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We continue the study of the so‐called Hausdorff–Zhu operators. In this paper, we study the behavior of the singular values of such operators. We prove the general fact that the sequence of singular numbers tends to zero, as well as we prove power‐type estimates for such behavior under additional conditions on the kernel of the operator. We give application of these results to the boundedness of Hausdorff–Zhu operators in general classes of analytic functions in the unit disc and also in some special classes of analytic functions defined in terms of conditions on Taylor or Fourier coefficients. [ABSTRACT FROM AUTHOR]

We introduce and study a special class of Hausdorff operators, we call this class “Hausdorff-Zhu operators”, on Banach spaces of analytic functions in the unit disc. Conditions of boundedness, compactness, and nuclearity of these operators are given. A special attention is paid to particular but important cases of analytic symbols and radial symbols. In particular it is shown that Hausdorff-Zhu operators with analytic symbols are at most two-dimensional and their spectra are computed. [ABSTRACT FROM AUTHOR]

Let $${{\bf D}_{\bf x} := \sum_{i=1}^n \frac{\partial}{\partial x_i} e_i}$$ be the Euclidean Dirac operator in $${\mathbb{R}^n}$$ and let P( X) = a X + . . . + a X + a be a polynomial with real coefficients. Differential equations of the form P( D) u( x) = 0 are called homogeneous polynomial Dirac equations with real coefficients. In this paper we treat Dirichlet type problems of the a slightly less general form P( D) u( x) = f( x) (where the roots are exclusively real) with prescribed boundary conditions that avoid blow-ups inside the domain. We set up analytic representation formulas for the solutions in terms of hypercomplex integral operators and give exact formulas for the integral kernels in the particular cases dealing with spherical and concentric annular domains. The Maxwell and the Klein-Gordon equation are included as special subcases in this context. [ABSTRACT FROM AUTHOR]

We characterize the boundedness of Volterra operators from Bergman spaces to Hardy spaces. Area integral operators and Carleson measures are heavily involved. [ABSTRACT FROM AUTHOR]

Abstract: Let be the Dirac operator in and let be a polynomial with complex coefficients. Differential equations of the form are called polynomial Dirac equations. In this paper we consider Hilbert spaces of Clifford algebra-valued functions that satisfy such a polynomial Dirac equation in annuli of the unit ball in . We determine an explicit formula for the Bergman kernel for solutions of complex polynomial Dirac equations of any degree m in the annulus of radii μ and 1 where . We further give formulas for the Szegö kernel for solutions to polynomial Dirac equations of degree in the annulus. This includes the Helmholtz and the Klein–Gordon equation as special cases. We further show the non-existence of the Szegö kernel for solutions to polynomial Dirac equations of degree in the annulus. As an application we give an explicit representation formula for the solutions of the Helmholtz and the Klein–Gordon equation in the annulus in terms of integral operators that involve the explicit formulas of the Bergman kernel that we computed. [Copyright &y& Elsevier]