Petersson norms and liftings of Hilbert modular forms

We prove the algebraicity of the ratio of the Petersson norm of a holomorphic Hilbert modular form over a totally real number field and the norm of its Saito-Kurokawa lift. We prove a similar result for the Ikeda lift of an elliptic modular form. In order to obtain these we combine some results on local symplectic groups to generalize a special value of the standard L-function attached to a Siegel-Hilbert cuspform.     

On the spectral norms of Cauchy–Toeplitz and Cauchy–Hankel matrices

In this study, we have found upper and lower bounds for the spectral norm of Cauchy–Toeplitz and Cauchy–Hankel matrices in the forms T_n=[1/(a+(i−j)b)]ⁿ_i,j=1, H_n=[1/(a+(i+j)b)]ⁿ_i,j=1.     

Essentially commuting Hankel and Toeplitz operators

We characterize when a Hankel operator and a Toeplitz operator have a compact commutator.     

Quasidirect decompositions of Hankel and Toeplitz matrices

Decompositions, over an algebraically closed field, of a Hankel matrix into a sum of Hankel matrices the sum of the ranks of which is equal to the rank of the original matrix, are completely described. Similar results hold for Toeplitz matrices.     

Some extensions of Hankel and Toeplitz matrices

New necessary and sufficient conditions are given for a nonsingular matrix to have Hankel or Toeplitz form. Two generalizations, termed conjugate-Toeplitz and Leslie-Toeplitz matrices, are introduced A number of properties of Toeplitz matrices are shown to carry over to these extensions     

Fast algorithms for Toeplitz and Hankel matrices

The paper gives a self-contained survey of fast algorithms for solving linear systems of equations with Toeplitz or Hankel coefficient matrices. It is written in the style of a textbook. Algorithms of Levinson-type and Schur-type are discussed. Their connections with triangular factorizations, Padè recursions and Lanczos methods are demonstrated. In the case in which the matrices possess additional symmetry properties, split algorithms are designed and their relations to butterfly factorizations are developed.     

Toeplitz plus Hankel Operators with Infinite Index

We study Toeplitz plus Hankel operators acting between Lebesgue spaces on the unit circle, and having symbols which contain standard almost periodic discontinuities. Conditions are obtained under which these operators are right-invertible and with infinite kernel dimension, left-invertible and with infinite cokernel dimension or simply not normally solvable.     

Efficient algorithm for Toeplitz plus Hankel matrices

The work of both authors was supported in part by the NSF grant DMS-8801961     

Several observations on Toeplitz and Hankel circulants

An application of the unitary similarity with the discrete Fourier transform to the algebra of diagonal matrices yields the algebra of circulants. It turns out that if, in this construction, the unitary similarity is replaced by the unitary congruence, then the class of the so-called Hankel circulants is obtained. The causes and certain effects of this fact are discussed. Bibliography: 2 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 334, 2006, pp. 121–127.     

Hankel and Toeplitz operators in hardy spaces

In this paper one considers some general theorems of the theory of Hankel and Toeplitz operators in spaces of analytic functions. Under natural restrictions on the spaces X, it is shown that the symbols of the Toeplitz operators, acting in X, are bounded. One describes completely the symbols of the Hankel and Toeplitz operators, acting from H^p into ¯H_q (into H^q) for 0 < p, q < .Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 141, pp. 165–175, 1985.The author expresses his gratitude to A. L. Vol'berg and S. V. Kislyakov for useful discussions.     

Hankel and Toeplitz operators on Dirichlet spaces

In this paper we study Hankel and Toeplitz operators on Dirichlet type spaces D. We obtain necessary and sufficient condition on the symbols for these operators to be bounded and to belong to the Schatten ideal S^p for certain and p.     

A characterization of the Toeplitz and Hankel circulants

A number of assertions of the following type are proved: A Toeplitz matrix T is a circulant if and only if T has an eigenvector e with unit components. These assertions characterize the circulants (and, more generally, the ϕ circulants), as well as their Hankel counterparts, in the sets of all Toeplitz and all Hankel matrices, respectively. Bibliography: 2 titles.     

Classifying anti-commuting pairs of Toeplitz and Hankel matrices

Conditions for commuting a Toeplitz matrix and a Hankel matrix were obtained relatively recently (in 2015). The solution to the problem of describing all anti-commuting pairs (T, H), where T is a Toeplitz matrix and H is a Hankel matrix, is sketched below.     

Toeplitz and Hankel operators on the Paley-Wiener space

The basic theory of Toeplitz and Hankel operators acting on the Paley-Weiner space is developed. This includes criteria for boundedness, compactness, being of finite rank, and membership in the Schatten-von Neumann ideals. Similar questions are considered for the related operators formed by commuting the discrete Hilbert transform with a multiplication operator.Supported in part by a grant from the National Science Foundation.     

Formal orthogonal polynomials and Hankel/Toeplitz duality

For classical polynomials orthogonal with respect to a positive measure supported on the real line, the moment matrix is Hankel and positive definite. The polynomials satisfy a three term recurrence relation. When the measure is supported on the complex unit circle, the moment matrix is positive definite and Toeplitz. Then they satisfy a coupled Szeg recurrence relation but also a three term recurrence relation. In this paper we study the generalization for formal polynomials orthogonal with respect to an arbitrary moment matrix and consider arbitrary Hankel and Toeplitz matrices as special cases. The relation with Padé approximation and with Krylov subspace iterative methods is also outlined.This research was supported by the National Fund for Scientific Research (NFWO), project Lanczos, grant #2.0042.93.     

On Hankel operators not in the Toeplitz algebra

In this paper we exhibit a class of Hankel operators, which is contained in the essential commutant of the unilateral shift, but disjoint from the Toeplitz algebra.

     

On conditions for permutability of Toeplitz and Hankel matrices

The problem of describing pairs of commuting matrices (T, H), where T and H are a Toeplitz and a Hankel matrix, respectively, is examined. Several families of such pairs are indicated.