Abstract
For complex Hilbert space H of d dimensions and for any number K ⩾ 1, we may define m( K, d) as the least number with the following property: if ‖ p( T)‖ ⩽ K for all polynomials p mapping the complex unit disk into itself, then the operator T may be made a contraction by changing to a new norm |·|, derived from an inner product, such that ‖h‖ ⩽ |h| ⩽m(K,d)‖h‖ (h∈H). It is a long-standing open question whether m( K, d) has a finite bound independent of d. The present paper studies this and related questions and provides, in particular, an explicit estimate for m( K, d)—which, however, grows with d.
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