2017H5^ ADVANCES IN MATHEMATICS (CHINA) May, 2017

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1 246fi23L < f Vol. 46, No H5^ ADVANCES IN MATHEMATICS (CHINA) May, , F (3) *ο»ff(+fiμ2χ ]w< (f^ma; ;ffi, f^, ff, ) doi: /sxjz b 3-: 9rjt d^kjg=qdl_, [ az±~raksbcb^ F (3) yytexv ΦmO>D9±Ξ, ΩP ;ufzcb^ E (3) YT Tingley sn> D? ou, Πh}McxDWN. Ω: EXvzlΨ; ;ufz; F (3) yyt MR(2010) 7!fflß: 46A22; 46B20 / 6"fflß : O177.3 $& ψfl: A $4± : (2017) r 30 G" Banach ff/ß*$,8n, -ρ += FvMY%2 +Yfl μtvb+0μ. Banach `ν# [1] uqρ3$ c e+-ρ (%2) 0q. %, $ +ßΦ@z0q#6O(MxeΠ2Fv-ρ +"zff"0q..ad»iu$ -ρ% 2?hY9z f (MYW} q+ff"0q dωkφνi A, M"z0q9 A T A = AA T = E), fifi_+l@ff YE*Iu$ H ΠT+Es2t. tg 6E*WßΦ+H FWl@eyfi8? S D, DPDΛ+ß*kd»H@$ -ρ +"zff"0qtae?!, hpο TxW8H@$ +@zπ2ymxe2t v"zfiv+yfltsk"+,fl, 6DC>, a_fw[6$ +%2-ρ +0q, 8χ y+ Tingley ο +flflott!ξx. ffi 1987 G Tingley `ν# [9] u±ρ3#μv>-ρ> ο F), 6Z]FvCfiG- 37:R?2 ffa2+tλ, *'3»4H #μv>9-ρ +ff"0qk"+fi_ (Ξ [3 4, 6]), ffi;q±flfl3'n+x4 Banach $ e+ Tingley ο [5]. $t, ν [5] ucπfiqρ, 8WEßΦH@+DΛ+ Banach $, sffl T!+ d, wp8w 2» +H@$, Cl;T#6+nψpg?.. fiο +1D'Dt, `;TßΦH@0q+U& χ,.als8 2»+H@$, ß=CgD8M#μV (>) 1TKWο flfl+±t?y pgf (T$) +Ξ8, s:}mtyw[38»ffi+. ν/-`ν [2] u8 T$» $, M# μv>0a :>Φ +ΠxU& 13Ξ8, *'3D,b+gTP=IJ+fi_. ν [7 8] u `@zyw#μqßt8»2d#52-2t+ωπχ, *'3DΛH@$ E (2) d E A (n) +% 2-ρ +0q. ß=wψDν [7 8] 6J3 F (Γ) $, h-;t 6ßΦ+H@0q, ffit'8ν [7 8], Λ *'3 F (3) $ %2-ρ +ff"0q. vo`l: MOv(`L: rffi)c: ^ ~]ψ;rffi (No , No ). liangxiaobin2004@126.com

2 3L 0+fl: F (3) /%,&3.ffQi Πffi/ fl. 2.1 j Γ={1, 2,,n} tdpqωu, aχ` {e s } n s=1 t Banach $ X +D # μq, π Γ Γ DDPh, x X, z s Γ x(s)e s, T (I) s Γ x(s)e s = s Γ x(s) e s = s Γ x(s)e π(s). ß=»9 fiωπ+ X F (n).$ (Γ tρ$uc!t+~+6j). F (n).$ t' %[B+, soe, MΠ(Iu$ H p- d@zt$»$ B,H sup, --. 6, g:ff>eχo)e F (n).+$, M±tZlt (pg}!fzk ) F (n). $. a: j X =span{e i } n, x = n x ie i, H x =( n λ i x i p ) 1 p, λi > 0, p 1, pg H x =(λ n x i p +(1 λ)max{ x 1 p, x 2 p,, x n p }) 1 p, p 2, 0 <λ<1. a_$ X l9 : (II) b y X, 8<P s Γ, T x(s) y(s), c x X T x y. bχ` s N, p x(s N ) < y(s N ), c x < y. (III) b x =1, s Γ (x(s))2 =1%T % x(s) rtdgffl 5.»fio Banach $ E (n).