Toral and Spherical Aluthge Transforms

Transcription

1 Toral and Spherical Aluthge Transforms (joint work with Jasang Yoon) Raúl E. Curto University of Iowa INFAS, Des Moines, IA November 11, 2017 (our first report on this research appeared in Comptes Rendus Acad. Sci. Paris (2016))

2 Overview 1 Unilateral Weighted Shifts 2 Aluthge Transforms 3 Multivariable Weighted Shifts 4 Toral Aluthge Transform 5 Spherical Aluthge Transform 6 Invariance of Hyponormality 7 Spherically Quasinormal Pairs

3 Hyponormality and Subnormality L(H): algebra of operators on a Hilbert space H T L(H) is normal if T T = TT subnormal if T = N H, where N is normal and NH H (We say that N is a lifting of T, or an extension of T. hyponormal if T T TT normal subnormal hyponormal

4 For S,T B(H), [S,T] := ST TS. An n-tuple T (T 1,...,T n ) is (jointly) hyponormal if [T1,T 1] [T2,T 1] [Tn,T 1 ] [T [T1,T] :=,T 2] [T2,T 2] [Tn,T 2 ] [T1,T n] [T2,T n] [Tn,T n ] For k 1, an operator T is k-hyponormal if (T,...,T k ) is (jointly) hyponormal, i.e., (Bram-Halmos): [T,T] [T k,t]..... [T,T k ] [T k,t k ] T subnormal T is k-hyponormal for all k 1. 0

5 Unilateral Weighted Shifts α {α k } k=0 l (Z + ), α k > 0 (all k 0) W α : l 2 (Z + ) l 2 (Z + ) W α e k := α k e k+1 (k 0) When α k = 1 (all k 0), W α = U +, the (unweighted) unilateral shift In general, W α = U + D α (polar decomposition) W α = sup k α k Wαe n k = α k α k+1 α k+n 1 e k+n, so Wα n n 1 = i=0 W β (i),

6 Weighted Shifts and Berger s Theorem Given a bounded sequence of positive numbers (weights) α α 0,α 1,α 2,..., the unilateral weighted shift on l 2 (Z + ) associated with α is W α e k := α k e k+1 (k 0). The moments of α are given as { γ k γ k (α) := 1 if k = 0 α α2 k 1 if k > 0 }. W α is never normal W α is hyponormal α k α k+1 (all k 0)

7 Berger Measures (Berger; Gellar-Wallen) W α is subnormal if and only if there exists a positive Borel measure ξ on [0, W α 2 ] such that γ k = t k dξ(t) (all k 0). ξ is the Berger measure of W α. For 0 < a < 1 we let S a := shift(a,1,1,...). The Berger measure of U + is δ 1. The Berger measure of S a is (1 a 2 )δ 0 +a 2 δ 1. The Berger measure of B + (the Bergman shift) is Lebesgue measure n+1 on the interval [0,1]; the weights of B + are α n := n+2 (n 0).

8 Spectral Pictures of Hyponormal U.W.S. WLOG, assume W α = 1. Observe that r(w α ) = 1 = supα. Thus, σ(w α ) = D σ e (W α ) = T ind(w α λ) = 1 for λ < 1. Therefore, all norm-one hyponormal weighted shifts are spectrally equivalent.

9 σ e(w α) index = 1 (0, 0) index = Figure 1. Spectral picture of a norm-one hyponormal weighted shift

10 (RC, 1990) W α is k-hyponormal if and only if the following Hankel moment matrices are positive for m = 0,1,2,... : γ m γ m+1 γ m+2... γ m+k γ m+1 γ m+2... γ m+k+1 γ m γ m+k γ m+k γ m+k+1... γ m+2k (Thus, an operator matrix condition is replaced by a scalar matrix condition.)

11 Aluthge Transform Let T be a Hilbert space operator, let P := T be its positive part, and let T = VP denote the canonical polar decomposition of T, with V a partial isometry and ker V = ker T = ker P. We define the Aluthge transform of T as ˆT := PV P. The iterates are ˆT n+1 := (ˆT) n (n 1). The Aluthge transform has been extensively studied, in terms of algebraic, structural and spectral properties.

