THREE KINDS OF NUMERICAL INDICES lp-SPACES

초록

In this paper, we investigate the polynomial numerical index n((k))(l(p)), the symmetric multilinear numerical index n(s)((k))(l(p)), and the multilinear numerical index n(m)((k))(l(p)) of l(p) spaces, for 1 <= p <= infinity. First we prove that n(s)((k))(l(1)) = n(m)((k))(l(1)) = 1, for every k >= 2. We show that for 1 < p < infinity, n(I)((k))(l(p)(j+1)) <= n(I)((k))(l(p)(j)), for every j is an element of N and n(I)((k))(l(p)) = lim(j ->infinity)n(I)((k))(l(p)(j)), for every I = s, m, where l(p)(j) = (C-j, parallel to center dot parallel to(p)) or (R-j, parallel to center dot parallel to(p)). We also show the following inequality between n(s)((k))(l(p)(j)) and n((k))(l(p)(j)): let 1 < p < infinity and k is an element of N he fixed. Then c(k : l(p)(j))(-1) n((k))(l(p)(j)) <= n(s)((k)) (l(p)(j)) <= n((k))(l(p)(j)), for every j is an element of N boolean OR {infinity}, where l(p)(infinity) := l(p), c(k : l(p)) = inf { M >0 : parallel to Q parallel to <= M parallel to Q parallel to, for every Q is an element of P((k)l(p))} and Q denotes the symmetric k-linear form associated with Q. From this inequality, we deduce that if l(p) is a complex space, then lim(j ->infinity)n(s)((j))(l(p)) = lim(j ->infinity)n(m)((j))(l(p)) = 0, for every 1 < p < infinity.

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