$, {e s } n s=1 t fi$ (a_ (III) L : (A)b x(s) sg1gffl 5, x =1, c s Γ (x(s))2 < 1. pg (B) b x(s) sg1gffl 5, x =1, c s Γ (x(s))2 > 1. ha+ E (n).$ Ffi» E A (n) p EB (n).$.) ν [7 8] +wbfi_tyw E (2) d E A (n).$ +. ±ν%2-ρ +ff"0qt` F (3).$ e*'+, ;TC>C Yfflfi. 2 1ff ffiff 0ρ 2.1 [8] j V t F (n).$ H'fi@ X, Y n +o-ρ%2, {e k } n k=1, {γ k} n k=1 Ffit X, Y $ (V (e 1 ),V(e 2 ),,V(e n )) T = A(γ 1,γ 2,,γ n ) T, A =(a ij ) t n ΩoCi, c A +ΠjR+A 1. 5ffi ν [8] u+l@t` E (n).$ e 1+, ;Mρ64L8 F (n) Ct /+. 0ρ 2.2 b A =(a ij ) t n ΩoCi, λe A = λ n + b 1 λ n b n 1 λ + b n, Mu b k -W A +DS k Ωw q (w:b A + i 1 i 2 i k 1Tt i 1 i 2 i k 4+ TZ [8μ V +14q) ndμf ( 1) k. z A[i 1 i 2 i k ] A + i 1 i 2 i k 1Fv4V + k Ωw q, i 1i 2 i k A[i 1 i 2 i k ] νi A + T k Ωw qnd, c b k =( 1) k i 1i 2 i k A[i 1 i 2 i k ]. 8 1 b A =(a ij ) F (n).$ H'fi@ X, Y n +o-ρ%2, {e k } n k=1, {γ k} n k=1 Ffit X, Y $ (V (e 1 ),V(e 2 ),,V(e n )) T = A(γ 1,γ 2,,γ n ) T. L F (n). $ T2t (I),!m n n n n n x i y i e i = x i e i y i e i = ((x k y k )a ki )γ i k=1 n n = ((x k y k )(a ki )) γ π(i) k=1 n n = ((x k y k )( a ki ))γ π(i), k=1 ffldχρ, νi A +_I4 (pg1) e TZ Ψ}JcE t-ρ%2, νi A _IΦ

3 426 ; e 46fi m14 (pg11) *'+-νiht-ρ%2. Fχß=z A νi A V +14q, z A ij t A +1 i 11 j 4Z +X q (w A jx1 i 11 j 4Z V +14q). flρ 2.1 j V t1ph'fi@z+ F (3).$ X, Y n +-ρ%2, c V dωkφνi. 5ffi j (V (e 1 ),V(e 2 ),V(e 3 )) T = A(γ 1,γ 2,γ 3 ) T, A =(a ij ) t 3 ΩoCi, {e k } 3 k=1, {γ k} 3 k=1 Ffit X, Y $ V t1ph'fi@z+ F (3).$ X, Y n +-ρ%2. (i) SM, 2.1, A +ΠjR+A 1,!j A = θ, θ = ±1, L A NzΩi, csgtdr 1 pg 1, Gm A TΠjR θ. GcpgdR θ, pgdr θ 61RUΞ, 7Y A = θ :9. Wtß=T θe A =0. SM, 2.2 T θ 3 +( 1)θ 2 i 1 A[i 1 ]+( 1) 2 θ i 1i 2 A[i 1 i 2 ]+( 1) 3 A =0. S A[1] = a 11,A[2] = a 22, A[3] = a 33 Fv A[12] = A 33, A[13] = A 22,A[23] = A 11, wt θ (a 11 + a 22 + a 33 )+θ(a 11 + A 22 + A 33 ) θ =0. (1) z A (1) A ΨJc *νi. S A (1) H -ρ%2, c A (1) H 6TΠj R+ 1 p 1, yi' A (1) = θ, fia0!m A (1) TΠjR θ. z i 1i 2 i k A (1) [i 1 i 2 i k ] νi A (1) + k Ωw qnd, SM, 2.2, VT θ i 1 A (1) [i 1 ] θ i 1i 2 A (1) [i 1 i 2 ]+θ =0. (2) yi' A (1) [1] = a 11, A (1) [2] = a 22, A (1) [3] = a 33, A (1) [12] = A 33, A (1) [13] = A 22, A (1) [23] = A 11. (3) S (2) (3) T S (1) (4)!