Localization and compactness of Operators on Fock Spaces

LOCALIZATION AND COMPACTNESS OF OPERATORS ON FOCK SPACES

arXiv:1712.07278v1 [math.CV] 20 Dec 2017

ZHANGJIAN HU1 , XIAOFEN LV1 , AND BRETT D. WICK2 Abstract. For 0 < p ≤ ∞, let Fϕp be the Fock space induced by a weight function ϕ satisfying ddc ϕ ≃ ω0 . In this paper, given p ∈ (0, 1] we introduce the concept of weakly localized operators on Fϕp , we characterize the compact operators in the algebra generated by weakly localized operators. As an application, for 0 < p < ∞ we prove that an operator T in the algebra generated by bounded Toeplitz operators with BMO symbols is compact on Fϕp if and only if its Berezin transform satisfies certain vanishing property at ∞. In the classical Fock space, we extend the Axler-Zheng condition on linear operators T , which ensures T is compact on Fαp for all possible 0 < p < ∞.

1. Introduction Let H(Cn ) be the collection of all entire√ functions on Cn , and let ω0 = ddc |z|2 be the Euclidean K¨ahler form on Cn , where dc = 4−1 (∂ − ∂). Set B(z, r) to be the Euclidean ball in Cn with center z and radius r, and B(z, r)c = Cn \B(z, r). Throughout the paper, we assume that ϕ ∈ C 2 (Cn ) is real-valued and there are two positive numbers M1 , M2 such that (1.1)

M1 ω0 ≤ ddc ϕ ≤ M2 ω0

in the sense of currents. The expression (1.1) will be denoted as ddc ϕ ≃ ω0 . Given 0 < p < ∞ and a positive Borel measure µ on Cn , let Lpϕ (µ) be the space defined by  Lpϕ (µ) = f is µ-measurable on Cn : f (·)e−ϕ(·) ∈ Lp (Cn , dµ) . When dµ = dV , the Lebesgue measure on Cn , we write Lpϕ for Lpϕ (µ) and set Z  p1 −ϕ(z) p kf kp,ϕ = f (z)e dV (z) . Cn

For 0 < p < ∞ the Fock space Fϕp is defined as Fϕp = Lpϕ ∩ H(Cn ), and   −ϕ(z) ∞ n 0, which induces the classical Fock space. For this particular special weight ϕ, Fϕp and k · kp,ϕ will be written as Fαp and k · kp,α, respectively. The space Fαp has been studied by Date: December 21, 2017. 2000 Mathematics Subject Classification. Primary: 47B35. Secondary: 32A37. Key words and phrases. Fock space, Weakly localized operator, Compactness. 1. This research is partially supported by the National Natural Science Foundation of China (11601149, 11771139, 11571105), Natural Science Foundation of Zhejiang province (LY15A010014). 2. Research supported in part by a National Science Foundation DMS grant #0955432. 1

2

Z. HU, X. LV, AND B. D. WICK

many authors, see [2, 5, 7, 18–21] and the references therein. Another special case is with ϕ(z) = α2 |z|2 − m2 ln(A + |z|2 ) with suitable A > 0, and then Fϕp is the Fock-Sobolev space Fαp,m studied in [3, 4]. It is well-known that Fϕ2 is a Hilbert space with inner product Z f (z)g(z)e−2ϕ(z) dV (z). hf, giFϕ2 = Cn

Given z, w ∈ Cn , the reproducing kernel of Fϕ2 will be denoted by Kz (w) = K(w, z). We write kz = kKKz kz2,ϕ to denote the normalized reproducing kernel. Given some bounded linear operator T on Fϕp , the Berezin transform of T is well defined as Te(z) = hT kz , kz iFϕ2 ,

since T kz ∈ Fϕp ⊂ Fϕ∞ and kz ∈ Fϕ1 . Set P to be the projection from L2ϕ to Fϕ2 , that is Z P f (z) = f (w)K(z, w)e−2ϕ(w)dV (w) for f ∈ L2ϕ . Cn

For a complex Borel measure µ on Cn and f ∈ Fϕp , we define the Toeplitz operator Tµ to be Z Tµ f (z) = f (w)K(z, w)e−2ϕ(w)dµ(w). Cn

If dµ = gdV , for short, we will use Tg to stand for the induced Toeplitz operator and will use that e g = Teg .

In the case of Fock spaces Fα2 , fixed g bounded on Cn , |hTg kz , kw i| as a function of (z, w) decays very fast off the diagonal of Cn × Cn , see [20, Proposition 4.1]. From this point of view, Xia and Zheng in [20] introduced the notion of “sufficiently localized” operators on Fα2 which include the algebra generated by Toeplitz operators with bounded symbols, and they proved that, if T is in the C ∗ -algebra generated by the class of sufficiently localized operators, T is compact on Fα2 if and only if its Berezin transform tends to zero when z goes to infinity. In [10], Isralowitz extended [20] to the generalized Fock space Fϕ2 with ddc ϕ ≃ ω0 . Isralowitz, Mitkovski and the third author extended Xia and Zheng’s idea further in [11] to what they called “weakly localized” operators on Fϕp with 1 < p < ∞. They showed that, if T is in the C ∗ -algebra generated by the class of weakly localized operators, T is compact on Fϕp if and only if its Berezin transform shares certain vanishing property near infinity. We would like to emphasize that the prior results in the area, for example [1, 2, 5, 8–10, 14, 15, 17, 19–22], depend strongly on two points. The first is the use of Weyl unitary operators induced by holomorphic self mappings of the domain; and the second is the restriction on the range of the exponent p, for example p = 2 or 1 < p < ∞, so that Banach space techniques are applicable. But on Fϕp with 0 < p < 1 and ddc ϕ ≃ ω0 these two points are not available. The main purpose of this work is, on Fϕp with 0 < p < 1 and ddc ϕ ≃ ω0 , to study the so called “weakly localized” operators WLϕp and to characterize those compact operators T ∈ WLϕp . The paper is divided into four sections. In Section 2, we introduce the concept of weakly localized operators WLϕp for 0 < p ≤ 1, we will characterize the compact operators in WLϕp , and furthermore give a quantity equivalent to the essential norm of an operator in WLϕp . Section 3 is devoted to the compactness of Toeplitz operators induced by

LOCALIZATION AND COMPACTNESS ON FOCK SPACES

3

BMO symbols acting on Fϕp for all 0 < p < ∞, our theorem shows an operator T in the algebra generated by bounded Toeplitz operators with BMO symbols is compact on Fϕp if and only if its Berezin transform satisfies a certain vanishing property at ∞ (more precisely, lim Te(z) = 0 when ϕ(z) = α2 |z|2 ). In Section 5, we extend Axler-Zheng’s condition on linear

z→∞

operators T , which insures T are bounded (or compact) on Fαp for all possible 0 < p < ∞. In the final section, we provide some remarks and point to some open problems.

