Linear isometries of some function spaces

Isometries Between Function Spaces

“…The isometries of the complex C1 [0,1] space with the above E-norm were described by Rao and Roy [20]. EXAMPLE 5.…”

“…Lip(X)). Rao and Roy [20] proved that any isometry from the complex ACp([0,1]) space, p = 1 or oo, onto itself is canonical (cf. Example 5) and asked whether the same holds for 1 < p < oo.…”

Isometries in semisimple, commutative Banach algebras

“…In 1965 M. Cambern [1] [7] proved that this holds for algebras of Lipschitz functions and continuously differentiable functions both with norm ||/|| = ll/lloo + ll/'lloo.…”

“…The wellknown Nagasawa theorem [4] states that A and B are isometric if and only if they are isomorphic in the category of algebras, and that every linear isometry from A onto B which preserves units is an isomorphism of algebras. In [1,2,[5][6][7] it has been proved that the Nagasawa theorem remains true for some other Banach algebras.…”

Linear isometries between spaces of functions of bounded variation

“…We use only straightforward concepts, always taking into account that the functions we deal with are not continuous in general. Related results are given for instance in [2,4] and [5], where the authors study the isometries between spaces of absolutely continuous functions, endowed with a similar norm. However, in these papers the fact that the functions are absolutely continuous is fundamental to carrying out their proofs, and no similar approach can be taken in our context.…”