Strictly singular operators between L p L q spaces and interpolation

Transcription

1 Strictly singular operators between L p L q spaces and interpolation Francisco L. Hernández Madrid Complutense University Positivity IX Conference, Alberta University joint work with E. Semenov and P. Tradacete

2 1 Preliminary 2 Strictly singular non-compact operator sets V p,q 3 Extensions of Krasnoselskii compact interpolation theorem to S

3 T : E F strictly singular (S) ( or Kato) if T is not invertible on any -dim. (closed) subspace E 1 of E.

4 T : E F strictly singular (S) ( or Kato) if T is not invertible on any -dim. (closed) subspace E 1 of E. if c > 0 s.t. c x Tx, x E 1 = dim E 1 <

5 T : E F strictly singular (S) ( or Kato) if T is not invertible on any -dim. (closed) subspace E 1 of E. if c > 0 s.t. c x Tx, x E 1 = dim E 1 < closed operator ideal S

6 T : E F strictly singular (S) ( or Kato) if T is not invertible on any -dim. (closed) subspace E 1 of E. if c > 0 s.t. c x Tx, x E 1 = dim E 1 < closed operator ideal S K compact operator ideal, K S

7 T : E F strictly singular (S) ( or Kato) if T is not invertible on any -dim. (closed) subspace E 1 of E. if c > 0 s.t. c x Tx, x E 1 = dim E 1 < closed operator ideal S K compact operator ideal, K S l p l q, 1 p < q, strictly singular non-compact

8 T : E F strictly singular (S) ( or Kato) if T is not invertible on any -dim. (closed) subspace E 1 of E. if c > 0 s.t. c x Tx, x E 1 = dim E 1 < closed operator ideal S K compact operator ideal, K S l p l q, 1 p < q, strictly singular non-compact L q [0, 1] = L q L p (1 p q < ) no strictly singular [(r n )] q [(r n )] p l 2

9 T : E F strictly singular (S) ( or Kato) if T is not invertible on any -dim. (closed) subspace E 1 of E. if c > 0 s.t. c x Tx, x E 1 = dim E 1 < closed operator ideal S K compact operator ideal, K S l p l q, 1 p < q, strictly singular non-compact L q [0, 1] = L q L p (1 p q < ) no strictly singular [(r n )] q [(r n )] p l 2 L L p strictly singular non-compact (Grothendieck)

10 T : E F strictly singular (S) ( or Kato) if T is not invertible on any -dim. (closed) subspace E 1 of E. if c > 0 s.t. c x Tx, x E 1 = dim E 1 < closed operator ideal S K compact operator ideal, K S l p l q, 1 p < q, strictly singular non-compact L q [0, 1] = L q L p (1 p q < ) no strictly singular [(r n )] q [(r n )] p l 2 L L p strictly singular non-compact (Grothendieck) T : E F strictly singular for every subspace E 1 E there exists E 2 E 1 s.t. T E2 compact

11 Spectral properties of strictly singular op. like compact: σ(t ) = {λ n }, σ(t )\{0} = σ p (T )\{0}, (σ(t )) {0}

12 Spectral properties of strictly singular op. like compact: σ(t ) = {λ n }, σ(t )\{0} = σ p (T )\{0}, (σ(t )) {0} Fredholm perturbation th., R Fredholm, T S R + T Fredholm.

13 Spectral properties of strictly singular op. like compact: σ(t ) = {λ n }, σ(t )\{0} = σ p (T )\{0}, (σ(t )) {0} Fredholm perturbation th., R Fredholm, T S R + T Fredholm. - Some bad properties of the class S :

14 Spectral properties of strictly singular op. like compact: σ(t ) = {λ n }, σ(t )\{0} = σ p (T )\{0}, (σ(t )) {0} Fredholm perturbation th., R Fredholm, T S R + T Fredholm. - Some bad properties of the class S : S is not stable by duality. (for operators T: L p L p duality holds, (V.Milman, L.Weiss 1977))