In what follows, C will denote a positive constant whose value may change from one occasion to another but does not depend on the functions or operators in consideration. For two positive quantities A and B, the expression A ≃ B means there is some C > 0 such that 1 B ≤ A ≤ CB. C 2. The operator class WLϕp with 0 < p ≤ 1 As a generalization of the “strongly localized” operators of Xia and Zheng in [20], Isralowitz, Mitkovski and the third author introduced “weakly localized” operators on Fϕp with 1 < p < ∞, see [11]. In this section, we first give the definition of weakly localized operators on Fϕp when 0 < p ≤ 1. We use D to stand for the linear span of all normalized reproducing kernel functions kz (·). It is obvious that D is dense in Fϕp . As in [11], we will assume that the domain of every linear operator T appearing in this paper contains D, and that the function z 7→ T Kz is conjugate holomorphic. We also assume the range of T is in Fϕ∞ . Then hT kz , kw iFϕ2 can make sense. Definition 2.1. Let 0 < p < ∞, set s = min{1, p}. A linear operator T from D to Fϕ∞ is called weakly localized for Fϕp if Z Z s s (2.1) sup hkz , T kw iFϕ2 dV (w) < ∞; hT kz , kw iFϕ2 dV (w) < ∞, sup z∈Cn

z∈Cn

Cn

and

(2.2)

(2.3)

lim sup

r→∞ z∈Cn

lim sup

r→∞ z∈Cn

Z

B(z,r)c

Z

B(z,r)c

Cn

s hT kz , kw iFϕ2 dV (w) = 0,

s hkz , T kw iFϕ2 dV (w) = 0.

The algebra generated by weakly localized operators for Fϕp will be denoted by WLϕp . For ϕ(z) = α2 |z|2 , we write WLϕp = WLαp for convenience. When 1 ≤ p < ∞ WLϕp = WLϕ1 by definition, and then Definition 2.1 was first introduced in [11]. Let Tpϕ denote the Toeplitz algebra on Fpϕ generated by L∞ symbols, and let K(Fϕp ) be the set of all compact operators on Fpϕ . We use kT ke,Fϕp to stand for the essential norm of a given operator T on Fϕp  kT ke,Fϕp = inf kT − AkFϕp →Fϕp : A ∈ K(Fϕp ) . The purpose of this section is to characterize compact operators in WLϕp , 0 < p ≤ 1. To carry out our analysis, we need some preliminary facts.

4

Z. HU, X. LV, AND B. D. WICK

Lemma 2.2 ([16]). Given ϕ as in the introduction, the Bergman kernel K(·, ·) for Fϕ2 satisfies the following estimates: (1) There exists C and θ > 0 such that |K(z, w)|e−ϕ(z) e−ϕ(w) ≤ Ce−θ|z−w| for z, w ∈ Cn . (2) There exists some r > 0 such that |K(z, w)| e−ϕ(z) e−ϕ(w) ≃ 1 whenever w ∈ B (z, r) and z ∈ Cn . (3) For 0 < p ≤ ∞ fixed, kK(·, z)kp,ϕ ≃ eϕ(z) ≃

p

K(z, z),

z ∈ Cn .

Lemma 2.3 ([8]). Suppose 0 < p < ∞ and r > 0. Then there exists C such that for f ∈ H(Cn ) and z ∈ Cn , we have f (z)e−ϕ(z) p ≤ C

Z

B(z,r)

f (w)e−ϕ(w) p dV (w)

and Z

Cn

f (z)e−ϕ(z) p dµ(z) ≤ C

Z

Cn

f (z)e−ϕ(z) p µ br (z)dV (z)

where µ is some given positive Borel measure and µ br (·) =

µ(B(·,r)) . V (B(·,r))

n + Let d(·, ·) be the Euclidean distance on Cn . Given some  domain Ω ⊆ 1C n, write Ω = {z ∈ + C : d(z, Ω) < 1}, and Ω is again a domain. Set L = a + bi : a, b ∈ 4 Z , L is countable so that we may write L = {z1 , z2 , · · · , zj , · · · }. It is obvious that L forms a 1/4-lattice in Cn (see [21] for the definition). For E ⊂ Cn , let χE be the characteristic function of E. We have some absolute constant N > 0 such that n

X

(2.4)

for w ∈ Cn .

χB(zj , 1 ) (w) ≤ N 2

zj ∈L

Lemma 2.4. For 0 < p ≤ 1 there is some constant C (depending only on p and n) such that for any domain Ω ⊂ Cn and f ∈ H(Cn ), Z

Ω

f (w)e−ϕ(w) dV (w)

p

≤C

Z

Ω+

f (w)e−ϕ(w) p dV (w).

LOCALIZATION AND COMPACTNESS ON FOCK SPACES

5

Proof. It is trivial to see that (u + v)p ≤ up + v p for positive u, v and 0 < p ≤ 1. Applying Lemma 2.3 and (2.4), for f ∈ H(Cn ) we have  p Z p XZ f (w)e−ϕ(w) dV (w) f (w)e−ϕ(w) dV (w) ≤  Ω

Ω∩B(zj ,1/4)

zj ∈L

X

≤ C

max

zj ∈L,d(zj ,Ω) 0 such that ∞ X

(2.6)

χFj+ (w) ≤

∞ X

χGj (w) ≤

χG+j (w) ≤ N

∀w ∈ Cn .

j=1

j=1

j=1

∞ X

This covering Fr can also be chosen in a simple way. For example, let {aj } be an enumeration of the lattice √2rn Z2n . And take Fj to be the cube with centers aj , side-length √2rn and n half of the boundary so that ∪∞ j=1 Fj = C , Fj ∩ Fk = ∅ if j 6= k. Proposition 2.8. Let 0 < p ≤ 1 and T ∈ WLϕp . Then for every ε > 0, there is some ∞ r > 0 sufficiently large such that, for the covering {Fj }∞ j=1 and {Gj }j=1 (associated to r) from Lemma 2.7, we have



T − P

(2.7)

∞ X j=1

!

MχFj T P MχGj

< ε.

Fϕp →Fϕp

Proof. Let T ∈ WLϕp be given. For ε > 0, we have some r > 0 sufficiently large (we may assume r > 10) such that Z

p

B(z, r−1)c

|hT kz , kw iFϕ2 | dV (w) < ε and

Z

B(z, r−1)c

|hkz , T kw iFϕ2 |p dV (w) < ε.

