On the shift semigroup on the Hardy space of Dirichlet series

@article{Olofsson2010OnTS,
  title={On the shift semigroup on the Hardy space of Dirichlet series},
  author={Anders Olofsson},
  journal={Acta Mathematica Hungarica},
  year={2010},
  volume={128},
  pages={265-286}
}
  • A. Olofsson
  • Published 8 May 2010
  • Mathematics
  • Acta Mathematica Hungarica
AbstractWe develop a Wold decomposition for the shift semigroup on the Hardy space $$ \mathcal{H}^2 $$ of square summable Dirichlet series convergent in the half-plane $$ \Re (s) > 1/2 $$. As an application we have that a shift invariant subspace of $$ \mathcal{H}^2 $$ is unitarily equivalent to $$ \mathcal{H}^2 $$ if and only if it has the form $$ \phi \mathcal{H}^2 $$ for some $$ \mathcal{H}^2 $$-inner function φ. 
View on Springer
maths.lth.se
11 Citations
Background Citations
6
Methods Citations
2

11 Citations

Citation Type

Citation Type

Dilation theory and analytic model theory for doubly commuting sequences of $C_{.0}$-contractions
Sz.-Nagy and Foias proved that each $C_{\cdot0}$-contraction has a dilation to a Hardy shift and thus established an elegant analytic functional model for contractions of class $C_{\cdot0}$. This has…
Interpolation and Semigroup on Hilbert and Hardy Spaces of Dirichlet Series
We investigate that a bounded sequence of points in the half plane σ > 1 2 is an interpolating sequence for Hilbert space of ordinary Dirichlet series with square summable coefficiens. Also, in the…
Power dilation systems $\{f(z^k)\}_{k\in\mathbb{N}}$ in Dirichlet-type spaces
In this paper, we concentrate on power dilation systems $\{f(z^k)\}_{k\in\mathbb{N}}$ in Dirichlet-type spaces $\mathcal{D}_t\ (t\in\mathbb{R})$. When $t\neq0$, we prove that…
On generalized Hardy classes of Dirichlet series
We generalize the Hardy class H^2 of Dirichlet series studied by Hedenmalm, Lindqvist, Olofsson, Olsen, Saksman, Seip and others to consider more general Dirichlet series. We prove some results on…
Splitting the Riesz basis condition for systems of dilated functions through Dirichlet series
Introducing the Polylogarithmic Hardy Space
In this paper we investigate the reproducing kernel Hilbert space where the polylogarithm appears as the kernel function. This investigation begins with the properties of functions in this space, and…
Local properties of Hilbert spaces of Dirichlet series
Toeplitzness of composition operators on a Hilbert space of Dirichlet series
Abstract In this paper, we investigate the -Toeplitzness and asymptotic -Toeplitzness of composition operators and their adjoints on the Hilbert space of Dirichlet series with square summable…
In a shadow of the RH: Cyclic vectors of Hardy spaces on the Hilbert multidisc
— Completeness of a dilation system (φ(nx))n>1 on the standard Lebesgue space L2(0, 1) is considered for 2-periodic functions φ. We show that the problem is equivalent to an open question on cyclic…
The periodic dilation completeness problem: cyclic vectors in the Hardy space over the infinite‐dimensional polydisk
  • Hui Dan, K. Guo
  • Mathematics
    Journal of the London Mathematical Society
  • 2020
The classical completeness problem raised by Beurling and independently by Wintner asks for which ψ∈L2(0,1) , the dilation system {ψ(kx):k=1,2,…} is complete in L2(0,1) , where ψ is identified with…
...
1
2
...

References

SHOWING 1-10 OF 23 REFERENCES
A Hilbert space of Dirichlet series and systems of dilated functions in $L^2(0,1)$
For a function $\varphi$ in $L^2(0,1)$, extended to the whole real line as an odd periodic function of period 2, we ask when the collection of dilates $\varphi(nx)$, $n=1,2,3,\ldots$, constitutes a…
Hardy Spaces of Dirichlet Series and Their Composition Operators
Abstract. In [9], Hedenmalm, Lindqvist and Seip introduce the Hilbert space of Dirichlet series with square summable coefficients , and begin its study, with modern functional and harmonic analysis…
Beurling's Theorem for the Bergman space
A celebrated theorem in operator theory is A. Beurling's description of the invariant subspaces in $H^2$ in terms of inner functions [Acta Math. {\bf81} (1949), 239--255; MR0027954 (10,381e)]. To do…
Local interpolation in Hilbert spaces of Dirichlet series
We denote by H the Hilbert space of ordinary Dirichlet series with square-summable coefficients. The main result is that a bounded sequence of points in the half-plane σ > 1/2 is an interpolating…
Integral means and boundary limits of Dirichlet series
This paper deals with the boundary behaviour of functions in the Hardy spaces ℋp for ordinary Dirichlet series. The main result, answering a question of Hedenmalm, shows that the classical Carlson…
Dirichlet series and functional analysis
The study of Dirichlet series of the form \( \sum\nolimits_{n = 1}^\infty {a_n n^{ - s} } \) has a long history beginning in the nineteenth century, and the interest was due mainly to the central…
A Wold-type decomposition for commuting isometric pairs
We obtain a Wold-type decomposition theorem for an arbitrary pair of commuting isometries V on a Hilbert space. More precisely, V can be uniquely decomposed into the orthogonal sum between a…
A von Neumann Wold decomposition of two-isometries, Acta Sci. Math. (Szeged) 70 (2004) 715-726
We present a von Neumann-Wold decomposition of a two- isometric operator on a general Hilbert space. A pure two-isometry is shown to be unitarily equivalent to a shift operator (multiplication by the…
Wold-type decompositions and wandering subspaces for operators close to isometries
For operators T satisfying certain inequalities we obtain a weak analog of the classical Wold-Kolmogorov decomposition theorem, representing T as a direct sum of a unitary operator and a shift…
Hilbert spaces of Dirichlet series and their multipliers
We consider various Hilbert spaces of Dirichlet series whose norms are given by weighted 2 norms of the Dirichlet coefficients. We describe the multiplier algebras of these spaces. The functions in…
...
1
2
3
...