12 For instance, (i) T = ˆT T is quasinormal; (ii) (Aluthge, 1990) If 0 < p < 1 2 and T is p-hyponormal, then ˆT is (p )-hyponormal; (iii) (Jung, Ko & Pearcy, 2000) If ˆT has a n.i.s., then T has a n.i.s. (iv) (Kim-Ko, 2005; Kimura, 2004) T has property (β) if and only if ˆT has property (β); and (v) (Ando, 2005) (T λ) 1 (ˆT λ) 1 (λ / σ(t)). (vi) Observe that if A := P and B := V P, then ˆT = AB and T = BA, and therefore σ(ˆt)\{0} = σ(t)\{0}.

13 On the other hand, G. Exner (IWOTA 2006 Lecture): subnormality is not preserved under the Aluthge transform. Concretely, Exner proved that the Aluthge transform of the weighted shift in the following example is not subnormal. Example (RC, Y. Poon and J. Yoon, 2005) Let 1 α α n := 2, if n = 0, 2 n n +1, if n 1 Then W α is subnormal, with 3-atomic Berger measure µ = 1 3 (δ 0 +δ 1/2 +δ 1 ).

14 (S.H. Lee, W.Y. Lee and J. Yoon, 2012) For k 2, the Aluthge transform, when acting on weighted shifts, need not preserve k-hyponormality. Note that the Aluthge transform of a weighted shift is again a weighted shift. Concretely, the weights of Ŵα are α0 α 1, α 1 α 2, α 2 α 3, α 3 α 4,. Define W α := shift ( α 0, α 1, α 2, ). Then Ŵα is the Schur product of W α and its restriction to the subspace {e1,e 2, }. Thus, a sufficient condition for the subnormality of Ŵα is the subnormality of W α.

15 Agler Shifts For j = 2,3,..., the j-th Agler shift A j is given by α j := 1 j, 2 j +1, 3 j +2,... It is well known that A j is subnormal, with Berger measure dµ j (t) = (j 1)(1 t) j 2 dt. Clearly, A 2 is the Bergman shift, and the remaining Agler shifts are the upper row shifts of the Drury-Arveson 2-variable weighted shift, which incidentally is a spherical complete hyperexpansion.

16 Weight Diagram of the Drury-Arveson Shift T 1 T 2 (0,1) (0,2) (0,3) (0,0) (1,0) (2,0) (3,0) δ dt (1 t)dt (1 t) 2 dt

17 Multivariable Weighted Shifts α k,β k l (Z 2 +), k (k 1,k 2 ) Z 2 + := Z + Z + l 2 (Z 2 +) = l 2 (Z + ) l 2 (Z + ). We define the 2-variable weighted shift T (T 1,T 2 ) by T 1 e k := α k e k+ε1 T 2 e k := β k e k+ε2, where ε 1 := (1,0) and ε 2 := (0,1). Clearly, T 1 T 2 = T 2 T 1 β k+ε1 α k = α k+ε2 β k (all k). α k1,k 2 +1 (k 1, k 2 + 1) (k 1 + 1, k 2 + 1) β k1,k 2 β k1 +1,k 2 α k1,k 2 (k 1, k 2) (k 1 + 1, k 2)

18 (0,3)... β 02 β 12 β 22 T 2 (0,2) α 02 α 12 α 22 β 01 β 11 β 21 (0,1) α 01 α 11 α 21 β 00 β 10 β 20 α 00 α 10 α 20 (0,0) (1,0) (2,0) (3,0) T 1

19 Recall the definition of joint hyponormality. An n-tuple T (T 1,...,T n ) is (jointly) hyponormal if [T1,T 1] [T2,T 1] [Tn,T 1 ] [T [T1,T] :=,T 2] [T2,T 2] [Tn,T 2 ]... [T1,T n] [T2,T n] [Tn,T n ] 0.

20 To detect hyponormality, there is a simple criterion: Theorem (RC, 1988) (Six-point Test) Let T (T 1,T 2 ) be a 2-variable weighted shift, with weight sequences α and β. Then ( ) αk+ε 2 T is hyponormal 1 αk 2 α k+ε2 β k+ε1 α k β k 0 α k+ε2 β k+ε1 α k β k βk+ε 2 2 βk 2 (all k Z 2 +).