15 Spectral properties of strictly singular op. like compact: σ(t ) = {λ n }, σ(t )\{0} = σ p (T )\{0}, (σ(t )) {0} Fredholm perturbation th., R Fredholm, T S R + T Fredholm. - Some bad properties of the class S : S is not stable by duality. (for operators T: L p L p duality holds, (V.Milman, L.Weiss 1977)) There is stric. singular op. without invariant subspaces (Read 1.999)

16 Spectral properties of strictly singular op. like compact: σ(t ) = {λ n }, σ(t )\{0} = σ p (T )\{0}, (σ(t )) {0} Fredholm perturbation th., R Fredholm, T S R + T Fredholm. - Some bad properties of the class S : S is not stable by duality. (for operators T: L p L p duality holds, (V.Milman, L.Weiss 1977)) There is stric. singular op. without invariant subspaces (Read 1.999) S is not suitable for interpolation (some results for operators with the same domain ( Heinrich 1980, Beucher ))

17 K (L p ) = S(L p ) p = 2. -Ex: if p > 2, P rad (f ) = n ( 1 0 f (x)r n(x)dx) r n T : L p L p Rad l 2 l p

18 K (L p ) = S(L p ) p = 2. -Ex: if p > 2, P rad (f ) = n ( 1 0 f (x)r n(x)dx) r n T : L p L p Rad l 2 l p K (l p ) = S(l p ), 1 p <

19 K (L p ) = S(L p ) p = 2. -Ex: if p > 2, P rad (f ) = n ( 1 0 f (x)r n(x)dx) r n T : L p L p Rad l 2 l p K (l p ) = S(l p ), 1 p < There is lattice function spaces E[0, 1]( L 2 ) s.t. K (E) = S(E)

20 K (L p ) = S(L p ) p = 2. -Ex: if p > 2, P rad (f ) = n ( 1 0 f (x)r n(x)dx) r n T : L p L p Rad l 2 l p K (l p ) = S(l p ), 1 p < There is lattice function spaces E[0, 1]( L 2 ) s.t. K (E) = S(E) K (L p, L q ) = S(L p, L q )?

21 M. Riesz-Thorin convexity theorem 1 = 1 θ + θ, p θ p 0 p 1 1 = 1 θ + θ q θ q 0 q 1 T pθ,q θ T 1 θ p 0,q 0 T θ p 1,q 1

22 M. Riesz-Thorin convexity theorem 1 = 1 θ + θ, p θ p 0 p 1 1 = 1 θ + θ q θ q 0 q 1 T pθ,q θ T 1 θ p 0,q 0 T θ p 1,q 1 Theorem (Krasnoselskii 1960) Let 1 p 0, p 1, q 0, q 1. If T : L p0 L q0 compact, T : L p1 L q1 bounded = T : L pθ L qθ compact, for 0 < θ < 1.

23 M. Riesz-Thorin convexity theorem 1 = 1 θ + θ, p θ p 0 p 1 1 = 1 θ + θ q θ q 0 q 1 T pθ,q θ T 1 θ p 0,q 0 T θ p 1,q 1 Theorem (Krasnoselskii 1960) Let 1 p 0, p 1, q 0, q 1. If T : L p0 L q0 compact, T : L p1 L q1 bounded = T : L pθ L qθ compact, for 0 < θ < 1. Many extensions to abstract interpolation methods and applications

24 M. Riesz-Thorin convexity theorem 1 = 1 θ + θ, p θ p 0 p 1 1 = 1 θ + θ q θ q 0 q 1 T pθ,q θ T 1 θ p 0,q 0 T θ p 1,q 1 Theorem (Krasnoselskii 1960) Let 1 p 0, p 1, q 0, q 1. If T : L p0 L q0 compact, T : L p1 L q1 bounded = T : L pθ L qθ compact, for 0 < θ < 1. Many extensions to abstract interpolation methods and applications QUESTION : - extensions of K.T to the class of strictly singular operators? - what about endomorphisms?