* θ ( a 11 + a 22 + a 33 ) θ(a 11 A 22 A 33 )+θ =0. (4) a 11 = θa 11. (5) S A = θ!m, b A -ρ%2, c T a 11 = A A 11. (ii) b8νi A +1 i 1d i 1 1Φm*'-νi, hf*'+-νi+1 i 1 1d i 2 1Φm, EflχX, os± A +1 i 1Φm'1D1; fia0eflφm4, ±1 j 4Φm' 1D4; hap* a ij μw h-νi+1d11d4, ß=z h+-νi A i,j. A i,j H t-ρ%2. T A i,j =( 1) i+j θ, F A i,j ToΠjR ( 1) i+j θ. 8νi A i,j viafi A +Ξ8. E (5) q, ß=T a ij =( 1) i+j θa i,j 11, ;Rο*' A i,j +ΨmC>,! A i,j 11 = A ij. LfiwT a ij =( 1) i+j A A ij, F A T A = AA T = E. w V + "zff"0q dωkφνi.

4 3L 0+fl: F (3) /%,&3.ffQi 427 #fi 2.1 a_ V td»iu$ R 3 e+o%2-ρ, c V YW} q+"zff" 0q dωkφνi. 8 2 YW%2-ρ +=, ±)t flfl Tingley ο. "`ß==", 6, 2.1 fi 8+IJ, /+K sg7!f}, h-qρ3dpffle*wd»iu$ Es2t;*'M %2-ρ +ff"0q?hy9z f hpfi8+dpy-l@, Tfil@C>'% ff, %, fic>8nz»u0nkttp=2+ (fin A +ΠjR+A 1, c A TΠjR +1 pg 1), $»z]n, fld+l@c>y 2μΠfi!TI_, 6 Jz»U0χ A ρfi 2t, l@μ ffl'fi. `»4+ν#pgΨ u, ß=m)Iu$ u%2-ρ YW} q+ff"0qtkφνi (A T A = AA T = E), Λ+l@t YE*WEs2t (pg}ts Es U&ρ)+ßΦ+ C;T_eν#TM l>pg±'l@uesωπte B+. "`ffiß=6, 2.1 +fi8!fχρ, BΩΠ!FΛc d»$ χ`h@8»q (w F (n).$ +2t (I), ;d»iu$! 9 fi), hctxwß=sk"0,fl%2-ρ dkφνi+yflfvh@8»q2t+λr. Iu$ o%2-ρ kφνi,?`), a IkΦνi7V fi$ YW} q+%2-ρ,!fl@fi$ Iu$, h!ft8iu$ +DP"i. ffifχffl 8ß=!Fχ' E (n) +.$ (ΠaH p, p 2 d@z H sup, -) μ ρ>ff` 9zΨm *'ßT'fiH@0qYp+-q. %, Cχ`M +T$»H@$, ffi[qλbd 9z *'ßT'fiH@0qYp+-q..a<»$ Fk Φ0 #μv> > +H@$. #fi 2.2 a_ V t1ph'fi@z+ E (3).$ X, Y +o%2-ρ, {e k } 3 k=1, {γ k } 3 k=1 Ffit X, Y $ c V 0q : V (e i)=θ i γ π(i), θ i = ±1, i =1, 2, 3. Mu π D {1, 2, 3} {1, 2, 3} +DDPh. 5ffi j (V (e 1 ),V(e 2 ),V(e 3 )) = (γ 1,γ 2,γ 3 )A T, c 1= V (e 1 ) = a 1j γ j, j=1 ; A T A = AA T = E, T (a 1j ) 2 =1. j=1 yi' E (3).$ T2t (III), F%T %QTDgffi8p 1 n /. M 1fi,!l. Lfi, ffl8*l. flρ 2.2 a_ V 0 t1ph'fi@z+ E (3).$ X, Y #μv> +o-ρ, {e k } 3 k=1, {γ k} 3 Ffit k=1 X, Y $ c V 0!> Y$ +-ρ%2 V 0 +0q : V 0 (e i )=θ i γ π(i), θ i = ±1,, 2, 3, Mu π D {1, 2, 3} {1, 2, 3} +DDP h. T x S(X), x= 3 x ie i,v 0 (x) = 3 θ ix i γ π(i). 5ffi B2. ~6 V 0 F> Y$ +%2 V, Sffl8 2.2, T V 0 (e i )=V (e i )=θ i γ π(i), θ i = ±1,, 2, 3,