21 (k 1, k 2 + 2) β k1,k 2 +1 (k 1, k 2 + 1) α k1,k 2 +1 (k 1 + 1, k 2 + 1) β k1,k 2 β k1 +1,k 2 α k1,k 2 α k1 +1,k 2 (k 1, k 2 ) (k 1 + 1, k 2 ) (k 1 + 2, k 2 ) Figure 3. Weight diagram used in the Six-point Test

22 We now recall the notion of moment of order k for a commuting pair (α,β). Given k Z 2 +, the moment of (α,β) of order k is γ k γ k (α,β) 1 if k = 0 α(0,0) 2 :=... α2 (k 1 1,0) if k 1 1 and k 2 = 0 β(0,0) 2... β2 (0,k 2 1) if k 1 = 0 and k 2 1 α(0,0) 2... α2 (k 1 1,0) β2 (k 1,0)... β2 (k 1,k 2 1) if k 1 1 and k 2 1. By commutativity, γ k can be computed using any nondecreasing path from (0,0) to (k 1,k 2 ).

23 (Jewell-Lubin) W α is subnormal γ k := = k 1 1 i=0 k 2 1 α(i,0) 2 j=0 β 2 (k 1 1,j) t k 1 1 tk 2 2 dµ(t 1,t 2 ) (all k 0). Thus, the study of subnormality for multivariable weighted shifts is intimately connected to multivariable real moment problems.

24 The Spectral Picture of Subnormal 2-variable Weighted Shifts For subnormal 2-variable weighted shifts, RC-K. Yan gave in 1995 a complete description of the spectral picture, by exploiting the groupoid machinery in Muhly-Renault and RC-Muhly, and the presence of the Berger measure, which was used to analyze the asymptotic behavior of sequences of weights. Notation µ : compactly supported finite positive Borel measure on C n (n 1) P 2 (µ) : norm closure in L 2 (µ) of C[z 1,,z n ] M z M (µ) z := (M (µ) z 1,,M (µ) z n ) : multiplication operators acting on P 2 (µ) M z on P 2 (µ) is the universal model for cyclic subnormal n-tuples

25 Theorem (RC-K. Yan, 1995) Let µ be a Reinhardt measure on C 2, and let C := log K. Assume that K (z 1 z 2 = 0) contains no 1-dimensional open disks. Then (i) σ T (M z,p 2 (µ)) = σ r (M z,p 2 (µ)) = K (ii) σ Te (M z,p 2 (µ)) = σ { re (M z,p 2 (µ)) = K 1 if λ int.( K) (iii) index(m z λ) = 0 if λ / int.( K) (vi) kerdm 1 = z λ rand0 M z λ for all λ int.( K).

26 z 2 (0, 1) σ T e (M z) (0, b) - index = 1 (0, 0) (a,0) (1, 0) z 1 Figure 5. Spectral picture of a typical subnormal 2-variable weighted shift

27 Toral Aluthge Transform We introduce the toral Aluthge transform of 2-variable weighted shifts W (α,β) (T 1,T 2 ). For i = 1,2, consider the polar decomposition T i U i T i. Then for a 2-variable weighted shift W (α,β) (T 1,T 2 ), we define the toral Aluthge transform of W (α,β) as follows: W (α,β) := (T ( ) 1,T 2 ) := T U1 T 1 1 2, T2 1 2 U2 T (1) Observe: commutativity of W (α,β) does not automatically follow from the commutativity of W (α,β) ; actually, the necessary and sufficient condition to preserve commutativity is α (k1,k 2 +1)α (k1 +1,k 2 +1) = α (k1 +1,k 2 )α (k1,k 2 +2) (for all k 1,k 2 0).

