Single-value extension property for anti-diagonal operator matrices and their square

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1 Journal of East China Normal University Natural Science No. 1 Jan. 215 : ,, :, Hilbert. : ; ; : O177.2 : A DOI: /j.issn Single-value extension property for anti-diagonal operator matrices and their square Abstract: CUI Miao-miao, CAO Xiao-hong Department of Mathematics and Information Science, Shaanxi Normal University, Xi an 71119, China In this paper, we mainly proved the equivalence of the perturbation of single-value extension property for anti-diagonal operator matrices and their square on an infinite dimensional separable Hilbert space. Key words: single-value extension property; compact perturbations; anti-diagonal operator matrices 1, H Hilbert. BH H, KH H. T BH, NT RT T. T Fredholm, RT nt nt, nt dim NT, nt dim NT T T. T indt nt nt. Wolf σ SF T σ SF T {λ C : T λi Fredholm }. : : , ; GK21317 :,,,. address: cuiye@snnu.edu.cn. :,,,,. address: xiaohongcao@snnu.edu.cn.

2 ρ SF TC\σ SF T T Fredholm. ρ a T{λ ρ SF T : nt λi}, ρ SF+ T{λ ρ SF T : nt λi < }. T Fredholm, < indt < +, T Fredholm ; indt T Weyl. T asct NT n NT n+1,, asct ; T dest RT n RT n+1,, dest. T Fredholm, T Browder. σ e T, Weyl σ w T σ e T {λ C : T λi Fredholm }, σ w T {λ C : T λi Weyl }. σ T {λ isoσt : T λi Browder }. 2, SVEP, T T SVEP. N. Dunford [1-3], Fredholm., [4-5].,, [6-8] [9].,., T intσ p T, Bishop s β, δ [7], σ p T T.. [1-11], T BH H,. 1 AB ; 2 BA ; 3 T AB intσ SF AB ρ SF AB [4], intσ SF BA., B δ µ σ SF BA. intσ SF AB, µ 1 B δ µ AB µ 1 I Fredholm. ρ SF AB, AB µ 1 I Browder, µ 2 µ 2 B δ µ AB µ 2 I, BA µ 2 I,. 2 ρ SF BA. : ρ SF BA ρba σ BA. ρ SF BA ρba σ BA. µ ρ SF BA, µ / ρba, µ ρ SF BA σba, B δ µ µ B δ µ BA µi Fredholm. intσ SF AB, µ 1 B δ µ AB µ 1 I Fredholm. ρ SF AB, AB µ 1 I Browder, µ 2 µ 2 B δ µ AB µ 2 I. ρab ρba, BA µ 2 I, µ σba. BA µ I Fredholm BA µ I n d, µ σba, BA µ I Browder [12], 4.9.

3 1, : 97 ρ SF AB, ρ SF AB ρab E, E C. ρab ρba, ρ SF AB ρ SF BA BA int σ SF BA ρ SF BA [4], intσ SF T 2., B δ µ σ SF T 2. intσ SF BA, µ 1 B δ µ BA µ 1 I Fredholm. ρ SF BA, BA µ 1 I Browder, µ 2 µ 2 B δ µ BA µ 2 I. ρab ρba, AB µ 2 I, T 2 µ 2 I [13], 3.9,. 2 ρ SF T 2. BA, 1 2 AB, intσ SF AB ρ SF AB [4], 1.3. AB SVEP BA SVEP, µ C, indab µi indba µi, ρ SF AB ρ SF+ AB, ρ SF BA ρ SF+ BA [5], 11. ρ SF AB ρ SF BA, ρ SF AB ρ SF+ AB ρab E 1, ρ SF BA ρ SF+ BA ρba E 2, E 1 C, E 2 C. ρab ρba, ρ SF AB ρ SF BA ρ SF T 2 ρ SF AB ρ SF BA T 2, K, intσ SF T 2 intσ SF T 2 + K ρ SF T 2 ρ SF T 2 + K [4], intσ SF AB., B δ µ σ SF AB. intσ SF T 2, µ 1 B δ µ T 2 µ 1 I Fredholm. ρ SF T 2, T 2 µ 1 I Browder, µ 2 µ 2 B δ µ T 2 µ 2 I, AB µ 2 I [13], 3.9,. 2 ρ SF AB. ρ SF AB, ρ SF AB Ω. Γ Ω. N Γ σ SF AB, K 1 AB + K 1, N σn σ SF N Γ [14], 2.1. N, Φ C\[σN\σ N], [15, 3.1] K 2 σn + K 2 σn Φ Ω, K2 Ω σn + K 2 \σ w N + K 2. K + K 1, K T 2 + K K AB + K BA N + K2 BA,. intσ SF T 2 + K, µ Ω T 2 + K µi Fredholm. ρ SF T 2 + K, µ µ Ω T 2 + K µ I, AB + K µ I [13], 3.9, N + K 2 µ I. N + K 2 µ I Weyl, N + K 2 µ I,. T, σ ω T σ ω AB σ ω BA, σ ω {σ, σ e, σ w } [13], T, Wolf.

