Algebraic Properties of Toeplitz Operators on the Pluri-harmonic Fock Space

We study some algebraic properties of Toeplitz operators with radial and quasi homogeneous symbols on the pluriharmonic Fock space over Cn. We determine when the product of two Toeplitz operators with radial symbols is a Toeplitz operator, the zero-product problem for the product of two Toeplitz operators. Next we characterize the commutativity of Toeplitz operators with quasi homogeneous symbols and finally we study finite rank of the product of Toeplitz operators with quasi homogeneous symbols.


Introduction
Let n ∈ N and consider a family {µ s } s>0 of normalized Gaussian measures on C n defined by (1) Here dv denotes the usual Lebesgue volume form on C n R 2n and we write | • | for the Euclidean norm on C n .The (Segal-Bargmann) Fock space H 2 s = H 2 (C n , µ s ) consists of all µ s -square integrable entire functions on C n .As is well known, H 2 s forms a closed subspace of L 2 s := L 2 (C n , µ s ) and we write P (s) : L 2 s → H 2 s for the orthogonal projection.We let P = P (1) , H 2 = H 2 1 .Let H 2 ah be tha space of all anti-holomorphic functions in L 2 = L 2 1 and we denote the pluriharmonic Fock space over C n by H 2 ph .We recall that H 2 ph consist of all C 2 −functions f in L 2 such that ∂ 2 f /∂z j ∂z k = 0 for all j, k.It is known that H 2 ah and H 2 ph are all closed subspaces of L 2 and they have reproducing kernels K and K ph respectively, where K is the reproducing kernel for H 2 .Moreover there are orthogonal projections P ah and P ph from L 2 onto H 2 ah and H 2 ph respectively.It is shown in (Englis, 2009), that the reproducing kernel for H 2 ph is given by K ph (z, w) = K(z, w) + K(z, w) − 1, z, w ∈ C n , and P ph = P + P ah − P C is the orthogonal projection operator of L 2 onto H 2 ph .For a function f ∈ L ∞ (C n ) = L ∞ , we define the Toeplitz operator T f : H 2 ph → H 2 ph , with symbol f , by For the product problem, on the Hardy space, it was shown in (Brown and Halmos, 1964), that if f and g are bounded functions on the unit circle, then T f T g is a Toeplitz operator if and only if either f or g is analytic.In the setting of the Bergman space, the condition that either f or g is analytic is still sufficient but it is no longer necessary.This problem becomes more complicated on the Hardy space over the unit sphere and Bergman space over unit ball in C n .In (Choe et al., 2007), this problem was resolved for Toeplitz operators with plurihamonic symbols on the Bergman space of the polydisk while (Dong and Zhou, 2009), were able to determine when the product of two radial Toeplitz operators is a Toeplitz operator.For the case of Fock space, things are a lot more different.It was shown in (Coburn, 2007), that if the symbols f, g are either polynomials in z, z or the symbols belong to the Bochner algebra, then T f T g is a Toeplitz operator with the symbols given explicitely.Furthermore, in (Bauer, 2009), it is shown that if the symbols f and g belong to the range of the Heat transform at some time t > 0 then T f T g is a Toeplitz operator.Recently, in (Agbor and Bauer, 2014), this result has been extended to finite products.
The "zero-product "problem for Toeplitz operators has captured people's attentions for a long time, we mention just a few here.For the case of the Hardy space, the problem has been solved in (Aleman and Vukotic, 2009), while an affirmative answer was given to the "zero-product "problem in (Axler and Cucković, 1991), for Toeplitz operators on the Bergman space over the unit disc (in the case of operator products of length 2) if the symbols are both bounded harmonic functions.
For the Fock space, and for products of length 2, the problem has been solved in (Bauer and Lee, 2011), in the case when one of the symbols is radial.
