Unitary equivalence and reductibility or invertibly weighted shifts

Source abstract

Let H be a complex Hilbert space and let { A 1 , A 2 , …} be a uniformly bounded sequence of invertible operators on H . The operator S on l 2 ( H ) = H ⊕ H ⊕ … given by S 〈 x 0 , x 1 , …〉 = 〈0, A 1 x 0 , A 2 x 1 , …〉 is called the invertibly veighted shift on l 2 ( H ) with weight sequence { A n }. A matricial description of the commutant of S is established and it is shown that S is unitarily equivalent to an invertibly weighted shift with positive weights. After establishing criteria for the reducibility of S the following result is proved: Let { B 1 , B 2 , …} be any sequence of operators on an infinite dimensional Hilbert space K . Then there is an operator T on K such that the lattice of reducing subspaces of T is isomorphic to the corresponding lattice of the W * algebra generated by { B 1 , B 2 , …}. Necessary and sufficient conditions are given for S to be completely reducible to scalar weighted shifts.