25 -the case of endomorphisms: Theorem (H.-Semenov-Tradacete, 2010) Let 1 p q If T : L p L p bounded, T : L q L q strictly singular = T : L pθ L pθ strictly singular for every q < p θ < p

26 -the case of endomorphisms: Theorem (H.-Semenov-Tradacete, 2010) Let 1 p q If T : L p L p bounded, T : L q L q strictly singular = T : L pθ L pθ strictly singular for every q < p θ < p Even more T : L pθ L pθ is compact (strong interpolation property)

27 -the case of endomorphisms: Theorem (H.-Semenov-Tradacete, 2010) Let 1 p q If T : L p L p bounded, T : L q L q strictly singular = T : L pθ L pθ strictly singular for every q < p θ < p Even more T : L pθ L pθ is compact (strong interpolation property) extrapolation property: If 1 < q, p < and T : L q L q, T : L p L p bounded.then T S(L r ) for some r (q, p) T K (L s ) for some s (q, p).

28 -the case of endomorphisms: Theorem (H.-Semenov-Tradacete, 2010) Let 1 p q If T : L p L p bounded, T : L q L q strictly singular = T : L pθ L pθ strictly singular for every q < p θ < p Even more T : L pθ L pθ is compact (strong interpolation property) extrapolation property: If 1 < q, p < and T : L q L q, T : L p L p bounded.then T S(L r ) for some r (q, p) T K (L s ) for some s (q, p). In particular it follows that T K (L s ) for every s (q, p)

29 -the case of endomorphisms: Theorem (H.-Semenov-Tradacete, 2010) Let 1 p q If T : L p L p bounded, T : L q L q strictly singular = T : L pθ L pθ strictly singular for every q < p θ < p Even more T : L pθ L pθ is compact (strong interpolation property) extrapolation property: If 1 < q, p < and T : L q L q, T : L p L p bounded.then T S(L r ) for some r (q, p) T K (L s ) for some s (q, p). In particular it follows that T K (L s ) for every s (q, p) Rigidity of special operator classes : -regular operators in L p -spaces ( Caselles-Gonzalez 1987) - composition operators on H p, Volterra operators (Tylli et al )

30 The proof uses properties of the sets S(L p )\K (L p ), Kadec-Pelczynski method, l p -disjointification Dor result,...

31 The proof uses properties of the sets S(L p )\K (L p ), Kadec-Pelczynski method, l p -disjointification Dor result,... Theorem (Dor, 1975) Let 1 p 2 <, 0 < θ 1, and (f i ) i=1 in L p[0, 1]. Assume that either: 1 1 p < 2, f i 1 and n i=1 a if i θ( n i=1 a i p ) 1/p for scalars (a i ) n i=1, and every n, or 2 2 < p <, f i 1 and n i=1 a if i θ 1 ( n i=1 a i p ) 1/p for scalars (a i ) n i=1 and every n. Then there exist disjoint measurable sets (A i ) i=1 s.t. f i χ Ai θ 2/ p 2.

32 the general case T : L p L q for p q :

33 the general case T : L p L q for p q : S(L p, L q ) = K (L p, L q ) q 2 p

34 the general case T : L p L q for p q : S(L p, L q ) = K (L p, L q ) q 2 p -If p < 2, consider T - if q > 2 take S L p T L q L p S L q P p l p l 2 J rad P rad l 2 l q J q

35 the general case T : L p L q for p q : S(L p, L q ) = K (L p, L q ) q 2 p -If p < 2, consider T - if q > 2 take S L p T L q L p S L q P p l p l 2 J rad P rad l 2 l q J q ( ) If F is a Banach lattice with lower 2-estimate. If T : l 2 F strictly singular compact (Flores,H.,Kalton,Tradacete 2009)

36 -If q 2 p. w Assume T / K (L p, L q ), there is (x i ) s.t. x i p = 1, x i 0 and Tx i q λ > 0. By Kadec-Pelczynski m. there is (x ik ) equivalent to the basis of l p or l 2. If (x ik ) equivalent to basis of l p, 1 λ n n q r k (t)tx ik dt T k=1 1 0 n p r k (t)x ik dt n 1 p. Thus, (x ik ) must be equivalent to basis of l 2.Now by ( ), T is not strictly singular k=1

