a~ 1.1 [4] x, y X. x + λy x, λ C, Ifi x 4 y Φ Birkhoff MIß, a~ 1.2 [8] ε [0, 1), x, y X. x + λy 2 x 2 2ε x λy, λ C, Ifi x 4

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1 fl45xfl4r ffi - R K Vol.45, No q7F ADVANCES IN MATHEMATICS (CHINA) July, 2016 d ju Birkhoff Πh`fff! " (~i,efl,ba <9<,E, 9<fl,/V, ~i, }χ, ) doi: /sxjz b Λ}: 1» ψff4, μffi#ffi) Birkhoff 2fiffl- ρχß, *ν#.0fflfl$, fl& 3%#ffi)fi ρ +- 2fiοffi)fi ρ - 2fi+,/' (ffi)fi Birkhoff 2fi+,/'. cg]: Birkhoff 2fi; ffi) Birkhoff 2fi; ffi)fi Birkhoff 2fi/' MR(2010) Ωvbl: 47D45; 46L40 / Ψwble: O177.1 y [sp: A yξze: (2016) fif.u($ Fß?6(Q, R-MI B- MI I- MIffMI(ffi5Λ%D7 4.U (G [1 2, 4, 6, 12, 14 17]). Sp^, *!+Lρ2f3 MI(ffi5, ομf-ssfflmi(ffi5. fl [8] H($ FR. fsffl B- MIßffi54(Q, fl [9 10] Ho= F. fsfflmiß ffi5, fl [11] HΞ($ FR7 fsffl I- MIßffi5, fl [12 13, 18] HΞ($ FRM f Sffl ρ + - MI Sffl ρ - MI4Sffl ρ- MIßffi5Φ.Ufs(Q. H3 -SMI(ffi5 R, B-MI> Birkhoff MIΩμf*!+Lß0T: fl [3] HΞ($ FR. f'd x 4 y Birkhoff MIß2,ffl0μH; fl [13] HΞ($ FR. f'd x 4 y Sffl Birkhoff MI ßffl0μH; fl [7] H)($ FR.Uf'd x 4 y Birkhoff MIßffl0μH, ρ2ffl [3] ßΩ&Lh; fl [5] H)($ FROlf ffi Birkhoff MI$(: ΦffWßfffi. Ω ff Lhßt#, ΞflH)($ FR,. f'd x 4 y Sffl Birkhoff MIß2,ffl0μH, Φu OlfSffl ρ + - MI4Sffl ρ - MI$(: χφsffl Birkhoff MI$(:. 1 YΦt io HΞflR, X, Y Y± )($ F, Z ± Ξ($ F, C ± )fic, R ± ΞfiC. a~ 1.1 [4] x, y X. x + λy x, λ C, Ifi x 4 y Φ Birkhoff MIß, Bfi x B y. a~ 1.2 [8] ε [0, 1), x, y X. x + λy 2 x 2 2ε x λy, λ C, Ifi x 4 y ΦSffl Birkhoff MIß, Bfi x ε B a~ y. 1.3 : [13] T : X Y, ε [0, 1). ψy4ß x, y X, x B y T (x) ε B T (y), Ifi T ΦSffl Birkhoff MIß. a~ 1.4 [12] x, y X, ρ ±(x, y) = lim t 0 ± x+ty 2 x 2, fifi$fi fi. Ψ+zr: )*+Ψνzr: ;N&m: }χλz@νxxz,; /VA9&m (No. 2012JM1018) 5}χΛKDνV&Z,/V&m (No. 15JK1221). kongliang2005@163.com 2t

