Toeplitz-Hausdorff Theorem - Convexity and Connectedness
Tin-Yau Tam
Department of Mathematics and Statistics Auburn University, USA & University of Nevada, Reno, USA, [email protected]
WONRA 2018 Technical University, Münich, Germany
June 13-17, 2018
1 / 27 100 Anniversary of Toeplitz-Hausdorff Theorem
The numerical range of A ∈ Cn×n is the set
W(A) = {x∗Ax : x ∈ Cn, x∗x = 1} ⊂ C.
W(A) is the image of the unit sphere of Cn under the map x 7→ x∗Ax.
→
Theorem (Toeplitz 1918, Hausdorff 1919) W(A) is convex.
• O. Toeplitz, Das algebraische Analogen zu einem Sätze von Fejer, Math. Z. 2 (1918), 187-197. • F. Hausdorff, Der Wertvorrat einer Bilinearform, Math. Z. 3 (1919), 314-316.
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3 / 27 Toeplitz and Hausdorff - From Wiki
Otto Toeplitz (1 August, 1881 - 15 February 1940) was a German mathematician working in functional analysis. ... In 1928 Toeplitz succeeded Eduard Study at Bonn University. In 1933, the Civil Service Law came into effect and professors of Jewish origin were removed from teaching. Initially, Toeplitz was able to retain his position due to an exception for those who had been appointed before 1914, but he was nonetheless dismissed in 1935. In 1939 he emigrated to Palestine, where he was scientific advisor to the rector of the Hebrew University of Jerusalem. He died in Jerusalem from tuberculosis a year later.
Felix Hausdorff (November 8, 1868 - January 26, 1942) was a German mathematician who is considered to be one of the founders of modern topology and who contributed significantly to set theory, descriptive set theory, measure theory, function theory, and functional analysis. Life became difficult for Hausdorff and his family after Kristallnacht in 1938. ... On 26 January 1942, Felix Hausdorff, along with his wife and his sister-in-law, committed suicide by taking an overdose of veronal, rather than comply with German orders to move to the Endenich camp, ...
4 / 27 Some generalized numerical ranges
(Halmos, Berger 1963) The k-numerical range (1 ≤ k ≤ n) { } ∑k ∗ n Wk(A) = xi Axi : x1,..., xk are orthonormal in C i=1 is convex. (Westwick 1975) The c-numerical range (c ∈ Rn) { } ∑n ∗ n Wc(A) = cixi Axi : x1,..., xn are orthonormal in C i=1 ∗ = {tr (CU AU): U ∈ U(n)} , where C = diag (c1,..., cn), is convex.
(Cheung and Tsing 1996) The C-numerical range (C ∈ Cn×n)
∗ WC(A) = {tr (CU AU): U ∈ U(n)}
is star-shaped.
5 / 27 Westwick’s proof
A = A1 + iA2 Hermitian decomposition. n C ∈ Cn×n Hermitian with eigenvalues c ∈ R . Then Wc(A) can be identified with
∗ ∗ 2 WC(A1, A2) = {(tr CU A1U, tr CU A2U): U ∈ U(n)} ⊂ R . Note that iθ iθ WC(e A) = e WC(A), ∀θ ∈ R. and
iθ e A = (cos θA1 − sin θA2) + i(sin θA1 + cos θA2) := (A1(θ), A2(θ)).
iθ So Wc(e A) is identified with WC(A1(θ), A2(θ)).
6 / 27 Note that
∗ WC(A) = WA(C), WC(U AU) = WC(A), ∀U ∈ U(n)
Hausdorff connectedness argument:
Let D(n) be the subgroup of diagonal matrices in U(n). May assume that B and C are diagonal matrices.
Westwick considered fB,C : U(n)/D(n) → R
∗ fB,C([U]) = tr CU BU, where B, C ∈ Cn×n are Hermitian and [U] = U/D(n) for U ∈ U(n).
7 / 27 Main ideas in Westwick’s proof:
When B and C have distinct eigenvalues, fB,C is a Morse function and its Hessian has even index at each critical point. make use of differential topology to conclude that f−1 c is connected B,C( ) for any c ∈ R. Claim: Connectedness holds for arbitrary Hermitian B and C.
Poon (1980) gave the first elementary and complete proof of Westwick’s result.
