Simultaneous zero inclusion property for spatial numerical ranges
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1 Simultaneous zero inclusion property for spatial numerical ranges Janko Bračič University of Ljubljana, Slovenia Joint work with Cristina Diogo WONRA, Munich, Germany, June 2018
2 X finite-dimensional complex normed space, usually: X = (C n, ); X the dual normed space,, : X X C unit spheres: the pairing; S X = {x X ; x = 1} S X = {ξ X ; ξ = 1} ; L(X ) linear operators on X, I usually: L(X ) = (M n, ); the identity operator.
3 By the Hahn-Banach theorem: x S X ξ S X : x, ξ = 1. Denote: D(x) = {ξ S X ; x, ξ = 1}. If X is strictly convex, then D(x) is a singleton. If X is a Hilbert space with inner product (, ), then D(x) = {(, x)}. If X = l 1 (n) and e j = (0,..., 1,... 0) T, then D(e j ) = {(γ 1,..., 1,..., γ n ) T ; γ 1 1,..., γ n 1} S l (n).
4 The spatial numerical range of A L(X ) is W X (A) = { Ax, ξ ; x S X, ξ D(x)}.
5 The spatial numerical range of A L(X ) is W X (A) = { Ax, ξ ; x S X, ξ D(x)}. Properties: σ(a) W X (A) D(0, A ); W X (A) is closed; W X (A + B) W X (A) + W X (B); W X (αa + βi) = αw X (A) + β; U a linear isometry on X : W X (U 1 AU) = W X (A).
6 The spatial numerical range of A L(X ) is W X (A) = { Ax, ξ ; x S X, ξ D(x)}. Properties: σ(a) W X (A) D(0, A ); W X (A) is closed; W X (A + B) W X (A) + W X (B); W X (αa + βi) = αw X (A) + β; U a linear isometry on X : W X (U 1 AU) = W X (A). If X = H (Hilbert space), then W H (A) is convex. This does not hold for every X.
7 Observation: X = H Hilbert space, A L(H ) invertible, then 0 W H (A) 0 W H (A 1 ). (1)
8 Observation: X = H Hilbert space, A L(H ) invertible, then 0 W H (A) 0 W H (A 1 ). (1) Proof. 0 W H (A) x S H : (Ax, x) = 0 y = 1 Ax Ax S H (Ax 0 because A is invertible) (A 1 y, y) = 1 Ax 2 (x, Ax) = 0.
9 Observation: X = H Hilbert space, A L(H ) invertible, then 0 W H (A) 0 W H (A 1 ). (1) Proof. 0 W H (A) x S H : (Ax, x) = 0 y = 1 Ax Ax S H (Ax 0 because A is invertible) (A 1 y, y) = 1 Ax 2 (x, Ax) = 0. Question: Does (1) hold in every normed space X?
10 Observation: X = H Hilbert space, A L(H ) invertible, then 0 W H (A) 0 W H (A 1 ). (1) Proof. 0 W H (A) x S H : (Ax, x) = 0 y = 1 Ax Ax S H (Ax 0 because A is invertible) (A 1 y, y) = 1 Ax 2 (x, Ax) = 0. Question: Does (1) hold in every normed space X? Answer: No.
11 Example: X = l 1 (2).
12 Example: X = l 1 (2). A = [ ] a b c d M2 ; W l1 (2)(A) D(a, c ), D(d, b ), is union of: [a + cε, d + bε] = {t(a + cε) + (1 t)(d + bε); 0 t 1}, line segment for each ε C, ε = 1.
13 d + bε d b d + b a a + cε c a + c
14 A = [ ] A 1 = [ W l1 (2)(A) = D(1, 1) and W l1 (2)(A 1 ) = D(1, 1 2 ); ] ; Im i Im i Re Re i i W l1 (2)(A) W l1 (2)(A 1 ) 0 W l1 (2)(A) and 0 W l1 (2)(A 1 ).
15 l 1 (n), n A 1 = W l1 (n)(a 1 ) and 0 W l1 (n)(a 1 1 ).
16 l 1 (n), n A 1 = W l1 (n)(a 1 ) and 0 W l1 (n)(a 1 1 ). l (n), n 2. W l (n)(s) = W l1 (n)(s T ) for every S M n ; 0 W l (n)(a T 1 ) and 0 W l (n)((a T 1 ) 1 )).
17 l p (n), n 2, 2 < p. There exists 0 < a p < 1 such that for a p a p 0 0 A p = W lp(n)(a p ) and 0 W lp(n)(a 1 p ).
18 l p (n), n 2, 2 < p. There exists 0 < a p < 1 such that for a p a p 0 0 A p = W lp(n)(a p ) and 0 W lp(n)(a 1 p ). l q (n), n 2, 1 < q < 2. p = q q 1 > 2; W lq(n)(s) = W lp(n)(s T ) for every S M n ; 0 W lq(n)(a T p) and 0 W lq(n)((a T p) 1 )).
19 Definition. Normed space X has the simultaneous zero inclusion property (S0I) if 0 W X (A) 0 W X (A 1 ) for every invertible A L(X ).
20 Definition. Normed space X has the simultaneous zero inclusion property (S0I) if 0 W X (A) 0 W X (A 1 ) for every invertible A L(X ). Hilbert space has S0I. l p (n), 1 p, p 2 does not have S0I.
21 X normed space, T L(X ) invertible; new norm: x T = Tx (x X ); X T = (X, T ). Theorem. The following are equivalent: X X X T has S0I; has S0I; has S0I.