28 (k 1,k 2 +2) T 2 (k 1,k 2 +1) α (k1,k 2 +1) α (k1 +1,k 2 +1) (k 1,k 2) (k 1 +1,k 2) (k 1 +2,k 2) T 1

29 (k 1,k 2 +2) α k1,k 2 +2 T 2 (k 1,k 2 +1) α (k1 +1,k 2 ) (k 1,k 2) (k 1 +1,k 2) (k 1 +2,k 2) T 1

30 (k 1,k 2 +2) α k1,k 2 +2 T 2 β (k1,k 2 +1) β (k1 +1,k 2 +1) (k 1,k 2 +1) α (k1,k 2 +1) α (k1 +1,k 2 +1) β (k1,k 2 ) β (k1 +1,k 2 ) α (k1,k 2 ) α (k1 +1,k 2 ) (k 1,k 2) (k 1 +1,k 2) (k 1 +2,k 2) T 1

31 Spherical Aluthge Transform Consider a (joint) polar decomposition of the form (T 1,T 2 ) (U 1 P,U 2 P). where P := T 1 T 1 +T 2 T 2. Now let Ŵ (α,β) := ( PU1 P, PU2 P ), (2) One can prove that U 1 U 1 +U 2 U 2 is a (joint) partial isometry, and that Ŵ (α,β) is commutative whenever W (α,β) is commutative.

32 The toral Aluthge transform does not preserve hyponormality:... ω 1 ω 1 ω 1 a ω 0 ω 1 ω 0 x 0 ω 0 ω 0 a ω 0 ω 1 a a x 0 ω 0 ω 1 ξ Let ξ be the Berger measure of shift(ω 0,ω 1, ) and let ρ := 1 s dξ(s). Theorem Assume ω 2 1 ρ < 2. Then: (i) (T 1,T 2 ) is subnormal; (ii) (T1,T 2 ) is not hyponormal. (The condition ω1 2 ρ < 2 can be satisfied with a 2-atomic measure ξ.)

33 Question When is hyponormality invariant under the toral Aluthge transform? Given a 1-variable weighted shift W ω, let Θ(W ω ) W (α,β) be the 2-variable weighted shift given by α (k1,k 2 ) = β (k1,k 2 ) := ω k1 +k 2 for k 1,k 2 0. We will say that Θ(W ω ) is a 2-variable embedding of the unilateral weighted shift W ω.

34 (0,3)... ω 2 ω 3 ω 4 (0,2) ω 2 ω 3 ω 4 T 2 ω 1 ω 2 ω 3 (0,1) ω 1 ω 2 ω 3 ω 0 ω 1 ω 2 ω 0 ω 1 ω 2 (0,0) (1,0) (2,0) (3,0) (i) T 1

35 (0,3) ω 2 ω 3 ω 4 ω2ω3 ω3ω 4 ω (0,2) ω 2 ω 3 ω 4 ω2ω 3 ω3ω 4 T 2 ω 1 ω 2 ω 3 T 2 ω1ω 2 ω2ω 3 ω (0,1) ω 1 ω 2 ω 3 ω1ω 2 ω2ω 3 ω 0 ω 1 ω 2 ω0ω 1 ω1ω 2 ω ω 0 ω 1 ω 2 (0,0) (1,0) (2,0) (3,0) ω0ω 1 ω1ω 2 (0,0) (1,0) (2,0) (i) T 1 (ii) T 1

36 Proposition Consider Θ(W ω ) (T 1,T 2 ) given as above. Then for k 1 Theorem W ω is k-hyponormal if and only if Θ(W ω ) is k-hyponormal. Suppose that Θ(W ω ) is hyponormal. Then the toral Aluthge transform ) Θ(W ω ) Θ ( Wω is also hyponormal. The same result holds for the spherical Aluthge transform.

37 Spherically quasinormal pairs Let T be a commuting pair with joint polar decomposition T i U i P, (i = 1,2). Recall that the spherical Aluthge transform preserves commutativity for 2-variable weighted shifts. We say that T is spherically quasinormal if T = T. Lemma Assume P injective. Then T is spherically quasinormal if and only if T i P = PT i (i = 1,2) if and only if U i P = PU i (i = 1,2). As a consequence, if T is spherically quasinormal then (U 1,U 2 ) is commuting. Proposition (RC-J. Yoon; 2015) A 2-variable weighted shift T is spherically quasinormal if and only if there exists C > 0 such that 1 CT is a spherical isometry, that is, T 1 T 1 +T 2 T 2 = I.