2 604 fl, Q J 45X a~ 1.5 [12] x, y X. ρ + (x, y) =0, Ifi x 4 y Φ ρ +- MIß, Bfi x ρ+ y. ρ (x, y) =0, Ifi x 4 y Φ ρ - MIß, Bfi x ρ y. a~ 1.6 [12] ε [0, 1), x, y X. ρ +(x, y) ε x y, Ifi x 4 y ΦSffl ρ + - MI ß, Bfi x ε ρ + y. ρ (x, y) ε x y, Ifi x 4 y ΦSffl ρ - MIß, Bfi x ε ρ y. a~ 1.7 : [12] T : X Y, ε [0, 1). ψy4ß x, y X, x ρ+ y T (x) ε ρ + T (y), Ifi T ΦSffl ρ + - MIß. ψy4ß x, y X, x ρ y T (x) ε ρ T (y), Ifi T Φ Sffl ρ - MIß.» fi, fl# [3] HΞ($ FRa;$fi fi. f'd x 4 y Birkhoff MIßffl0μH: am 1.1 x, y Z, α R, I x B (y αx) ρ (x, y) α x 2 ρ + (x, y). x B y ρ (x, y) 0 ρ + (x, y). fl# [13] HΞ($ FRρ2fffi` 1.1 ßLh, ομf'd x 4 y Sffl Birkhoff MIß ffl0μh: am 1.2 ε [0, 1), x, y Z, α R, I» fi, x ε B (y αx) ρ (x, y) ε x y αx α x 2 ρ + (x, y)+ε x y αx. x ε B y ρ (x, y) ε x y 0 ρ + (x, y)+ε x y. 8Wwßffi Φ: H)($ FRΦ'1>_fflffi` 1.1 4ffi` 1.2 ßLh? fl# [7] H) ($ FR!.Uf$fi fiß(q (ffi` 1.3), P". f x 4 y Birkhoff MIßffl0μH (ffi` 1.4). am 1.3 x, y X, α, β C, I (1) ρ (x, y) = ρ + (x, y) = ρ + ( x, y); (2) ρ ±(x, x) = x 2 ; (3) ρ ± (αx, βy) = αβ ρ ± (x, ei(θ ω) y), sr α = α e iω, β = β e iθ ; (4) ρ ± (x, αx + y) =Re(α) x 2 + ρ ± (x, y); (5) ρ ±(x, y) x y. am 1.4 x, y X, I» fi, x B y ρ (x, e iθ y) 0 ρ +(x, e iθ y), θ R. ΞflH)($ FR. 'd x 4 y Sffl Birkhoff MIßffl0μH, ομfψkßlh. am 1.5 (y αx)»uo» ε [0, 1), x, y X, α C, I x ε B ρ (x, e iθ y) ε x y αx Re(e iθ α) x 2 ρ +(x, e iθ y)+ε x y αx, θ R. (1) x ε B y ρ (x, e iθ y) ε x y 0 ρ +(x, e iθ y)+ε x y, θ R.

3 4r ]e: 1AT Birkhoff NJρUC io_ r fifffflffi` 1.5 ßOl,!M?,7`. m 2.1 ε [0, 1), x, y X. x ε B y, I x ε B ±q =ffi5 ( y). 1.2 [P. m 2.2 ε [0, 1), x, y X, I x ε B y x + te iθ y 2 x 2 2ε x te iθ y, t 0, θ R x + se iθ y 2 x 2 2ε x se iθ y, s, θ R. ±q =ffi5 1.2 [P. m 2.3 u, v X, a R, ffi5: ϕ : R R fi ϕ(s) = u + sv 2 + a u sv. ϕ(0) <ϕ(s), s (b 1,b 2 ) \{0}, sr b 1 < 0, b 2 > 0, I ϕ(0) ϕ(s), s R. ±q 6fi s 1,s 2 R, ( ) s1 + s 2 ϕ = 2 u + s 1 + s 2 2 v 2 + a u s 1 + s 2 v 2 ( u 2 + s 1 2 v u s ) 2 2 ( 2 v s 1 + a u 2 v s ) v ( u s 1 2 v 2 u s ) 2 2 v 2 + a u s 1 2 v s 2 + a u 2 v = ϕ(s 1) 2 + ϕ(s 2), 2 3 ϕ fiß3fi. s R \{0}, v σ (0, 1) Π (1 σ)0 + σs (b 1,b 2 ) \{0}, I= ϕ Φß3fiο ϕ(0) <ϕ((1 σ)0 + σs) (1 σ)ϕ(0) + σϕ(s). " ϕ(0) <ϕ(s), s R \{0}. / ϕ(0) ϕ(s), s R. am 1.5 ^±q Π0(. (1) t 0, θ R, = x ε B (y αx) 47` 2.2 x + te iθ (y αx) 2 x 2 2ε x te iθ (y αx) = x 2 2ε x t(y αx), "=ffi` 1.3 (4) ο ρ + (x, x + te iθ (y αx) 2 x 2 eiθ (y αx)) = lim ε x y αx, t 0 + 2t ε x y αx ρ + (x, eiθ (y αx)) = ρ +(x, e iθ αx +e iθ y) =Re( e iθ α) x 2 + ρ + (x, eiθ y) = Re(e iθ α) x 2 + ρ +(x, e iθ y).

