Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userA mean counting function for Dirichlet series and compact composition operators
67
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We introduce a mean counting function for Dirichlet series, which plays the same role in the function theory of Hardy spaces of Dirichlet series as the Nevanlinna counting function does in the classical theory. The existence of the mean counting function is related to Jessen and Tornehaves resolution of the Lagrange mean motion problem. We use the mean counting function to describe all compact composition operators with Dirichlet series symbols on the Hardy--Hilbert space of Dirichlet series, thus resolving a problem which has been open since the bounded composition operators were described by Gordon and Hedenmalm. The main result is that such a composition operator is compact if and only if the mean counting function of its symbol satisfies a decay condition at the boundary of a half-plane.
rate research
Read More
We consider composition operators $mathscr{C}_varphi$ on the Hardy space of Dirichlet series $mathscr{H}^2$, generated by Dirichlet series symbols $varphi$. We prove two different subordination principles for such operators. One concerns affine symbols only, and is based on an arithmetical condition on the coefficients of $varphi$. The other concerns general symbols, and is based on a geometrical condition on the boundary values of $varphi$. Both principles are strict, in the sense that they characterize the composition operators of maximal norm generated by symbols having given mapping properties. In particular, we generalize a result of J. H. Shapiro on the norm of composition operators on the classical Hardy space of the unit disc. Based on our techniques, we also improve the recently established upper and lower norm bounds in the special case that $varphi(s) = c + r2^{-s}$. A number of other examples are given.
Let $mathscr{H}^2$ denote the Hilbert space of Dirichlet series with square-summable coefficients. We study composition operators $mathscr{C}_varphi$ on $mathscr{H}^2$ which are generated by symbols of the form $varphi(s) = c_0s + sum_{ngeq1} c_n n^{-s}$, in the case that $c_0 geq 1$. If only a subset $mathbb{P}$ of prime numbers features in the Dirichlet series of $varphi$, then the operator $mathscr{C}_varphi$ admits an associated orthogonal decomposition. Under sparseness assumptions on $mathbb{P}$ we use this to asymptotically estimate the approximation numbers of $mathscr{C}_varphi$. Furthermore, in the case that $varphi$ is supported on a single prime number, we affirmatively settle the problem of describing the compactness of $mathscr{C}_varphi$ in terms of the ordinary Nevanlinna counting function. We give detailed applications of our results to affine symbols and to angle maps.
We investigate (uniform) mean ergodicity of (weighted) composition operators on the space of smooth functions and the space of distributions, respectively, both over an open subset of the real line. Among other things, we prove that a composition operator with a real analytic diffeomorphic symbol is mean ergodic on the space of distributions if and only if it is periodic (with period 2). Our results are based on a characterization of mean ergodicity in terms of Ces`aro boundedness and a growth property of the orbits for operators on Montel spaces which is of independent interest.
We establish the necessary and sufficient conditions for those symbols $b$ on the Heisenberg group $mathbb H^{n}$ for which the commutator with the Riesz transform is of Schatten class. Our main result generalises classical results of Peller, Janson--Wolff and Rochberg--Semmes, which address the same question in the Euclidean setting. Moreover, the approach that we develop bypasses the use of Fourier analysis, and can be applied to characterise that the commutator is of the Schatten class in other settings beyond Euclidean.
Volterra integral operators ${cal A}=sum_{k=0}^m {cal A}_k$, $({cal A}_k f)(x)= a_k (x)int_0^x t^k f(t) ,dt$, are studied acting between weighted $L_2$ spaces on $(0,+infty)$. Under certain conditions on the weights and functions $a_k$, it is shown that $cal A$ is bounded if and only if each ${cal A}_k$ is bounded. This result is then applied to describe spaces of pointwise multipliers in weighted Sobolev spaces on $(0,+infty)$.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
