Essential positivity for Toeplitz operators on the Fock space

In this short note, we discuss essential positivity of Toeplitz operators on the Fock space, as motivated by a recent question of Per\"al\"a and Virtanen. We give a proper characterization of essential positivity in terms of limit operators. A conjectured characterization of essential positivity of Per\"al\"a and Virtanen is disproven when the assumption of radiality is dropped. Nevertheless, when the symbol of the Toeplitz operator is of vanishing mean oscillation, we show that the conjecture of Per\"al\"a and Virtanen holds true, even without radiality.


Introduction
In their recent note [9], Perälä and Virtanen study the notion of essential positivity and describe a class of Toeplitz operators which are essentially positive.A bounded self-adjoint operator A on the Hilbert space H is said to be essentially positive provided σ ess (A) ⊆ [0, ∞).Here, σ ess (A) is the essential spectrum: After discussing some characterizations of essential positivity ([9, Theorem 3]), they go on to characterize essential positivity of certain Toeplitz operators on the Hardy space and the Bergman space of the disc.Their result on the Bergman space of the disc is the following: Theorem ([9, Theorem 9]).Let µ be a radial real-valued Borel measure on D such that |µ| is a Carleson measure for the Bergman space A 2 (D).Suppose that the limit L = lim |z|→1 μ(z) exists.Then, T µ is essentially positive on A 2 (D) if and only if L ≥ 0.
Further, Perälä and Virtanen conjectured that essential positivity of Toeplitz operators on the Bergman space of the disc might in general by classified by the property that lim inf |z|→1 f (z) ≥ 0.More specifically, they suggested: Conjecture.Let f ∈ L ∞ (D) be real-valued and radial.Then, T f : A 2 (D) → A 2 (D) is essentially positive if and only if lim inf |z|→1 f (z) ≥ 0.
Perälä and Virtanen make crucial use of a certain Tauberian theorem in the proof of their above theorem.At the end of their work, they ask if an analogous result holds true for Toeplitz operators on the Fock space, pointing out that a suitable substitute for the Tauberian theorem needs to be found.
In this short note, we explain how to obtain an analogous result on the Fock space through the methods of limit operators.We prove that their conjecture is (at least when the assumption of radiality is dropped) wrong on the Fock space, but it is true if the symbol is assumed to be of vanishing mean oscillation.We provide a general characterization of essential positivity in terms of limit operators.As a special case, we obtain a result analogous to that of Perälä and Virtanen but without the assumption of the symbol being radial.