+ c ∞ Take {Fj }∞ j=1 and {Gj }j=1 to be as in Lemma 2.7 with r. For w ∈ Fj and u ∈ Gj we have |u − w| > r − 1, then u ∈ B(w, r − 1)c . That is Gcj ⊂ B(w, r − 1)c whenever w ∈ Fj+ . Hence, for w ∈ Fj+ ,

  T P MχGc f (w) = hP MχGc f, T ∗ Kw iFϕ2 j j ∗ = hMχGc f, T Kw iFϕ2 j Z ≤ |f (u)| hKu , T ∗Kw iFϕ2 e−2ϕ(u) dV (u) Gcj

≤

Z

B(w,r−1)c

∗ |f (u)| hKu , T Kw iFϕ2 e−2ϕ(u) dV (u).

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Z. HU, X. LV, AND B. D. WICK

Set S = T P −

∞ P

j=1

MχFj T P MχGj . Then |P Sf (z)|p Z p −2ϕ(w) Sf (w)K(z, w)e dV (w) ≤ Cn !p Z X ∞ MχFj T P MχGc f (w) |K(z, w)|e−2ϕ(w) dV (w) = j n C j=1 !p ∞ Z X = MχFj T P MχGc f (w) |K(z, w)|e−2ϕ(w) dV (w) j

Cn

j=1

!p Z ∞ X ≤ T P MχGc f (w) |K(z, w)|e−2ϕ(w) dV (w) . Fj

j=1

j

Notice that |K(z, w)| = |K(w, z)|, applying Lemma 2.4 twice to above, we get |P Sf (z)|p ∞ Z p X ≤C T P MχGc f (w) |K(w, z)|p e−2pϕ(w) dV (w) ≤C

≤C

+ j=1 Fj ∞ Z X

Fj+

j=1 ∞ Z X j=1

j

p −pϕ(w)

|K(w, z)| e

p −pϕ(w)

Fj+

|K(w, z)| e

Z

Z

B(w,r−1)c

B(w,r−2)c

p −ϕ(u) hT ku , kw iFϕ2 dV (u) dV (w) f (u)e

 p −ϕ(u) hT ku , kw iFϕ2 dV (u) dV (w). f (u)e

By Fubini’s Theorem and (2.6), we get kP Sf kpp,ϕ is no more than C

∞ Z X j=1

× ≤ CN ≤

Z

Cn

Z

Cn

f (u)e−ϕ(u) p

Z

Fj+

p χB(u,r−1)c (w) hT ku , kw iFϕ2 e−pϕ(w)

|K(w, z)|p e−pϕ(z) dV (z)dV (w)dV (u)  Z p f (u)e−ϕ(u) p hT ku , kw iFϕ2 dV (w) dV (u)

Cn Cεkf kpp,ϕ.

B(u,r−1)c

The constants C above are independent of ε. Notice that P T P = T on Fϕp , so P S = ! ∞ P T −P MχFj T P MχGj is well defined on Fϕp and the estimate (2.7) is proved under the j=1

restriction that T ∈ WLϕp .



Lemma 2.9. Given 0 < p ≤ 1, there is some constant C such that for all bounded linear ∞ operator T on Fϕp and {Fj }∞ j=1 , {Gj }j=1 associated to r > 1 as in Lemma 2.7 and each

LOCALIZATION AND COMPACTNESS ON FOCK SPACES

9

positive integer m, we have lim sup kP Tm kFϕp →Fϕp ≤ C lim sup

(2.8)

m→∞

where Tm =

P

j>m

sup

m→∞ w∈∪j>m G+ j

kT kw kp,ϕ ,

MχFj T P MχGj .

Proof. First, we are going to show (2.9)



sup T P p f ∈Fϕ \{0}

χGj f kχG+j f kp,ϕ

In fact, given f ∈ Fϕp not identically zero, set gj = P

!



≤ C sup kT kw kp,ϕ . w∈G+ j

p,ϕ

χGj f kχG+ f kp,ϕ j

!

.

Then gj (z) =

Z

Gj

f (w)K(z, w)e−2ϕ(w) dV (w). kχG+j f kp,ϕ

It is trivial to see that gj ∈ Fϕp because of the compactness of Gj . Since T is bounded on Fϕp , then Z |f (w)||T Kw (z)|e−2ϕ(w) |T (gj )(z)| ≤ dV (w) kχG+ f kp,ϕ Gj j

Note that T Kw is conjugate holomorphic respecting to w. From Lemma 2.4 we have kT (gj )kpp,ϕ ≤

Z

Cn

Z

 Z 

Gj

 Z 

|f (w)| T Kw (z) e−2ϕ(w)

dV (w) e−pϕ(z) dV (z)

f kp,ϕ p p |f (w)| T Kw (z) e−2pϕ(w) kχ

G+ j

p



dV (w) e−pϕ(z) dV (z) kχG+j f kpp,ϕ p Z  Z p f (w)e−ϕ(w) −ϕ(z) T kw (z)e dV (z) dV (w) ≤C p + f kp,ϕ kχ n C G+ G j j p Z f (w)e−ϕ(w) dV (w) p ≤ C sup kT kw kp,ϕ kχG+ f kpp,ϕ G+ w∈G+ j ≤C

Cn

G+ j

j

j

= C sup

w∈G+ j

kT kw kpp,ϕ .

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Z. HU, X. LV, AND B. D. WICK

This gives (2.9). To prove (2.8), we have from Lemma 2.4 that ! p Z 2 (T k )(·) hf, k i w w Fϕ dV (w) (z) P χFj (·) kχG+j f kp,ϕ Gj Z p Z 2 (T kw )(u) hf, k i w F ϕ −2ϕ(u) K(z, u)e ≤ dV (w)dV (u) Fj kχG+ f kp,ϕ Gj j Z Z hf, kw iF 2 (T kw )(u) p ϕ p −2pϕ(u) ≤C |K(z, u)| e dV (w) dV (u) p Gj kχG+j f kp,ϕ Fj+ ! Z Z p p 2 | |(T kw )(u)| |hf, k i w F ϕ dV (w) dV (u). ≤C |K(z, u)|p e−2pϕ(u) p + + f k kχ p,ϕ Fj Gj G+ j Hence, integrating both sides and interchanging the order of integrations we obtain

  p

P MχFj T gj p,ϕ Z Z p Z |hf, kw iFϕ2 | p −2pϕ(u) ≤C |K(z, u)|p e−pϕ(z) dV (z)dV (u)dV (w) |(T kw )(u)| e p + kχ + f kp,ϕ + Gj Cn Fj Gj ! Z Z |hf, kw iFϕ2 |p |(T kw )(u)|p e−pϕ(u) dV (u) dV (w). ≤C p + + f kp,ϕ kχ F G+ Gj j j This gives

  p

P MχFj T gj ≤ C p,ϕ

sup w∈G+ j

kT kw kpp,ϕ

!Z

|hf, kw iFϕ2 |p kχG+ f kpp,ϕ

G+ j

dV (w) = C sup kT kw kpp,ϕ . w∈G+ j

j

Therefore, (2.6) yields

kP Tm f kpp,ϕ ≤

X

j>m

=

X

j>m

≤C

kP MχFj T P MχGj f kpp,ϕ   kP MχFj T gj kpp,ϕ kχG+j f kpp,ϕ

X

sup kT kw kpp,ϕ kχG+j f kpp,ϕ +

j>m w∈Gj

≤ CN

sup w∈∪j>m G+ j

From this, (2.8) follows.

kT kw kpp,ϕ

!

kf kpp,ϕ . 

