Weak Product Spaces of Dirichlet Series
(2016) In Integral Equations and Operator Theory 86(4). p.453473 Abstract
Let (Formula presented.) denote the space of ordinary Dirichlet series with square summable coefficients, and let (Formula presented.) denote its subspace consisting of series vanishing at (Formula presented.). We investigate the weak product spaces (Formula presented.) and (Formula presented.), finding that several pertinent problems are more tractable for the latter space. This surprising phenomenon is related to the fact that (Formula presented.) does not contain the infinitedimensional subspace of (Formula presented.) of series which lift to linear functions on the infinite polydisc. The problems considered stem from questions about the dual spaces of these weak product spaces, and are therefore naturally phrased in terms of... (More)
Let (Formula presented.) denote the space of ordinary Dirichlet series with square summable coefficients, and let (Formula presented.) denote its subspace consisting of series vanishing at (Formula presented.). We investigate the weak product spaces (Formula presented.) and (Formula presented.), finding that several pertinent problems are more tractable for the latter space. This surprising phenomenon is related to the fact that (Formula presented.) does not contain the infinitedimensional subspace of (Formula presented.) of series which lift to linear functions on the infinite polydisc. The problems considered stem from questions about the dual spaces of these weak product spaces, and are therefore naturally phrased in terms of multiplicative Hankel forms. We show that there are bounded, even Schatten class, multiplicative Hankel forms on (Formula presented.) whose analytic symbols are not in (Formula presented.). Based on this result we examine Nehari’s theorem for such Hankel forms. We define also the skew product spaces associated with (Formula presented.) and (Formula presented.), with respect to both halfplane and polydisc differentiation, the latter arising from Bohr’s point of view. In the process we supply square function characterizations of the Hardy spaces (Formula presented.), for (Formula presented.), from the viewpoints of both types of differentiation. Finally we compare the skew product spaces to the weak product spaces, leading naturally to an interesting Schur multiplier problem.
(Less)
 author
 Brevig, Ole Fredrik and Perfekt, Karl Mikael ^{LU}
 organization
 publishing date
 201612
 type
 Contribution to journal
 publication status
 published
 subject
 keywords
 Dirichlet series, Hankel form, Square function, Weak product space
 in
 Integral Equations and Operator Theory
 volume
 86
 issue
 4
 pages
 21 pages
 publisher
 Springer
 external identifiers

 wos:000389606800001
 scopus:84991738210
 ISSN
 0378620X
 DOI
 10.1007/s0002001623203
 language
 English
 LU publication?
 yes
 id
 b77a3b5c92af43c39727f8f7a86941f8
 date added to LUP
 20161109 11:00:22
 date last changed
 20241005 05:20:25
@article{b77a3b5c92af43c39727f8f7a86941f8, abstract = {{<p>Let (Formula presented.) denote the space of ordinary Dirichlet series with square summable coefficients, and let (Formula presented.) denote its subspace consisting of series vanishing at (Formula presented.). We investigate the weak product spaces (Formula presented.) and (Formula presented.), finding that several pertinent problems are more tractable for the latter space. This surprising phenomenon is related to the fact that (Formula presented.) does not contain the infinitedimensional subspace of (Formula presented.) of series which lift to linear functions on the infinite polydisc. The problems considered stem from questions about the dual spaces of these weak product spaces, and are therefore naturally phrased in terms of multiplicative Hankel forms. We show that there are bounded, even Schatten class, multiplicative Hankel forms on (Formula presented.) whose analytic symbols are not in (Formula presented.). Based on this result we examine Nehari’s theorem for such Hankel forms. We define also the skew product spaces associated with (Formula presented.) and (Formula presented.), with respect to both halfplane and polydisc differentiation, the latter arising from Bohr’s point of view. In the process we supply square function characterizations of the Hardy spaces (Formula presented.), for (Formula presented.), from the viewpoints of both types of differentiation. Finally we compare the skew product spaces to the weak product spaces, leading naturally to an interesting Schur multiplier problem.</p>}}, author = {{Brevig, Ole Fredrik and Perfekt, Karl Mikael}}, issn = {{0378620X}}, keywords = {{Dirichlet series; Hankel form; Square function; Weak product space}}, language = {{eng}}, number = {{4}}, pages = {{453473}}, publisher = {{Springer}}, series = {{Integral Equations and Operator Theory}}, title = {{Weak Product Spaces of Dirichlet Series}}, url = {{http://dx.doi.org/10.1007/s0002001623203}}, doi = {{10.1007/s0002001623203}}, volume = {{86}}, year = {{2016}}, }