Browsing by Subject "46E22"

Now showing 1 - 4 of 4

Open Access
Aspects of multivariable operator theory on weighted symmetric Fock spaces
(World Scientific Publishing, 2014) Kaptanoğlu, H. T.
We obtain all Dirichlet spaces Fq, q ∈ ℝ, of holomorphic functions on the unit ball of ℂN as weighted symmetric Fock spaces over ℂN. We develop the basics of operator theory on these spaces related to shift operators. We do a complete analysis of the effect of q ∈ ℝ in the topics we touch upon. Our approach is concrete and explicit. We use more function theory and reduce many proofs to checking results on diagonal operators on the Fq. We pick out the analytic Hilbert modules from among the Fq. We obtain von Neumann inequalities for row contractions on a Hilbert space with respect to each Fq. We determine the commutants and investigate the almost normality of the shift operators. We prove that the C∗-algebras generated by the shift operators on the Fq fit in exact sequences that are in the same Ext class. We identify the groups K0 and K1 of the Toeplitz algebras on the Fq arising in K-theory. Radial differential operators are prominent throughout. Some of our results, especially those pertaining to lower negative values of q, are new even for N = 1. Many of our results are valid in the more general weighted symmetric Fock spaces Fb that depend on a weight sequence b. © World Scientific Publishing Company.
Open Access
Dilations of some VH-spaces operator valued invariant Kernels
(Springer, 2012) Gheondea, A.
We investigate VH-spaces (Vector Hilbert spaces, or Loynes spaces) operator valued Hermitian kernels that are invariant under actions of *-semigroups from the point of view of generation of *-representations, linearizations (Kolmogorov decompositions), and reproducing kernel spaces. We obtain a general dilation theorem in both Kolmogorov and reproducing kernel space representations, that unifies many dilation results, in particular B. Sz.-Nagy's and Stinesprings' dilation type theorems. © 2012 Springer Basel.
Open Access
Harmonic Besov spaces on the ball
(World Scientific Publishing, 2016) Gergün, S.; Kaptanoğlu, H. T.; Üreyen, A. E.
We initiate a detailed study of two-parameter Besov spaces on the unit ball of ℝn consisting of harmonic functions whose sufficiently high-order radial derivatives lie in harmonic Bergman spaces. We compute the reproducing kernels of those Besov spaces that are Hilbert spaces. The kernels are weighted infinite sums of zonal harmonics and natural radial fractional derivatives of the Poisson kernel. Estimates of the growth of kernels lead to characterization of integral transformations on Lebesgue classes. The transformations allow us to conclude that the order of the radial derivative is not a characteristic of a Besov space as long as it is above a certain threshold. Using kernels, we define generalized Bergman projections and characterize those that are bounded from Lebesgue classes onto Besov spaces. The projections provide integral representations for the functions in these spaces and also lead to characterizations of the functions in the spaces using partial derivatives. Several other applications follow from the integral representations such as atomic decomposition, growth at the boundary and of Fourier coefficients, inclusions among them, duality and interpolation relations, and a solution to the Gleason problem. © 2016 World Scientific Publishing Company.
Open Access
Operator models for hilbert locally c*-modules
(Element D.O.O., 2017) Gheondea, A.
We single out the concept of concrete Hilbert module over a locally C*-algebra by means of locally bounded operators on certain strictly inductive limits of Hilbert spaces. Using this concept, we construct an operator model for all Hilbert locally C*-modules and, as an application, we obtain a direct construction of the exterior tensor product of Hilbert locally C*-modules. These are obtained as consequences of a general dilation theorem for positive semidefinite kernels invariant under an action of a ∗-semigroup with values locally bounded operators. As a by-product, we obtain two Stinespring type theorems for completely positive maps on locally C*-algebras and with values locally bounded operators. © 2017, Element D.O.O. All rights reserved.