An application of Hadamard multiplication to operators on weighted Hardy spaces
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.
linear transformation on the space of analytic polynomials into
linear transformation on the space of analytic polynomials into the space of formal power series
scientific article; zbMATH DE number 4146900