T 1 Th T 1. 1 T 1 h Th

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5 T H c c > 0 Tx c x x H H K T : H K T T K T h H T 1 > 0 h = T 1 Th T 1 Th 1 T 1 h Th h H T : T h H,h 0 Th = 0 T c > 0 c h Th = 0. c h > 0 T : (Th n ) n T (h n ) n H T h n h m 1 c Th n Th m c > 0 (h n ) n n h 0 H

6 (Th n ) n n Th 0 Im(T ) T T Im(T ) = Im(T ) = K T T T H Λ(T ) := {λ T λ } T Π(T ) := {λ T λ } T Γ(T ) := {λ T λ } T Λ(T ) = Π(T ) Γ(T ) T H λ Π(T ) (x n ) n H lim (T λ)x n = 0. λ Π(T ) T λ c > 0 x H (T λ)x < c x n x n H (T λ)x n < 1 n x n lim 1 n x n x n = 0 lim (T λ) 1 x n x n = 0, ( 1 x n x ) n n (x n ) n H lim (T λ)x n = 0 c > 0 n (T λ)x n < c x n c > 0 x (x n ) n H (T λ)x < c x. T λ

7 T H Π 0 (T ) := {λ x H, x = 1 : Tx = λx} T T H Π 0 (T ) Π(T ) λ Π 0 (T ) x H Tx = λx (x n ) n x n = x n (T λ)(x n ) = 0 n λ T H λ : ρ(λ) := (T λ) 1. ρ ρ T H ρ ρ = ρ T U f : U x 0 U P(x) = a n (x x 0 ) n, n=0 (a n ) n x 0 f (x) f U f U = f U H ϕ : U L (H ) x U f,д H λ ϕ(λ)f д ϕ x U

8 ψ (λ) = ϕ(1/λ) ϕ ϕ ψ T H T < 1 (1 T ) (1 T ) 1 = T n. T < 1 n=0 T n (1 T ) ( 1 + T + T T ) k = 1 + T + T T k ( T + T T k + T k+1) = 1 T k+1, n=0 T < 1 (1 T ) k n=0 T n k+1 k = 1 T 1. (1 T ) T n = 1. n=0 n=0 T n T H λ 0 T T λ 0 λ λ 0 (T λ) (T λ 0 ) T λ = (T λ 0 ) (λ λ 0 ) = (T λ 0 ) 1 (T λ 0 ) 1 (λ λ 0 ). λ λ 0 1 λ λ 0 < (T λ 0 ) 1, (T λ 0) 1 (λ λ 0 ) < 1

9 1 (T λ 0 ) 1 (λ λ 0 ) (1 (T λ 0 ) 1 (λ λ 0 )) 1 = ((T λ 0 ) 1 (λ λ 0 )) n. λ λ 0 T λ (T λ) 1 = (T λ 0 ) 1 ((T λ 0 ) 1 (λ λ 0 )) n. n=0 λ λ 0 ρ(λ) f,д H ρ(λ)f д = (T λ0 ) n 1 f д (λ λ 0 ) n. n=0 ρ λ 0 λ 0 ρ : λ λ 0 n=0 T 1 λ = 1 (1 λt ), λ T 1 λ (1 λt ) 1 0 < λ < T ( 1 ) τ (λ) := ρ = λ (1 λt ) 1 = (λt ) n. n=0 ( T 1 λ ) 1 = λ (1 λt ) 1, 0 < λ < 1 T τ (λ) = λ (λt ) n. λ τ (0) = 0 ρ n=0

10 H T H λ 0 \ Π(T ) T λ 0 c > 0 Tx λ 0 x c x c H λx λx Tx λx + λx λ 0 x Tx λ 0 x c x λ x H Tx λx c x λ λ 0 x = (c λ λ 0 ) x λ x H λ λ 0 λ B c 2 (λ 0 ) T λ \ Π(T ) Π(T ) H T H λ T λ Π(T ) T λ T λ (λ n ) n \ Λ(T ) lim λ n = λ x H x 0 T λ (x n ) n x n = 1 x n := (T λ n) 1 x (T λ n ) 1 x. lim ((T λ) (T λ n))x n = lim (λ n λ)x n = 0. (T λ)x n T λ (T λ n )x n = x (T λ n ) 1 x

