Schur multipliers: recent developments

Transcription

1 Schur multipliers: recent developments Anna Skripka University of New Mexico Brazos Analysis Seminar 2017 Texas A&M University

2 Linear Schur multipliers on matrices

3 Schur multiplier Definition The Schur multiplier M m by a matrix m = (m ij ) d i,j=1 is the operator x x 1d m 11 x m 1d x 1d x d1... x dd m d1 x d1... m dd x dd Remark: The function ϕ(i, j) := m ij is also called a Schur multiplier. Example Triangular truncation: m =

4 Norm of a Schur multiplier on S 2 d Notation B(l 2 d ) is the space of bounded linear operators on l2 d ( ) 1/2 X 2 = x ij 2 is the Hilbert-Schmidt norm i,j=1 S 2 d is the space B(l2 d ) equipped with the norm 2 Proposition Mm : Sd 2 S2 d = max m ij. 1 i,j d Proof. ( M m (X ) 2 = i,j=1 m ij x ij 2 ) 1/2 max 1 i,j d m ij X 2. Mm : Sd 2 S2 d M m (E ij ) 2 = m ij.

5 Norm of a Schur multiplier via factorizations Proposition If m ij = n k=1 α i(k)β j (k), then ( M m : B(l 2 d ) B(l2 d ) That is, sup 1 i d n k=1 α i (k) 2 sup 1 i d n ) 1/2 β j (k) 2. k=1 m ij = α i, β j Mm sup 1 i d α i sup 1 j d β j. Proof. Denote D α(k) = diag(α 1 (k),..., α d (k)). Then, n n M m (X )f, g = D α(k) XD β(k) f, g XD β(k) f Dα(k) g k=1 k=1 n 1/2 n 1/2 X Dβ(k) D β(k)f, f D α(k) Dα(k) g, g X k=1 k=1 n D α(k) Dα(k) 1/2 n Dβ(k) D β(k) k=1 k=1 1/2 f g.

6 Norm of the triangular truncation Notation T is the triangular truncation Example [Davidson 88] { 1 i j if i j x ij = 0 if i = j, X π, T (X ) lim 2/3. d log d Theorem [Angelos, Cowen, Narayan 92]; also [Kwapien, Pelczynski 70] T : B(l 2 d ) B(l2 d ) = 1 log d + O(1) as d. π Related contributions: [Haagerup 80], [Paulsen, Power, Smith 89].

7 Schur multiplier in different bases of eigenvectors Matrix: Eigenvalues: Orthonormal basis of eigenvectors: A = A {λ i } d i=1 {g i } d i=1 B = B {µ j } d j=1 {h j } d j=1 Notation P gi :=, g i g i Definition Let ϕ : R 2 C and m = ( ϕ(λ i, µ j ) ) d A,B. Then T i,j=1 ϕ : B(l 2 d ) B(l2 d ) given by d d Tϕ A,B (X ) := ϕ(λ i, µ j ) P gi X P hj i=1 j=1 is the Schur multiplier by m in the bases { {g i } d i=1, {h j } d j=1}.

8 Schur multiplier w.r.t. spectral measures Matrix: Distinct eigenvalues: Spectral measure: A = A {λ i } d 1 i=1 E A B = B {µ j } d 2 j=1 E B E A ({λ i }) is the projection onto the eigenspace corresponding to λ i Proposition Let ϕ : R 2 C. Then, d 1 d 2 Tϕ A,B (X ) = ϕ(λ i, µ j ) E A ({λ i })X E B ({µ j }). i=1 j=1

9 Schur multiplier in matrix perturbation theory Definition For f differentiable on R, its divided difference (difference quotient) is f [1] (x, y) := Proposition [Löwner 34] { f (y) f (x) y x if y x f (x) if y = x. Let A = A, B = B B(l 2 d ), f Lip(R). Then, f (B) f (A) = T B,A (B A). f [1] Proof. Applying the spectral theorem gives (f (B) f (A))gi, h j = gi, f (B)h j f (A)gi, h j = (f (µ j ) f (λ i )) g i, h j µj λ i µ j λ i = f [1] (λ i, µ j ) (B A)g i, h j.