38 Definition A commuting pair T is a spherical isometry if T 1 T 1 +T 2 T 2 = I. Lemma (RC-J. Yoon; 2015) W α,β is a spherical isometry if and only if α 2 k +β2 k = 1 for all k Z2 +. Theorem (Athavale; JOT, 1990) A spherical isometry is always subnormal. Corollary (RC-J. Yoon; 2016) A spherically quasinormal 2-variable weighted shift is subnormal.

39 Corollary (RC-J. Yoon; 2016) Let T be a spherically quasinormal pair, and assume that P is injective. Then T is hyponormal. Theorem (A. Athavale - S. Poddar; 2015 and S. Chavan - V. Sholapurkar; 2013) Let T be a spherically quasinormal pair. Then T is subnormal. Theorem (V. Müller - M. Ptak; 1999) Spherical isometries are hyperreflexive. Theorem (J. Eschmeier - M. Putinar; 2000) For every n 3 there exists a non-normal spherical isometry T such that the polynomially convex hull of σ T (T) is contained in the unit sphere.

40 Construction of spher. isom. 2-var. w.s. (0,3) (0,2) T 2 (0,1) α 00 α 10 α 20 (0,0) (1,0) (2,0) (3,0)

41 Construction of spher. isom. 2-var. w.s. (0,3) (0,2) T 2 (0,1) β 00 α 00 α 10 α 20 (0,0) (1,0) (2,0) (3,0)

42 Construction of spher. isom. 2-var. w.s. (0,3) (0,2) T 2 (0,1) β 00 β 10 β 20 α 00 α 10 α 20 (0,0) (1,0) (2,0) (3,0)

43 Construction of spher. isom. 2-var. w.s. (0,3). (0,2) T 2 (0,1) α 01 β 00 β 10 β 20 α 00 α 10 α 20 (0,0) (1,0) (2,0) (3,0)

44 Construction of spher. isom. 2-var. w.s. (0,3)... (0,2) T 2 (0,1) α 01 α 11 α 21 β 00 β 10 β 20 α 00 α 10 α 20 (0,0) (1,0) (2,0) (3,0)

45 Construction of spher. isom. 2-var. w.s. (0,3)... (0,2) T 2 β 01 β 11 β 21 (0,1) α 01 α 11 α 21 β 00 β 10 β 20 α 00 α 10 α 20 (0,0) (1,0) (2,0) (3,0)

46 Construction of spher. isom. 2-var. w.s. (0,3)... β 02 β 12 β 22 (0,2) α 02 α 12 α 22 T 2 β 01 β 11 β 21 (0,1) α 01 α 11 α 21 β 00 β 10 β 20 α 00 α 10 α 20 (0,0) (1,0) (2,0) (3,0)

47 Recall that a unilateral weighted shift is recursively generated if the sequence of moments satisfy a linear relation γ n+k = ϕ 0 γ n +ϕ 1 γ n+1 + +ϕ k 1 γ n+k 1 (k 1, n 0). Theorem (RC-Fialkow; IEOT, 1993) A subnormal weighted shift is recursively generated if and only if its Berger measure is finitely atomic.

48 (0,3) τ... β 02 β 12 β 22 T 2 (0,2) α 02 α 12 α 22 β 01 β 11 β 21 (0,1) α 01 α 11 α 21 β 00 β 10 β 20 α 00 α 10 α 20 (0,0) (1,0) (2,0) (3,0) σ T Raúl E. Curto (Univ. of Iowa) Multivariable Aluthge 1 Transforms INFAS, Nov 11, 2017

49 Theorem (RC-Yoon; 2016) Let W (α,β) be a spherical isometry, and assume that the zero-th row is subnormal with finitely atomic Berger measure σ. (i) Each horizontal row is recursively generated, and its moments satisfy the same linear relation as the zero-th row. (ii) Each vertical column is recursively generated, and its moments satisfy the linear relation obtained from (ii) which appropriately reflects the condition αk 2 +β2 k = 1 (k Z2 +). (iii) The Berger measure of W (α,β) is finitely atomic, with support contained in the Cartesian product of σ and τ, where τ is the Berger measure of the zero-th column of W (α,β).

