Geometry of Spaces of Non-Homogeneous Polynomials

Transcription

1 July 2014 Workshop

2 Duality in Linear Operators c o l 1 l

3 Duality in Linear Operators c o l 1 l H Hilbert space K(H) N(H) B(H).

4 E Banach space, n N P : E K is a polynomial of degree n if P is continuous and the restriction of P to each affine line of E is a polynomial of degree n

5 E Banach space, n N P : E K is a polynomial of degree n if P is continuous and the restriction of P to each affine line of E is a polynomial of degree n P( n E) the space of all polynomials of degree at most n

6 E Banach space, n N P : E K is a polynomial of degree n if P is continuous and the restriction of P to each affine line of E is a polynomial of degree n P( n E) the space of all polynomials of degree at most n P( n E) the space of all n-homogeneous polynomials P(λx) = λ n P(x)

7 Geometry of Duality in P w ( n E) P A ( n E) s,n,ɛ E b s,n,π E bb P N ( n E b ) P I ( n E b ) P( n E bb )

8 Duality in of degree at most n P w ( n E) P A ( n E) P N ( n E b ) P I ( n E b ) P( n E bb )

9 Norm on P( n E) The natural norm on P( n E) (and hence P A ( n E)) is P = sup P(x). x 1

10 Approximable Space of approximable polynomials is closure of finite rank polynomials in the operator norm

11 Approximable Space of approximable polynomials is closure of finite rank polynomials in the operator norm Denoted by P A ( n E)

12 Given ϕ in E, n N we let ϕ n (x) = ϕ(x) n Integral

13 Given ϕ in E, n N we let ϕ n (x) = ϕ(x) n P P I ( n E) if P(x) = Integral B E ϕ(x) n dµ(ϕ) for some measure µ on (B E, σ(e, E))

14 P P I ( n E) (P is integral) if P(x) = Integral B E j=0 n ϕ(x) j dµ(ϕ) for some measure µ on (B E, σ(e, E))

15 P P I ( n E) (P is integral) if P(x) = Integral B E j=0 n ϕ(x) j dµ(ϕ) for some measure µ on (B E, σ(e, E)) P I = inf{ µ : µ represents P}

16 P P I ( n E) (P is integral) if P(x) = Integral B E j=0 n ϕ(x) j dµ(ϕ) for some measure µ on (B E, σ(e, E)) P I = inf{ µ : µ represents P} Truncation of idea of integral holomorphic function (Dimant, Galindo, Maestre & Zalduendo)

17 Geometry of Integral E real Banach space

18 Geometry of Integral E real Banach space ( & Ryan, Dimant, Galicer & R. García) The set of extreme points of unit ball of P I ( n E) is {φ n : φ E, φ = 1}

19 Geometry of Integral E real Banach space ( & Ryan, Dimant, Galicer & R. García) The set of extreme points of unit ball of P I ( n E) is {φ n : φ E, φ = 1} The set of extreme points of unit ball of P I ( n E) is {φ n : φ E, φ 1}

20 Unit Ball of P( 2 R)

21 Unit Ball of P( 2 R)

22 ( & S. Lassalle) Isometries of Approximable (Real Spaces) E and F be real Banach spaces, let n 2 an isometric isomorphism T : P A ( n E) P A ( n F)

23 ( & S. Lassalle) Isometries of Approximable (Real Spaces) E and F be real Banach spaces, let n 2 T : P A ( n E) P A ( n F) an isometric isomorphism There exists an isometric isomorphism r : E F such that T(P)(y) = ±P r t J F (y) for all P P A ( n E),

24 ( & S. Lassalle) Isometries of Approximable (Real Spaces) E and F be real Banach spaces, let n 2 T : P A ( n E) P A ( n F) an isometric isomorphism There exists an isometric isomorphism r : E F such that T(P)(y) = ±P r t J F (y) for all P P A ( n E), where P is the Aron-Berner extension of P, r t is the transpose of r and J F is the canonical inclusion of F in F

25 Isometries of Approximable (Real Spaces) Analogous result true in the non homogeneous case

26 Isometries of Approximable (Real Spaces) Analogous result true in the non homogeneous case E and F be real Banach spaces, let n 2 an isometric isomorphism T : P A ( n E) P A ( n F)

27 Isometries of Approximable (Real Spaces) Analogous result true in the non homogeneous case E and F be real Banach spaces, let n 2 T : P A ( n E) P A ( n F) an isometric isomorphism There exists an isometric isomorphism r : E F such that T(P)(y) = ±P r t J F (y) for all P P A ( n E),

28 Isometries of Approximable (Complex Spaces) ( & S. Lassalle) E and F be complex Banach spaces, let n be a positive integer T : P A ( n E) P A ( n F) an isometric isomorphism (Assume additional condition on extreme points of ball of P I ( n E))

