Berezin symbol and invertibility of operators on the functional Hilbert spaces

We give in terms of reproducing kernel and Berezin symbol the sufficient conditions ensuring the invertibility of some linear bounded operators on some functional Hilbert spaces.     

Some results related with statistical convergence and Berezin symbols

The concept of discrete statistical Abel convergence is introduced. In terms of Berezin symbols we present necessary and sufficient condition under which a series with bounded sequence {an}_n?0 of complex numbers is discrete statistically Abel convergent. By using concept of statistical convergence we also give slight strengthening of a result of Gokhberg and Krein on compact operators.     

On the Riccati equations

In this article we give, in terms of so-called Berezin symbols, some necessary conditions for the solvability of the Riccati equation on the set of all Toeplitz operators on the Hardy space . Author’s address: Department of Technical Programs, Isparta (MYO) Vocational School, Suleyman Demirel University, 32260 Isparta, Turkey     

The Completeness of the Bergman Distance of Planar Domains has a Local Character

It is shown that the property of completeness of the Bergman metric of a planar domain whose complement is not a polar set, is a local one.     

The essential norm of a composition operator on the bergman space of the unit ball

In this paper, we express the essential norm of a composition operator on Bergman spaces of the unit ball in terms of an asymptotic upper bound of a quantity involving the pull-back measure induced by the inducing map. The result could be generalized to convex domains of finite type and to the weighted situation.     

Orthogonality in Classes

 Let denote the von Neumann–Schatten class, its norm and let be an elementary operator defined by . We shall characterize those operators which are orthogonal to the range of in the sense that for all . The main results of this paper are: If and (i) if A, C, respectively B, D are commuting normal operators with , or (ii) if A, B are contractions and , then is orthogonal to the range of if and only of S is in the kernel of . Furthermore, in both cases, the algebraic direct sum satisfies . (Received 9 February 2000; in revised form 21 February, 2001)     

Berezin transform on¶compact Hermitian symmetric spaces

Let X=G ^* be a compact Hermitian symmetric space. We study the Berezin transform on L ²(X) and calculate its spectrum under the decomposition of L ²(X) into the irreducible representations of G ^*. As applications we find the expansion of powers of the canonical polynomial (Bergman reproducing kernel for the canonical line bundle) in terms of the spherical polynomials on X, and we find the irreducible decomposition of tensor products of Bergman spaces on X. Received: 10 September 1996 / Revised version: 10 September 1997     

Products of Toeplitz Operators on the Polydisk

This paper studies products of Toeplitz operators on the Hardy space of the polydisk. We show that T _f T _g = 0 if and only if T _f T _g is a finite rank if and only if T _f or T _g is zero. The product T _f T _g is still a Toeplitz operator if and only if there is a h $ \in $ L $ \infty $ (T ⁿ ) such that T _f T _g - T _h is a finite rank operator. We also show that there are no compact simi-commutators with symbols pluriharmonic on the polydisk. Submitted: October 5, 2000     

Self-similar solutions and large time asymptotics for the dissipative quasi-geostrophic equation

We analyze the well-posedness of the initial value problem for the dissipative quasi-geostrophic equations in the subcritical case. Mild solutions are obtained in several spaces with the right homogeneity to allow the existence of self-similar solutions. While the only small self-similar solution in the strong space is the null solution, infinitely many self-similar solutions do exist in weak- spaces and in a recently introduced [7] space of tempered distributions. The asymptotic stability of solutions is obtained in both spaces, and as a consequence, a criterion of self-similarity persistence at large times is obtained.     

单位球上Bergman空间上的紧算子

当S满足一些可积条件时,我们证明了Lap(B)(p>1)上的有界算子S是紧的当且仅当S的Berezin变换在单位球B的边界趋近于零.     

A Note on Wiener-Hopf Determinants and the Borodin-Okounkov Identity

The continuous analogue of a Toeplitz determinant identity for Wiener-Hopf operators is proved. An example which arises from random matrix theory is studied and an error term for the asymptotics of the determinant is computed. Submitted: October 15, 2001? Revised: January 3, 2002.     

Essential norm of the difference of weighted composition operators

We study differences of weighted composition operators between weighted Banach spaces H _ν ^∞ of analytic functions with weighted sup-norms and give an expression for the essential norm of these differences. We apply our result to estimate the essential norm of differences of composition operators acting on Bloch-type spaces. Authors’ addresses: Mikael Lindstr?m, Department of Mathematics, Abo Akademi University, FIN 20500 Abo, Finland; Elke Wolf, Mathematical Institute, University of Paderborn, D-33095 Paderborn, Germany     

Bergman Space Structure,Commutative Algebras of Toeplitz Operators,and Hyperbolic Geometry

We exhibit a surprising but natural connection among the Bergman space structure, commutative algebras of Toeplitz operators and pencils of hyperbolic straight lines. The commutative C*-algebras of Toeplitz operators on the unit disk can be classified as follows. Each pencil of hyperbolic straight lines determines the set of symbols consisting of functions which are constant on corresponding cycles, the orthogonal trajectories to lines forming a pencil. It turns out that the C*-algebra generated by Toeplitz operators with this class of symbols is commutative. Submitted: January 15, 2001?Revised: February 7, 2002     

Berezin transform and Toeplitz operators on harmonic Bergman spaces

For an operator which is a finite sum of products of finitely many Toeplitz operators on the harmonic Bergman space over the half-space, we study the problem: Does the boundary vanishing property of the Berezin transform imply compactness? This is motivated by the Axler-Zheng theorem for analytic Bergman spaces, but the answer would not be yes in general, because the Berezin transform annihilates the commutator of any pair of Toeplitz operators. Nevertheless we show that the answer is yes for certain subclasses of operators. In order to do so, we first find a sufficient condition on general operators and use it to reduce the problem to whether the Berezin transform is one-to-one on related subclasses.     

Some interior regularity results for solutions of Hessian equations

We prove monotonicity formulae related to degenerate k-Hessian equations which yield Morrey type estimates for certain integrands involving the second derivatives of the solution. In the special case k=2 we deduce that weak solutions in , , have locally H?lder continuous gradients. In the nondegenerate case we also show that weak solutions in , , have locally bounded second derivatives. Received February 25, 1999 / Accepted June 11, 1999 / Published online April 6, 2000     

Berezin transform on harmonic Bergman spaces on the real ball

We provide the full asymptotic expansion of the harmonic Berezin transform on the unit ball in Rⁿ${\mathbb{R}}^n$ purely by means of transformations of hypergeometric functions and function?s “hypergeometrization”.     

Weakly closed Jordan ideals in nest algebras on Banach spaces

We show that weakly closed Jordan ideals in nest algebras on Banach spaces are associative ideals. The decomposability of finite-rank operators in Jordan ideals and the commutants of bimodules are also investigated. Author’s address: J. Li and F. Lu, Department of Mathematics, Suzhou University, Suzhou 215006, People’s Republic of China This research was supported by NNSFC (No. 10771154) and PNSFJ (NO. BK2007049).     

Some results on supersolvability of finite groups

We get new characterizations of finite supersolvable groups based on the structure of the generalized Fitting subgroup. Then we extend our results to the saturated formations containing the class of all supersolvable groups. Authors’ addresses: M. Asaad, Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt; Piroska Cs?rg?, Department of Algebra and Number Theory, E?tv?s University, Pázmány Péter sétány 1/c, H-1117 Budapest, Hungary