Cayley transform and the Kronecker product of Hermitian matrices

We consider the conditions under which the Cayley transform of the Kronecker product of two Hermitian matrices can be again presented as a Kronecker product of two matrices and, if so, if it is a product of the Cayley transforms of the two Hermitian matrices. We also study the related question: given two matrices, which matrix under the Cayley transform yields the Kronecker product of their Cayley transforms.     

Some aspects of Hermitian Jacobi forms

We introduce a certain differential (heat) operator on the space of Hermitian Jacobi forms of degree 1, show its commutation with certain Hecke operators and use it to construct a map from elliptic cusp forms to Hermitian Jacobi cusp forms. We construct Hermitian Jacobi forms as the image of the tensor product of two copies of Jacobi forms and also from the differentiation of the variables. We determine the number of Fourier coefficients that determine a Hermitian Jacobi form and use the differential operator to embed a certain subspace of Hermitian Jacobi forms into a direct sum of modular forms for the full modular group.     

Differential operators on Hermitian Jacobi forms

We study multilinear differential operators on a space of Hermitian Jacobi forms as well as on a space of Hermitian modular forms of degree 2. First we define a heat operator and construct multilinear differential operators on a space of Hermitian Jacobi forms of degree 2. As a special case of these operators, we also study Rankin-Cohen type differential operators on a space of Hermitian Jacobi forms. And we construct multilinear differential operators on a space of Hermitian modular forms of degree 2 as an application of multilinear differential operators on Hermitian Jacobi forms.     

A Cauchy kernel for the Hermitian submonogenic system

Hermitian monogenic functions are the null solutions of two complex Dirac type operators. The system of these complex Dirac operators is overdetermined and may be reduced to constraints for the Cauchy datum together with what we called the Hermitian submonogenic system (see [8], [9]). This last system is no longer overdetermined and it has properties that are similar to those of the standard Dirac operator in Euclidean space, such as a Cauchy–Kowalevski extension theorem and Vekua type solutions. In this paper, we investigate plane wave solutions of the Hermitian submonogenic system, leading to the construction of a Cauchy kernel. We also establish a Stokes type formula that, when applied to the Cauchy kernel provides an integral representation formula for Hermitian submonogenic functions.     

Polar decomposition in e-rings

An e-ring is a generalization of the ring of bounded linear operators on a Hilbert space together with the subset consisting of all effect operators on that space. Associated with an e-ring is a partially ordered abelian group, called its directed group, that generalizes the additive group of bounded Hermitian operators on the Hilbert space. We prove that every element of the directed group of an e-ring has a polar decomposition if and only if every element has a carrier projection and is split by a projection into a positive and a negative part.     

Product numerical range in a space with tensor product structure

We study operators acting on a tensor product Hilbert space and investigate their product numerical range, product numerical radius and separable numerical range. Concrete bounds for the product numerical range for Hermitian operators are derived. Product numerical range of a non-Hermitian operator forms a subset of the standard numerical range containing the barycenter of the spectrum. While the latter set is convex, the product range needs not to be convex nor simply connected. The product numerical range of a tensor product is equal to the Minkowski product of numerical ranges of individual factors.     

Differential operators for Hermitian Jacobi forms and Hermitian modular forms

Kim (Arch Math (Basel) 79(3):208–215, 2002) constructs multilinear differential operators for Hermitian Jacobi forms and Hermitian modular forms. However, her work relies on incorrect actions of differential operators on spaces of Hermitian Jacobi forms and Hermitian modular forms. In particular, her results are incorrect if the underlying field is the Gaussian number field. We consider more general spaces of Hermitian Jacobi forms and Hermitian modular forms over \(\mathbb {Q}(i)\), which allow us to correct the corresponding results in Kim (2002).     

On the tensor operators in parallelizable spaces

The tensor-operators over parallelizable spaces are defined. Their Hermitian conjugation is defined by introducing an inner product based on the tensor-integral Some properties of the ordinary operators are generalized for the tensor-operators     

Abelian Hermitian geometry

We study the structure of Lie groups admitting left invariant abelian complex structures in terms of commutative associative algebras. If, in addition, the Lie group is equipped with a left invariant Hermitian structure, it turns out that such a Hermitian structure is Kähler if and only if the Lie group is the direct product of several copies of the real hyperbolic plane by a Euclidean factor. Moreover, we show that if a left invariant Hermitian metric on a Lie group with an abelian complex structure has flat first canonical connection, then the Lie group is abelian.     

Hecke Operators on Linear Ordinary Differential Equations

We construct Hecke operators acting on the space of certain linear ordinary differential equations, and describe a Hermitian inner product on the space of such differential equations. We also determine the adjoint of the Hecke operator with respect to this inner product, and prove that the space of ordinary differential equations associated to an automorphic form for a certain discrete subgroup of SL(2, R) has a basis consisting of common eigenvectors of a class of Hecke operators.     