22 Definition. S0I holds at invertible A L(X ) if 0 W X (A) 0 W X (A 1 ).
23 Definition. S0I holds at invertible A L(X ) if 0 W X (A) 0 W X (A 1 ). Zenger: conv ( σ(a)) W X (A).
24 Definition. S0I holds at invertible A L(X ) if 0 W X (A) 0 W X (A 1 ). Zenger: conv ( σ(a)) W X (A). Theorem. A L(X ) invertible. If 0 conv ( σ(a) ), then S0I holds at A.
25 Definition. S0I holds at invertible A L(X ) if 0 W X (A) 0 W X (A 1 ). Zenger: conv ( σ(a)) W X (A). Theorem. A L(X ) invertible. If 0 conv ( σ(a) ), then S0I holds at A. Proof. σ(a) = {λ 1,..., λ k } σ(a 1 ) = { 1 λ 1,..., 1 λ k }; t 1 λ t k λ k = 0, 0 t 1,..., t k 1, t t k = 1; for j = 1,..., k, s j = t j λ j 2 t 1 λ t k λ k 2 ; 0 s 1,..., s k 1 and s s k = 1; s 1 1 λ s k 1 λ k = 0.
26 Algebra numerical range of A L(X ) is V X (A) W L(X ) (L A ) where L A : T AT, T L(X ).
27 Algebra numerical range of A L(X ) is V X (A) W L(X ) (L A ) where L A : T AT, T L(X ). V X (A) = conv ( W X (A) ) ; Williams: λ V X (A) z λ zi A, z C.
28 Algebra numerical range of A L(X ) is V X (A) W L(X ) (L A ) where L A : T AT, T L(X ). V X (A) = conv ( W X (A) ) ; Williams: λ V X (A) z λ zi A, z C. Theorem. If A L(X ) is an invertible contraction and e iϕ I A 1 < 1, then S0I holds at A.
29 Algebra numerical range of A L(X ) is V X (A) W L(X ) (L A ) where L A : T AT, T L(X ). V X (A) = conv ( W X (A) ) ; Williams: λ V X (A) z λ zi A, z C. Theorem. If A L(X ) is an invertible contraction and e iϕ I A 1 < 1, then S0I holds at A. Proof. e iϕ I A 1 < 1 0 V X (A 1 ) W X (A 1 ); e iϕ I A e iϕ A e iϕ I A 1 < 1 0 V X (A) W X (A).
30 H(C) set of all non-empty compact subsets of C endowed with Hausdorff metric h.
31 H(C) set of all non-empty compact subsets of C endowed with Hausdorff metric h. Theorem. Mapping W X : L(X ) H(C), A W X (A) is continuous.
32 H(C) set of all non-empty compact subsets of C endowed with Hausdorff metric h. Theorem. Mapping W X : L(X ) H(C), A W X (A) is continuous. G out (X ) = {A L(X ); G in (X ) = {A L(X ); A invertible and 0 W X (A) W X (A 1 )}, A invertible and 0 W X (A) W X (A 1 )};
33 H(C) set of all non-empty compact subsets of C endowed with Hausdorff metric h. Theorem. Mapping W X : L(X ) H(C), A W X (A) is continuous. G out (X ) = {A L(X ); G in (X ) = {A L(X ); A invertible and 0 W X (A) W X (A 1 )}, A invertible and 0 W X (A) W X (A 1 )}; X has S0I G out (X ) G in (X ) all invertible;
34 H(C) set of all non-empty compact subsets of C endowed with Hausdorff metric h. Theorem. Mapping W X : L(X ) H(C), A W X (A) is continuous. G out (X ) = {A L(X ); G in (X ) = {A L(X ); A invertible and 0 W X (A) W X (A 1 )}, A invertible and 0 W X (A) W X (A 1 )}; X has S0I G out (X ) G in (X ) all invertible; Theorem. G out (X ) is open and G in (X ) is relatively closed subset in the set of all invertible operators.
35 What if in the definition of S0I W X is replaced by V X?
36 What if in the definition of S0I W X is replaced by V X? X = H : V H (A) = W H (A) for every A B(H ); X = l 1 (n): for A 1 = , W l1 (n)(a 1 ) and W l1 (n)(a 1 1 ) are convex W l1 (n)(a 1 ) = V l1 (n)(a 1 ) and W l1 (n)(a 1 1 ) = V l 1 (n)(a 1 1 ) 0 V l1 (n)(a 1 ) and 0 V l1 (n)(a 1 1 );
37 What if in the definition of S0I W X is replaced by V X? X = H : V H (A) = W H (A) for every A B(H ); X = l 1 (n): for A 1 = , W l1 (n)(a 1 ) and W l1 (n)(a 1 1 ) are convex W l1 (n)(a 1 ) = V l1 (n)(a 1 ) and W l1 (n)(a 1 1 ) = V l 1 (n)(a 1 1 ) 0 V l1 (n)(a 1 ) and 0 V l1 (n)(a 1 1 ); X = l p (n), 1 < p <, p 2: we do not know.
38 ϕ 0 [0, π 2 ): Σ(ϕ 0) = {re iϕ ; r 0, ϕ 0 ϕ ϕ 0 }. Im 0 Σ(ϕ 0 ) Re Definition. X has the simultaneous sector inclusion property (SSI) if, for every invertible A L(X ), V X (A) Σ(ϕ 0 ) for some ϕ 0 V X (A 1 ) Σ(ϕ 0 ).
39 ϕ 0 [0, π 2 ): Σ(ϕ 0) = {re iϕ ; r 0, ϕ 0 ϕ ϕ 0 }. Im 0 Σ(ϕ 0 ) Re Definition. X has the simultaneous sector inclusion property (SSI) if, for every invertible A L(X ), V X (A) Σ(ϕ 0 ) for some ϕ 0 V X (A 1 ) Σ(ϕ 0 ). Hilbert space has this property; l 1 (n) does not have: A 1 is a counter example. Is there any relation between SSI and S0I?
40 Thank you!
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