10 Schur multipliers on infinite dimensional operators

11 Schur multiplier on B(l 2 ) Definition Let m = (m ij ) i,j=1 C be such that M m : B(l 2 ) B(l 2 ) defined by M m (X ) = ( m ij x ij ) i,j=1 is bounded. Then, M m is called a Schur multiplier. Theorem [Grothendieck 56]; see [Pisier 96] The following assertions are equivalent. M m is a Schur multiplier with M m : B(l 2 ) B(l 2 ) C. a Hilbert space H and {α i } i=1, {β j } j=1 H such that Moreover, m ij = α i, β j and sup i M m : B(l 2 ) B(l 2 ) = inf α,β α i sup β j C. j { sup i α i sup β j }. j

12 Recall Schatten classes Definition Let 1 p <. The Schatten p-class is the ideal S p := { ( ) 1/p } A is compact : A p := σ p k <, k=1 where {σ k } k=1 0 is the sequence of eigenvalues of A. Proposition 1 p < q < S p S q

13 Double operator integral (Schur multiplier S 2 S 2 ) Notation (Ω, A) is a measurable space H is a separable Hilbert space E and F are spectral measures on A with values in B(H) [ (E F )( 1 2 ) ] (X ) := E( 1 )XF ( 2 ), for X S 2 (H) Proposition E F extends to a B(S 2 )-valued spectral measure on A A. Definition [Birman, Solomyak 60s] Given ϕ L (E F ), the bounded linear operator Tϕ E,F = ϕ(s, t) d(e F )(s, t) Ω Ω on S 2 is called a double operator integral (Schur multiplier on S 2 ).

14 Schur multiplier associated with spectral measures Theorem [Peller 85] The following assertions are equivalent. T E,F ϕ is a bounded Schur multiplier on B(H) (or S 1 ). a measure space (Σ, µ) and measurable functions α and β on Ω Σ such that ϕ(s, t) = α(s, w) β(t, w) dµ(w), (s, t) Ω Ω, Σ α(, w) 2 d µ (w) β(, w) 2 d µ (w) <. L (F ) Ω L (E) Ω

15 Triangular truncation w.r.t. spectral measures Definition { 1 if s t Let E, F be spectral measures on R and let ϕ(s, t) = 0 if s > t. Then, Tϕ E,F = ϕ(s, t) d(e F )(s, t) is the triangular truncation R R on S 2 with respect to E and F. Theorem see [Gohberg, Krein 67] The triangular truncation is bounded on S p for 1 < p <, unbounded on S 1 and B(H).

16 Schur multiplier in perturbation theory Notation A = A and B = B are closed operators densely defined in H E A is the spectral measure of A, E B is the spectral measure of B Theorem [Birman, Solomyak 73] If f Lip(R) and B A S 2, then f (B) f (A) = T E B,E A (B A). f [1] Question [Krein 64] Does B A S p f (B) f (A) S p for every f Lip(R)? Theorem [Farforovskaya 72] No when p = 1. [Potapov, Sukochev, 11] Yes when 1 < p <.

17 Schur multiplier by divided differences of modulus Recall that B A = T B,A x [1] (B A). Theorem [Davies 88] A = A, B = B B(l 2 d ), where γ p = B A p c γ p B A p, { p if 2 p < p p 1 if 1 < p 2. A = A, B = B B(l 2 d ), B A 1 c 1 log d B A 1. A = A, B = B B(l 2 2d ) such that σ(a) = σ(b) and B A 1 c 2 log d B A 1.

18 Multilinear Schur multipliers on matrices

19 Multilinear Schur multiplier Definition [Effros, Ruan 90] The Schur multiplier M m(n) by m(n) = (m i1 i 2...i n+1 ) d i 1,i 2,...,i n+1 =1 C is the operator on ( B(l 2 d )) n given by ( d (X 1, X 2,..., X n ) i1 i 2...i n+1 (X 1 ) i1 i 2 (X 2 ) i2 i 3... (X n ) inin+1)d. i 2,...,i n=1m i 1,i n+1 =1 Examples If m i1 i 2...i n+1 1, then M m(n) (X 1, X 2,..., X n ) = X 1 X 2... X n. { 1 if i k j If m ijk =, then 0 otherwise x 11 y 11 x 11 y 12 + x 12 y d k=1 x 1ky kd 0 x 22 y d k=2 M m(2) (X, Y ) = x 2ky kd x dd y dd

20 Multilinear via linear Schur multipliers Notation m(n) = (m i1 i 2...i n+1 ) d i 1,i 2,...,i n+1 =1 m k(1) = (m i1 k in+1 ) d i 1,i n+1 =1, where k = ( k,..., k ), k {1,..., d}. }{{} n 1 Theorem [Potapov, S, Sukochev, Tomskova] If p 1,..., p n, p [1, ] satisfy 1 p p n = 1 p, then M m(n) : S p 1 d... Spn d Sp d max 1 k d M m k (1) : Sd 1 Sp d. Theorem [Coine, Le Merdy, Potapov, Sukochev, Tomskova 16] M m(2) : Sd 2 S2 d S1 d = max 1 k d M m k(1) : Sd 1 S1 d.

