Loewner Theory in Several Complex Variables and Related Problems

Proposition 1.2.1. e ´ t 0 m(A(τ))dτ ≤∥U(t)∥≤e ´ t 0 k(A(τ))dτ, t ∈[0, ∞) and e− ´ t 0 k(A(τ))dτ ≤∥U(t)−1∥≤e− ´ t 0 m(A(τ))dτ, t ∈[0, ∞).

Proposition 1.2.3. (i) N0 = N (ii) If h ∈N0 then m(Dh(0)) ≥0 and if h ∈N then m(Dh(0)) > 0. (iii) If h ∈N0 and A = Dh(0) then Re ⟨A(z), z⟩1 −∥z∥ 1 + ∥z∥≤Re ⟨h(z), z⟩≤Re ⟨A(z), z⟩1 + ∥z∥ 1 −∥z∥, z ∈Bn (1.2.10) and ∥h(z)∥≤ 4 ∥z∥ (1 −∥z∥)2 |V (A)| ≤ 4 ∥z∥ (1 −∥z∥)2 ∥A∥, z ∈Bn. (1.2.11)

Lemma 1.2.4. Let f be a holomorphic mapping on Bn such that ∥f(z)∥≤Mr, ∥z∥≤r. Then there exists a constant Cr (depending on M but not on f) such that ∥f(z) −f(w)∥≤Cr ∥z −w∥, ∥z∥, ∥w∥≤r.

Proposition 1.2.5. ([HJ90] Theorem 7.2.1) A Hermitian matrix A ∈Mn is positive semidefinite if and only if all of its eigenvalues are nonnegative. It is positive definite if and only if all of its eigenvalues are positive. Let A, B ∈Mn be Hermitian matrices. We write A ≥B (A > B) if the matrix A−B is positive semidefinite (positive definite). For a matrix A ∈Mn, ρ(A) denotes its spectral radius.

Theorem 1.2.6. ([HJ90] Theorem 7.7.3) Let A, B ∈Mn be Hermitian matrices, and suppose A is positive definite and B is positive semidefinite. Then A ≥B if and only if ρ(BA−1) ≤1, and A > B if and only if ρ(BA−1) < 1.

Corollary 1.2.7. ([HJ90] Corollary 7.7.4) If A, B ∈Mn are positive definite, then: 1. A ≥B if and only if B−1 ≥A−1;

Theorem 1.2.8. ([HJ90] Corollary 7.3.3) If A ∈Mn, then it may be written in the form A = PU where P is positive semidefinite and U is unitary. The matrix P is always uniquely determined as P = (AA∗)1/2; if A is nonsingular , then U is uniquely determined as U = P −1A. If A is real, then P and U may be taken to be real. 1.3 Loewner chains without normalization If f (z, t) is a Loewner chain and v (z, s, t) is its transition mapping then from the defi- nitions it immediately follows that v (·, s, t)

Theorem 8.1.8]). For this we will need a consequence of the following result.

Proposition 1.3.1. Let f and g be holomorphic mappings on Bn such that f(0) = g(0) = 0 and Dg (0) is invertible. If f ≺g then Pf ≤Pg, where Pf and Pg are the unique positive semidefinite matrices from the polar decomposition of f and respectively g.

Corollary 1.3.2. If f (z, t) is a Loewner chain then Df (0, ·)−1 is bounded and there exists a constant C such that Df (0, t1)−1 −Df (0, t2)−1 ≤C ∥Df (0, t1) −Df (0, t2)∥.

Proposition 1.3.4. Let f(z, t) be a Loewner chain and let v(z, s, t) be its transition mapping. Then the following estimates hold: (i) For all ∥z∥≤r, 0 ≤s ≤t1, t2 we have that ∥v(z, s, t1) −v(z, s, t2)∥ ≤ Cr ∥Df(0, t1) −Df(0, t2)∥. (ii) For all ∥z∥≤r, 0 ≤s1 ≤s2 ≤t we have that ∥v(z, s1, t) −v(z, s2, t)∥ ≤ Cr ∥v(z, s1, s1) −v(z, s1, s2)∥ ≤ Cr ∥Df(0, s1) −Df(0, s2)∥.

