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Register a new userNorms of composition operators on the $H^2$ space of Dirichlet series
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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.
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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 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.
In this paper, we study the reducing subspaces for the multiplication operator by a finite Blaschke product $phi$ on the Dirichlet space $D$. We prove that any two distinct nontrivial minimal reducing subspaces of $M_phi$ are orthogonal. When the order $n$ of $phi$ is $2$ or $3$, we show that $M_phi$ is reducible on $D$ if and only if $phi$ is equivalent to $z^n$. When the order of $phi$ is $4$, we determine the reducing subspaces for $M_phi$, and we see that in this case $M_phi$ can be reducible on $D$ when $phi$ is not equivalent to $z^4$. The same phenomenon happens when the order $n$ of $phi$ is not a prime number. Furthermore, we show that $M_phi$ is unitarily equivalent to $M_{z^n} (n > 1)$ on $D$ if and only if $phi = az^n$ for some unimodular constant $a$.
In this paper we propose a different (and equivalent) norm on $S^{2} ({mathbb{D}})$ which consists of functions whose derivatives are in the Hardy space of unit disk. The reproducing kernel of $S^{2}({mathbb{D}})$ in this norm admits an explicit form, and it is a complete Nevanlinna-Pick kernel. Furthermore, there is a surprising connection of this norm with $3$ -isometries. We then study composition and multiplication operators on this space. Specifically, we obtain an upper bound for the norm of $C_{varphi}$ for a class of composition operators. We completely characterize multiplication operators which are $m$-isometries. As an application of the 3-isometry, we describe the reducing subspaces of $M_{varphi}$ on $S^{2}({mathbb{D}})$ when $varphi$ is a finite Blaschke product of order 2.
This paper is devoted to the study of reducing subspaces for multiplication operator $M_phi$ on the Dirichlet space with symbol of finite Blaschke product. The reducing subspaces of $M_phi$ on the Dirichlet space and Bergman space are related. Our strategy is to use local inverses and Riemann surface to study the reducing subspaces of $M_phi$ on the Bergman space, and we discover a new way to study the Riemann surface for $phi^{-1}circphi$. By this means, we determine the reducing subspaces of $M_phi$ on the Dirichlet space when the order of $phi$ is $5$; $6$; $7$ and answer some questions of Douglas-Putinar-Wang cite{DPW12}.
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