50 Theorem (cont.) (iv) If Λ (0) and (0) Λ are the Riesz functional of the zero-th row and zero-th column of W (α,β), resp., then (0) Λ(p(t)) = Λ (0) (p(1 t)) for every polynomial p. As a result, supp τ = 1 supp σ.

51 Spectral Properties We aim to calculate the spectral picture of W α,β (T 1,T 2 ) and of its toral and spherical Aluthge transforms. This entails finding the Taylor spectrum, the Taylor essential spectrum, and the Fredholm index. We focus on the case when the pair has a core of tensor form. The class of pairs with core of tensor form is large, and has been used to exhibit structural and spectral behavior of 2-variable weighted shifts, not found in the classical theory of unilateral weighted shifts.

52 Special Case (tensor core): Given ξ,η,σ,τ,a, study sp. picture of W α,β. M N... y 2 β 2 β 2 T 2 τ aβ 1 y 1 α 1 α 2 (T 1,T 2 ) M N ξ η y 1 β 1 β 1 a α 1 α 2 y 0 ay 0 x 0 ay 0 α 1 x 0 x 1 x 0 x 1 x 2 (0,0) (1,0) (2,0) (3,0) T 1 σ

53 Spectral Properties, cont. Lemma (i) Let H 1 and H 2 be Hilbert spaces, and let A i B(H 1 ), C i B(H 2 ) and B i B(H 1,H 2 ),(i = 1,,n) be such that ( ) (( ) ( )) A 0 A1 0 An 0 :=,..., B C B 1 C 1 B n C n ( ) A 0 is commuting. Assume that A and are Taylor invertible. B C Then, C is Taylor ( invertible. ) Furthermore, if A and C are Taylor A 0 invertible, then is Taylor invertible. B C

54 Lemma (cont.) (ii) For A and B two commuting n-tuples of bounded operators on Hilbert space, we have: σ T (A I,I B) = σ T (A) σ T (B) and σ Te (A I,I B) = σ Te (A) σ T (B) σ T (A) σ Te (B).

55 To use the Lemma, we split l 2 (Z 2 +) as the orthogonal direct sum of the 0-th row and the rest. For the Taylor essential spectrum, we use the fact that compressions of W (α,β) and W (α,β) differ by a compact perturbation, when W (α,β) is hyponormal. H 2 H 1 (0,0) (1,0) (2,0) (3,0)

56 Theorem Consider a hyponormal 2-variable weighted shift W (α,β) (T 1,T 2 ). Then σ T ( W(α,β) ) = ( Wω D W τ D ) and ( ) ( σ Te W(α,β) = Wω T W τ D ) ( W ω D W τ T ). (3) Here D denotes the closure of the open unit disk D and T the unit circle.

57 We next consider the Taylor essential spectrum σ Te (T 1,T 2 ) of W (α,β) (T 1,T 2 ). To prove the result for σ Te, observe that W ω (2) is a compact perturbation of W ω (1) and W ω (0). perturbations of I 1 and I 2, respectively. Theorem ω 0 y 0 x 0 x 1 I and τ 0 I are also compact Consider a hyponormal 2-variable weighted shift W (α,β) (T 1,T 2 ). Then, we have σ T ( W(α,β) ) = ( W ω D W τ D ) and σ Te ( W(α,β) ) = ( W ω T W τ D ) ( W ω D W τ T ). A similar result holds for the spherical Aluthge transform. (4)

58 Consider now the Drury-Arveson 2-shift, denoted by DA. As usual, DA is the toral Aluthge transform of DA and DA is the spherical Aluthge transform of DA. Also, it is well known that DA is essentially normal. Theorem (i) DA is a compact perturbation of DA. (ii) DA is a compact perturbation of DA. Corollary DA, DA and DA all share the same Taylor spectral picture; that is, (i) σ T (DA) = B 2, (ii) σ Te (DA) = B 2, and (iii) index DA = index DA = index DA. (Here B 2 denotes the open unit ball in C 2, and B 2 its topological boundary.)

59 Thank you!