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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Similarity of certain operators in $ l\sp{p}$


Author: Shmuel Kantorovitz
Journal: Proc. Amer. Math. Soc. 67 (1977), 99-104
MSC: Primary 47B37
MathSciNet review: 0454719
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Abstract: Let M be the multiplication operator in $ {l^p},1 \leqslant p \leqslant \infty $, i.e., $ M:x = \{ {x_k}\} \to \{ k{x_k}\} $. Let $ w = \{ {w_j}\} _{j = 0}^\infty $ be a weight, i.e., a positive sequence such that $ {w_1} < {w_0} = 1$ and $ {w_{n + m}} \leqslant {w_n}{w_m}$. For $ \zeta \in C$, define $ N_w^\zeta $ on $ {l^p}$ by

$\displaystyle {(N_w^\zeta x)_k} = \sum\limits_{j = 1}^k {\left( {\frac{{{w_k}}}... ... - 1 + k - j} \\ {k - j} \\ \end{array} } \right){x_j}\quad (k = 1,2, \ldots ).$

Then $ \{ N_w^\zeta ;\zeta \in C\} $ is a holomorphic group of operators, and for any function g holomorphic on the spectrum of $ N_w^\zeta ,M + g(N_w^\zeta )$ is similar to $ M + g(1)I$.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1977-0454719-4
PII: S 0002-9939(1977)0454719-4
Keywords: Holomorphic group of operators, closed operator, similarity, spectral operator
Article copyright: © Copyright 1977 American Mathematical Society