The commuting problem for Toeplitz operators on the Hardy, Bergman and Fock spaces have generated a lot of research in recent years (see for example Brown and Halmos (1964), Axler and Cucković (1991), Cucković and Rao (1998), Dong and Zhou (2009), Choe and Lee (1993), Guan et al (2013), Yan and Liu (2013), Bauer and Le (2011), Bauer and Lee (2011), Bauer and Issa (2012) and Bauer et al (2015)).In (Brown and Halmos, 1964), it is shown for the Hardy space on the unit circle that the Toeplitz operators T f and T g commute if and only if either (1) f and g are holomorphic or (2) f and g are anti-holomorphic or (3) one is a linear function of the other.It was shown that the same result holds true in the Bergman space over the unit disk in (Axler and Cucković, 1991).Furthermore, ( Cucković and Rao, 1998), studied the commutivity of two Toeplitz operators on the Bergman space and described those Toeplitz operators which commute with the Toeplitz operator T e iθ r m for p, m ∈ N. Also, (Guan et al., 2013), studied and characterized commuting Toeplitz operator with quasihomogeneous and separately quasihomogeneous symbols on the pluriharmonic Bergman space over the unit ball of C n .For the case of the Fock space, (Bauer and Lee, 2011), showed that if f and g have atmost exponential growth and f is radial and non-constant then T f and T g commute implies that g is radial.In (Bauer et al., 2015), the case when the symbols are pluri-harmonic and satifies some exponential growth condition has been studied.
For the finite rank problem, we note that for the case of a single operator on the Bergman space over domains Ω ⊂ C n this problem had been considered for a long time, and positive answers were given in the case were Ω is bounded or where Ω = C n , see for example (Luecking (2008), Rozenblum andShirokov (2010)).For the case of products of Toeplitz operators and commutators we mention the work in (Yang et al., 2013), on the Pluri-harmonic Bergman space.
Motivated by the results in (Guan et al. (2013), Yang et al. ( 2013)), on the pluriharmonic Bergman space on the unit ball and the results in (Bauer and Le (2011), Bauer and Lee (2011)), on the Fock space, we study the product of two Toeplitz operators with radial symbols on the pluriharmonic Fock space.We also study the zero-product problem, where we show that the main result in (Bauer and Lee, 2011), also holds in the setting of the pluriharmonic Fock space.Next we consider the commuting problem with quasihomogeneous symbols and finally we consider the finite rank problem for product of two Toeplitz operators with quasihomogeneous symbols and a commutator of Toeplitz operators with such symbols.
The paper is organized as follows: In section 2 we will study some preliminaries.In section 3 we will study the product problem.In section 4, we consider zero product problem while in section 5, we consider the commuting problem.Finally, in section 6, we study the finite rank problem.

Preliminaries
0 } is a basis for the Fock space and {e α : α ∈ N n 0 } is a basis for the space H 2 ah .It follows that {e α : α ∈ N n 0 } ∪ {e α : α ∈ N n 0 } is a basis for the pluri harmonic Fock space H 2 ph .Now we recall the definition of the symbol space Sym >0 (C n ) in (Bauer, 2009), together with the construction of a scale of Banach spaces on which the class of Toeplitz operators T (s)  f with f ∈ Sym >0 (C n ) acts by a "finite order shift".For each c ∈ R we set Equipped with the norm ) turns into a Banach space.Consider the following strictly increasing sequence It is easy to check that D c is continuously embedded into L 2 (C n , µ s ) whenever c < 1/(2s).Hence we obtain an increasing scale of Banach spaces Let k ∈ N, then we write O s (k) for the space of all linear operators A : D s → D s that continuously map D c j (s) into D c j+k (s) for all j ∈ N 0 .We say that operators in O s (k) act on the scale (2) by an order shift k.The algebra of operators on D s acting on (2) by a finite order shift is given by Lemma 2.1.The orthogonal projection P ph : L 2 → H 2 ph defines an element in O(1) = O 1 (1).Moreover for j ∈ N 0 and g ∈ D c j we have Proof It is sufficient to prove (3).Observe that By Lemma 10 of (Bauer, 2009) we have A similar computation also shows that Also, a direct calculation shows that which gives the desired result. 2 In order to define Toeplitz operators acting on the scale (2) we restrict ourselves to the following space of (i.g.unbounded) symbols: Clearly, with respect to pointwise multiplication and complex conjugation (5) turns into a * -algebra which in particular contains all polynomials in the complex variables z and z.Furthermore, we remark that for f ∈ Sym >0 (C n ) and all s > 0 the multiplication operator M f defines an element in O s (1).By Lemma 2.1 it follows that and the restriction of this operator product to (D ∩ H 2 ) + (D ∩ H 2 ah ) is the Toeplitz operator with symbol f .Since L fos (D) is an algebra, all finite products of Toeplitz operators T f with symbols f ∈ Sym >0 (C n ) are well-defined and can be interpreted as densely defined (i.g.unbounded) operators on the Hilbert space H 2 ph .Similarly we can consider the spaces We obtain a scale of Banach spaces: and Now, we set for each j ∈ N, Then we obtain the following scale of Banach spaces: Completely analogous to our definition above we can consider the algebra L fos (H ph ) of all operators acting on (8) by a finite order shift.