8 / 27 Raïs’ setting for compact Lie groups
K compact Lie group with Lie algebra k ⟨·, ·⟩ is a Ad K-invariant inner product on k. (Raïs) For X, Y, C ∈ k, the C-numerical range of (X, Y) is
WC(X, Y) = {(⟨X, Ad (k)C⟩, ⟨Y, Ad (k)C⟩): k ∈ K}.
Adjoint orbit of C:
Ad (K)C := {Ad (k)C : k ∈ K}.
Theorem (Tam 2002)
Given X, Y, C ∈ k, WC(X, Y) is convex.
9 / 27 Corollary
1 (Westwick) Let G = U(n) or SU(n). The C-numerical range
∗ ∗ WC(A1, A2) = {(tr A1UCU , tr A2UCU ): U ∈ G}
is convex, where A1, A2 and C are Hermitian matrices. 2 The set
T T WC(A1, A2) = {(tr A1OCO , tr A2OCO ): O ∈ SO(n)}
is convex, where A1, A2, and C are real skew symmetric matrices. 3 The set
∗ ∗ WC(A1, A2) = {(tr A1UCU , tr A2UCU ): U ∈ Sp(n)}
is convex, where A1, A2, C ∈ sp(n).
10 / 27 The ingredients in Tam’s proof are Kirillov-Kostant-Souriau: Co-adjoint orbits are symplectic manifold. Atiyah: Let M be a compact connected symplectic manifold and let f : M → R whose Hamiltonian vector field is almost periodic. Then all fibres f−1(c) are connected.
The connectedness of the fibres of the map πC : Ad (K)X → R defined by
πC(Y) = ⟨C, Y⟩, for all Y ∈ Ad (K)X is established. The convexity of WC(X1, X2) then follows through rotation.
• T.Y. Tam, Convexity of generalized numerical range associated with a compact Lie group, J. Austral. Math. Soc. 72 (2002), 57-66.
11 / 27 Markus-Tam’s result
Markus-Tam gave another proof of the convexity of WC(X1, X2) via connectedness. Without symplectic technique, they proved
Theorem (Markus-Tam, 2011)
The connectedness of the fibres of fC,X : K → R for all C, X ∈ k:
fC,X(k) = ⟨C, Ad (k)X⟩, for all k ∈ K. are connected.
Again, the convexity of WC(X1, X2) then follows through rotation.
• A. Markus and T.Y. Tam, Connectedness of some fibers on a compact connected Lie group, Linear and Multilinear Algebra, 59 (2011), 1121–1126.
12 / 27 Markus-Tam’s fibre connectedness result (2011) in K is stronger than Tam’s fibre connectedness result (2002) in the adjoint orbit Ad (K)X:
Ad (·)X / K > Ad (K)X > w >> ww >> ww f >> ww πC C,X {ww R since the map Ad (·)X : K → Ad (K)X is continuous. We will give a third convexity proof via a connectedness result (on Ad (K)X) of Atiyah and a Hessian index result of Duistermaat, Kolk and Varadarajan.
13 / 27 Complex semisimple Lie algebras and star-shapedness conjecture
G complex semisimple Lie group with Lie algebra g g = k ⊕ ik Cartan decomposition K connected subgroup of G with Lie algebra k
Bθ(·, ·) inner product on g induced by the Killing form B(·, ·) and θ For X, C ∈ g, the C-numerical range of X is
WC(X) = {Bθ(C, Ad (k)X): k ∈ K}. Recall
WC(X) := {Bθ(C, Ad (k)X): k ∈ K}, C, X ∈ g.
Conjecture (Tam 2001, still open)
Given C, X ∈ g, WC(X) is star-shaped with origin as the star center for all complex semisimple Lie algebras g.
• T.Y. Tam, On the shape of numerical ranges associated with Lie groups, Taiwanese J. Math., 5 (2001), 497-506. 14 / 27 Development of the conjecture
Theorem (Cheung-Tsing 1996, Djoković-Tam 2003, Cheung-Tam 2011)
WC(X) is star-shaped for complex simple Lie algebras of type A, B, D, E6 and E7.
The conjecture is valid for simple Lie algebras of
1 type A (sln(C), Cheung-Tsing, 1996).
2 type D, i.e., so2n(C), E6 and E7 (Djokovic and Tam, 2003).
