TAILIEUCHUNG - Báo cáo toán học: "Unicellular 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 thay đổi về không gian Banach. | J. OPERATOR THEORY 8 1982 157-165 Copyright by INCREST 1982 UNICELLULAR SHIFTS ON BANACH SPACES SANDY GRABINER INTRODUCTION Recall that a bounded operator on a Banach space is unicellular if its lattice of closed invariant subspaces is totally ordered. In 6 Theorem p. 21 we showed that every separable Banach space has a unicellular unilateral weighted shift. In the present paper we show that every separable Banach space also has a unicellular backward shift Theorem and a unicellular bilateral shift Theorem . We also prove an analogous result for diagonal operators Theorem . We construct a unicellular backward shift by an extension of the method we used in 6 Theorem p. 27 to construct unicellular forward shifts. Our construction of a unicellular bilateral shift is based partly on the existence of unicellular forward and backward shifts and partly on a theorem of Domar about scalar sequences 2 Theorem 5 Domar used his theorem to construct unicellular bilateral shifts on 2 Theorem 2 . Our shifts and diagonal operators will all be defined with respect to M-bases. Recall that an M-basis or Markushevich basis for a locally convex space X is a biorthogonal sequence x x 0 for which the span of x s is dense in X and the linear functionals X are total over X. Since has dense span the associated linear functionals are completely determined so that x alone is sometimes called an M-basis. We can use any countably infinite set not just the non-negative integers as an index set for an M-basis. Notice that if x X is an M-basis and if c is a sequence of non-zero scalars then c x x c is also an M-basis. In the case that X is a Banach space the M-basis x X always satisfies x 11 11 x x 1- The M-basis is said to be bounded if in addition there is an M 0 with x x II M for all n. It is easy to construct an M-basis in any separable Banach space 10 Proposition p. 43 and one can in fact always construct a bounded M-basis 10 Theorem p. 44 11 . 158 SANDY .

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