An application of Hadamard multiplication to operators on weighted Hardy spaces
type: Item
instance of: scholarly article
title: An application of Hadamard multiplication to operators on weighted Hardy spaces
zbMATH Open document ID: 0699.47018
DOI: 10.1016/0024-3795(90)90238-8
author: Carl C. Cowen
published in: Linear Algebra and its Applications
publication date: 1990-01-01T00:00:00Z
review text: Let D denote the open unit disc of the complex plane. If a function f analytic on D is representable as a power series by \(f(z)=\sum^{\infty}_{j=0}\hat f(j)z^ j\), and if \(\beta =\{\beta (j)\), \(j=0,1,2,...\}\) is a sequence of positive numbers, then the inner product \(<f,g>\) is defined by \(<f,g>=\sum^{\infty}_{j=0}\hat f(j)\overline{\hat g(j)}\beta (j)^ 2.\) The weighted Hardy space \(H^ 2_{\beta}\) is defined to be \(\{\) f: \(\| f\|^ 2_{\beta}<\infty \}\). If \(A=(a_{ij})\), \(B=(b_{ij})\) are matrices, then the Hadamard product A.B of A and B is defined by \((A.B)=(a_{ij}b_{ij}).\) One of the main results of this paper states that if R is a linear transformation on the space of analytic polynomials into the space of formal power series which induces an operator A on \(H^ 2_{\alpha}\) and an operator B on \(H^ 2_{\beta}\), then the matrices representing A and B are related by \(B=M.A\), where \(M=(m_{ij})\), \(m_{ij}=(\beta (i)\alpha (j)/\beta (j)\alpha (i).\) Other statements of the paper involve the application of a theorem of Schur for determining positive definiteness of some weighted Toeplitz operators.
Mathematics Subject Classification ID: 47B38
Mathematics Subject Classification ID: 47B35
Mathematics Subject Classification ID: 46J15
Mathematics Subject Classification ID: 46B15
zbMATH DE Number: 4146900
zbMATH Keywords: positive definite matrices
zbMATH Keywords: Toeplitz matrices
zbMATH Keywords: weighted Hardy space
zbMATH Keywords: Hadamard product
zbMATH Keywords: linear transformation on the space of analytic polynomials into
zbMATH Keywords: the space of formal power series
zbMATH Keywords: theorem of Schur
zbMATH Keywords: positive definiteness
zbMATH Keywords: some weighted Toeplitz operators
zbMATH Keywords: linear transformation on the space of analytic polynomials into the space of formal power series
reviewed by: George O. Okikiolu
MaRDI profile type: Publication
full work available at URL: 0024-3795(90)90238-8
OpenAlex ID: W1998758785
cites work: Q4741262
cites work: Q5660202
cites work: Q5727906
cites work: Matrix Analysis
cites work: The spectra of composition operators on H\(^p\)
cites work: Q3881402
cites work: Q4722624
cites work: The essential norm of a composition operator
cites work: Q4059478
cites work: Q5607489
cites work: Q3398686
description: scientific article; zbMATH DE number 4146900