Jordan structures of alternating matrix polynomials

Alternating matrix polynomials, that is, polynomials whose coefficients alternate between symmetric and skew-symmetric matrices, generalize the notions of even and odd scalar polynomials. We investigate the Smith forms of alternating matrix polynomials, showing that each invariant factor is an even or odd scalar polynomial. Necessary and sufficient conditions are derived for a given Smith form to be that of an alternating matrix polynomial. These conditions allow a characterization of the possible Jordan structures of alternating matrix polynomials, and also lead to necessary and sufficient conditions for the existence of structure-preserving strong linearizations. Most of the results are applicable to singular as well as regular matrix polynomials.     

Reverse order laws in C-algebras

In this paper, we offer purely algebraic necessary and sufficient conditions for reverse order laws for generalized inverses in C^∗-algebras, extending rank conditions for matrices and range conditions for Hilbert space operators.     

Determinantal divisors of products of matrices over Dedekind domains

Given three lists of ideals of a Dedekind domain, the question is raised whether there exist two matrices A and B with entries in the given Dedekind domain, such that the given lists of ideals are the determinantal divisors of A, B, and AB, respectively. To answer this question, necessary and sufficient conditions are developed in this article.     

A Jacobi matrix inverse eigenvalue problem with mixed data

In this paper, the inverse eigenvalue problem of reconstructing a Jacobi matrix from part of its eigenvalues and its leading principal submatrix is considered. The necessary and sufficient conditions for the existence and uniqueness of the solution are derived. Furthermore, a numerical algorithm and some numerical examples are given.     

On zero-pattern invariant properties of structured matrices

It is interesting that inverse M-matrices are zero-pattern (power) invariant. The main contribution of the present work is that we characterize some structured matrices that are zero-pattern (power) invariant. Consequently, we provide necessary and sufficient conditions for these structured matrices to be inverse M-matrices. In particular, to check if a given circulant or symmetric Toeplitz matrix is an inverse M-matrix, we only need to consider its pattern structure and verify that one of its principal submatrices is an inverse M-matrix.     

HGAH矩阵的逆特征值问题

In this paper, the inverse eigenvalue problem of Hermitian generalized anti-Hamihonian matrices and relevant optimal approximate problem are considered. The necessary and sufficient conditions of the solvability for inverse eigenvalue problem and an expression of the general solution of the problem are derived. The solution of the relevant optimal approximate problem is given.     

Reducing subspaces of compressed analytic Toeplitz operators on the Hardy space

In this paper we discuss necessary conditions and sufficient conditions for the compression of an analytic Toeplitz operator onto a shift coinvariant subspace to have nontrivial reducing subspaces. We give necessary and sufficient conditions for the kernel of a Toeplitz operator whose symbol is the quotient of two inner functions to be nontrivial and obtain examples of reducing subspaces from these kernels. Motivated by this result we give necessary conditions and sufficient conditions for the kernel of a Toeplitz operator whose symbol is the quotient of two inner functions to be nontrivial in terms of the supports of the two inner functions. By studying the commutant of a compression, we are able to give a necessary condition for the existence of reducing subspaces on certain shift coinvariant subspaces.     

Bases in semilinear spaces over zerosumfree semirings

This paper investigates the cardinality of a basis and the characterizations of a basis in semilinear space of n-dimensional vectors over zerosumfree semirings. First, it discusses the cardinality of a basis and gives some necessary and sufficient conditions that each basis has the same number of elements. Then it presents some conditions that a set of vectors is a basis and that a set of linearly independent vectors can be extended to a basis. In the end, it shows a necessary and sufficient condition that two semilinear spaces are isomorphic.     

Positive diagonal scaling of a nonnegative tensor to one with prescribed slice sums

In this paper we give necessary and sufficient conditions on a nonnegative tensor to be diagonally equivalent to a tensor with prescribed slice sums. These conditions are variations of Bapat-Raghavan and Franklin-Lorenz conditions.     

