On linear isometries and ε-isometries between Banach spaces

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作者：Yu Zhou ; ^{roczhou_fly@126.com" class="auth_mail" title="E-mail the corresponding author} ; Zihou Zhang ^{zhz@sues.edu.com" class="auth_mail" title="E-mail the corresponding author} ; Chunyan Liu ^{cyl@sues.edu.com" class="auth_mail" title="E-mail the corresponding author}
关键词：Linear isometry ; ε-Isometry ; Stability ; Banach space
刊名：Journal of Mathematical Analysis and Applications
出版年：2016
出版时间：1 March 2016
年：2016
卷：435
期：1
页码：754-764
全文大小：331 K

文摘

Let oc=1&_issn=0022247X&md5=b9169c8cd224892fe94d3cc5a9276fda" title="Click to view the MathML source">X,Y

X, Y

be two Banach spaces, and oc=1&_issn=0022247X&md5=9b16393f420e684cd2495107cbd241d5" title="Click to view the MathML source">f:X→Y

f : X \to Y

be a standard ε -isometry for some oc=1&_issn=0022247X&md5=ca9edb8f0ada63a1f29adf8de416d133" title="Click to view the MathML source">ε≥0

ε \geq 0

. Recently, Cheng et al. showed that if oc=1&_issn=0022247X&md5=ea55e3f12461299d68b3f5220fa993fa">

\overline{co} [f (X) \cup - f (X)] = Y

, then there exists a surjective linear operator oc=1&_issn=0022247X&md5=cb7a2696ddf404016d27f54e30475622" title="Click to view the MathML source">T:Y→X

T : Y \to X

with oc=1&_issn=0022247X&md5=5f4e9701269ffab66c9df85020853a99" title="Click to view the MathML source">‖T‖=1

‖ T ‖ = 1

such that the following sharp inequality holds:

oc=1&_issn=0022247X&md5=107ae92e62d579b236e18963820bfa8c" title="Click to view the MathML source">‖Tf(x)−x‖≤2ε for all x∈X.

‖ T f (x) - x ‖ \leq 2 ε for all x \in X .

Making use of the above result, we prove the following results: Suppose that oc=1&_issn=0022247X&md5=ea55e3f12461299d68b3f5220fa993fa">

\overline{co} [f (X) \cup - f (X)] = Y

. Then

(1): if there is a linear isometry oc=1&_issn=0022247X&md5=c99550fdc820b49451b36c2d90bf68bf" title="Click to view the MathML source">S:X→Y $S : X \to Y$ such that oc=1&_issn=0022247X&md5=087859084d6eda7db94aeb9cc22cb86a" title="Click to view the MathML source">TS=Id_X $T S = {Id}_{X}$ , then oc=1&_issn=0022247X&md5=a9e4c5cfabbe563846883a8526932682" title="Click to view the MathML source">T^⁎S^⁎:Y^⁎→T^⁎(X^⁎) $T^{⁎} S^{⁎} : Y^{⁎} \to T^{⁎} (X^{⁎})$ is a oc=1&_issn=0022247X&md5=8cf9802ff546f2f88cb92b4b027bbcbb" title="Click to view the MathML source">w^⁎ $w^{⁎}$ -to-oc=1&_issn=0022247X&md5=8cf9802ff546f2f88cb92b4b027bbcbb" title="Click to view the MathML source">w^⁎ $w^{⁎}$ continuous linear projection with oc=1&_issn=0022247X&md5=9d791e4ea689b9e6dba7ec68be06822d" title="Click to view the MathML source">‖T^⁎S^⁎‖=1 $‖ T^{⁎} S^{⁎} ‖ = 1$ ,
(2): if there exists a oc=1&_issn=0022247X&md5=8cf9802ff546f2f88cb92b4b027bbcbb" title="Click to view the MathML source">w^⁎ $w^{⁎}$ -to-oc=1&_issn=0022247X&md5=8cf9802ff546f2f88cb92b4b027bbcbb" title="Click to view the MathML source">w^⁎ $w^{⁎}$ continuous linear projection oc=1&_issn=0022247X&md5=959a70ae8d4bd79b9427ceb84ebcb2f7" title="Click to view the MathML source">P:Y^⁎→T^⁎(X^⁎) $P : Y^{⁎} \to T^{⁎} (X^{⁎})$ with oc=1&_issn=0022247X&md5=da51881972597e83d5c3b9cec619b85e" title="Click to view the MathML source">‖P‖=1 $‖ P ‖ = 1$ , then there is an unique linear isometry oc=1&_issn=0022247X&md5=f24b5c577cea06986003dfc8b69cd2d4" title="Click to view the MathML source">S(P):X→Y $S (P) : X \to Y$ such that oc=1&_issn=0022247X&md5=c73f7222bd141d26928797e941f4589a" title="Click to view the MathML source">TS(P)=Id_X $T S (P) = {Id}_{X}$ and oc=1&_issn=0022247X&md5=6884b803ade205e6b8dc29683cd0e53c" title="Click to view the MathML source">P=T^⁎S(P)^⁎ $P = T^{⁎} S {(P)}^{⁎}$ . Furthermore, if oc=1&_issn=0022247X&md5=538b6dfa1bb651c4779a71a820e03ed8" title="Click to view the MathML source">P₁≠P₂ $P_{1} \neq P_{2}$ are two oc=1&_issn=0022247X&md5=8cf9802ff546f2f88cb92b4b027bbcbb" title="Click to view the MathML source">w^⁎ $w^{⁎}$ -to-oc=1&_issn=0022247X&md5=8cf9802ff546f2f88cb92b4b027bbcbb" title="Click to view the MathML source">w^⁎ $w^{⁎}$ continuous linear projection from oc=1&_issn=0022247X&md5=324755bbb5e47619121ff1ae6ca57c2f" title="Click to view the MathML source">Y^⁎ $Y^{⁎}$ onto oc=1&_issn=0022247X&md5=d981b6b2f7e3f0c26ffeb15918c279fe" title="Click to view the MathML source">T^⁎(X^⁎) $T^{⁎} (X^{⁎})$ with oc=1&_issn=0022247X&md5=d4237e171b900232b9e082fa37e248d2" title="Click to view the MathML source">‖P₁‖=‖P₂‖=1 $‖ P_{1} ‖ = ‖ P_{2} ‖ = 1$ , then oc=1&_issn=0022247X&md5=d8696eb44c03cac6a141d365be030dba" title="Click to view the MathML source">S(P₁)≠S(P₂) $S (P_{1}) \neq S (P_{2})$ .

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.

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