Volume 5, Issue 1

In the Drury-Arveson space, we consider the subspace of functions whose Taylor coefficients are supported in a set Y⊂ ℕ d with the property that ℕ\X + e j ⊂ ℕ\X for all j = 1, . . . , d. This is an easy example of shift-invariant subspace, which can be considered as a RKHS in is own right, with a kernel that can be explicitly calculated for specific choices of X. Every such a space can be seen as an intersection of kernels of Hankel operators with explicit symbols. Finally, this is the right space on which Drury’s inequality can be optimally adapted to a sub-family of the commuting and contractive operators originally considered by Drury.

We present some recent results on Hardy spaces of generalized analytic functions on D specifying their link with the analytic Hardy spaces. Their definition can be extended to more general domains Ω . We discuss the way to extend such definitions to more general domains that depends on the regularity of the boundary of the domain ∂Ω. The generalization over general domains leads to the study of the invertibility of composition operators between Hardy spaces of generalized analytic functions; at the end of the paper, we discuss invertibility and Fredholm property of the composition operator C ∅ on Hardy spaces of generalized analytic functions on a simply connected Dini-smooth domain for an analytic symbol ∅.

The Hilbert spaces ℋw consisiting of Dirichlet series F(s)=∑n=1∞ann-s$F(s) = \sum\nolimits_{n = 1}^\infty {{a_n}{n^{ - s}}}$ that satisfty ∑n=1∞|an|2/wn<∞${\sum\nolimits_{n = 1}^\infty {\left| {{a_n}} \right|} ^2}/{w_n} < \infty $ with { wn } n of average order log j n (the j -fold logarithm of n ), can be embedded into certain small Bergman spaces. Using this embedding, we study the Gordon–Hedenmalm theorem on such ℋw from an iterative point of view. By that theorem, the composition operators are generated by functions of the form Φ ( s ) = c 0 s + ϕ ( s ), where c 0 is a nonnegative integer and ϕ is a Dirichlet series with certain convergence and mapping properties. The iterative phenomenon takes place when c 0 = 0. It is verified for every integer j ⩾ 1, real α > 0 and { wn } n having average order (logj+n)α${(\log _j^ + n)^\alpha }$, that the composition operators map ℋw into a scale of ℋw ’ with w ’ n having average order (logj+1+n)α${(\log _{j + 1}^ + n)^\alpha }$. The case j = 1 can be deduced from the proof of the main theorem of a recent paper of Bailleul and Brevig, and we adopt the same method to study the general iterative step.

This paper explores the continuity and differentiability properties for the distribution function for a polynomial

This paper concerns the analysis of the structure of bi-contractive projections on spaces of vector valued continuous functions and presents results that extend the characterization of bi-contractive projections given by the first author. It also includes a partial generalization of these results to affine and vector valued continuous functions from a Choquet simplex into a Hilbert space.