4 T BH H, 1 T λi Fredholm, AB λ 2 I BA λ 2 I Fredholm indab λ 2 I indba λ 2 I indt λi; 2 σ SF T σ SF T 2. 1 λ, T 2 A Fredholm, AB BA BA Fredholm, A B Fredholm indab indba. 2 indt indt 2 indab+indba, indab indba indt. λi λ, 1: NT λi NAB λ 2 x I. NT λi, B y x λi λix + Ay Bx λy, AB λ 2 Ix, NAB λ 2 I. y B λx λ 2, x NAB λ 2 x + ABx λi I, T λi, x Bx B NT λi. 2: nab λ 2 I nt λi. x 1, x 2,, x n NAB λ 2 I λx1 λx2 λxn, 1,,, NT λi, Bx 1 Bx 2 Bx n nab λ 2 I nt λi. ξ 1 ξ 2 ξ m,,, NT λi, 1 x 1, x 2,, η 1 η 2 η m x m NAB λ 2 ξi λxi λx1 λx2 I, i 1,, m.,,, η i Bx i Bx 1 Bx 2 λxm. a 1 x 1 +a 2 x 2 + +a m x m, a 1, a 2,, a m C, a 1 Bx 1 + Bx m λx 1 λx 2 a 2 Bx 2 + +a m Bx m a 1 λx 1 +a 2 λx 2 + +a m λx m, a 1 +a 2 + Bx 1 Bx 2 λxm +a m, a 1 a 2 a m, x 1, x 2,, x m NAB λ 2 I Bx m. nab λ 2 I nt λi. 3: RAB λ 2 I. AB λ 2 Ix n yn, T 2 λ 2 x n I y x n y n. T +λi z n, T λiz n n. T λi x y x Fredholm RT λi, T λi, y y y AB λ 2 x λ. RAB λ2 I. : I BA λ 2 I Fredholm nba λ 2 I nt λi, dab λ 2 I dba λ 2 I dt λi. T λi Fredholm, AB λ 2 I BA λ 2 I

5 1, : 99 Fredholm indab λ 2 I indba λ 2 I indt λi; II T λi Fredholm, AB λ 2 I, BA λ 2 I Fredholm indab λ 2 I indba λ 2 I indt λi. 2 λ / σ SF T 2, T 2 λ 2 I Fredholm, T ± λi Fredholm, λ / σ SF T., λ / σ SF T, 1 AB λ 2 I BA λ 2 I Fredholm, T 2 λ 2 I Fredholm, λ / σ SF T T, T 2 T. S1 S 2. : S SVEP, S 1 SVEP. S 3 U C, f 1 : U H S 1 λif 1 λ. fλ f 1 λ, S1 λi S 2 f1 λ S λifλ S 1 λif 1 λ. S 3 λi S SVEP, f, f 1, S 1 SVEP. : 1 intσ SF T., B ε λ σ SF T. Γ B ε λ. Γ σ SF T, N K 1 T + K 1, N σn σ SF N Γ [14], 2.1. N, Φ C\[σN\σ N], [15, 3.1] K 2 σn + K 2 σn Φ Ω, Ω σn + K2 K 2 \σ w N + K 2. K + K 1, T 2 + K T + K 2 N + K2 2 K TK + KT + K 2. T 2, 1 N + K 2 2 SVEP. σn Γ, I λ, λ B ε λ, λ / Γ; II λ, ε / B ε λ, λ λ B ε λ, λ / Γ. λ 1 B ε λ λ 1 / Γ, N λ 1 I, N + λ 1 I, N + K 2 2 λ 2 1I Weyl. N + K 2 2 SVEP, N + K 2 2 λ 2 1I Browder [4], 15, N + K 2 λ 1 I Browder, λ 2 Ω N + K 2 λ 2 I,. 2ρ SF T 2., ρ SF T 2, ρ SF T 2 Ω. Γ N Ω. Γ σ SF T, K 1 T 2 + K 1, N σn σ SF N Γ [14], 2.1. N, Φ C\[σN\σ N], [15, 3.1] K 2 σn + K 2 σn Φ Ω,,