5 428 ; e 46fi Mu π D {1, 2, 3} {1, 2, 3} +DDPh. b x S(X), x= 3 x ie i, c ( ) V 0 (x) =V (x) =V x i e i = x i V (e i )=x i V 0 (e i )= θ i x i γ π(i). οf2. x E (3), 7 0, x =0, ( ) V (x) = x x V 0, x 0. x! V V 0 +>, S2t (I) d (II) ffld@l V H %2-ρ. Φ) tg8w_i F (n).$ +-ρ%2 7T±ν+fi_? Ψν%' [1] Banach, S., Théorie des Opérations Linéaires, Warszawa: Z Subwencji Funduszu Kultury Narodowej, 1932 (in French). [2] Cheng, L.X. and Dong, Y.B., On a generalized Mazur-Ulam question: extension of isometries between unit spheres of Banach spaces, J. Math. Anal. Appl., 2011, 377(2): [3] Ding, G.G., The isometric extension problem in the unit spheres of l p(γ) (p >1) type spaces, Sci. China Math., 2002, 32(11): [4] Ding, G.G., The representation of onto isometric mappings between two spheres of l -type spaces and application on isometric extension problem, Sci. Sin. Math., 2004, 34(2): (in Chinese). [5] Ding, G.G., On the linearly isometric extension problem, Sci. Sin. Math., 2015, 45(1): 1-8 (in Chinese). [6] Fang, X.N. and Wang, J.H., Extension of isometries on the unit sphere of l p(γ) space, Sci. China Math., 2010, 53(4): [7] Liang, X.B. and Huang, S.X., On the representation of linear isometries between the E (2) type real spaces, ActaMath.Sci.Ser.AChin.Ed., 2010, 30(4): (in Chinese). [8] Liang, X.B. and Xie, X.H., On the representation of linear isometries between the En A type real spaces, Pure Appl. Math. (Xi an), 2014, 30(2): (in Chinese). [9] Tingley, D., Isometries of the unit sphere, Geom. Dedicata, 1987, 22(3): On the Representation of Isometric Linear Operators in the F (3) Type Spaces LIANG Xiaobin (Department of Mathematics, Shangrao University, Shangrao, Jiangxi, , P. R. China) Abstract: In the paper, we introduce the concept of normed symmetric bases. By using its properties and algebraic techniques, we get the representation theorem of isometric linear operators in F (3) type Banach spaces, so as to reveal the nature that the linear isometric operators are the synthesis of reflections and rotations in 3-dimensional Euclidean space. Finally, we also use the representation theorem to obtain a necessary and sufficient condition of Tingley issues in E (3) space, which is a new result. Keywords: isometric linear operators; representation theorem; F (3) type space