On Foguel's Answer to Nagy's Question
نویسنده
چکیده
Nagy's question is whether or not every power-bounded operator is similar to a contraction [3]. ("Power-bounded" means that the norms of the positive powers are bounded.) Foguel's answer is no [l]. The purpose of this note is to look at Foguel's ingenious counterexample from a point of view somewhat different from his own. The advantage of the new look is that it is less computational; its drawback is that the intuitive motivation is less transparent. Let Ho he a Hubert space with an orthonormal basis {e0,ei,e2, ■ ■ ■}, and let 5 be the unilateral shift on H0 (Sen = en+i, ra = 0, 1, 2, • • • )• Let / be an infinite set of natural numbers that is "sparse" in the sense that if i and j belong to / and * <j, then 2i <j. (Example : / can be the set of positive integral powers of 3.) Let Q he the projection from Ho onto the span of all the e/s with j in /. If H is the direct sum of two copies of H0 (the set of all ordered pairs (/, g ) with / and g in Ho), then every operator on H is given by a two-by-two matrix whose entries are operators on H0. Principal assertion: if
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