11 T λ : (T λ)x n (T λ n )x n 2 = (T λ)x n 2 + (T λ n )x n 2 (T λ)x n 2. lim (T λ)x n = 0 λ Π(T ) T H Λ(T ) {λ λ T }. λ λ 0 \ Λ(T ) ρ T T λ = (λ 0 λ) + (T λ 0 ) = (T λ 0 )(1 (T λ 0 ) 1 (λ λ 0 )) = (T λ 0 )(1 ρ(λ 0 )(λ λ 0 )). λ λ 0 < 1/ ρ(λ 0 ) (1 ρ(λ 0 )(λ λ 0 )) 1/ ρ(λ 0 ) λ 0 λ 0 \Λ(T ) \ Λ(T ) \ Λ(T ) Λ(T ) λ > T T λ = λ T 1 λ 1 T 1 λ < 1 T λ λ Λ(T ) Λ(T ) {λ λ T }. f : f (z) = n=0 a n z n z f f f 0 f (z) < M M z B r (0) 0 r a n = f (n) (0) n! = 1 f (ζ ) 2πi B r (0) ζ n+1dζ

12 a n < M r n. a n = 0 f T H ρ T T ρ T f д λ (T λ) 1 f д f,д H ρ T ( ) = 0 ρ T f д ρ T f д ρ T ( ) = 0 (T λ) 1 f д = 0 f,д H λ (T λ) 1 (T λ)f f = f f = 0 f H, f 0 f 0

13 T H : Π 0 (T ) = Γ(T ) Π 0 (T ) = Γ(T ). Λ(T ) = Π(T ) Π(T ). λ Π 0 (T ) x H x 0 (T λ)x = 0 T λ Im(T λ) ker(a) = Im(A ) A H Im(T λ) = Im(T λ) H T λ λ Γ(T ) Π 0 (T ) = Γ(T ) T T Π 0 (T ) = Γ(T ) Λ(T ) = Π(T ) Γ(T ). Γ(T ) = Π 0 (T ) Λ(T ) = Π(T ) Π 0 (T ). Π 0 (T ) Π(T ) Λ(T ) Π(T ) Π(T ). Π(T ) Λ(T ) λ Π(T ) T λ (T λ) = T λ λ Λ(T ) T Π(T ) Π(T ) Λ(T )

14 H AB A B AB A B AB A B x H x 0 ABx = 0 ker(b) = {0} ker(a) {0} (x n ) n H lim AB(x n ) = 0 lim B(x n ) 0 lim A(Bx n ) = 0 Im(AB) H Im(A) H Im(B) H T H p : Π 0 (p(t )) = p (Π 0 (T )). Π (p(t )) = p (Π(T )). Γ (p(t )) = p (Γ(T )). deg(p) > 0 1 λ 0 Π 0 (T ) p(z) p(λ 0 ) z λ 0 p(λ 0 ) Π 0 (p(t )) p (Π 0 (T )) Π 0 (p(t )) α Π 0 (p(t )) p(z) α z λ α = p(λ ) λ Π 0 (T ) α p (Π 0 (T )) Π 0 (p(t )) p (Π 0 (T )) Π 0 (p(t )) = p (Π 0 (T )) Γ (p(t )) = p (Γ(T ) )

15 H f f (z) = 1/z : Π 0 (f (T )) = f (Π 0 (T )). Π(f (T )) = f (Π(T )). Γ(f (T )) = f (Γ(T )). λ Π 0 (T ) λ 0 Tx = λx x H x 0 : Tx = λx T 1 Tx = λt 1 x T 1 x = 1 λ x, 1 Π 0 (T ) Π 0(T 1 ) T 1 1 x = λx Π 0 (T 1 ) Π 0(T ) Π 0 (T 1 ) 1 Π 0 (T ) Π 0(T 1 ) = 1 Π 0 (T ) f (Π 0 (T )) = 1 Π 0 (T ) = Π 0(T 1 ) = Π 0 (f (T )). λ Π(T ) λ 0 lim Tx n λx n = 0 (x n ) n H T 1 lim Tx n λx n = 0 lim x n λt 1 1 x n = 0 lim λ x n T 1 x n = 0 1 Π(T ) Π(T 1 ) T 1 f (Π(T )) = 1 Π(T ) = Π(T 1 ) = Π(f (T )). : f (Γ(T )) = f (Γ(T ) ) = f (Π 0 (T )) = Π 0 (f (T )) = Γ(f (T )). A B P P 1 AP = B

