Abstract
In this paper, some recent advances and open problems on perturbations and extensions of the isometric operators are presented. It is shown that for the spaces $B(L^\beta(\mu) \longrightarrow L^\beta(\nu)) (0 \lt \beta \lt 1), B(L^\infty(\mu) \longrightarrow L^\infty(\nu))$ (with some special measure spaces), $B(E_{(2)} \longrightarrow L^1(\mu)), B(E_{(n)} \longrightarrow C(\Omega))$ and $B(X \longrightarrow L^\infty(\mu))$, where $X$ is a uniformly smooth Banach space with $\dim X=1$ or $\infty$ and $(\Omega, \mu)$ is a purely atomic or purely non-atomic finite measure space and so on, the answer to the problem of the isometric approximations is positive. However, for the spaces $B(l^1 \times l^\infty), B(L^1(\mu) \longrightarrow L^\infty(\nu))$ and $B(L^1(\mu) \longrightarrow C_b(\Delta))$, the answer is negative.
Citation
Guanggui Ding. "ON PERTURBATIONS AND EXTENSIONS OF ISOMETRIC OPERATORS." Taiwanese J. Math. 5 (1) 109 - 115, 2001. https://doi.org/10.11650/twjm/1500574890
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