3 type B, i.e., so2n+1(C), type D (Cheung and Tam, 2011).
4 Unknown for type C, i.e., spn(C), E8, F4, G2. Question: Is the conjecture true?
• W.S. Cheung and N.K. Tsing, The C-numerical range of matrices is star-shaped, Linear Multilinear Algebra, 41 (1996), 245-250. • D.Z. Djoković and T.Y. Tam, Some questions about semisimple Lie groups originating in matrix theory, Canad. Math. Bull., 46 (2003), 332-343. • W.S. Cheung and T.Y. Tam, Star-shapedness and K-orbits in complex semisimple Lie algebras, Canad. Math. Bull., 54 (2011), 44-55.
15 / 27 Setting for real semisimple Lie algebras
g = k ⊕ p Cartan decomposition of real semisimple g The Killing form B is positive definite on p G connected Lie group with Lie algebra g K connected subgroup of G with Lie algebra k For C, X, Y ∈ p, the C-numerical range of (X, Y) is
WC(X, Y) = {(B(C, Ad (k)X), B(C, Ad (k)Y)) : k ∈ K}.
Example: When g = sln(R), up to a scalar multiple,
T T WC(X, Y) = {(tr COXO , tr COYO ): O ∈ SO(n)}
where C, X, Y ∈ Rn×n are symmetric matrices.
16 / 27 Li-Tam’s results
Theorem (Li-Tam 2000)
WC(X, Y) is convex for all classical real simple Lie algebras except sl2(R).
Li-Tam’s approach is case-by-case computation.
Questions: Is there a unified proof without case-by-case computation? How about exceptional cases?
• C.K. Li and T.Y. Tam, Numerical ranges arising from simple Lie algebras, J. Canad. Math. Soc., 52 (2000), 141–171.
17 / 27 Classical real simple Lie algebras
The classical real simple Lie algebras are isomorphic to one of the following real forms h ⊂ g and gR (the realification of g). 1 g = sln(C), n ≥ 2 1 h = sln(R) 2 h = slm(H), n = 2m n 3 (p , p q n) h = sup,q = 0, 1,..., [ 2 ] + =
2 g = so2n+1(C), n ≥ 2
1 h = sop,q (p = 0, 1,..., n, p + q = 2n + 1)
3 C ≥ g = sp2n( ), n = 2m, m 3 1 R h = sp2n( ), n = 2m 2 h = spp,q,(p = 0, 1,..., [m/2], p + q = m)
4 g = so2n(C), n ≥ 4
1 h = sop,q,(p = 0, 1,..., n, p + q = 2n) ∗ 2 h = so (2n).
18 / 27 sl2(R)
Example B(X, Y) = tr XY θ(X) = −X⊤ and K SO k = so{((2) )= (2) } {( ) } a b a 0 p = : a, b ∈ R and a = : a ∈ R b −a 0 −a ( ) ( ) 1 0 0 1 Pick C = X = ∈ a and Y = ∈ p 0 −1 1 0 ( ) cos θ sin θ −1 For k = ∈ K, fC X(k) = tr CkXk = 2 cos(2θ). − sin θ cos θ , {( ) ( )} − 1 0 −1 0 The fibre f 1 (2) = , is not connected. C,X 0 1 0 −1 2 WC(X, Y) is a circle in R .
19 / 27 Duistermaat-Kolk-Varadarajan
For C, X ∈ p, define fC,X : K → R by
fC,X(k) = B(C, Ad (k)X).
One may assume C, X ∈ a since p = ∪k∈KAd (k)a.
W = M′/M Weyl group of G, where M centralizer of a in K and M′ is the normalizer of a in K.
For each X ∈ a, let KX and WX denote the centralizers of X in K and in W, respectively.
Lemma (DKV 1983) ∪ The critical set of fC,X is KC,X = KCWKX = KCwKX, where the
w∈WC\W/WX union is disjoint.