16 H : Λ(A) = Λ(B). Π 0 (A) = Π 0 (B). Π(A) = Π(B). Γ(A) = Γ(B). A λ P 1 (A λ) P = P 1 AP λ = B λ λ Π 0 (A) f H f 0 Af = λf λ B P 1 f = P 1 AP P 1 f = λ P 1 f. λ Π(A) lim Af n λf n = 0 (f n ) n H lim B P 1 f n λ P 1 f n = lim P 1 AP P 1 f n λ P 1 f n = lim P 1 (Af n λf n ) = 0. 1 = 1 = f n = PP 1 f n P P 1 f n 1 P P 1 f n, P 1 f n 1 P P 1 f n ( P lim P 1 1 ) ( f n P 1 ) f n AP P 1 f n λ P 1 f n = 0. B λ = P 1 AP λ = P 1 (A λ)p P 1 (Im(A λ)) A λ P 1 (A λ)p H A B = P 1 AP P 1 AP P 1 A P

17 T H Γ(T ) = Π 0 (T ) Λ(T ) = Π(T ). λ T (T λ) (T λ)x = (T λ )x x H λ Π 0 (T ) Tx λx = 0 x H x 0 T x λ x = 0 Π 0 (T ) = (Π 0 (T ) ) H (α n ) n (e n ) n H A H Ae n = α n e n n (α n ) n (α n ) n A A H (α n ) n A = sup α n. n (e n ) n H A Ae n = α n e n (α n ) n α n = α n e n = Ae n A e n = A.

18 (α n ) n sup α n A n 2 2 A ξ j e j = α j ξ j e j = j=0 j=0 ( 2 αj ξ j j=0 sup n α n ) 2 ( 2 ξj = j=0 sup n 2 2 α n ) ξ j e j j=0 ξ j j A sup α n n H (α n ) n Π 0 (A) = Γ(A) = {α n n }. : A λ λ = α j j α j e j (e n ) n H (e n ) n H f = j ξ je j ξ j j λ α j j (A λ)f = j (α j λ)ξ j e j = 0 ξ j = 0 j f = 0 λ Π 0 (A) = Γ(A) = {α n n }. H (α n ) n Π(A) = Λ(A) = {α n n }. Π 0 (A) Λ(A) λ Π 0 (A) λ (e n ) n H B H Be j = (α j λ) 1 e j (α n ) n (α n λ) 1 n B B (A λ) λ Π 0 (A) = Λ(A)

19 M µ ϕ ϕ T L 2 (µ) (Af ) (x) = ϕ(x)f (x) x M f L 2 (µ) ϕ T L 2 (µ). ϕ Im ess (ϕ) : {λ λ : ϕ 1 (U ) }. M σ µ T ϕ : M : Π 0 (T ) = Γ(T ) = {λ µ ϕ 1 ({λ}) > 0}. Π(T ) = Λ(T ) = Im ess (ϕ). f L 2 (M, µ) λ Π 0 (T ) ϕ(x)f (x) = λf (x) x \ ker(f ) ϕ(x) = λ x µ ϕ 1 ({λ}) > 0 λ {λ µ ϕ 1 ({λ}) > 0} M σ B ϕ(x) = λ x B f B ϕ f = λf λ T Π o (T ) = Γ(T ) T λ λ ϕ λ δ > 0 ϕ λ δ δ > 0 ϕ λ < δ (ϕ λ) 1 (U δ (0)) δu δ (0) 0 Im ess (ϕ λ) λ Im ess (ϕ) Λ(T ) = Im ess (ϕ)

20 l 2 () U l 2 () U (ξ 0, ξ 1, ξ 2,... ) = (0, ξ 0, ξ 1, ξ 2,... ) (ξ n ) n l 2 () U l 2 () U (ξ 0, ξ 1, ξ 2,... ) 2 = (0, ξ 0, ξ 1, ξ 2,... ) 2 = (ξ 0, ξ 1, ξ 2,... ) 2, U = 1 e n (ξ 0, ξ 1, ξ 2,... ) ξ n = 1 ξ i = 0 i n n = 0, 1, 2,... e n l 2 () Ue n = e n+1 (n = 0, 1, 2,... ). U (e n ) n U U U e i = e i 1 (i = 1, 2, 3,... ), i, j = 0, 1, 2,... U e i ej = ei U ej = ei ej+1 = δi, j+1 U e 0 = 0. i > 0 δ i, j+1 = δ i 1, j = e i 1 ej

