ON PERTURBATIONS AND EXTENSIONS OF ISOMETRIC OPERATORS

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.