theorem 6.15. Let A ∈ T Ft . Then, A is Fredholm iff A x is invertible for every x ∈ ∂C n .
Proof. We already know that A being Fredholm is equivalent to A being invertible in T Ft modulo T K. Clearly, invertibility in T Ft modulo T K of A implies that A x is invertible for every x ∈ ∂C n . Only the other implication needs a proof.
Let A x be invertible for every x ∈ ∂C n . Then, all the inverses lie in T Ft . Since α z (A −1 x ) is an inverse to α z (A x ), we obtain α z (A −1 x ) = A −1 τ−z(x) . Further, x → A −1 x is continuous in strong operator topology: This is a consequence of the second resolvent identity and the uniform boundedness of the A −1 y , which follows from the previous proposition. Finally, note that {z → α z (A −1 x ) : x ∈ ∂C n } is uniformly equicontinuous. This is a consequence of the standard estimate following from the standard Neumann series argument, using (α z (A x )) −1 = α z (A −1 x ), and is valid for each z with α z (A x ) − A x < 1 sup y∈∂C n A −1 y . These facts show that γ(x) = A −1 x is a compatible family of limit operators. Therefore, Theorem 6.11 shows that there exists B ∈ T Ft with B x = A −1 x . Hence, B x A x = A x B x = I for every x ∈ ∂C n , the compactness characterization for operators in T Ft yields that A is Fredholm.
Here are two consequences, which were already presented in [10] in the reflexive cases. The statements can now be obtained for arbitrary F t with identical proofs. Here, σ ess and σ denote the essential spectrum and the spectrum, respectively.
(2) Let f ∈ VMO ∂ (C n ). Then, We want to emphazise that results concerning σ ess (T f ) with f ∈ VMO ∂ (C n ) are available for a significantly larger class of Fock spaces, e.g. non-reflexive Fock spaces with non-Gaussian weights. We refer the interested reader to the literature, see e.g. [13,14].

Acknowledgement
The author wishes to thank Raffael Hagger for a useful discussion on the Fredholm property.