21 Multilinear Schur multipliers in different bases Operator: Distinct eigenvalues: Spectral measure: A r = A r {λ (r) i r } dr i r =1 E r Definition Let ϕ : R n+1 C and m(n) = { ϕ(λ (1) i 1, λ (2),..., λ (n+1) i n+1 ) }. i 1,i 2,...,i n+1 The transformation T A 1,A 2,...,A n+1 ϕ on B(l 2 d )... B(l2 d ) }{{} n T A 1,A 2,...,A n+1 ϕ (X 1, X 2,..., X n ) := n+1 d r ϕ ( λ (1) i 1, λ (2) r=1 i r =1 i 2..., λ (n+1) i n+1 ) E1 (λ (1) i 2 i 1 )X 1 E 2 (λ (2) i 2 given by )X 2... X n E n+1 (λ (n+1) i n+1 ) is the multilinear Schur multiplier by m(n) with respect to the spectral measures of A 1, A 2,..., A n+1.

22 Multilinear Schur multiplier in matrix perturbation Notation n 1 1 d k R n,f,a,b := f (B) f (A) k! dt k f (A + t(b A)) t=0, k=1 where the derivative is evaluated in the operator norm. Proposition Let A = A, B = B B(l 2 d ), f C n (R). Then, R n,f,a,b = T B,A,...,A ( B A,..., B A ), f [n] }{{} n where the nth order divided difference f [n] is defined recursively by f [k] = (f [k 1] ) [1].

23 Multilinear Schur multipliers on infinite dimensional operators

24 Triple operator integral (S. m. S 2 S 2 S 2 ) Notation (C, A) is a Borel space A, B, C are normal operators in a separable Hilbert space λ A, λ B, λ C are positive finite measures on A equivalent to the spectral measures of A, B, C, respectively Definition [Coine, Le Merdy, Sukochev] Let ϕ L (λ A λ B λ C ). The triple operator integral T A,B,C ϕ : S 2 S 2 S 2 is defined as follows. If f 1, f 2, f 3 are simple functions on (C, A), then T A,B,C f 1 f 2 f 3 (X, Y ) := f 1 (A) X f 2 (B)Y f 3 (C). A general ϕ is approximated by f 1 f 2 f 3 in w -topology, and T A,B,C ϕ is defined as the respective limit.

25 Schur multiplier S 2 S 2 S 1 Theorem [Coine, Le Merdy, Sukochev] The following assertions are equivalent. T A,B,C ϕ is a bounded Schur multiplier on S 2 S 2 S 1. a Hilbert space H and functions α L (λ A λ B, H), β L (λ B λ C, H) such that Moreover, ϕ(s, t, w) = α(s, t), β(t, w). T ϕ A,B,C : S 2 S 2 S 1 { } = inf α β. α,β

26 Multiple operator integral Notation E is the spectral measure of A = A E l,m = E ([ l m, l+1 )) m ϕ : R n+1 C is a bounded Borel function Definition [Potapov, S, Sukochev 13] The transformation T A,...,A ϕ : S p 1... S pn S p given by T A,...,A ϕ (X 1,..., X n ) := lim m l 1,...,l n+1 Z ( l1 ϕ m,..., l ) n+1 E l1,m X 1 E l2,m... E ln,m X n E ln+1,m m (provided the limit exists) is a multiple operator integral. Theorem [Potapov, S, Sukochev 13] limit for a broad set of ϕ, including ϕ = f [n] with f Cc n+1 (R).

27 Multilinear Schur multiplier in perturbation theory Assumptions Theorem A = A and B = B B(H) n N, n 2 For sufficiently nice f and B A, R n,f,a,b = T B,A,...,A ( B A,..., B A ). f [n] }{{} n Question Let p n. Does B A S p R n,f,a,b S p/n for every f C n (R)? Theorem [Potapov, S, Sukochev, Tomskova] No when p = n. Yes when p > n.

28 Multilinear Schur multiplier arising from modulus Theorem [Potapov, S, Sukochev, Tomskova] { x n 1 x ( log log x 1 ) 1/2 if x 0 Let f (x) = 0 if x = 0. Then, there exist A = A, B = B B(l 2 4d(n+1) 2 ) with σ(a), σ(b) [ ɛ e 1, e 1 + ɛ] such that R n,f,a,b 1 c log d B A n.

29 Open problems

30 Description of Schur multipliers Problem Describe m(n) for which the following Schur multiplier is bounded: M m(1) : S p S p, p / {1, 2, } M m(2) : S p 1 S p 2 S p, (p 1, p 2, p) / {(2, 2, 2), (2, 2, 1)} M m(n) : S p 1... S pn S p

31 Image space of an operator remainder Problem If A = A, B = B B(H) satisfy B A S n and f C n (R), n 2, does R n,f,a,b S 1,? Definition S 1, := { A is compact : A 1, := sup k σ k < }, k N where {σ k } k=1 0 is the sequence of eigenvalues of A. Theorem [Caspers, Potapov, Sukochev, Zanin] If f Lip(R), then B A S 1 f (B) f (A) S 1,.

32 Thank You!