In the case of 1 ≤ p < ∞, the projection P is bounded from Lpϕ to Fϕp , and so is P MχE when E ⊂ Cn is measurable. But P is not bounded on Lpϕ if 0 < p < 1. The following lemma, Lemma 2.10, says P MχE is still bounded on Fϕp . Lemma 2.10. Suppose 0 < p ≤ 1. There exists some constant C such that for any domain E in Cn we have kP MχE kFϕp →Fϕp ≤ C.

LOCALIZATION AND COMPACTNESS ON FOCK SPACES

11

Proof. Suppose E ∈ Cn is a Lemma 2.4 and Lemma 2.2, Z p kP MχE f kp,ϕ =

domain. For f ∈ Fϕp , we have |f (w)K(z, w)| = |f (w)K(w, z)|. estimate (3) gives Z p −pϕ(z) −2ϕ(w) f (w)K(z, w)e e dV (w) dV (z) Cn E  Z Z −2ϕ(w) p ≤C f (w)K(w, z)e dV (w) e−pϕ(z) dV (z) n + C E Z  Z p −2pϕ(w) −ϕ(z) p K(w, z)e dV (z) dV (w). =C |f (w)| e ≤

E+ Ckf kpp,ϕ .

Cn

 Lemma 2.11. Suppose 0 < p ≤ 1 and T ∈ K(Fϕp ). Then lim kP MχB(0,R) T − T kFϕp →Fϕp = 0.

R→∞

Proof. Notice that, P T = T on Fϕp . For f ∈ Fϕp with kf kp,ϕ ≤ 1, we get



  p  p



P M T − P T f = P M T − T f

χB(0,R) χB(0,R) p,ϕ p,ϕ p Z Z −2ϕ(w) = T f (w)K(z, w)e dV (w) e−pϕ(z) dV (z). Cn

Then by Lemma 2.4, Z

  p

P MχB(0,R) T − T f ≤ C p,ϕ

=

Z

Cn

|w|≥R

Z

|w|≥R−1

≤C

Z

 p −2ϕ(w) T f (w)K(w, z)e dV (w) e−pϕ(z) dV (z) |w|≥R−1  Z p −pϕ(z) T f (w)e−2ϕ(w) p |K(w, z)| e dV (z) dV (w) Cn

|w|≥R−1

T f (w)e−ϕ(w) p dV (w).

 Since T ∈ K(Fϕp ), T f : f ∈ Fϕp with kf kp,ϕ ≤ 1 ⊂ Fϕp is relatively compact. By [8, Lemma 3.2], for each ε > 0 there is some R > 0 such that Z T f (w)e−ϕ(w) p dV (w) < εp . sup f ∈Fϕp ,kf kp,ϕ ≤1

|w|>R−1

Therefore,

kP MχB(0,R) T − T k

Fϕp →Fϕp

=

sup

f ∈Fϕp ,kf kp,ϕ ≤1

where C is independent of ε.

 

P MχB(0,R) T − T f

Lemma 2.12. Suppose 0 < p ≤ 1. Then for T bounded on Fϕp we have kT ke,Fϕp ≃ lim sup kP MχB(0,R)c T kFϕp →Fϕp . R→∞

p,ϕ

< Cε, 

12

Z. HU, X. LV, AND B. D. WICK

Proof. For any R > 0, P MχB(0,R) is a Toeplitz operator induced by χB(0,R) , Lemma 2.9 from [8] tells us it is compact on Fϕp . Given T bounded on Fϕp , P MχB(0,R) T is compact. Thus, kT ke,Fϕp ≤ kT − P MχB(0,R) T kFϕp →Fϕp . This yields kT ke,Fϕp ≤ lim sup kP MχB(0,R)c T kFϕp →Fϕp . R→∞

On the other hand, for any A ∈ K(Fϕp ), Lemma 2.11 shows lim sup kP MχB(0,R)c AkFϕp →Fϕp = 0. R→∞

From Lemma 2.10, we know lim sup kP MχB(0,R)c T kFϕp →Fϕp = lim sup kP MχB(0,R)c (T − A)kFϕp →Fϕp R→∞

R→∞

≤ lim sup kP MχB(0,R)c kFϕp →Fϕp kT − AkFϕp →Fϕp R→∞

≤ CkT − AkFϕp →Fϕp . Hence, lim sup kP MχB(0,R)c T kFϕp →Fϕp ≤ CkT ke,Fϕp . R→∞

 Now we are in the position to characterize those compact operators in WLϕp with 0 < p ≤ 1, which extends the main results in [10, 11, 20] to the small exponential case. Theorem 2.13. Let 0 < p ≤ 1 and T ∈ WLϕp . The following statements are equivalent: (A) T ∈ K(Fϕp ); (B) lim sup hT kz , kw iFϕ2 = 0 for any r > 0; z→∞ w∈B(z,r) (C) lim sup hT kz , kw iFϕ2 = 0; z→∞ w∈Cn

(D) lim kT kz kp,ϕ = 0. z→∞

Proof. It is trivial that (C)⇒(B). We will show the implication (B)⇒(D) under the hypothesis T ∈ WLϕp . In fact, for any ε > 0, by (2.2) we have some r > 0 such that sup z∈Cn

Z

B(z,r)c

p hT kz , kw iFϕ2 dV (w) < ε.

LOCALIZATION AND COMPACTNESS ON FOCK SPACES

13

Combining the above inequality with (B), we get Z p p kT kz kp,ϕ = hT kz , kw iFϕ2 dV (w) n C  Z Z p + = hT kz , kw iFϕ2 dV (w) c B(z,r) B(z,r) p ≤ ε + A(B(z, r)) sup hT kz , kw iFϕ2 w∈B(z,r)

! p sup hT kz , kw iFϕ2

≤ ε + Cr 2n

w∈B(z,r)

< 2ε

whenever |z| is sufficiently large. Therefore, (B) implies (D). Suppose T satisfies (D). By Lemma 2.3 we know (2.10)  p1 Z −ϕ(u) p −ϕ(w) T kz (u)e dV (u) ≤ CkT kz kp,ϕ . ≤C hT kz , kw iFϕ2 = T kz (w)e B(w,1)

Then,

sup hT kz , kw iFϕ2 ≤ CkT kz kp,ϕ

w∈Cn

which gives the implication (D)⇒(C). To prove (D)⇒(A), given ε > 0 we pick some r > 10 with sets {Fj }j and {Gj }j as in Proposition 2.8 so that

! ∞

X

MχFj T P MχGj < ε.

