<abstract xmlns="http://www.w3.org/1999/xhtml">

The aim of these notes is to discuss the completeness of the dilated systems in a most general framework of an arbitrary sequence lattice <italic>X</italic>, including weighted <italic>ℓ</italic>p spaces. In particular, general multiplicative and completely multiplicative sequences are treated. After the Fourier–Bohr transformation, we deal with the cyclicity property in function spaces on the corresponding infinite dimensional Reinhardt domain 
<inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2023-0017_ieq_001.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi>𝔻</mi></mrow><mi>X</mi><mo>∞</mo></msubsup></mrow></math>
<tex-math>\mathbb{D}_X^\infty</tex-math>
</alternatives>
</inline-formula>
. Functions with (weakly) dominating free term and (in particular) linearly factorable functions are considered. The most attention is paid to the cases of the polydiscs 
<inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2023-0017_ieq_002.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi>𝔻</mi></mrow><mi>X</mi><mo>∞</mo></msubsup><mo>,</mo><mo>|</mo><msup><mrow><mi>ℂ</mi></mrow><mi>N</mi></msup><mo>=</mo><msup><mrow><mi>𝔻</mi></mrow><mi>N</mi></msup></mrow></math>
<tex-math>\mathbb{D}_X^\infty ,|{\mathbb{C}^N} = {\mathbb{D}^N}</tex-math>
</alternatives>
</inline-formula>
 and the <italic>ℓ</italic>p-unit balls 
<inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2023-0017_ieq_003.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi>𝔻</mi></mrow><mi>X</mi><mo>∞</mo></msubsup><mo>,</mo><mo>|</mo><msup><mrow><mi>ℂ</mi></mrow><mi>N</mi></msup><mo>=</mo><msubsup><mrow><mi>𝔹</mi></mrow><mi>p</mi><mi>N</mi></msubsup></mrow></math>
<tex-math>\mathbb{D}_X^\infty ,|{\mathbb{C}^N} = \mathbb{B}_p^N</tex-math>
</alternatives>
</inline-formula>
, in particular to Dirichlet-type and Dirichlet–Drury–Arveson-type spaces and algebras, as 
<inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2023-0017_ieq_004.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mi>X</mi><mo>=</mo><msup><mrow><mi>ℓ</mi></mrow><mi>p</mi></msup><mrow><mo>(</mo><mrow><msubsup><mrow><mi>ℤ</mi></mrow><mo>+</mo><mi>N</mi></msubsup><mo>,</mo><msup><mrow><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mi>α</mi></mrow><mo>)</mo></mrow></mrow></mrow><mi>s</mi></msup></mrow><mo>)</mo></mrow></mrow></math>
<tex-math>X = {\ell ^p}\left( {_ + ^N,{{\left( {1 + \alpha } \right)}^s}} \right))</tex-math>
</alternatives>
</inline-formula>, <italic>s</italic> = (<italic>s</italic>1, <italic>s</italic>2, … ) and <inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2023-0017_ieq_005.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mi>X</mi><mo>=</mo><msup><mrow><mi>ℓ</mi></mrow><mi>p</mi></msup><mrow><mo>(</mo><mrow><msubsup><mrow><mi>ℤ</mi></mrow><mo>+</mo><mi>N</mi></msubsup><mo>,</mo><mi> </mi><mi> </mi><msup><mrow><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><mi>α</mi><mo>!</mo></mrow><mrow><mrow><mo>|</mo> <mi>α</mi> <mo>|</mo></mrow><mo>!</mo></mrow></mfrac></mrow><mo>)</mo></mrow></mrow></mrow><mi>t</mi></msup><msup><mrow><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mrow><mo>|</mo> <mi>α</mi> <mo>|</mo></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mi>s</mi></msup></mrow><mo>)</mo></mrow></mrow></math>
<tex-math>X = {\ell ^p}\left( {\mathbb{Z}_ + ^N,\,\,{{\left( {{{\alpha !} \over {\left| \alpha \right|!}}} \right)}^t}{{\left( {1 + \left| \alpha \right|} \right)}^s}} \right)</tex-math>
</alternatives>
</inline-formula>, <italic>s,t</italic> ≥ 0, as well as to their infinite variables analogues. We priviledged the largest possible scale of spaces and the most elementary instruments used.
</abstract>

The aim of these notes is to discuss the completeness of the dilated systems in a most general framework of an arbitrary sequence lattice X, including weighted ℓp spaces. In particular, general multiplicative and completely multiplicative sequences are treated. After the Fourier–Bohr transformation, we deal with the cyclicity property in function spaces on the corresponding infinite dimensional Reinhardt domain 



𝔻X∞
\mathbb{D}_X^\infty


. Functions with (weakly) dominating free term and (in particular) linearly factorable functions are considered. The most attention is paid to the cases of the polydiscs 



𝔻X∞,|ℂN=𝔻N
\mathbb{D}_X^\infty ,|{\mathbb{C}^N} = {\mathbb{D}^N}


 and the ℓp-unit balls 



𝔻X∞,|ℂN=𝔹pN
\mathbb{D}_X^\infty ,|{\mathbb{C}^N} = \mathbb{B}_p^N


, in particular to Dirichlet-type and Dirichlet–Drury–Arveson-type spaces and algebras, as 



X=ℓp(ℤ+N,(1+α)s)
X = {\ell ^p}\left( {_ + ^N,{{\left( {1 + \alpha } \right)}^s}} \right))

, s = (s1, s2, … ) and 


X=ℓp(ℤ+N,  (α!| α |!)t(1+| α |)s)
X = {\ell ^p}\left( {\mathbb{Z}_ + ^N,\,\,{{\left( {{{\alpha !} \over {\left| \alpha  \right|!}}} \right)}^t}{{\left( {1 + \left| \alpha  \right|} \right)}^s}} \right)

, s,t ≥ 0, as well as to their infinite variables analogues. We priviledged the largest possible scale of spaces and the most elementary instruments used.

On the Cyclicity of Dilated Systems in Lattices: Multiplicative Sequences, Polynomials, Dirichlet-type Spaces and Algebras

Moroccan Journal of Pure and Applied Analysis

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

The aim of these notes is to discuss the completeness of the dilated systems in a most general framework of an arbitrary sequence lattice X, including weighted...

On the Cyclicity of Dilated Systems in Lattices: Multiplicative Sequences, Polynomials, Dirichlet-type Spaces and Algebras

Published Online: Jun 07, 2023

Page range: 238 - 275

Received: Nov 02, 2022

Accepted: Mar 17, 2023

DOI: https://doi.org/10.2478/mjpaa-2023-0017

Keywords
sequence Banach lattices, completeness of dilations, Hilbert multidisc, Bohr transform, Dirichlet type spaces, Drury-Arveson type spaces

© 2023 Nikolai Nikolski, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

On the Cyclicity of Dilated Systems in Lattices: Multiplicative Sequences, Polynomials, Dirichlet-type Spaces and Algebras

Published Online: Jun 07, 2023

Page range: 238 - 275

Received: Nov 02, 2022

Accepted: Mar 17, 2023

DOI: https://doi.org/10.2478/mjpaa-2023-0017

Keywordssequence Banach lattices, completeness of dilations, Hilbert multidisc, Bohr transform, Dirichlet type spaces, Drury-Arveson type spaces

© 2023 Nikolai Nikolski, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Keywords
sequence Banach lattices, completeness of dilations, Hilbert multidisc, Bohr transform, Dirichlet type spaces, Drury-Arveson type spaces