Fock-Sobolev空间上有界、紧与S_p-类算子

本文研究Fock-Sobolev空间上稠密定义算子,将这些算子统一表示成积分算子,利用积分算子的方法得到了它们的一个充分条件,并构造反例说此充分条件是非必要的,还得到这些算子为紧算子的两个充分条件.最后构造符号函数在复平面上每一点处本性无界的紧和Sp-类(0p∞)Toeplitz算子.     

A GENERALIZATION OF FINSLER'S THEOREM FOR QUADRATIC INEQUALITIES AND EQUALITIES

ABSTRACT

A generalization is given of Finsler's theorem on the positivity of a quadratic form subject to a quadratic equality. The generalized result is, under a certain assumption, a set of necessary and sufficient conditions for non-negativity of a quadratic form subject to an arbitrary number of quadratic inequality and equality constraints. Certain properties of matrix inverses are deduced using the conditions.     

An inverse eigenvalue problem for periodic Jacobi matrices in Minkowski spaces

The spectral properties of periodic Jacobi matrices in Minkowski spaces are studied. An inverse problem for these matrices is investigated, and necessary and sufficient conditions under which the problem is solvable are presented. Uniqueness results are also discussed, and an algorithm to construct the solutions and illustrative examples is provided.     

On some matrix equalities for generalized inverses with applications

Necessary and sufficient conditions are derived for the matrix equality A^-=PN^-Q to hold, where A^- and N^- are generalized inverses of matrices. Some consequences and applications are also given. In particular, necessary and sufficient conditions are derived for the additive decompositions C^-=A^-+B^- and to hold.     

Compact differences of composition operators

A characterization of compact difference is given for composition operators acting on the standard weighted Bergman spaces and necessary conditions are given on a larger scale of weighted Dirichlet spaces. Conditions are given under which a composition operator can be written as a finite sum of composition operators modulo the compacts. The additive structure of the space of composition operators modulo the compact operators is investigated further and a sufficient condition is given to insure that two composition operators lie in the same component.     

Lipschitz stability of controlled invariant subspaces

Let (A,B)∈C^n×n×C^n×m and M be an (A, B)-invariant subspace. In this paper the following results are presented: (i) If M∩ImB={0}, necessary and sufficient conditions for the Lipschitz stability of M are given. (ii) If M contains the controllability subspace of the pair (A, B), sufficient conditions for the Lipschitz stability of the subspace M are given.     

The normed finiteness property of compact contraction operators

We study the joint spectral radius given by a finite set of compact operators on a Hilbert space. It is shown that the normed finiteness property holds in this case, that is, if all the compact operators are contractions and the joint spectral radius is equal to 1 then there exists a finite product that has a spectral radius equal to 1. We prove an additional statement in that the requirement that the joint spectral radius be equal to 1 can be relaxed to the asking that the maximum norm of finite products of a length norm is equal to 1. The length of this product is related to the dimension of the subspace on which the set of operators is norm preserving.     

Toeplitz operators on Dirichlet spaces

In this paper we consider Toeplitz operators on Dirichlet spaces of the unit disk in whose symbols are nonnegative measures. We obtain necessry and sufficient conditions on the symbols for the operator to be bounded and compact. If the symbols are supported in a cone we also get necessary and sufficient conditions for the operators to belong to the Schatten p-class. Application to the Hankel operators are indicated.This work supported in part by NSF grant DMS 8701271     

Characterizations of inverse M-matrices with special zero patterns

In this paper, we provide some characterizations of inverse M-matrices with special zero patterns. In particular, we give necessary and sufficient conditions for k-diagonal matrices and symmetric k-diagonal matrices to be inverse M-matrices. In addition, results for triadic matrices, tridiagonal matrices and symmetric 5-diagonal matrices are presented as corollaries.     

Symplectic Self-adjointness of Infinite Dimensional Hamiltonian Operators

Symplectic self-adjointness of infinite dimensional Hamiltonian operators is studied, the necessary and sufficient conditions are given. Using the relatively bounded perturbation, the sufficient conditions about symplectic self-adjointness are shown.