4 606 fl, Q J 45X / Re(e iθ α) x 2 ρ + (x, eiθ y)+ε x y αx, θ R. (2) t 0, θ R, = x ε B (y αx) 47` 2.1 ο, x ε B I " x + te iθ (y αx) 2 = x +( t)e iθ [ (y αx)] 2 x 2 2ε x te iθ (y αx) = x 2 2ε x t(y αx), x + te iθ (y αx) 2 x 2 ε x y αx, t <0, 2t [ (y αx)], G=7` 2.2 [P ρ (x, x + te iθ (y αx) 2 x 2 eiθ (y αx)) = lim ε x y αx. t 0 2t 8;ffi` 1.3 (4), _ffl (1) ßOl[ο, Re(e iθ α) x 2 ρ (x, e iθ y) ε x y αx, θ R. / ρ (x, eiθ y) ε x y αx Re(e iθ α) x 2 ρ + (x, eiθ y)+ε x y αx, θ R. ffl&(. Ψ%E x 0, y αx 0. (1) ±flb, I θ R, =ffi` 1.3 (4) P Re(e iθ α) x 2 ρ +(x, e iθ y)+ε x y αx. ε x y αx ρ + (x, eiθ y) Re(e iθ α) x 2 = ρ +(x, e iθ (y αx)), nj x + se iθ (y αx) 2 x 2 2ε x y αx 2 lim, θ R. s 0 + 2s ψy4ß γ (0, 1), x + se iθ (y αx) 2 x 2 2(ε + γ) x y αx < lim, θ R, s 0 + s " H δ 2 > 0, Πο s (0,δ 2 ), 2(ε + γ) x s(y αx) < x + se iθ (y αx) 2 x 2, θ s (0,δ 2 ), x 2 < x + se iθ (y αx) 2 +2(ε + γ) x s(y αx), θ R.?= ρ (x, e iθ y) ε x y αx Re(e iθ α) x 2 4ffi` 1.3 (4) [ο ρ (x, e iθ (y αx)) ε x y αx, θ R,

5 4r ]e: 1AT Birkhoff NJρUC 607 I x + se iθ (y αx) 2 x 2 2 lim 2ε x y αx, θ R. s 0 2s ψ@ kß γ (0, 1), ο` H δ 1 < 0, Πο s (δ 1, 0), / s (δ 1, 0) (0,δ 2 ), x 2 < x + se iθ (y αx) 2 +2(ε + γ) x s(y αx), θ R, x 2 < x + se iθ (y αx) 2 +2(ε + γ) x s(y αx), θ R. ffi5: ϕ : R R fi ϕ (s) = x + se iθ (y αx) 2 +2(ε + γ) x s(y αx), I G=7` 2.3 ο ϕ (0) <ϕ (s), s (δ 1, 0) (0,δ 2 ), x 2 x + se iθ (y αx) 2 +2(ε + γ) x s(y αx), s, θ R. (2) = γ ßy4(, H (2) ±Rg γ 0 + ο x 2 x + se iθ (y αx) 2 +2ε x s(y αx), s, θ R, " x + se iθ (y αx) 2 x 2 2ε x se iθ (y αx), s, θ R. =7` 2.2 [P x ε B (y αx). m 2.4 T : X Y Φ$(:, I T ΦSffl ρ + - MI$(:»uO» T Φ Sffl ρ - MI$(:. ±q Π0(. x ρ y, I=ffi` 1.3 (1) ο ( x) ρ+ y.?= T ΦSffl ρ + - MI $(: ο ( T (x)) ε ρ + T (y), "=ffi` 1.3 (1) 4ffi5 1.6 ο T (x) ε ρ T (y), / T ΦSffl ρ - MI$(:. ffl&(ο`[o. xn 2.1 T : X Y ΦSffl ρ + - MI:Sffl ρ - MI$(:, I T ΦSffl Birkhoff MI$(:. ±q x X \{0}, =ffi` 1.3 (4) ο, θ R, ρ ± ( x, ρ ± (x, ) eiθ y) x 2 x +e iθ y = ρ ± (x, eiθ y) x 2 x 2 + ρ ± (x, eiθ y)=0. 6fi T ΦSffl ρ + - MI:Sffl ρ - MI$(:, =7` 2.4 [P ( T (x) ε ρ ± ρ ±(x, e iθ ) y) x 2 T (x)+e iθ T (y), θ R,