We recall the definition of the Mellin Transform: Given a function ψ on the half line (0, ∞) the Mellin transform M[ψ](z) of the complex parameter z is defined by whenever the improper integral exists.Also, each M[ψ] is complex analytic on a strip in the complex plane parallel to the imaginary axis.Moreover, the Mellin transform is injective.We also recall that if f, g : R + → C and x > 0 the Mellin convolution is defined by )dy.
The Mellin convolution theorem implies on a certain strip in the complex plane parallel to the imaginary axis.
Let A denote the class of complex valued functions u for which there exist constants The following is Proposition 4.8 of (Bauer and Lee, 2011).
The following lemma replaces the Blaschke condition in the unit disc and this is Proposition 4.11 of (Bauer and Lee, 2011).
Lemma 2.3.Suppose that u ∈ A and a ∈ (0, 2].If there is an m ∈ N such that x ak e −x dx = 0 for all k ≥ m then u = 0 a.e. on R + . We recall that for a suitable holomorphic function Ψ(z) on the right half-plane Rez > δ, the inverse Mellin transform is given by Then we have the following which is Lemma 4.5 of (Bauer and Lee, 2011).
and analytic on K n by the Dominated convergence Theorem and Morera's Theorem.
For n = 1 and f locally integrable on C we see that where f (r) = 1 π ∫ 2π 0 f (re iθ )dθ, establishing a relationship between F ( f ) and the Mellin transform.The next proposition is Proposition 3.1 of (Bauer and Le, 2011).
In one dimension, the following holds: Corollary 2.6.Let ψ, ϕ ∈ S ym >0 (C n ) be radial function, k a positive integer and suppose vanishes for all z with Re(z) > 0.
Proof.The function F(z) := Γ(z + k) Φ(z) vanishes on the set {n + 1, n + 2, • • • } and thus the function vanishes in N and then by Proposition 2.5 F vanishes on K. 2 The following Lemma will be useful to us: Lemma 2.7.Suppose f and g are holomorphic in the plane Rez > 0 and satisfies f (z and thus f (z + mp)g(z) = f (z)g(z + mp), as required.2

Product of Toeplitz Operators with Radial Symbols
In this section we will study the product problem, that is, when is the product of two Toeplitz operators a Toeplitz operator, and what is the resultant symbol.
We will frequently use the following lemma.
Proof.Let α ∈ N n 0 .Then for any β ∈ N n 0 we have that and is a basis for the pluriharmonic Fock space , we have By a similar argument we have that 2 As an immediate consequence of Lemma 3.1 we have: Proposition 3.2.Suppose f, g ∈ S are radial, then T f T g = T g T f on the span The next theorem is well known and the proof is similar to that in the Pluriharmonic Bergman space, see Theorem 5 of (Yang et al., 2013).
Then the following are equivalent.
Proof.By (11) we have It follows from Theorem 3.3 that h is radial.2 Theorem 3.5.Let f, g ∈ A be radial functions.Then T f T g is equal to the Toeplitz operator T h if and only if h is the solution of the equation Proof Let α be any multi-index of non-negative integers.Then T f T g = T h implies It follows from ( 16) that Thus, the function is an entire function which vanishes on {k : k ∈ N}.By Proposition 2.6 there exist functions h 1 , h 2 ∈ A such that ](2z) and by Lemma 2.3 we have that h 1 = h 2 on R + .This implies ( f e −r 2 ) * (ge −r 2 ) = 2(e −r 2 ) * (he −r 2 ), as required.
