Abstract:
We prove Carleson embeddings for Bergman spaces of tube domains over symmetric cones, we apply them to characterize symbols of bounded Ces\`aro-type operators from weighted Bergman spaces to weighted Besov spaces. We also obtain Schatten class criteria of Toeplitz operators and Ces\`aro-type operators on weighted Hilbert-Bergman spaces.

Abstract:
In 1993, Peloso introduced a kind of operators on the Bergman space of the unit ball that generalizes the classical Hankel operator. In this paper, we estimate the essential norm of the generalized Hankel operators on the Bergman space of the unit ball and give an equivalent form of the essential norm. 1. Introduction Let be the open unit ball in , the Lebesgue measure on normalized so that , denotes the class of all holomorphic functions on . The Bergman space is the Banach space of all holomorphic functions on such that . It is easy to show that is a closed subspace of . There is an orthogonal projection of onto , denoted by and where is the Bergman kernel on . For a function , define the Hankel operator with symbol by where is the identity operator. Since the Hankel operator is connected with the Toeplitz operator, the commutator, the Bloch space, and the Besov space, it has been extensively studied. Important papers in this context are [1, 2] for the case and [3–5] for the case . It is known that is bounded on if and only if and is compact if and only if , where is the radial derivative of defined by is called the Bloch space, and is called the little Bloch space. For , ？？( is the open unit disc), is in the Schatten class if and only if ; if and only if is a constant, where and is the invariant volume measure on , is called the Besov space on . This theorem expresses that there is a cutoff of at . For , , if and only if , if and only if is a constant, where and is the invariant volume measure on . is called the Besov space on . Then, the cutoff phenomenon of appears at . If denotes the value of “cutoff,” then Obviously, depends on the dimension of the unit ball. In 1993, Peloso [3] replaced with to define a kind of generalized Hankel operator: Here, . Clearly, if , and are just the classical Hankel operator . He proved that has the same boundedness and compactness properties as , but the Schatten class property of is different from that of . If , , if and only if ; if , if and only if is a polynomial of degree at most . So the value of “cutoff” of is ; this means that the cutoff constant depends not only on the dimension but also on the degree of the polynomial and we are able to lower the cutoff constant by increasing . The cutoff phenomenon expressed that the generalized Hankel operator defined by Peloso and the classical Hankel operator are different. In the present paper, we will consider the generalized Hankel operators defined by Peloso on the Bergman space which is the Banach space of all holomorphic functions on such that , for . For , is a

Abstract:
In this paper, we show that on the weighted Bergman space of the unit disk the essential norm of a noncompact Hankel operator equals its distance to the set of compact Hankel operators and is realized by infinitely many compact Hankel operators, which is analogous to the theorem of Axler, Berg, Jewell and Shields on the Hardy space in Axler et al. (1979); moreover, the distance is realized by infinitely many compact Hankel operators with symbols continuous on the closure of the unit disk and vanishing on the unit circle.

Abstract:
Hankel operators with anti-holomorphic symbols are studied for a large class of weighted Fock spaces on $\cn$. The weights defining these Hilbert spaces are radial and subject to a mild smoothness condition. In addition, it is assumed that the weights decay at least as fast as the classical Gaussian weight. The main result of the paper says that a Hankel operator on such a Fock space is bounded if and only if the symbol belongs to a certain BMOA space, defined via the Berezin transform. The latter space coincides with a corresponding Bloch space which is defined by means of the Bergman metric. This characterization of boundedness relies on certain precise estimates for the Bergman kernel and the Bergman metric. Characterizations of compact Hankel operators and Schatten class Hankel operators are also given. In the latter case, results on Carleson measures and Toeplitz operators along with H\"{o}rmander's $L^2$ estimates for the $\bar{\partial}$ operator are key ingredients in the proof.

Abstract:
We obtain asymptotics for Toeplitz, Hankel, and Toeplitz+Hankel determinants whose symbols possess Fisher-Hartwig singularities. Details of the proofs will be presented in another publication.

Abstract:
We characterize the boundedness and compactness of a Toeplitz-type operator on weighted Bergman spaces satisfying the Bekollé-Bonami condition in terms of the Berezin transform.

Abstract:
We give various equivalent formulations to the (partially) open problem about $L^p$-boundedness of Bergman projections in tubes over cones. Namely, we show that such boundedness is equivalent to the duality identity between Bergman spaces, $A^{p'}=(A^p)^*$, and also to a Hardy type inequality related to the wave operator. We introduce analytic Besov spaces in tubes over cones, for which such Hardy inequalities play an important role. For $p\geq 2$ we identify as a Besov space the range of the Bergman projection acting on $L^p$, and also the dual of $A^{p'}$. For the Bloch space $\SB^\infty$ we give in addition new necessary conditions on the number of derivatives required in its definition.

Abstract:
In this paper, we study the product of a Hankel operator and a Toeplitz operator on the Hardy space. We give necessary and sufficient conditions of when such a product $H_f T_g$ is compact.

Abstract:
We study the asymptotics in n for n-dimensional Toeplitz determinants whose symbols possess Fisher-Hartwig singularities on a smooth background. We prove the general non-degenerate asymptotic behavior as conjectured by Basor and Tracy. We also obtain asymptotics of Hankel determinants on a finite interval as well as determinants of Toeplitz+Hankel type. Our analysis is based on a study of the related system of orthogonal polynomials on the unit circle using the Riemann-Hilbert approach.