T − P

p p j=1

Fϕ →Fϕ

For each positive integer m, set Tm =

P

j>m

compact on Fϕp , we get (2.11)

kT kpe,Fϕp



≤ T − P

m X j=1

MχFj T P MχGj . Since P

! p

MχFj T P MχGj

p

Fϕ →Fϕp

m P

j=1

MχFj T P MχGj

!

is

< εp + kP Tm kpFϕp →Fϕp .

Suppose T satisfies (D), then there exists t > 0 such that kT kz kp,ϕ < ε for |z| ≥ t. Notice c that, ∪j>m G+ j ⊂ B(0, t) whenever m is large enough. So, (2.8) in Lemma 2.9 and (2.11) imply kT ke,Fϕp = 0 which gives the compactness of T . To finish our proof, we only need to prove the implication (A)⇒(B). Given T ∈ K(Fϕp ), Lemma 2.11 tells us (2.12)

lim kP MχB(0,R) T − T kFϕp →Fϕp = 0.

R→∞

First, we claim that (2.13)

lim

sup hP MχB(0,R) T kz , kw iFϕ2 = 0.

z→∞ w∈B(z,r)

14

Z. HU, X. LV, AND B. D. WICK

In fact, Lemma 2.10 shows P MχB(0,R) T kz ∈ Fϕp ⊂ Fϕ2 , we obtain hP MχB(0,R) T kz , kw iFϕ2 = hMχB(0,R) T kz , kw iFϕ2 Z ≤ T kz (u)kw (u) e−2ϕ(u) dV (u) B(0,R) Z ≤ kT kz k∞,ϕ |kw (u)| e−ϕ(u) dV (u) B(0,R)

sup |kw (u)| e−ϕ(u)

≤ CkT kz kp,ϕ

u∈B(0,R)

≤ CkT kFϕp →Fϕp kkz kp,ϕ e−θ|w|

≤ Ce−θ|w| ,

where the constants C are independent of z and w. Hence, (2.13) is true. Using (3) in Lemma 2.2 and (2.12) to get that D  E  

T − P Mχ T k , k ≤ T − P M T kz kkw k1,ϕ

z w χB(0,R) B(0,R) Fϕ2 ∞,ϕ

 

≤ C T − P MχB(0,R) T kz p,ϕ

≤ CkT − P MχB(0,R) T kFϕp →Fϕp kkz kp,ϕ ≤ CkT − P MχB(0,R) T kFϕp →Fϕp → 0

as R → ∞. Combining the above with (2.13), we obtain D D E  E + sup T − P Mχ T kz , kw sup hT kz , kw iFϕ2 ≤ sup P MχB(0,R) T kz , kw B(0,R) F2 F2

w∈B(z,r)

w∈B(z,r)

ϕ

w∈B(z,r)

goes to 0 as z → ∞.

ϕ



If 1 < p < ∞, kz → 0 weakly on Fϕp , which implies lim kT (kz )kp,ϕ = 0 for T ∈ K(Fϕp ). z→∞

Theorem 1.2 in [11] tells us that the equivalence from (A) to (D) remains true for T ∈ WLϕp if 1 < p < ∞. For our later applications, we exhibit the following result. Theorem 2.14. Let 0 < p < ∞ and T ∈ WLϕp . The following statements are equivalent: (A) T ∈ K(Fϕp ); 2 (B) lim sup hT kz , kw iFϕ = 0 for any r > 0; z→∞ w∈B(z,r) (C) lim sup hT kz , kw iFϕ2 = 0; z→∞ w∈Cn

(D) lim kT kz kp,ϕ = 0. z→∞

From Theorem 2.13, it is natural to ask whether the essential norm of T ∈ WLϕp can be dominated by its behavior on normalized reproducing kernel kz ? This problem has attracted much interest, see [10, Section 6] for example. Our Theorem 2.15 says the answer is affirmative when 0 < p ≤ 1. Theorem 2.15. Suppose 0 < p ≤ 1. Then for T ∈ WLϕp we have (2.14)

kT ke,Fϕp ≃ lim sup kT kz kp,ϕ . z→∞

LOCALIZATION AND COMPACTNESS ON FOCK SPACES

15

Proof. Suppose T ∈ WLϕp . From Lemma 2.5 we know T is bounded on Fϕp which implies lim sup kT kz kp,ϕ < ∞. By Theorem 2.13, kT ke,Fϕp = 0 if lim sup kT kz kp,ϕ = 0. So, we may z→∞

z→∞

assume lim sup kT kz kp,ϕ = ε1 > 0. From Proposition 2.8, we have two sequences of sets z→∞

{Fj }j and {Gj }j so that



T − P

∞ X j=1

!

MχFj T P MχGj

< ε1 .

Fϕp →Fϕp

Then, for m = 1, 2, · · · , from (2.8) and (2.11) we have kT ke,Fϕp ≤ ε1 + kP Tm kFϕp →Fϕp ≤ ε1 + C

sup z∈∪j>m G+ j

kT kz kp,ϕ .

Since 0 < p ≤ 1, Lemma 2.9 tells us that the constants C above do not depend on the precise choice of {Fj }j and {Gj }j , and hence do not depend on T . Let m → ∞, we have the desired estimate kT kpe,Fϕp ≤ εp1 + C lim sup kT kz kpp,ϕ = C lim sup kT kz kpp,ϕ . z→∞

z→∞

On the other hand, fixed R > 0, notice that P MχB(0,R) is a Toeplitz operator induced by χB(0,R) , which is a bounded function. So P MχB(0,R) ∈ WLϕp and P MχB(0,R) T ∈ WLϕp , because WLϕp is a algebra. Since P MχB(0,R) is compact and T is bounded on Fϕp (see Lemma 2.5), we get that P MχB(0,R) T is compact on Fϕp . Theorem 2.13 tells us