The geometrical properties of parity and time reversal operators in two dimensional spaces

The parity operator P and time reversal operator T are two important operators in the quantum theory, in particular, in the PT-symmetric quantum theory. By using the concrete forms of P and T, we discuss their geometrical properties in two dimensional spaces. It is showed that if T is given, then all P links with the quadric surfaces; if P is given, then all T links with the quadric curves. Moreover, we give out the generalized unbroken PT-symmetric condition of an operator. The unbroken PT-symmetry of a Hermitian operator is also showed in this way.     

A generalization of Yoshida–Nicolaescu theorem using partial signatures

We give a simplified proof of the Yoshida–Nicolaescu Theorem in the product case using the theory of partial signatures as in Giambò et al. (2004). The theorem gives the equality of the spectral flow of a family of first order self-adjoint differential operators defined on sections of a Hermitian vector bundle over a partitioned manifold and the Maslov index of the corresponding pair of Cauchy data spaces. No nondegeneracy assumption is made on the endpoints of the path of differential operators.     

The Cauchy-Kovalevskaya extension theorem in Hermitian Clifford analysis

Hermitian Clifford analysis is a higher dimensional function theory centered around the simultaneous null solutions, called Hermitian monogenic functions, of two Hermitian conjugate complex Dirac operators. As an essential step towards the construction of an orthogonal basis of Hermitian monogenic polynomials, in this paper a Cauchy-Kovalevskaya extension theorem is established for such polynomials. The minimal number of initial polynomials needed to obtain a unique Hermitian monogenic extension is determined, along with the compatibility conditions they have to satisfy. The Cauchy-Kovalevskaya extension principle then allows for a dimensional analysis of the spaces of spherical Hermitian monogenics, i.e. homogeneous Hermitian monogenic polynomials. A version of this extension theorem for specific real-analytic functions is also obtained.     

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.     

The Teodorescu Operator in Clifford Analysis

Euclidean Clifford analysis is a higher dimensional function theory centred around monogenic functions,i.e.,null solutions to a first order vector valued rotation invariant differential operator (θ) ca...     

Canonical Extensions of Hermitian Operators

The concept of canonical extensions of Hermitian operators is introduced. Not only are such extensions of interest on their own merits, but they also have significant applications (Theorem 3 in particular) in constructing spaces of boundary values of Hermitian operators with various defect numbers. In recent years boundary-value spaces have found important applications in the study of various classes of extensions of Hermitian operators and in scattering theory.     

On Tensor Products of l-Operators and bo-Operators

It is proved that the tensor product of two linear operators is a cone summing operator (respectively, order bounded operator) if and only if both operators are cone summing (respectively, order bounded).     

Hermitian Weighted Composition Operators and Bergman Extremal Functions

Weighted composition operators have been related to products of composition operators and their adjoints and to isometries of Hardy spaces. In this paper, Hermitian weighted composition operators on weighted Hardy spaces of the unit disk are studied. In particular, necessary conditions are provided for a weighted composition operator to be Hermitian on such spaces. On weighted Hardy spaces for which the kernel functions are ${(1 - \overline{w}z)^{-\kappa}}$ for κ ≥ 1, including the standard weight Bergman spaces, the Hermitian weighted composition operators are explicitly identified and their spectra and spectral decompositions are described. Some of these Hermitian operators are part of a family of closely related normal weighted composition operators. In addition, as a consequence of the properties of weighted composition operators, we compute the extremal functions for the subspaces associated with the usual atomic inner functions for these weighted Bergman spaces and we also get explicit formulas for the projections of the kernel functions on these subspaces.     

Hermitian generalization of the Rarita-Schwinger operators

We introduce two new linear differential operators which are invariant with respect to the unitary group SU（n）. They constitute analogues of the twistor and the Rarita-Schwinger operator in the orthogonal case. The natural setting for doing this is Hermitian Clifford Analysis. Such operators are constructed by twisting the two versions of the Hermitian Dirac operator 6z_ and 6z_ and then projecting on irreducible modules for the unitary group. We then study some properties of their spaces of nullsolutions and we find a formulation of the Hermitian Rarita-Schwinger operators in terms of Hermitian monogenic polynomials.     

Hermitian Symmetric Polynomials and CR Complexity

Properties of Hermitian forms are used to investigate several natural questions from CR geometry. To each Hermitian symmetric polynomial we assign a Hermitian form. We study how the signature pairs of two Hermitian forms behave under the polynomial product. We show, except for three trivial cases, that every signature pair can be obtained from the product of two indefinite forms. We provide several new applications to the complexity theory of rational mappings between hyperquadrics, including a stability result about the existence of non-trivial rational mappings from a sphere to a hyperquadric with a given signature pair.