Asymptotic Toeplitz and Hankel operators on the Bergman space

In this paper the concept of asymptotic Toeplitz and asymptotic Hankel operators on the Bergman space are introduced and properties of these classes of operators are studied. The importance of this notion is that it associates with a class of operators a Toeplitz operator and with a class of operators a Hankel operator where the original operators are not even Toeplitz or Hankel. Thus it is possible to assign a symbol to an operator that is not Toeplitz or Hankel and hence a symbol calculus is obtained. Further a relation between Toeplitz operators and little Hankel operators on the Bergman space is established in some asymptotic sense.     

A commutator method for the diagonalization of Hankel operators

A method for the explicit diagonalization of some Hankel operators is presented. This method makes it possible to give new proofs of classical results on the diagonalization of Hankel operators with absolutely continuous spectrum and obtain new results. The approach relies on the commutation of a Hankel operator with a certain second-order differential operator.     

Convolution Hankel Transforms on the Zemanian Spaces

In this paper we investigate the convolution Hankel transforms on the Zemanian spaces of Hankel transformable functions and distributions. The convolution Hankel transform is defined on generalized functions by using the adjoint method. Our new definition includes as special cases other known definitions of the convolution Hankel transform of distributions. Finally we establish a distributional inversion formula for the transformation under consideration involving Bessel differential operators.     

Algebraic properties of Toeplitz and small Hankel operators on the harmonic Bergman space

In this paper,we discuss some algebraic properties of Toeplitz operators and small Hankel operators with radial and quasihomogeneous symbols on the harmonic Bergman space of the unit disk in the complex plane C.We solve the product problem of quasihomogeneous Toeplitz operator and quasihomogeneous small Hankel operator.Meanwhile,we characterize the commutativity of quasihomogeneous Toeplitz operator and quasihomogeneous small Hankel operator.     

Hankel operators, the Segal-Bargmann space, and symmetrically-normed ideals

Starting with the Segal-Bargmann space, we investigate the Hankel operators with symbol functions in a certain linear space. Given an appropriate symbol function, we consider the associated Hankel operator together with the Hankel operator associated with that symbol function's complex conjugate. We give a necessary and sufficient condition for the simultaneous membership of these two operators in the symmetrically-normed ideal associated with any given symmetric norming function.     

The finite rank perturbations of the product of Hankel and Toeplitz operators

In this paper we completely characterize when the product of a Hankel operator and a Toeplitz operator on the Hardy space is a finite rank perturbation of a Hankel operator, and when the commutator of a Hankel operator and a Toeplitz operators has finite rank.     

Hankel convolution operators on entire functions and distributions

In this paper we study the Hankel convolution operators on the space of even and entire functions and on Schwartz distribution spaces. We characterize the Hankel convolution operators as those ones that commute with Hankel translations and with a Bessel operator. Also we prove that the Hankel convolution operators are hypercyclic and chaotic on the spaces under consideration.     

Compact adjoint operators and approximation properties

This paper is concerned with the space of all compact adjoint operators from dual spaces of Banach spaces into dual spaces of Banach spaces and approximation properties. For some topology on the space of all bounded linear operators from separable dual spaces of Banach spaces into dual spaces of Banach spaces, it is shown that if a bounded linear operator is approximated by a net of compact adjoint operators, then the operator can be approximated by a sequence of compact adjoint operators whose operator norms are less than or equal to the operator norm of the operator. Also we obtain applications of the theory and, in particular, apply the theory to approximation properties.     

Sums of Products of Toeplitz and Hankel Operators on the Dirichlet Space

On the Dirichlet space of the unit disk, we consider operators that are finite sums of Toeplitz products, Hankel products or products of a Toeplitz operator and a Hankel operator. We characterize when such operators are equal to zero. Our results extend several known results using completely different arguments.     

On generalized hankel operators induced by sums and products

Vectorial Hankel operators are studied, in particular the ranges of Hankel operators induced by sums and products of matrix functions defined on the unit circle are determined. The analytical tools involve factorization theorems for operator valued analytic functions and the spectral analysis of operators that intertwine restricted shifts.     

