# Journal of Operator Theory

Volume 34, Issue 2, Fall 1995 pp. 217-238.

On polynomially bounded weighted shifts. II**Authors**: Srdjan Petrovic

**Author institution:**Department of Mathematics, Indiana University, Bloomington, IN 47405-5701, U.S.A.

**Summary:**Let T be an operator-weighted shift whose weights are 2-by-2 matrices. We say that, given $\varepsilon > 0$, T is in the e-canonical form if each weight is an upper triangular matrix (a_{i,j}), with $0 \le a_{11} ,a_{22} \le 1$ and $a_{12} \ne 0$ implies $a_{11} ,a_{22} < \varepsilon $. We generalize this concept to operator-weighted shifts whose weights are n-by-n matrices and we show that every polynomially bounded weighted shift, whose weights are finite-dimensional matrices of the fixed dimension n, is similar to an operator in the $\varepsilon$-canonical form. This enables us to prove that every polynomially bounded weihghted shift with finite dimensional weights is similar to a contraction.

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