ON OPERATOR VALUED WEIGHTED SHIFTS COMMUTING WITH \(U_{+}\)

Abstract

In this paper we consider some properties of the operators \(K_{U_+}\) and of the operators on \(K_A\) where \(A\in K_{U_+}\). Also we shall give some information about the operators that commutes with \(U_+\). The computations show that operator \(X\in B(\ell^2(H))\) which commutes with \(U_+\) is the formal power series

\[
U_+^*\overline{A}_0+\sum_{i=0}^{\infty}U_+^i\overline{A}_{i+1},
\qquad
\overline{A}_i\in K_{U_+}.
\]

At the end we shall prove that if \((\|A_i\|)_{i=0}^{\infty}\in\ell^1\) then the formal power series is a bounded operator and is the limit in uniform operator topology of a sequence of polinomials in \(U_+\).