6 1 215 K 2 Ω σn + K 2 \σ w N + K 2. K + K 1, T 2 + K T + K 2 N + K 2 2, K TK+KT +K 2. T 2, T 2 +K, intσ SF T 2 +K ρ SF T 2 +K [4], 1.3. λ Ω, λ / σ SF T. 2.2 T 2 +K λ 2 I Fredholm, ρ SF T 2 +K, T 2 +K λ 2 I Browder, T +K λ I Browder, ascn +K 2 λ I <. N + K 2 λ I Weyl, N + K 2 λ I Browder. 1 :.. ρ SF T ρ SF T 2 [ρ SF T] 2. fx x 2, ρ SF ft ρ SF T 2 [ρ SF T] 2 fρ SF T. ρ SF T 2, A, B C ρ SF T 2 A B, [A B] [A B]. ρ SF T 2 [ρ SF T] 2 ρ SF T f 1 A B, f 1 A B A B f. [f 1 A f 1 B] [f 1 A f 1 B] [f 1 A f 1 B] [f 1 A f 1 B] f 1 A B f 1 A B f 1 [A B A B], ρ SF T. intσ SF T 2., B δ µ σ SF T 2. µ λ 2, 2.2 λ σ SF T. intσ SF T, λ n λ T λ n I, T + λ n I λ 2 n λ 2 [13], , T 2 λ 2 n I, µ intσ SF T T, T 2 T. 2.1 : 2.1 AB BA T, A B T. : 1 T A B., Ax 1, x 2, x 3,, x 1, x 2,, Bx 1, x 2, x 3, x 2, x 3, x 4,, ABx 1, x 2, x 3,, x 2, x 3,, BAx 1, x 2, x 3, x 1, x 2, x 3,, : σa σb D, σ SF A σ SF B D AB 2 AB, σt 2 {, 1}. T 2 [4], 1.3, 2.1 T. A B [4], A B T., A 1 x 1, x 2, x 3,, x 1, x 2,, A 2 x 1, x 2, x 3, x 2, x 3, x 4,, B 1 x 1, x 2, x 3,, x 1,, x 2,, B 2 x 1, x 2, x 3, x 2, x 4, x 6,,

7 1, : 11 1 A, B Bl 2 l 2, T Bl 2 l 2 l 2 l 2 A A 2 B1, T. B 2 A1 B 2 A T 2 2 B 1 BA B1 A 2, B 2 A 1 A 2 A1 A 2, B 2 B1 B 2. I I, B A 1 A 2 x 1, x 2, x 3,, x 2, x 3, x 4,, B 1 B 2 x 1, x 2, x 3,, x 2,, x 4,, A 1 B 2 x 1, x 2, x 3,, x 2, x 4, x 6,, A 2 B 1 x 1, x 2, x 3, x 1,, x 2,,, B 1 A 2 x 1, x 2, x 3,, x 2,, x 3,, B 2 A 1 x 1, x 2, x 3, x 1, x 3, x 5,. : 1 σa 1 B 2 D, σt 2 σ SF T 2 D. T 2 [4], 1.3, 2.1 T. 2 σa 1 A 2 σb 1 B 2 {, 1}, σa σb {, 1, 1}, A B [4], T, A B, T. 2.2 T, 1 A SVEP, B AB BA, BA SVEP. 2 A, B, BK KB K, BA + K SVEP. 1 U C, f : U H BA λifλ, f. B, γ 1, γ 2,, γ k B γ 1 B γ 2 B γ k, k. p j λ λ γ 1 λ γ 2 λ γ j, j 1, 2,, k. : p j Bfλ, j 1, 2,, k. BA λifλ, B γ k Afλ + γ k A λfλ. AB BA, p k BAfλ + γ k A λp k 1 Bfλ, γ k A λp k 1 Bfλ. A SVEP, p k 1 Bfλ. p j Bfλ, j 1, 2,, k, p 1 Bfλ B γ 1 fλ, B γ 1 Afλ. BA λifλ, B γ 1 Afλ + γ 1 A λfλ, γ 1 A λfλ. A SVEP, f. : BA SVEP. 2 BK KB K, U C, f : U H BA + K λifλ, f.

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