29 Isometries of Approximable (Complex Spaces) ( & S. Lassalle) E and F be complex Banach spaces, let n be a positive integer T : P A ( n E) P A ( n F) an isometric isomorphism (Assume additional condition on extreme points of ball of P I ( n E)) There exists an isometric isomorphism r : E F such that T(P)(y) = P r t J F (y) for all P P A ( n E),

30 Isometries of Approximable (Complex Spaces) E be a complex Banach spaces, let n N Then T(P)(y) = P r t J F (y) for all P P A ( n E), is an isometry of P A ( n E)

31 Isometries of Approximable (Complex Spaces) E be a complex Banach spaces, let n N Then T(P)(y) = P r t J F (y) for all P P A ( n E), is an isometry of P A ( n E) T(P A ( j E)) = P A ( j E)

32 New Type of Isometry E complex Banach space then P A ( n E) = P A ( n E C) n R : P j P j=0

33 New Type of Isometry E complex Banach space then P A ( n E) = P A ( n E C) n R : P j P j=0 P(x, λ) = n j=0 λn j P j (x)

34 Geometry of P A ( n E) R P A ( n E C) T New Type of Isometry P A ( n E) R 1 T P A ( n E C)

35 Geometry of P A ( n E) R P A ( n E C) T New Type of Isometry P A ( n E) R 1 T P A ( n E C) T(R(P)) = R(P) s J E C

36 s(x, λ) = (s 1 (x), e iθ λ) get same isometries as homogeneous case New Type of Isometry

37 s(x, λ) = (s 1 (x), e iθ λ) get same isometries as homogeneous case New Type of Isometry E = X C write x as (x 1, µ) with x 1 X, µ C

38 s(x, λ) = (s 1 (x), e iθ λ) get same isometries as homogeneous case New Type of Isometry E = X C write x as (x 1, µ) with x 1 X, µ C consider the isometry s(x 1, µ, λ) = (r(x 1 ), λ, µ)

39 For j = 1, 2,..., n 1: T((φ 2, µ) j ) = n q=n j+1 + ( j ) ( n q ) q q q+j n (0, µ j n j ) n j, q k=1 New Type of Isometry ( e 2π(n j)k q i φ 2 r, e 2πk q ) i µ n q q n j where r is an isometry of X. for j = 0: T((φ 2, µ) 0 ) = (0, 1) n. for j = n T (φ n ) (x, λ) = φ n (x)λ n n = φ(x) n = (φ, 0) n (x, λ),

40 Consider T : P( 8 l 5 ) P( 8 l 5 ) defined by T(P)(x 1,..., x 5 ) = 8 j=0 k 1 + +k 5 =j Isometries of P( 8 l 5 ) α k1,k 2,8 j,k 4,k 5 x k 1 1 xk 2 2 xk 3 3 xk 4 4 xk 5 5.

41 Consider T : P( 8 l 5 ) P( 8 l 5 ) defined by T(P)(x 1,..., x 5 ) = 8 j=0 k 1 + +k 5 =j Isometries of P( 8 l 5 ) α k1,k 2,8 j,k 4,k 5 x k 1 1 xk 2 2 xk 3 3 xk 4 4 xk 5 5. If P 1 (x 1, x 2, x 3, x 4, x 5 ) = 1 then T(P 1 )(x 1,..., x 5 ) = x 8 3,

42 If P 2 (x 1,..., x 5 ) = x 1 then T(P 2 )(x 1,..., x 5 ) = x 1 x 7 3, Isometries of P( 8 l 5 )

43 If P 2 (x 1,..., x 5 ) = x 1 then T(P 2 )(x 1,..., x 5 ) = x 1 x 7 3, Isometries of P( 8 l 5 ) If P 3 (x 1,..., x 5 ) = x 3 then T(P 3 )(x 1,..., x 5 ) = x 7 3,

44 If P 4 (x 1,..., x 5 ) = x 1 x 2 then T(P 4 )(x 1,..., x 5 ) = x 1 x 2 x 6 3, Isometries of P( 8 l 5 )

45 If P 4 (x 1,..., x 5 ) = x 1 x 2 then T(P 4 )(x 1,..., x 5 ) = x 1 x 2 x 6 3, Isometries of P( 8 l 5 ) If P 5 (x 1,..., x 5 ) = x 3 x 5 then T(P 5 )(x 1,..., x 5 ) = x 6 3 x 5,

46 If P 4 (x 1,..., x 5 ) = x 1 x 2 then T(P 4 )(x 1,..., x 5 ) = x 1 x 2 x 6 3, Isometries of P( 8 l 5 ) If P 5 (x 1,..., x 5 ) = x 3 x 5 then T(P 5 )(x 1,..., x 5 ) = x 6 3 x 5, If P 6 (x 1,..., x 5 ) = x 2 3 then T(P 6 )(x 1,..., x 5 ) = x 6 3,

47 For isometries, T, of the second type we have New Type of Isometry T(P A ( j E)) P A ( n j E)