(2.15) lim P MχB(0,R) T kz = 0. z→∞

p,ϕ

Therefore, Lemma 2.12, (2.15) and the fact that P T = T yield

kT kpe,Fϕp ≃ lim sup kP MχB(0,R)c T kpFϕp →Fϕp R→∞

p

≥ lim sup lim sup P MχB(0,R)c T kz p,ϕ z→∞ R→∞ 

p 

p ≥ lim sup lim sup kT kz kp,ϕ − P MχB(0,R) T kz R→∞

≥

z→∞

p,ϕ

lim sup kT kz kpp,ϕ . z→∞

 3. Toeplitz Operators with BMO Symbols In this section, we are going to discuss the characterizations on Toeplitz operators with BMO symbols. First, we will characterize the boundedness (and the compactness) of Toeplitz operators Tf on Fϕp with BMO symbols f . Furthermore, we will characterize those compact operators on Fϕp which are in the algebra generated by bounded Toeplitz operators with BMO symbols. For this purpose, we need some more auxiliary function spaces. Fixed r > 0, recall that B(·, r) = {w ∈ Cn : |w − ·| < r}. Given a locally Lebesgue integrable function f on Cn (written as f ∈ L1loc (Cn )), write ωr (f )(·) = sup {|f (w) − f (·)| : w ∈ B(·, r)}

16

and

Z. HU, X. LV, AND B. D. WICK

1 MOr (f )(·) = V (B(·, r))

where fbr (·) =

Z

1 V (B(·, r))

B(·,r)

Z

f − fbr (·) dV f dV.

B(·,r)

For f on Cn with f (·)|kz (·)|2 ∈ L1ϕ for all z ∈ Cn , the Berezin transform of f is defined as Z e f (z) = f (w) |kz (w)|2 e−2ϕ(w) dV (w). Cn

Let BOr be the collection of all continuous functions f on Cn such that ωr (f ) is bounded. c| and We use BAr and BMOr to denote respectively the set of all f ∈ L1loc (Cn ) such that |f r MOr (f ) are bounded on Cn . The space BMO is the family of all measurable function f on Cn satisfying f (·)|kz (·)|2 ∈ L1ϕ for z ∈ Cn and Z kf kBMO = sup f (w) − fe(z) |kz (w)|2e−2ϕ(w) dV (w) < ∞. z∈Cn

Cn

By Lemma 3.33 in [21], we obtain that the spaces BOr and BAr are independent of r, they will be denoted as BO and BA below. The next lemma says BMOr is independent of r as well. Lemma 3.1. Suppose f ∈ L1loc (Cn ). The following three statements are equivalent: (A) f ∈ BMOr for some (or any) r > 0; (B) f ∈ BMO; (C) f = f1 + f2 , where f1 ∈ BA and f2 ∈ BO.

Proof. For n = 1 and ϕ(z) = α2 |z|2 , this is Theorem 3.34 from [21]. For general n and ϕ satisfying ddc ϕ ≃ ω0 , the proof can be carried out as that of [21] with a little modification. The details will be omitted here.  For f ∈ BMO, say f = f1 + f2 with f1 ∈ BA and f2 ∈ BO, similar to [7, Lemma 4.1], we know the Toeplitz operator Tf1 is well defined on Fϕp . From [21], |f2 (z)| ≤ a|z| + b with constants a, b > 0, Tf2 is also well defined on Fϕp . Thus, Tf is well defined on Fϕp , where 0 < p < ∞. Moreover, Z (3.1) hTf kz , kw i = kz (u)kw (u)f (u)e−2ϕ(u) dV (u). Cn

Coburn, Isralowitz and Li [5] proved that Tf (f ∈ BMO) is compact on the classical 2 Fock space F1/2 if and only if the Berezin transform fe vanished at the infinity. The first p two authors extended this result to the setting of F1/2 with 0 < p < ∞ in [8]. Under the assumption that S is a linear combination of operators of form Tf1 · · · Tfm with each function fj | bounded, Isralowitz proved that S is compact on F 2 if and only if Se fj satisfying |f 1/2 vanishes at the infinity, see [9] for details. In all these references, the Weyl unitary operators acting on Fα2 by Wz f (·) = kz f (· − z) and the involutive unitary operators Uz f (·) = kz f (z − ·) play as a very crucial role. Unfortunately, there are not these kinds of unitary operators on our generalized Fock space Fϕp .

LOCALIZATION AND COMPACTNESS ON FOCK SPACES

17

We will use Bpϕ to denote the collection of all linear combination of the form Tf1 Tf2 · · · Tfm , where each function fj ∈ BMO and fej is bounded on Cn .

Theorem 3.2. Let 0 < p < ∞.

(A) If f ∈ BMO, then Tf is bounded on Fϕp if and only if fe is bounded on Cn ; Tf is compact on Fϕp if and only if (3.2) lim sup hTf kz , kw iFϕ2 = 0 ∀r > 0. z→∞ w∈B(z,r)

(B) If S ∈ Bpϕ , then S is compact on Fϕp if and only if (3.3) lim sup hSkz , kw iFϕ2 = 0 z→∞ w∈B(z,r)

∀r > 0.

Proof. We claim Tf ∈ WLϕp if f ∈ BMO and fe remains bounded. In fact, similar to [5, Lemma 1] it is trivial to verify f|(z) ≤ kf kBMO + sup |fe(z)| < ∞. sup |f

(3.4)

z∈Cn

z∈Cn

By [8, Theorem 3.5], |f |dV is a Fock-Carleson measure. Hence, Z |kz (u)kw (u)| e−2ϕ(u) |f (u)|dV (u) hTf kz , kw iFϕ2 ≤ n C Z c |kz (u)kw (u)| e−2ϕ(u) dV (u) ≤ C sup |f |r (u) n n u∈C ZC f|(u) e−θ|z−u|−θ|w−u|dV (u) ≤ C sup |f u∈Cn

− 2θ |z−w|

≤ Ce

Cn

.

This implies Tf ∈ WLϕp for any p ∈ (0, ∞) (and also, Tf is strongly localized in the sense of Xia and Zheng, see [20]). (A). Suppose f ∈ BMO. If fe is bounded on Cn , then Tf ∈ WLϕp which implies Tf is bounded on Fϕp for any p ∈ (0, ∞). Conversely, the condition that Tf is bounded implies fe is bounded, which can be proved in a standard way with fe(z) = hTf kz , kz iFϕ2 . Now we deal with the compactness of Tf . If (3.2) holds, then fe(z) = hTf kz , kz iF 2 is ϕ

bounded, hence Tf ∈ WLϕp . Therefore, by (3.2) and Theorem 2.13, Tf is compact on Fϕp for all 0 < p < ∞. Conversely, if Tf is compact on Fϕp for some 0 < p < ∞. If 1 < p < ∞, we have lim kTf kz kp,ϕ = 0 because kz tends to zero weakly, from which (3.2) follows for any z→∞

r > 0. If 0 < p ≤ 1, we know fe to be bounded. Then, Tf ∈ WLϕp . Now the estimate (3.2) comes from Theorem 2.13. (B) Since each Tfj ∈ WLϕp , we have Bpϕ ⊂ WLϕp for 0 < p < ∞. Now the conclusion follows from Theorem 2.13.  As shown by Isralowitz in [10, Proposition 1.5], on the classical Fock space Fαp the estimate e = 0. Therefore, Theorem 3.2 extends [5, 8]. (3.3) is equivalent to lim S(z) z→∞

18

Z. HU, X. LV, AND B. D. WICK

f| bounded. As in [9], set BT to be the collection of all measurable functions f on Cn with |f ϕ As shown in the proof of Theorem 4.2, Tf ∈ WLp if f ∈ BT. We have Corollary 3.3 at once.