37 -If q 2 p. w Assume T / K (L p, L q ), there is (x i ) s.t. x i p = 1, x i 0 and Tx i q λ > 0. By Kadec-Pelczynski m. there is (x ik ) equivalent to the basis of l p or l 2. If (x ik ) equivalent to basis of l p, 1 λ n n q r k (t)tx ik dt T k=1 1 0 n p r k (t)x ik dt n 1 p. Thus, (x ik ) must be equivalent to basis of l 2.Now by ( ), T is not strictly singular k=1

38 -If q 2 p. w Assume T / K (L p, L q ), there is (x i ) s.t. x i p = 1, x i 0 and Tx i q λ > 0. By Kadec-Pelczynski m. there is (x ik ) equivalent to the basis of l p or l 2. If (x ik ) equivalent to basis of l p, 1 λ n n q r k (t)tx ik dt T k=1 1 0 n p r k (t)x ik dt n 1 p. Thus, (x ik ) must be equivalent to basis of l 2.Now by ( ), T is not strictly singular k=1 If q 2 p, every T : L p L q is either compact or fixes a l 2 -isomorphic copy

39 -If q 2 p. w Assume T / K (L p, L q ), there is (x i ) s.t. x i p = 1, x i 0 and Tx i q λ > 0. By Kadec-Pelczynski m. there is (x ik ) equivalent to the basis of l p or l 2. If (x ik ) equivalent to basis of l p, 1 λ n n q r k (t)tx ik dt T k=1 1 0 n p r k (t)x ik dt n 1 p. Thus, (x ik ) must be equivalent to basis of l 2.Now by ( ), T is not strictly singular k=1 If q 2 p, every T : L p L q is either compact or fixes a l 2 -isomorphic copy If E B. lattice with type 2 and unconditional basis and F B. lattice with a lower 2-estimate K (E, F ) = S(E, F )

40 Given 1 p, q < V p,q := S(L p, L q )\K (L p, L q ).

41 Given 1 p, q < V p,q := S(L p, L q )\K (L p, L q ). the case 2 < q p <. If T : L p L q and T V p,q = there exists normalized (y k ) in L p s.t.: - (y k ) is equivalent to the basis of l 2 with ( y k ) equi-measurable, - (Ty k ) is equivalent to the basis of l q, - [(y k )] is complemented in L p.

42 Given 1 p, q < V p,q := S(L p, L q )\K (L p, L q ). the case 2 < q p <. If T : L p L q and T V p,q = there exists normalized (y k ) in L p s.t.: - (y k ) is equivalent to the basis of l 2 with ( y k ) equi-measurable, - (Ty k ) is equivalent to the basis of l q, - [(y k )] is complemented in L p.

43 Given 1 p, q < V p,q := S(L p, L q )\K (L p, L q ). the case 2 < q p <. If T : L p L q and T V p,q = there exists normalized (y k ) in L p s.t.: - (y k ) is equivalent to the basis of l 2 with ( y k ) equi-measurable, - (Ty k ) is equivalent to the basis of l q, - [(y k )] is complemented in L p.

44 Given 1 p, q < V p,q := S(L p, L q )\K (L p, L q ). the case 2 < q p <. If T : L p L q and T V p,q = there exists normalized (y k ) in L p s.t.: - (y k ) is equivalent to the basis of l 2 with ( y k ) equi-measurable, - (Ty k ) is equivalent to the basis of l q, - [(y k )] is complemented in L p. There is strictly singular T : L p L q s. t. T is not strictly singular 2 < q < p.