21 U e i = e i 1 (i = 1, 2, 3,... ), (ξ n ) n l 2 () U (ξ 0, ξ 1, ξ 2,... ) = (ξ 1, ξ 2, ξ 3,... ). U l 2 () Λ(U ) = {λ λ 1}, Π 0 (U ) =, Π(U ) = {λ λ = 1} Γ(U ) = {λ λ < 1}. Λ(U ) = Π(U ) = {λ λ 1}, Π 0 (U ) = {λ λ < 1} Γ(U ) =. U f = λf f l 2 () f = (ξ 0, ξ 1, ξ 2,... ) λ (0, ξ 0, ξ 1, ξ 2,... ) = (λξ 0, λξ 1, λξ 2,... ), 0 = λξ 0 ξ n = λξ n+1 n ξ n = 0 n λ = 0 λ 0 Π 0 (U ) = Γ(U ) = U f = λf f l 2 () f = (ξ 0, ξ 1, ξ 2,... ) λ (ξ 1, ξ 2, ξ 3,... ) = (λξ 0, λξ 1, λξ 2,... ), ξ n+1 = λξ n ξ n+1 = λ n ξ 0 n ξ 0 = 0 f = 0 ξ 0 0 λ < 1 Π 0 (U ) Γ(U ) U U = 1 U U Λ(U ) Λ(U )

22 Π(U ) Π(U ) λ < 1 U f λf 2 U f 2 λf 2 = 1 λ f 2 f l 2 () U λ λ < 1 Π(U ) Π 0 Π Π 0 (U ) Π(U ) l 2 () l 2 () (..., ξ 2, ξ 1, (ξ 0 ), ξ 1, ξ 2,... ), W l 2 () W (..., ξ 2, ξ 1, (ξ 0 ), ξ 1, ξ 2,... ) = (..., ξ 3, ξ 2, (ξ 1 ), ξ 0, ξ 1,... ) (ξ n ) n l 2 () W l 2 () W l 2 () = l 2 () W = 1 e n (..., ξ 2, ξ 1, (ξ 0 ), ξ 1, ξ 2,... ) ξ n = 1 ξ i = 0 i n (n = 0, ±1, ±2,... ) e n l 2 () W W e n = e n+1 (n = 0, ±1, ±2,... ). W

23 W W W 1 W W 1 (..., ξ 2, ξ 1, (ξ 0 ), ξ 1, ξ 2,... ) = (..., ξ 1, ξ 0, (ξ 1 ), ξ 2, ξ 3,... ) (ξ n ) n l 2 () (e n ) n l 2 () W 1 e n = e n 1 (n = 0, ±1, ±2, ±3,... ). W W 1 P Pe n = e n (n = 0, ±1, ±2, ±3,... ), W W 1 P 1 W P = W 1 : P 1 W P(..., ξ 2, ξ 1, (ξ 0 ), ξ 1, ξ 2,... ) = P 1 W (..., ξ 2, ξ 1, (ξ 0 ), ξ 1, ξ 2,... ) = P 1 (..., ξ 2, (ξ 1 ), ξ 0, ξ 1, ξ 2,... ) = (..., ξ 2, ξ 1, ξ 0, (ξ 1 ), ξ 2,... ) = W 1 (..., ξ 2, ξ 1, (ξ 0 ), ξ 1, ξ 2,... ). W l 2 () Λ(W ) = Π(W ) = {λ λ = 1} Π 0 (W ) = Γ(W ) =. W W W f = λf f l 2 () f = (..., ξ 1, (ξ 0 ), ξ 1,... ) λ λ 0 (..., ξ 3, ξ 2, (ξ 1 ), ξ 0, ξ 1,... ) = (..., λξ 2, λξ 1, (λξ 0 ), λξ 1, λξ 2,... ), ξ n = λξ n+1 n ξ n = λ j ξ n+j n, j λ = 1 ξ n = c c n (ξ n ) n l 2 () λ < 1 ξ n = 1 λ j ξ n j 1 λ j > 1 j = 1, 2, 3,... (ξ n ) n l 2 ()