Corollary 3.3. Let 0 < p < ∞, and let S be in the family all linear combination of the form Tf1 Tf2 · · · Tfm , where each function fj ∈BT. Then S is compact on Fϕp if and only if one of the following three statements holds: (A) lim sup |hT kz , kw i| = 0 for any r > 0; z→∞ w∈B(z,r)

(B) lim sup |hT kz , kw i| = 0; z→∞ w∈Cn

(C) lim kT kz kp,ϕ = 0. z→∞

While ϕ(z) = 41 |z|2 , p = 2 and S is a linear combination of operators of the form Tf1 Tf2 · · · Tfm with each fj ∈ BT, Corollary 3.3 gives the main result of [9]. 4. Operators Satisfying Axler and Zheng’s Condition In this section, we will restrict ourselves to the classical Fock space Fαp , that is ϕ(z) = α2 |z|2 with α > 0. We are going to characterize the boundedness and compactness of linear operators with the Axler-Zheng condition on Fαp . Let φz be the holomorphic self-map of Cn , φz (·) = z − ·. Uz is the operator on Fαp defined by Uz f = (f ◦ φz )kz . Given some linear operator S on Fαp , define Sz = Uz SUz∗ . In the context of Bergman space A2 (D) on the unit disc D, with φz (w) = (f ◦ φz )φ′ (z), Axler-Zheng introduced the condition

w−z 1−zw

and Uz f =

sup kSz 1kAp < ∞ with some p > 2 z∈D

in [1]. The work in [5, 6, 13, 15, 22] also explored the condition kSz 1kAp ≤ C. In the Fock space setting, Wang, Cao, and Zhu carried out related research in [19] to obtain that, if there exist some p > 2 such that sup kSz 1kp, 2α < ∞

z∈Cn

p

(or kSz 1kp, 2α → 0 as z → ∞), p

the operator S is bounded (or compact) on Fα2 . Theorem 4.1. Suppose S is a linear operator defined on D. If there are some 0 < σ < p < ∞ such that Z ασ 2 |Sz 1(u)|pe− 2 |u| dV (u) < ∞, (4.1) M = sup z∈Cn

then

Cn

α(p−σ) hSkz , kw iF 2 ≤ CM p1 e− 2p |z−w|2 , α

so S is bounded on Fαs for all 0 < s < ∞. Furthermore, if both (4.1) and Z ασ 2 |Sz 1(u)|pe− 2 |u| dV (u) = 0 (4.2) lim z→∞

Cn

hold, then S is compact on Fαs for 0 < s < ∞.

LOCALIZATION AND COMPACTNESS ON FOCK SPACES

19

Proof. Since K(·, ·) = eαh·,·i , it is easy to verify kz (z − u)kz (u) = 1 and K(z − u, z − u) = K(z, z)K(u, u)|K(u, z)|−2. By the equality Sz 1(u) = kz (u)(Skz )(z − u) (see [21]) and Lemma 2.3 we have Z Z ασ 2 2 |u| p − ασ dV (u) = |kz (u)(Skz )(z − u)|p e− 2 |u| dV (u) |Sz 1(u)| e 2 n Cn ZC ασ 2 = |kz (z − u)(Skz )(u)|pe− 2 |u−z| dV (u) n ZC ασ 2 = |(Skz )(u)|p |kz (u)|−p e− 2 |u−z| dV (u) n ZC ασ 2 ≥ |(Skz )(u)|p |kz (u)|−(p−σ) e− 2 |u| dV (u) B(w,1)

≥ C|(Skz )(w)|p|kz (w)|−(p−σ) e− = C|hSkz , kw iFα2 |p e

α(p−σ) |z−w|2 2

ασ |w|2 2

.

From the above inequalities and (4.1), we get 1

|hSkz , kw iFα2 | ≤ CM p e−

α(p−σ) |z−w|2 2p

.

Since p − σ > 0, S is weakly localized for Fαs , so S is bounded on Fαs for all 0 < s < ∞. Furthermore, if both (4.1) and (4.2) are valid, from the proof above we have S ∈ WLαs . And also, for p ∈ (0, ∞) there is some constant Cr such that Z  p1 ασ 2 p − |u| |Sz 1(u)| e 2 sup hSkz , kz iFα2 ≤ Cr →0 dV (u) w∈B(z,r)

Cn

as z → ∞. By Theorem 2.13 S is compact on Fαs for all 0 < s < ∞. Remark. [19].



If p > σ = 2 and s = 2, then Theorem 4.1 reduces to Theorems A and B in

5. Further Remarks An important theme in analysis on function spaces is to characterize when a given operator is compact. In the setting of the Bergman space Ap (Bn ) on the unit ball Bn , 1 < p < ∞, in 2007 Su´arez proved, see [17], that a bounded operator S is compact if and only if S is in the Toeplitz algebra and the Berezin transform of S vanishes on the boundary. Later on, Mitkovski, Su´arez and the third author [14] extended [17] to the weighted Bergman space Apα (Bn ). On the classical Fock space Fαp for 1 < p < ∞, in [2] Bauer and Isralowitz showed that Su´arez’s characterization on compact operators is valid. For general ϕ with ddc ϕ ≃ ω0 and 1 < p < ∞, most recently in [10] Isralowitz obtained K(Fϕp ) = Tϕp (Cc∞ (Cn )), which implies the results in [2]. For Toeplitz operators Tµ with positive Borel measures µ as symbols, the boundedness (or compactness) on Fϕp with 0 < p ≤ 1 can be characterized with the same condition as that on Fϕq with q > 1. Unfortunately, some differences appear when we talk about the structure of K(Fϕp ). For example, we find K(Fϕp )\Tϕp 6= ∅ if 0 < p ≤ 1. To see this, from [21, Lemma

20

Z. HU, X. LV, AND B. D. WICK

4.39] (or [12]) we take a separated sequence {zj }∞ j=1 which is an interpolating sequence for ∞ ∞ Fα . Hence, we have some f ∈ Fα such that (5.1)

α

2

f (zk )e− 2 |zk | = 1,

∀k ∈ N.