45 extrapolation property : Theorem Let 1 < p 0, p 1, q 0, q 1 < with q 0 q 1, p 0 p 1. Suppose mín{ q 0 p 0, q 1 p 1 } 1, or mín{ q 0 p 0, q 1 p 1 } > 1 and q 1 q 0 p 1 p 0 < 0. If T : L p0 L q0 bounded, T : L p1 L q1 bounded, and T : L pθ L qθ strictly singular for some 0 < θ < 1 = T K (L pτ, L qτ ) for every 0 < τ < 1.

46 Given T : L L 1, the V -characteristic set, {( 1 V (T ) := p, 1 ) q } : T V p,q = S(L p, L q )\K (L p, L q ).

47 Given T : L L 1, the V -characteristic set, {( 1 V (T ) := p, 1 ) q } : T V p,q = S(L p, L q )\K (L p, L q ). V (T ) L(T ) where L(T ) is the characteristic set (Krasnoselskii and Zabreiko ) {( 1 L(T ) := p, 1 ) q } [0, 1] [0, 1] : T : L p L q bounded.

48 Given T : L L 1, the V -characteristic set, {( 1 V (T ) := p, 1 ) q } : T V p,q = S(L p, L q )\K (L p, L q ). V (T ) L(T ) where L(T ) is the characteristic set (Krasnoselskii and Zabreiko ) {( 1 L(T ) := p, 1 ) q } [0, 1] [0, 1] : T : L p L q bounded. L(T ) is a monotone convex F σ set

49 Given T : L L 1, the V -characteristic set, {( 1 V (T ) := p, 1 ) q } : T V p,q = S(L p, L q )\K (L p, L q ). V (T ) L(T ) where L(T ) is the characteristic set (Krasnoselskii and Zabreiko ) {( 1 L(T ) := p, 1 ) q } [0, 1] [0, 1] : T : L p L q bounded. L(T ) is a monotone convex F σ set Corollary V (T ) L(T ).

50 interpolation property ( extension of Krasnoselskii Thm.): Theorem Let 1 p 0, p 1, q 0, q 1 <. If T : L p0 L q0 strictly singular, T : L p1 L q1 bounded = T : L pθ L qθ strictly singular, for 0 < θ < 1 and 1 = 1 θ + θ, p θ p 0 p 1 1 q θ = 1 θ q 0 + θ q 1

51 interpolation property ( extension of Krasnoselskii Thm.): Theorem Let 1 p 0, p 1, q 0, q 1 <. If T : L p0 L q0 strictly singular, T : L p1 L q1 bounded = T : L pθ L qθ strictly singular, for 0 < θ < 1 and 1 = 1 θ + θ, p θ p 0 p 1 1 = 1 θ + θ q θ q 0 q 1 p 0 < is a necessary condition

52 strong interpolation property : Proposition Let 1 < p 0, p 1, q 0, q 1 < with p 0 p 1, q 0 q 1 Suppose mín{ q 0 p 0, q 1 p 1 } 1, or mín{ q 0 p 0, q 1 p 1 } > 1 and q 1 q 0 p 1 p 0 < 0. If T : L p0 L q0 strictly singular T : L p1 L q1 bounded = T K (L pθ, L qθ ) for 0 < θ < 1.

53 strong interpolation property : Proposition Let 1 < p 0, p 1, q 0, q 1 < with p 0 p 1, q 0 q 1 Suppose mín{ q 0 p 0, q 1 p 1 } 1, or mín{ q 0 p 0, q 1 p 1 } > 1 and q 1 q 0 p 1 p 0 < 0. If T : L p0 L q0 strictly singular T : L p1 L q1 bounded = T K (L pθ, L qθ ) for 0 < θ < 1. Domain restrictions are necessary:

54 Fixed 0 < λ < 1, consider the segment S λ = {(p, q) : 1 q = 1 p λ} Proposition There is operators T V p,q for every (p, q) S λ

55 Fixed 0 < λ < 1, consider the segment S λ = {(p, q) : 1 q = 1 p λ} Proposition There is operators T V p,q for every (p, q) S λ Take the fractional op. T λ = T, for a pairwise disjoint (A k ) k N T (f ) = ( ) 1 µ(a k ) 1 λ fdµ χ Ak. k N A k

56 Fixed 0 < λ < 1, consider the segment S λ = {(p, q) : 1 q = 1 p λ} Proposition There is operators T V p,q for every (p, q) S λ Take the fractional op. T λ = T, for a pairwise disjoint (A k ) k N T (f ) = ( ) 1 µ(a k ) 1 λ fdµ χ Ak. k N A k T is L p L q bounded for 1 p < q < and 1 q = 1 p λ.