Little Hankel operators on the half plane

In this paper we consider a class of weighted integral operators onL ² (0, ) and show that they are unitarily equivalent to Hankel operators on weighted Bergman spaces of the right half plane. We discuss conditions for the Hankel integral operator to be finite rank, Hilbert-Schmidt, nuclear and compact, expressed in terms of the kernel of the integral operator. For a particular class of weights these operators are shown to be unitarily equivalent to little Hankel operators on weighted Bergman spaces of the disc, and the symbol correspondence is given. Finally the special case of the unweighted Bergman space is considered and for this case, motivated by approximation problems in systems theory, some asymptotic results on the singular values of Hankel integral operators are provided.     

Big Hankel operators of higher weight

Known results concerning the smoothness and boundedness of «big» Hankel operators (Hankel operators in the sense of Axler) are generalized to the case of higher weight (in the sense of representation theory). The key result is a certain estimate for thes-numbers of a particular such operator, involving a combinatorial sum.     

Asymptotic and Partial Asymptotic Hankel Operators on $$H^2(\mathbb{D}^n)$$ H 2 ( D n ) )

In this paper, we generalize the concept of asymptotic Hankel operators on H²(D) to the Hardy space H²(Dⁿ) (over polydisk) in terms of asymptotic Hankel and partial asymptotic Hankel operators and investigate some properties in case of its weak and strong convergence. Meanwhile, we introduce i^th-partial Hankel operators on H²(Dⁿ) and obtain a characterization of its compactness for n > 1. Our main results include the containment of Toeplitz algebra in the collection of all strong partial asymptotic Hankel operators on H²(Dⁿ). It is also shown that a Toeplitz operator with symbol φ is asymptotic Hankel if and only if φ is holomorphic function in L^∞(Tⁿ).     

Difference operators,Green's matrix,sampling theory and applications in signal processing

This paper introduces sampling representations for discrete signals arising from self adjoint difference operators with mixed boundary conditions. The theory of linear operators on finite-dimensional inner product spaces is employed to study the second-order difference operators. We give necessary and sufficient conditions that make the operators self adjoint. The equivalence between the difference operator and a Hermitian Green's matrix is established. Sampling theorems are derived for discrete transforms associated with the difference operator. The results are exhibited via illustrative examples, involving sampling representations for the discrete Hartley transform. Families of discrete fractional Fourier-type transforms are introduced with an application to image encryption.     

Essential Norm of Toeplitz Operators and Hankel Operators on the Weighted Bergman Space

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. (Ann Math 109:601–612, 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.     

Hankel Operators on Fock Spaces and Related Bergman Kernel Estimates

Hankel operators with anti-holomorphic symbols are studied for a large class of weighted Fock spaces on ? ⁿ . 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 Hörmander’s L ² estimates for the $\bar{\partial}$ operator are key ingredients in the proof.     

On determining the domain of the adjoint operator

A theorem that is of aid in computing the domain of the adjoint operator is provided. It may serve e.g. as a criterion for selfadjointness of a symmetric operator, for normality of a formally normal operator or for H-selfadjointness of an H-symmetric operator. Differential operators and operators given by an infinite matrix are considered as examples.     

On weak supercyclicity II

This paper considers weak supercyclicity for bounded linear operators on a normed space. On the one hand, weak supercyclicity is investigated for classes of Hilbertspace operators: (i) self-adjoint operators are not weakly supercyclic, (ii) diagonalizable operators are not weakly l-sequentially supercyclic, and (iii) weak l-sequential supercyclicity is preserved between a unitary operator and its adjoint. On the other hand, weak supercyclicity is investigated for classes of normed-space operators: (iv) the point spectrum of the normed-space adjoint of a power bounded supercyclic operator is either empty or is a singleton in the open unit disk, (v) weak l-sequential supercyclicity coincides with supercyclicity for compact operators, and (vi) every compact weakly l-sequentially supercyclic operator is quasinilpotent.     

Unbounded quantum graphs with unbounded boundary conditions

We consider metric graphs with a uniform lower bound on the edge lengths but no further restrictions. We discuss how to describe every local self‐adjoint Laplace operator on such graphs by boundary conditions in the vertices given by projections and self‐adjoint operators. We then characterize the lower bounded self‐adjoint Laplacians and determine their associated quadratic form in terms of the operator families encoding the boundary conditions.     

完备随机内积模上稠定模同态的共轭算子

对定义在完备随机内积模的稠子模上的模同态引入了共轭算子的概念并讨论其基本性质,尤其证明了共轭算子的闭性.