24 Π 0 (W ) = Γ(W ) = W W Γ(W ) = Π 0 (W ) = Π(W ) = Λ(W ) λ < 1 W f λf 2 W f 2 λf 2 = 1 λ f 2 f l 2 () W λ Λ(W ) {λ λ 1} \ {λ λ < 1} = {λ λ = 1}. λ = 1 f n f n := 1 (..., 0, 0, 1, ( λ ) ), λ 2,..., λ n 2, λ n 1, 0, 0,... n n f n 2 = 1 f n l 2 () f n n 1/ n (W λ)f n 2 2 = 1 ( ) n..., 0, 0, λ, 0, 0,..., 0, λ n 1 2, 0, 0,... 2 = 1 ( λ 2 + n λ n 1 ) 2 = 2 n 0. λ Π(W ) {λ λ = 1} Λ(W ), Π(W ) = Π(W ) = Λ(W ) = Λ(W ) = {λ λ = 1}.

25 T r(t ) = sup{ λ : λ Λ(T )}. 0 r(t ) T Λ(T ) 0 T r(t n ) = (r(t )) n n T r(t ) = lim T n 1 n. T H (r(t )) n r(t ) T n 1 n n = r(t n ) T n r(t ) lim inf T n 1 n. ( 1 ) τ (λ) := ρ = λ ( T 1 λ ) 1, τ (λ) = λ(1 λt ) 1 λ 0 1 λ 1 λ > r(t ) τ (T ) = λ λ n T n. λ < 1 r (T ) n=0 τ f д f,д H

26 λ λ n T n f д. n=0 ( (λt ) n f д ) n λ λ < 1 r (T ) ( λ n T n ) n λ n T n α α > 0 n λ T n 1 n α 1 n, λ lim sup T n 1 n lim sup α n 1, λ lim sup T n 1 n 1. λ λ < 1 r (T ) lim sup T n 1 n r(t ). lim sup T n 1 n r(t ) lim inf T n 1 n, r(t ) = lim T n 1 n. A B r(ab) = lim (AB) n 1 n r(ab) r(a)r(b). ( lim (A) n 1 n (B) n 1 n = ) ( ) lim (A) n 1 n lim (B) n 1 n = r(a)r(b).

27 A B H A AB f H ABf f r(b) Af f f H n = 0 n n + 1: ABf f 2n AB 2n f f Af f 2n 1. ABf f 2n+1 = ( ABf f 2n ) 2 I.V. ( AB 2n f f Af f ) 2n 1 2 AB 2 n f 2 Af Af f 2 n+1 2 = AB 2 n f AB 2 n f Af Af Af f 2 n+1 2 = AB 2n f B 2n f Af f Af f 2n+1 2 = B 2n AB 2n f f Af f 2n+1 1. : B k A = AB k k k = 0 k k + 1: B k+1 A = B B k A = B AB k = (AB) B k = ABBk = AB k+1. ABf f 2n+1 AB 2n+1 f f Af f 2n+1 1 ABf f 2n AB 2n f f Af f 2n 1 A B 2n f 2 Af f 2n 1, ( ABf f 2 n ) 1 2 n ( A B 2n f 2 Af f 2n 1 ) 1 2 n

28 ABf f A 1 2 n B 2n 1 2 n ( f 2) 1 2 n ( Af f 2n 1 ) 1 2 n. lim ABf f lim ( A 1 2 n 1 B 2n 2 n ( f 2) 1 2 n ( Af f ) ) 2n n ABf f r(b) Af f. B r(b) (e n ) n l 2 () l 2 () n = 0, 1, 2,... n = 0, ±1, ±2,... (α n ) n n = 0, 1, 2,... n = 0, ±1, ±2,... SP S Se n = e n+1 (α n ) n Pe n = α n e n n α n A l 2 () l 2 () α n A = sup α n r(a) = lim n sup k n k 1 i=0 α n+i 1 k. A A A = sup α n n Ae n = α n e n+1 A 2 e n = α n α n+1 e n+2 A 3 e n = α n α n+1 α n+2 e n+3 A k α

29 α n Ak k 1 = sup α n+i (k = 0, 1, 2, 3,... ). n i=0

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