Essential positivity for Toeplitz operators on the Fock space
On C n we consider the family of probability measures µ t given by where | • | is the Euclidean norm, dz the standard Lebesgue measure and t > 0 a fixed real number.We define the Fock space F 2 t by where Hol(C n ) denotes the entire functions on C n .The standard reference for the Fock space and its properties is [11].This space is always endowed with its natural inner product, i.e.
F 2 t is well-known to be a reproducing kernel Hilbert space, the reproducing kernels being given by The normalized reproducing kernels are now defined as For A ∈ L(F 2 t ), we define its Berezin transform as the function A(z) = Ak t z , k t z , where z ∈ C n .Then, A is always a bounded and uniformly continuous function on C n .
We of course have the well-known orthogonal projection P t ∈ L(L 2 (C n , µ t )) mapping onto F 2 t by For any f ∈ L ∞ (C n ) the Toeplitz operator T t f given by is well defined and bounded, satisfying T t f ≤ f ∞ .Given some signed Borel measure ν on C n , one defines the Toeplitz operator with symbol ν formally as Of course, in general this expression is not well-defined.The class of positive measures which define in this way a bounded linear operator are wellunderstood.They are usually called Fock-Carleson measures and characterized by the following properties: Theorem ([8, Theorem 3.1]).Let ν be a positive Borel measure on C n .Then, the following are equivalent: (ii) For some (equivalently: every) R > 0 there exists a constant When these properties are satisfied, µ agrees with the Berezin transform of Now, if ν is any signed Borel measure on C n , we can consider its Hahn-Jordan decomposition and the total variation measure If one assumes that |ν| is a Carleson measure, then clearly both ν + and ν − satisfy (ii) in the above theorem.Hence, both T t ν+ and T t ν− are bounded operators on F 2 t , which yields that their linear combination T t ν = T t ν+ − T t ν− is also a bounded linear operator.
For z ∈ C n we define the Weyl operator W t z by These operators satisfy the following well-known properties, which we fix as a lemma: (3) z → W t z is continuous in strong operator topology over F 2 t .
It has been proven very fruitful in the study of Toeplitz operators on the Fock space to consider the shift action of C n on operators, defined as α z (A) = W z AW −z for A ∈ L(F 2 t ) and z ∈ C n , cf. [3,4].In particular, we can consider the C * -algebra Then, we have the following important consequence: The following equalities hold true: Here, BUC(C n ) denotes the bounded, uniformly continuous functions on C n .Now, if ν is a signed Borel measure on C n such that |ν| is Fock-Carleson, [1, Theorem 3.7]1 shows that T t ν can be approximated in norm by Toeplitz operators with bounded symbols.We therefore obtain: We denote by M = M(BUC(C n )) the maximal ideal space of the commutative unital C * -algebra BUC(C n ), which we understand as a compactification of C n .Then, as is described in [3,4,5], for every A ∈ C 1 the map which is continuous with respect to SOT * , i.e. the map is continuous in strong operator topology for both the operator and the adjoint.The operators α x (A) for x ∈ M \ C n are usually referred to as the limit operators of A. One then has the following result from [5], cf. also [4] which puts it more into the perspective of the algebra C 1 .Here, we denote by σ(B) the spectrum of the operator B.
We now return to the problem of essential positivity on the Fock space.Let A ∈ C 1 be self-adjoint.Then, α z (A) = W z AW −z is self-adjoint for any z ∈ C n .
Since for each f ∈ F 2 t the map z → α z (A)f, f is real valued and extends continuously to we obtain that α x (A) is self-adjoint for every x ∈ M. Now, by the above theorem, we obtain: This characterizes essential positivity in largest generality for elements from the Toeplitz algebra.Let us reformulate the result in one particular case, which involves the notion of vanishing oscillation.We recall that a continuous bounded function f : C n → C is said to be of vanishing oscillation (at infinity), abbreviated as The key fact we will use is the following: Lemma 2.6.Let A ∈ C 1 .Then, A is of vanishing oscillation if and only if for every x ∈ M \ C n there exists some c x ∈ C such that α x (A) = c x I. Proof.By the assumption, α x (A) = c x I for every x ∈ M \ C n .Since c x = α x (A)1, 1 continuously depends on x ∈ M, it is not hard to see that In this case, we have These facts show the result.
We want to emphasize that any Toeplitz operator T t f the symbol of which is of vanishing mean oscillation satisfies the previous result.This class also contains discontinuous symbols.
Specialising the previous result to A = T t ν with |ν| Fock-Carleson yields a result which is in analogy to [9,Theorem 9] without the assumption of radiality.Indeed, if one assumes that A ∈ C 1 such that lim |z|→∞ A(z) exists, then this has the rather strong implication that A ∈ K(H) + CI.In particular, for A self-adjoint with lim |z|→∞ A(z) ≥ 0, this clearly shows essential positivity of A.
It is well-known that there are certain implications between properties of the symbol, essential positivity of the operator and properties of the Berezin transform.We shortly summarize them for completeness.These facts are certainly well-known and can be proven by different means.Just for the fun of it, we sketch a proof which goes by considerations of limit operators and limit functions.
Then, it is not hard to see that α zγ (f ) converges to some function α x (f ) ∈ BUC(C n ), and the convergence is uniformly on compact subsets (cf.[3,4] for details).Further, was arbitrary, this shows that all limit operators are positive.
Similarly, one proves the following: We will now prove that the conjecture of Perälä and Virtanen is in general false when the assumption of radiality is dropped.We need the following preparatory lemma.Lemma 2.10.There exists no constant M > 0 such that Proof.We first want to emphasize that the proof is a straightforward adaptation of [2, Corollary 1] since the right-hand side goes faster to 0 than the left-hand side as |z| → ∞.
Proposition 2.11.Let f ∈ BUC(C n ) be real-valued.Then, for T t f being essentially positive, it is not sufficient that lim inf |w|→∞ f (w) ≥ 0.
Proof.The proof is an adaptation of [10,Theorem 2.3] for essential positivity.
We assume the contrary, i.e. we assume that lim inf |w|→∞ f (w) ≥ 0 would imply essential positivity for every real-valued f ∈ BUC(C n ).Consider such real-valued f .Then, letting Here, (C ∓ f ) ∼ denotes the Berezin transform of the functions C ∓ f .By assumption, we therefore obtain that T t C∓f = CI ∓ T t f is essentially positive, hence: This now implies: Since this would hold true for any g ∈ BUC(C n ), this violates the previous lemma.
In light of this result, it would be surprising if the conjecture of Perälä and Virtanen would turn out to hold true: Our discussion has shown that it is not the assumption of radiality, but the implicit assumption of vanishing oscillation which yields their result [9,Theorem 9].Nevertheless, we cannot entirely rule out the possibility that some strange effects show up under the additional assumption of radiality.Inspecting the previous arguments, it turns out that to disprove the conjecture, one needs to find a sequence of radial functions This would then imply a version of Lemma 2.10 for radial symbols, which in turn would (by arguments analogous to the previous result) show that the conjecture is false.We consider the special case n = 1 and t = 2.When diagonalizing the radial operator by the standard orthonormal basis, the task of finding such radial functions f k ∈ L ∞ (C) is equivalent to the problem of finding a sequence of functions f k ∈ L ∞ ([0, ∞)) such that we have lim sup m→∞ Remark 2.12.The methods of limit operators is also available for the Bergman space of the disc or even bounded symmetric domains, cf.[6,7].There, the algebra C 1 has to be replaced by the algebra of band-dominated operators.One obtains that a self-adjoint band-dominated operator A is essentially positive if and only if all of its limit operators are positive.Improvements of this result are available, in similar ways, when the Berezin transform of the operator is of vanishing oscillation, cf.[7,Theorem 36].Further, note that reasoning similar to Lemma 2.10 and Proposition 2.11 work on the Bergman space.Hence, without the assumption of radiality, the conjecture of Perälä and Virtanen is wrong also on the Bergman space.

Proof.
Using the correspondence theorem, [3, Theorem 2.21] or [4, Theorem 3.1], together with an adaptation of [7, Theorem 36], shows that for A ∈ C 1 , A ∈ VO ∂ (C n ) is equivalent to α x (A) being a constant multiple of the identity for every x ∈ M \ C n .Corollary 2.7.Let A ∈ C 1 be self-adjoint such that A is of vanishing oscillation.Then, A is essentially positive if and only if lim inf |z|→∞ A(z) ≥ 0.