Toeplitz Operators with Positive Operator-Valued Symbols on Vector-Valued Generalized Fock Spaces
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Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences, 2020
http://actams.wipm.ac.cn
TOEPLITZ OPERATORS WITH POSITIVE OPERATOR-VALUED SYMBOLS ON VECTOR-VALUED GENERALIZED FOCK SPACES∗
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Jianjun CHEN (
School of Mathematics and Statistics, Zhaoqing University, Zhaoqing 526061, China E-mail : [email protected]
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Xiaofeng WANG (
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g 0 and z ∈ Cn . One more word about the average function G us that it is meaningful if kGkB(L(H)) exists or G ∈ T (L(H)) by Lemma 4 in [4].
In 2010, J.Isralowitz and K.Zhu [5] had taken the lead in characterizing the Toeplitz operators with positive measures symbols on the scalar Fock space. They had studied the boundedness, compactness, and Schatten-p class (0 < p ≤ ∞) of the Toeplitz operators. In the end of Introduction Part in [5], the authors had made a comment on that the results would be extended to higher dimension as long as one can handle the related lattices. Afterwards, Z.Hu, X.Lv [6] and T.Mengestie [7] had accomplished the goal, and moreover, they had researched the boundedness and compactness of Toeplitz operators on one Fock space to another in terms of the average function and t-Berezin transform. Furthermore, Z.Hu and X.Lv [8] had extended their results on the generalized Fock space. However, the rest about Schatten-p class was supplied by L.Xiao [9]. At last, X.Wang and R.Cho had used the same ideas to obtain the similar results on the Fock-Sobolev spaces [10] and Fock-Sobolev type spaces [11], respectively. The well-informed reader will notice that nonnegative measure symbols of Toeplitz operators are considered by all described literatures above. Particularly in the scalar setting, if given a nonnegative measurable function f , we know that there exists a nonnegative measure µ such that dµ = f dv. Unfortunately, an important fact about the Bochner integral is that the Radon-Nikodym theorem fails to hold in general. See [3] for more details. In other words, we can not find a measure µ such that dµ = Gdv for an operator-valued function G generally. Therefore, our results would not be extended to the case of general measure symbols except the Radon-Nikodym property holds. The organization of this article is as follows. In Section 2, we will consider the definitions and basic of the Fock-Carleson condition for vector-valued generalized Fock spaces Fϕ2 (H), and then we also give some geometric equivalent conditions, from which we know that the results in [2, 5] remain valid in the vector-valued case. Section 3 will be dedicated to characterizing those positive operator-valued symbols G for which the induced Toeplitz operators TG are bounded or compact on Fϕ2 (H). In the sequence, we will provide the similar characterizations of the Schatten class membership of these Toeplitz operators. Remember that unlike the classical Fock space setting for which one could use the explicit formulas for the reproducing kernel, we instead have to rely on some known estimates on the behavior of the reproducing kernel (see [2, 8] for more details). The proofs of
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