Conversely, let f, g ∈ A and suppose there is an h ∈ A which satisfies (17).Then from ( 16) we have that Similarly, one can show that

The Zero Product Problem
We begin this section by observing that if f ∈ D c , c < 1 then T f = 0 on H 2 ph if and only if f ≡ 0 a.e on C n .Indeed, for any h ∈ H 2 we have that It follows that P( f h)(0) = 0 and thus P( f h)(0) = 0. Thus (20) shows that P( f h)(z) + P( f h)(z) = 0 for all z ∈ C n and P( f h) ≡ 0 for all h ∈ H 2 since P( f h)(0) = 0.But this shows that the Toeplitz operator Q f : H 2 → H 2 given by Q f g = P( f g) for g ∈ H 2 vanishes on H 2 and this implies that f = 0 a.e on C n by (Folland, 1989), page 140.
The following is Lemma 3.2 of (Bauer and Le, 2011).
For g ∈ L 2 (C n , dµ) radial and s ∈ C such that Re(s) > −n, we set ω(g, s) = M[2ge −r 2 ](2s + 2n)/Γ(s + n).Then ω(g, s) is analytic on its domain and It is shown in (Bauer and Le, 2011), that if one of the functions is non-constant, then either Z( f, g) = ∅ or Z( f, g) = {d} for some integer d.Also, if f is non-constant then Z( f, f ) = {0}.Our next result was given in (Bauer and Le, 2011), in the case of the Fock space, and we show here that the result remains true on the Pluri-harmonic space.
Using (1)-( 4) and the fact that T f T h = T h T g , we have the following: Let m and k be two fixed multi-indices in N n 0 , with α = m + l and β = k + l for l ∈ N n 0 .Then by ( 21) we have In a similar manner using (2 ′ )-(4 ′ ) we define G i (l), i = 2, 3, 4 for l ∈ N n 0 .Now, it is easily seen that for each i = 1, 2, 3, 4, the function (m + k + l)!G i (l) satisfies the hypothesis of Proposition 2.5.Therefore, G i (l) = 0 for all l ∈ N n 0 if and only if G i (l) = 0 for all l ∈ K n .This by analyticity, is equivalent to either ω( f, |k| + |l|) = ω(g, |m| + |l|) for all l ∈ K n or for l ∈ K n each of the following holds: Since one of the functions f, g is non-constant, we have the following two possibilities.This shows that ⟨T f T h e α − T h T g e α , e β + e γ ⟩ = 0 and ⟨T f T h e α − T h T g e α , e λ + e β ⟩ = 0 for every α, β, γ, λ ∈ N n 0 .It follows that T f T h e α = T h T g e α and T f T h e α = T h T g e α for every α ∈ N. We therefore conclude that ph then either f or g must be zero a.e on C n .
Proof If f is a constant, then T f T g and T g T f are constant multiples of T g .Now T g = 0, implies g(z) = 0 for a.e z ∈ C n , by remark in the begining of this section.Now suppose f is a non-constant function.Take h(z) = 0 for all z ∈ C n .If T f T g = 0 then T f T g = T g T h and since Z( f, h) = ∅ we see by Theorem 4.2 that g must be zero.The case T g T f = 0 is similar. 2

Commuting Toeplitz Operators
We will study the commutators of two Toeplitz operators with quasi homogeneous symbols.Our next theorem is a corollary of Theorem 4.2.
Theorem 5.1.Let f ∈ S ym >0 (C n ) be a radial and non-constant function.Then for any g ∈ S ym >0 (C n ), T f and T g commute if and only if g(γz) = g(z), for a.e.γ ∈ T and a.e z ∈ C n .
Proof Since Z( f, f ) = {0}, Theorem 4.2 implies that T f T g = T g T f if and only if g(γz) = g(z), for a.e.γ ∈ T and a.e z ∈ C n . 2 Recall that Theorem 5.1 also holds for the setting of the Fock space, see (Bauer and Lee, 2011), for the details.Based on the above theorem, we have the following corollary.With Theorem 5.1 one easily shows that the result on commutativity of Toeplitz operators given in (Bauer and Lee, 2011), for the Fock space also holds for the Pluriharmonic Fock space.We present here the result for completeness.Let n = 1 and define D c on C. The following is Lemma 2 of (Bauer and Issa, 2012).