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Báo cáo toán học: "Non-unicellular strictly cyclic quasinilpotent shifts on Banach spaces "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Đơn bào không đúng theo chu kỳ quasinilpotent thay đổi về không gian Banach. | J. OPERATOR THEORY 13 1985 163- 170 Copyright by INCREST 1985 NON-ƯNICELLULAR STRICTLY CYCLIC QUASINILPOTENT SHIFTS ON BANACH SPACES SANDY GRABINER and MARC p. THOMAS 1. INTRODUCTION The sequence of unit vectors e s in the Banach space X is a normalized M-basis if its span is dense and if there is a total sequence of linear functionals e o for which e em 5 im. The sequence e is uniquely determined by the above conditions. The usual basis of ỉp 1 p oo is a normalized Af-basis and it is easy to construct normalized Af-bases on any separable Banach space 8 Proposition l.f.3 p. 43 Suppose that w o is a sequence of non-zero scalars. The bounded operator T on X is a weighted shift with weights iv i w with respect to the normalized Af-basis if T w e w 1e 1 for all n 0. On tp the sequence w determines such a shift if and only if the sequence of weights is bounded but for completely arbitrary normalized Af-bases it may be necessary to require that X K 1 w le converge. It has long been recognized 5 9 10 that a convenient way to study weighted shifts and their invariant subspaces is to replace the space X by a Banach space B of formal power series called the space of power series determined by w in such a way that the shift T is represented by multiplication by the indeterminate z. Explicitly we identify z with w e and more generally we identify X in X with X w z in B see 5 pp. 19 20 for a more detailed description of this identification . When X ỉp with the usual basis the space of power series determined by w is just the weighted ip space p w of all power series f 2 z 0 for which the norm ll ll ll llp CO 11 p Xi l nW p is finite. The weighted shift is 0 said to be strictly cyclic when the space B of power series determined by w is an algebra. Our use of the term strictly cyclic weighted shift is equivalent to the usual meaning of the term in operator theory 7 10 Proposition 31 p. 94 4 pp. 83-85 . 164 SANDY GRABINER and MARC p. THOMAS Any subalgebra B of the algebra of .