20 / 27 Theorem (DKV 1983)
Let k = uxwv with u ∈ KC, v ∈ KX, and xw a representative of w in K. The Hessian Hk of fC,X at k ∑ 2 Hk(Z, Z) = − α(X)(w · α)(C)∥Fα(Ad (v)Z)∥ , ∀Z ∈ k α∈Σ+ where Fα : k → k ∩ (gα ⊕ g−α) is an orthogonal projection. In particular, fC,X is a Morse-Bott function and its index at k is ∑ dim gα. α∈Σ+,α(X)(w·α)(C)>0
• J.J. Duistermaat, J.A.C. Kolk, and V.S. Varadarajan, Functions, flows and oscillatory integrals on flag manifolds and conjugacy classes in real semisimple Lie groups, Compositio Math., 49 (1983), 309–398.
21 / 27 Example: sln(C)
View g = sln(C) as a real semisimple Lie algebra. B(X, Y) = 4n Re tr XY for all X, Y ∈ g. The Hermitian decomposition is the Cartan decomposition. Let a ⊂ p be the subspace of (real) diagonal matrices. Root space decomposition of g: ⊕ g = (a ⊕ ia) ⊕ CEij, i≠ j
where Eij is the matrix with 1 at the (i, j)-entry and 0 elsewhere. ∗ The root system is Σ = {ei − ej : 1 ≤ i ≠ j ≤ n}, where ei ∈ a sends A ∈ a to the i-th diagonal entry of A.
22 / 27 ∼ The Weyl group W = Pn group of permutation matrices. The centralizers KC (resp., KX) of C (resp., X) in K consists of all matrices in SU(n) that commute with C (resp., X).
The critical set of fC,X is KC,X = KCPnKX.
For each k = UPV with U ∈ KC, P ∈ Pn, and V ∈ KX, the Hessian of fC,X at k is ∑ 2 − (w · α)(C)α(X)∥Fα(Ad (v)Z)∥ ∈ + α ∑Σ −1 −1 2 = −8n tr P(ciEii − cjEjj)P (xiEii − xjEjj) · |(VZV )ij| i The index of fC,X at k is ∑ dimR CEij − − ((PCP 1)ii−(PCP 1)jj)(xi−xj)>0 i 23 / 27 Atiyah’s connectedness result Theorem (Atiyah 1982) Let f : M → R be a Morse-Bott function on a compact connected manifold M. If neither f nor −f has a critical manifold of index 1, then f−1(c) is connected (or empty) for every c ∈ R. • M.F. Atiyah, Convexity and commuting Hamiltonians, Bull. London Math. Soc., 308 (1982), 1–15. 24 / 27 sl3(R) even index sufficient but not necessary ⊤ G = SL3(R), g = sl3(R), θ(X) = −X k = so(3), K = SO(3) p = traceless real symmetric matrices a = traceless real diagonal matrices M = real diagonal matrices in SO(3) M′ = generalized permutation matrices in SO(3) whose nonzero entries are 1 ′ ∼ ∼ Weyl group W = M /M = S3 = P3 25 / 27 Pick C = diag (c1, c2, c3) ∈ a+ and X = diag (x1, x2, x3) ∈ a+ ′ KC = KX = M, KC,X = KCWKX = M The index ∑ dim gα α∈Σ+,α(X)(w·α)(C)>0 is equal to the number of positive roots sent to positive roots by w ∈ W. Since WX = WC is trivial, each w ∈ W can appear for some k ∈ KC,X. Indices are 3,2,2,0,1,1 for the six Weyl group elements. Since dim K = dim SO(3) = 3, neither even index condition nor Atiyah’s condition is satisfied, but WC(X, Y) is still convex. • X. Liu and T.Y. Tam, Connectedness, convexity and generalized numerical range, The Natalia Bebiano Anniversary Volume, Textos de Matematica, 44 (2013), 107-119. 26 / 27 Connectedness is sufficient but not necessary: sl3(R) Example Let g = sl3(R) and let C = diag (c, 0, −c) with c > 0 and X = diag (1, 0, −1). Then the image of fC,X : SO(3) → R is the interval [−2c, 2c] ⊂ R. f−1 c {I diag − − diag − − diag − − } C,X(2 ) = 3, (1, 1, 1), ( 1, 1, 1), ( 1, 1, 1) is evidently disconnected in SO(3). However if we consider sl3(C) with the same X and C, the corresponding f−1 c is C,X(2 ) iθ1 iθ2 iθ3 {diag (e , e , e ): θ1 + θ2 + θ3 = 0} ⊂ SU(3) which is connected. 27 / 27 Thank You! Questions?