Although [21] is only concerned with one variable interpolation, take {zj } ⊂ C and f ∈ H(C) satisfying the interpolation above, extend f to Cn with the equation f (z, z ′ ) = f (z) for (z, z ′ ) ∈ C × Cn−1 , we will have f satisfying (5.1) in Cn . Furthermore, for 0 < p ≤ 1 take g ∈ Fαp so that g(0) 6= 0. Define the operator T on Fαp as (5.2)

T (·) = h·, f iFα2 g.

T is bounded and of rank 1, so T is compact on Fαp . Also, hT kz , kw iF 2 = f (zk )e− α2 |zk |2 g(w)e− α2 |w|2 . k α

Because {zj }∞ j=1 is separated, we have lim zj = ∞. For each r > 0, as k is large enough we j→∞

have from Lemma 2.3 that Z Z p |w|2 −α hT kz , kw iF 2 p dV (w) = 2 dV (w) g(w)e k α B(zk ,r)c B(zk ,r)c Z α 2 p ≥ g(w)e− 2 |w| dV (w) B(0,1)

≥ C |g(0)|p .

T ∈ / WLαp by Definition 2.1. Hence, K(Fαp )\Tαp 6= ∅ for 0 < p ≤ 1. This tells us the characterization of compact operators T on Fϕp with 0 < p ≤ 1 is quite different from that with 1 < p < ∞. For 0 < p ≤ 1, {kz : z ∈ Cn } does not converge weakly to zero in Fαp as z goes to ∞. In fact, take f ∈ Fα∞ satisfying (5.1), since the dual space of Fαp is Fα∞ under the pairing hg, f iFα2 (see [21]), we know that Ff = h·, f iFα2 is a bounded linear functional on Fαp . However, Ff (kzk ) = hkzk , f iFα2 = 1 for all k. The operator T defined as (5.2) also shows lim kT kz kp,α 6= 0, because T kzj = g for z→∞ j = 1, 2, · · · , which says T kz need not converge to 0 in Fαp even if T is compact while 0 < p ≤ 1. So, the hypothesis T ∈ WLϕp both in Theorem 2.13 and 2.14 can not be removed. But for the Berezin transform, we have the following Proposition 5.1. Proposition 5.1. Suppose 0 < p ≤ 1 and T ∈ K(Fϕp ). Then Te(z) → 0 as z → ∞.

LOCALIZATION AND COMPACTNESS ON FOCK SPACES

Proof. For R > 0 fixed, P MχB(0,R) T kz ∈ Fϕp ⊂ Fϕ2 . Lemma estimate (3) give Z hP MχB(0,R) T kz , kz iFϕ2 = hMχB(0,R) T kz , kz iFϕ2 ≤

21

2.2 estimate (1) and Lemma 2.2

T kz (u)kz (u) e−2ϕ(u) dV (u) B(0,R) Z ≤ kT kz k∞,ϕ |kz (u)| e−ϕ(u) dV (u) B(0,R)

≤ CkT kz kp,ϕ sup |kz (u)| e−ϕ(u) |u|≤R

≤ kT kFϕp →Fϕp kkz kp,ϕ e−θ|z|

≤ Ce−θ|z| → 0

as z → ∞. Since T ∈ K(Fϕp ), Lemma 2.11 tells us D  E  

T − P Mχ kkz k1,ϕ T kz , kz ≤ T − P MχB(0,R) T kz B(0,R) 2 ∞,ϕ Fϕ

 

≤ C T − P MχB(0,R) T kz p,ϕ

≤ CkT − P MχB(0,R) T kFϕp →Fϕp kkz kp,ϕ ,

≤ CkT − P MχB(0,R) T kFϕp →Fϕp → 0

as R → ∞. Therefore, taking z → ∞, D E e |T (z)| = hT kz , kz iFϕ2 ≤ P MχB(0,R) T kz , kz

D  E + T − P Mχ → 0. T kz , kz B(0,R) F2 F2 ϕ

ϕ



Summarizing the discussion above we put forward the following problem.

Problem 5.2. For 0 < p ≤ 1, what are the necessary and sufficient conditions to characterize the membership in K(Fϕp )? Under the restriction 0 < p ≤ 1, we dominate the essential norm of T ∈ WLϕp by its behavior on kz , see Theorem 2.14. Our second problem is whether the estimate (2.14) still holds for 1 < p < ∞?. Problem 5.3. Suppose 1 < p < ∞. Does kTf ke,Fϕp ≃ lim sup kTf kz kp,ϕ z→∞

n

hold for bounded f on C ? In the previous section, with f ∈ BMO we have obtained the compactness of Toeplitz operators Tf on Fϕp . However, to consider the compactness of finite product Tf1 Tf2 · · · Tfm of Toeplitz operators with BMO symbols we have assumed each symbol fj has a bounded Berezin transform. Is this hypothesis necessary in the statement (B) of Theorem 3.2? Problem 5.4. Suppose 0 < p < ∞, and T is in the set of all linear combination of the form Tf1 Tf2 · · · Tfm , where each function fj ∈ BMO. Can we conclude that T is compact on Fϕp if and only if lim sup hT kz , kw iFϕ2 = 0 z→∞ w∈B(z,r)

22

Z. HU, X. LV, AND B. D. WICK

holds for each r > 0? We also point to the general question of how the story is similar, or different, in the case of the Bergman space Ap (D) when 0 < p < 1. Acknowledgments. good suggestions.

The authors would like to thank the referees for making some References

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[18] Chunjie Wang and Jie Xiao, Addendum to ”Gaussian integral means of entire functions”, Complex Anal. Oper. Theory 10 (2016), no. 3, 495–503. ↑2 [19] Xiaofeng Wang, Guangfu Cao, and Kehe Zhu, Boundedness and compactness of operators on the Fock space, Integral Equations Operator Theory 77 (2013), no. 3, 355–370. ↑2, 18, 19 [20] Jingbo Xia and Dechao Zheng, Localization and Berezin transform on the Fock space, J. Funct. Anal. 264 (2013), no. 1, 97–117. ↑2, 3, 12, 17 [21] Kehe Zhu, Analysis on Fock spaces, Graduate Texts in Mathematics, vol. 263, Springer, New York, 2012. ↑2, 4, 16, 19, 20 [22] Nina Zorboska, Toeplitz operators with BMO symbols and the Berezin transform, Int. J. Math. Math. Sci. 46 (2003), 2929–2945. ↑2, 18 Z. Hu, Department of Mathematics, Huzhou University, Huzhou, Zhejiang, 313000, China E-mail address: [email protected] X. Lv, Department of Mathematics, Huzhou University, Huzhou, Zhejiang, 313000, China E-mail address: [email protected] B. D. Wick, Department of Mathematics, Washington University - St. Louis, One Brookings Drive, St. Louis, MO USA 63110. E-mail address: [email protected]