On weak stability of ε-isometries on wedges and its applications

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• 作者：Lixin Cheng¹ ; ^{lxcheng@xmu.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Kun Tu ^{182065417@qq.com" class="auth_mail" title="E-mail the corresponding author} ; Wen Zhang ; ² ; ^{wenzhang@xmu.edu.cn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：ε-isometry ; Stability ; Weak stability ; Banach space
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2016
• 出版时间：15 January 2016
• 年：2016
• 卷：433
• 期：2
• 页码：1673-1689
• 全文大小：412 K

文摘

In this paper, we study weak stability properties of an ε-isometry defined on a wedge W of a Banach space X, instead of the whole space X  . As a result, we show that if class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X1500760X&_mathId=si1.gif&_user=111111111&_pii=S0022247X1500760X&_rdoc=1&_issn=0022247X&md5=7a524f70cecbd3cc748ac79ec77e35f3" title="Click to view the MathML source">f:W→Yclass="mathContainer hidden">class="mathCode"> is an ε  -isometry with class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X1500760X&_mathId=si2.gif&_user=111111111&_pii=S0022247X1500760X&_rdoc=1&_issn=0022247X&md5=6f47966b498b6841ce016481d04f6af7" title="Click to view the MathML source">f(0)=0class="mathContainer hidden">class="mathCode"> for some Banach space Y  , then there exists a class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X1500760X&_mathId=si3.gif&_user=111111111&_pii=S0022247X1500760X&_rdoc=1&_issn=0022247X&md5=5c57248395052000bfb911201e864cd8" title="Click to view the MathML source">w^⁎class="mathContainer hidden">class="mathCode">-compact absolutely convex set class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X1500760X&_mathId=si120.gif&_user=111111111&_pii=S0022247X1500760X&_rdoc=1&_issn=0022247X&md5=7e57573c133b3d8f1c31dd771d679c87" title="Click to view the MathML source">B⊂B_X^⁎class="mathContainer hidden">class="mathCode"> satisfying that (a) class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X1500760X&_mathId=si35.gif&_user=111111111&_pii=S0022247X1500760X&_rdoc=1&_issn=0022247X&md5=16ce1b3c3d9f85ae8681baf4a82dc868" title="Click to view the MathML source">p(x)≡sup_x^⁎∈B⁡〈x^⁎,x〉=‖x‖class="mathContainer hidden">class="mathCode"> for all class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X1500760X&_mathId=si6.gif&_user=111111111&_pii=S0022247X1500760X&_rdoc=1&_issn=0022247X&md5=3c8ed623012adfb896c42e4a21c5b18e" title="Click to view the MathML source">x∈W∪−Wclass="mathContainer hidden">class="mathCode">; and (b) for every class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X1500760X&_mathId=si123.gif&_user=111111111&_pii=S0022247X1500760X&_rdoc=1&_issn=0022247X&md5=9e254289ddf1e499e99b4e71342241dd" title="Click to view the MathML source">x^⁎∈Bclass="mathContainer hidden">class="mathCode">, there is class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X1500760X&_mathId=si124.gif&_user=111111111&_pii=S0022247X1500760X&_rdoc=1&_issn=0022247X&md5=73042084f40a79f2c1c495ad3213832e" title="Click to view the MathML source">ϕ∈B_Y^⁎class="mathContainer hidden">class="mathCode"> so that This is a generalization of a recent result so called a universal theorem for stability of ε-isometries (but the proof is more technical). As its application, we prove that if the ε-isometry f is defined on the positive cone W   of a class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X1500760X&_mathId=si10.gif&_user=111111111&_pii=S0022247X1500760X&_rdoc=1&_issn=0022247X&md5=d69ca0c7ba8210e098c4f7d2ab5ba3eb" title="Click to view the MathML source">C(K)class="mathContainer hidden">class="mathCode">-space, or, an abstract M  -space with a strong unit (in particular, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X1500760X&_mathId=si11.gif&_user=111111111&_pii=S0022247X1500760X&_rdoc=1&_issn=0022247X&md5=424b3780cc10fc1f3af8c925cc74f1ed" title="Click to view the MathML source">ℓ_∞(Γ)class="mathContainer hidden">class="mathCode">, and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X1500760X&_mathId=si12.gif&_user=111111111&_pii=S0022247X1500760X&_rdoc=1&_issn=0022247X&md5=7252d66712a7bc5e77f298112e3d0070" title="Click to view the MathML source">L_∞(μ)class="mathContainer hidden">class="mathCode"> for a finite measure μ), then we can choose the set B   to be class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X1500760X&_mathId=si13.gif&_user=111111111&_pii=S0022247X1500760X&_rdoc=1&_issn=0022247X&md5=7721dd7fded495ab9269ecb107fd8059" title="Click to view the MathML source">B_X^⁎class="mathContainer hidden">class="mathCode">; the closed unit ball of the dual class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X1500760X&_mathId=si14.gif&_user=111111111&_pii=S0022247X1500760X&_rdoc=1&_issn=0022247X&md5=1e60c76f543eadbf805bc6a615373ff7" title="Click to view the MathML source">X^⁎class="mathContainer hidden">class="mathCode">; and further show that class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X1500760X&_mathId=si15.gif&_user=111111111&_pii=S0022247X1500760X&_rdoc=1&_issn=0022247X&md5=477764790b307a127eb5bae01a7ae535" title="Click to view the MathML source">X^⁎⁎class="mathContainer hidden">class="mathCode"> is class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X1500760X&_mathId=si3.gif&_user=111111111&_pii=S0022247X1500760X&_rdoc=1&_issn=0022247X&md5=5c57248395052000bfb911201e864cd8" title="Click to view the MathML source">w^⁎class="mathContainer hidden">class="mathCode">-to-class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X1500760X&_mathId=si3.gif&_user=111111111&_pii=S0022247X1500760X&_rdoc=1&_issn=0022247X&md5=5c57248395052000bfb911201e864cd8" title="Click to view the MathML source">w^⁎class="mathContainer hidden">class="mathCode"> continuously isometric to a subspace of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X1500760X&_mathId=si16.gif&_user=111111111&_pii=S0022247X1500760X&_rdoc=1&_issn=0022247X&md5=ed72ea427c79a39e5b58e05fe8592f04" title="Click to view the MathML source">Y^⁎⁎class="mathContainer hidden">class="mathCode">.