On linear isometries and ε-isometries between Banach spaces

Let X,YX,Y be two Banach spaces, and f:X→Yf:X→Y be a standard ε  -isometry for some ε≥0ε≥0. Recently, Cheng et al. showed that if co‾[f(X)∪−f(X)]=Y, then there exists a surjective linear operator T:Y→XT:Y→X with ‖T‖=1‖T‖=1 such that the following sharp inequality holds:‖Tf(x)−x‖≤2ε for all x∈X.‖Tf(x)−x‖≤2ε for all x∈X. Making use of the above result, we prove the following results: Suppose that co‾[f(X)∪−f(X)]=Y. Then(1)if there is a linear isometry S:X→YS:X→Y such that TS=IdXTS=IdX, then T⁎S⁎:Y⁎→T⁎(X⁎)T⁎S⁎:Y⁎→T⁎(X⁎) is a w⁎w⁎-to-w⁎w⁎ continuous linear projection with ‖T⁎S⁎‖=1‖T⁎S⁎‖=1,(2)if there exists a w⁎w⁎-to-w⁎w⁎ continuous linear projection P:Y⁎→T⁎(X⁎)P:Y⁎→T⁎(X⁎) with ‖P‖=1‖P‖=1, then there is an unique linear isometry S(P):X→YS(P):X→Y such that TS(P)=IdXTS(P)=IdX and P=T⁎S(P)⁎P=T⁎S(P)⁎. Furthermore, if P1≠P2P1≠P2 are two w⁎w⁎-to-w⁎w⁎ continuous linear projection from Y⁎Y⁎ onto T⁎(X⁎)T⁎(X⁎) with ‖P1‖=‖P2‖=1‖P1‖=‖P2‖=1, then S(P1)≠S(P2)S(P1)≠S(P2). We apply these results to provide an alternative proof of a recent theorem, which gives an affirmative answer of a question proposed by Vestfrid. We also unify several known theorems concerning the stability of ε-isometries.