6 608 fl, Q J 45X I θ R, ρ ± ( T (x), ρ ±(x, e iθ y) x 2 T (x)+e iθ T (y)) ε T (x) ρ ± (x, eiθ y) x 2 T (x)+e iθ T (y) T (x) 2 ε T (x) T (y) + ε ρ x 2 ± (x, e iθ y). "=ffi` 1.3 (4) ο, θ R, ρ ± (T (x), T eiθ (x) 2 T (y)) x 2 ρ ± (x, eiθ y) I> ε T (x) T (y) + ε T (x) 2 x 2 ρ ± (x, eiθ y), T (x) 2 x 2 ρ +(x, e iθ T (x) 2 y) ε ρ x 2 + (x, e iθ y) ε T (x) T (y) ρ + (T (x), e iθ T (y)), θ R (3) 4 ρ (T (x), e iθ T (x) 2 T (y)) ε T (x) T (y) + ε ρ x 2 (x, e iθ y) T (x) 2 + x 2 ρ (x, e iθ y), θ R. (4) u, w X, u =0: w =0, ILh"wflb. u, w X \{0}, u B w, =ffi` 1.4 ο, ρ (u, e iθ w) 0 ρ +(u, e iθ w), θ ρ (u, eiθ w) = ρ (u, eiθ w), c7 (3) ±4 (4) ±ο ρ + (u, eiθ w) = ρ + (u, eiθ w), θ R. T (u) 2 (1 ε) u 2 ρ +(u, e iθ w) ε T (u) T (w) ρ +(T (u), e iθ T (w)), θ R (5) 4 ρ (T (u), e iθ T (u) 2 T (w)) ε T (u) T (w) +(1 ε) u 2 ρ (u, e iθ w), θ R. (6) = (5) ±4 (6) ±[ο I ε T (u) T (w) ρ +(T (u), e iθ T (w)), ρ (T (u), e iθ T (w)) ε T (u) T (w), θ R, ρ (T (u), e iθ T (w)) ε T (u) T (w) 0 ρ +(T (u), e iθ T (w)) + ε T (u) T (w), θ R. "=ffi` 1.5 P T (u) ε B T (w), / T ΦSffl Birkhoff MI$(:.

7 4r ]e: 1AT Birkhoff NJρUC 609 kz [1] Alonso, J., Martini, H. and Wu, S.L., On Birkhoff orthogonality and isosceles orthogonality in normed linear spaces, Aequationes Math., 2012, 83(1): [2] Alonso, J. and Soriano, M.L., On height orthogonality in normed linear spaces, Rocky Mountain J. Math., 1999, 29(4): [3] Amir, D., Characterizations of Inner Product Spaces, Operator Theory: Advances and Applications, Vol. 20, Basel: Birkhäuser, 1986, 33. [4] Birkhoff, G., Orthogonality in linear metric space, Duke Math. J., 1935, 1(2): [5] Blanco, A. and Turnšek, A., On maps that preserve orthogonality in normed spaces, Proc.Roy.Soc. Edinburgh Sect. A, 2006, 136(4): [6] Carlsson, S.O., Orthogonality in normed linear spaces, Ark. Mat., 1962, 4(4): [7] Chen, C.Q., Linear maps preserving orthogonality, M.S. Dissertation, Suzhou: Soochow University, 2013 (in Chinese). [8] Chmieliński, J., On an ε-birkhoff orthogonality, J. Inequal. Pure Appl. Math., 2005, 6(3): Article 79, 7 pages. [9] Chmieliński, J., Linear mappings approximately preserving orthogonality, J. Math. Anal. Appl., 2005, 304(1): [10] Chmieliński, J., Stability of the orthogonality preserving property in finite-dimensional inner product spaces, J. Math. Anal. Appl., 2006, 318(2): [11] Chmieliński, J. and Wójcik, P., Isosceles-orthogonality preserving property and its stability, Nonlinear Anal., 2010, 72(3/4): [12] Chmieliński, J. and Wójcik, P., On a ρ-orthogonality, Aequationes Math., 2010, 80(1): [13] Chmieliński, J. and Wójcik, P., ρ-orthogonality and its preservation revisited, In: Recent Developments in Functional Equations and Inequalities (Brzdęk, J., Chmieliński, J., Ciepliński, K., Ger, R., Páles, Z. and Zdun, M.C. eds.), Banach Center Publ., Vol. 99, Warszawa: Pol. Acad. Sci., 2013, [14] Diminnie, C.R., A new orthogonality relation for normed linear spaces, Math. Nachr., 1983, 114(1): [15] James, R.C., Orthogonality in normed linear spaces, Duke Math. J., 1945, 12(2): [16] Mazaheri, H. and Vaezpour, S.M., Orthogonality and ε-orthogonality in Banach spaces, Aust. J. Math. Anal. Appl., 2005, 2(1): Article 10, 1-5. [17] Saidi, F.B., An extension of the notion of orthogonality to Banach spaces, J. Math. Anal. Appl., 2002, 267(1): [18] Wójcik, P., Linear mappings preserving ρ-orthogonality, J. Math. Anal. Appl., 2012, 386(1): Notes on Approximate Birkhoff Orthogonality KONG Liang (College of Mathematics and Computer Application, Institute of Applied Mathematics, Shangluo University, Shangluo, Shaanxi, , P. R. China) Abstract: In a complex normed space, a sufficient and necessary condition for approximate Birkhoff orthogonality is given and it is a generalization of the known results. Also, it is proved that approximate ρ + -orthogonality and approximate ρ -orthogonality preserving linear mappings are all approximate Birkhoff orthogonality preserving mapping. Keywords: Birkhoff orthogonality; approximate Birkhoff orthogonality; approximate Birkhoff orthogonality preserving mapping

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