57 Fixed 0 < λ < 1, consider the segment S λ = {(p, q) : 1 q = 1 p λ} Proposition There is operators T V p,q for every (p, q) S λ Take the fractional op. T λ = T, for a pairwise disjoint (A k ) k N T (f ) = ( ) 1 µ(a k ) 1 λ fdµ χ Ak. k N A k T is L p L q bounded for 1 p < q < and 1 q = 1 p λ. T factors through l p l q (so it is strictly singular), since T L p P L q i l p l q T is not compact ( T (µ(a k ) 1 p χak ) = µ(a k ) 1 q χak ) Q

58 Example: operators T : L L 1 with V (T ) is 1 q 1 2 ( 1 p 1, 1 q 1 ) ( 1 p 0, 1 q 0 ) 1 2 > 1 p,

59 Example: operators T : L L 1 with V (T ) is 1 q 1 2 ( 1 p 1, 1 q 1 ) ( 1 p 0, 1 q 0 ) 1 2 p 0 < q 0, p 1 < q 1 s.t. 1 q 0 1 p 0 = 1 q 1 1 p 1 = λ, and p 1 < 2 < q 0 > 1 p,

60 Example: operators T : L L 1 with V (T ) is 1 q 1 2 ( 1 p 1, 1 q 1 ) ( 1 p 0, 1 q 0 ) 1 2 p 0 < q 0, p 1 < q 1 s.t. 1 q 0 1 p 0 = 1 q 1 1 p 1 = λ, and p 1 < 2 < q 0 > 1 p T = T 1 + T 2 + T 3 T 1 : L p [0, 1/3] L q0 [0, 1/3] factorizing by i 2,q0 : l 2 l q0 T 2 : L p [ 1 3, 2 3 ] L q[ 1 3, 2 3 ] λ-fractional on [1/3, 2/3] T 3 : L p1 [2/3] L q [2/3, 1] factorizing by i p1,2 : l p1 l 2,

61 Example: operators T : L L 1 with V (T ) is 1 q 1 2 ( 1 p 1, 1 q 1 ) ( 1 p 0, 1 q 0 ) 1 2 p 0 < q 0, p 1 < q 1 s.t. 1 q 0 1 p 0 = 1 q 1 1 p 1 = λ, and p 1 < 2 < q 0 > 1 p T = T 1 + T 2 + T 3 T 1 : L p [0, 1/3] L q0 [0, 1/3] factorizing by i 2,q0 : l 2 l q0 T 2 : L p [ 1 3, 2 3 ] L q[ 1 3, 2 3 ] λ-fractional on [1/3, 2/3] T 3 : L p1 [2/3] L q [2/3, 1] factorizing by i p1,2 : l p1 l 2, Question: which is the shape of V (T ) in general??

62 Proposition super strictly singular L p L q operator class is also interpolated

63 Proposition super strictly singular L p L q operator class is also interpolated T : X Y super strictly singular (or finitely strictly singular) if it does not exist c > 0 and subspaces (F n ) with dim F n = n s.t. Tx c x for all x n F n

64 Proposition super strictly singular L p L q operator class is also interpolated T : X Y super strictly singular (or finitely strictly singular) if it does not exist c > 0 and subspaces (F n ) with dim F n = n s.t. Tx c x for all x n F n Bernstein numbers { { Tx } } b n (T ) = sup ínf x : x 0 F : F X, dim(f ) = n. T super strictly singular b n (T ) n 0 T : X Y super strictly singular ultraoperator T U : X U Y U is strictly singular for every ultrafilter U,