An application of Hadamard multiplication to operators on weighted Hardy spaces (Q913213)

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.