Corollary 1.3.5. Let f(z, t) be a Loewner chain and v(z, s, t) be its transition mapping. Then the following statements are equivalent: (i) Df (0, ·) is continuous (of local bounded variation, locally absolutely continuous, locally Lipschitz) on [0, ∞). (ii) f(z, ·) is continuous (of local bounded variation, locally absolutely continuous, lo- cally Lipschitz) on [0, ∞), locally uniformly with respect to z. (iii) For all s ≥0, v(z, s, ·) is continuous (of local bounded variation, locally absolute

Proposition 1.3.6. Let f(z, t) be a Loewner chain such that Df (0, ·) is of local bounded variation. Then ∂tf(·, t) exists and is holomorphic on Bn for a.e. t ≥0 and

Lemma 1.4.1. Let G be a domain biholomorphic to the unit ball Bn and let w ∈G. Then there exists a unique biholomorphism f : Bn →G such that f(0) = w and Df(0) > 0.

Proposition 1.4.2. Let {Gk} be a sequence of domains containing 0 and which are biholomorphic to Bn. Furthermore, assume that Gk →G where G is also biholomorphic to Bn. Let {fk} be the sequence of biholomorphisms fk : Bn →Gk such that fk(0) = 0 and Dfk(0) > 0 and let f be the biholomorphism f : Bn →G such that f(0) = 0 and Df(0) > 0. If {Dfk(0)−1fk} is locally uniformly bounded on Bn then fk →f locally uniformly on Bn.

Proposition 1.4.3. Let {Gt}t≥0 be a family of domains containing 0 which are biholo- morphic to Bn and such that Gs ⊂Gt for every 0 ≤s ≤t and Gt →Gt0 as t →t0 for every t ≥t0. Let f(z, t) be such that ft are biholomorphisms of Bn onto Gt and fur- ther satisfying f(0, t) = 0 and Df(0, t) > 0 for all t ≥t0. Then f(z, t) is a Loewner chain such that Df (0, ·) is continuous and of local bounded variation. Furthermore

Proposition 1.5.1. Let h ∈H0 be such that A (t) is locally Lebesgue integrable (locally Lipschitz). Then the initial value problem (1.1.2) has a unique solution v(z, s, t) such that v(·, s, t) is a univalent Schwarz mapping, v(z, s, ·) is locally absolutely continuous (locally Lipschitz) on [s, ∞) locally uniformly with respect to z ∈Bn and the following relations hold: ∥v(z, s, t)∥ (1 −∥v(z, s, t)∥)2 ≤e− ´ t s m(A(τ))dτ ∥z∥ (1 −∥z∥)2, z ∈Bn, t ≥s ≥0; (1.5.4) ∥v(z, s, t)∥ (1 + ∥v(z, s, t)∥)2 ≥e−

Proposition 1.5.2. Let h ∈H0 be such that A (t) is locally Lebesgue integrable and let v(z, s, t) be the unique locally absolutely continuous solution of the initial value problem (1.1.2). Assume that sup s≥0 ˆ ∞ s ∥A(t)∥∥V (s, t)−1∥e−2 ´ t s m(A(τ))dτdt < ∞. (1.5.6) Then the limit lim t→∞V (t)−1v(z, s, t) = f(z, s) (1.5.7)

Proposition 1.5.3. Assume A(·) is bounded and satisfies the condition (1.5.11). Let f : Bn×[0, ∞) →Cn be such that f(·, t) ∈H(Bn), f(0, t) = 0 and f(z, ·) is locally absolutely

Proposition 1.5.1. Assume that f (z, t) is a bounded solution of (1.1.2). Then f(z, t) is a Loewner chain with transition mapping v(z, s, t) and f(z, s) = lim t→∞V (t)−1v(z, s, t) locally uniformly on Bn for s ≥0, where v(z, s, t) is as in Proposition 1.5.1.

Proposition 1.5.4. Let f(z, t) be a Loewner chain that is locally absolutely continuous in t, locally uniformly in z and let g(z, t) be a subordination chain that is locally absolutely continuous in t, locally uniformly with respect to z. Suppose that both chains satisfy the same Loewner chain equation. Then g(z, t) = φ(f(z, t)) where φ is a holomorphic mapping on ∪ft(Bn). Furthermore, g(z, t) is a Loewner chain if and only if φ is a biholomorphism.

Proposition 1.6.1 will show that an A-normalized polynomially bounded solution of (1.1.1) can be recovered from its first n0 coefficients and the solution of (1.1.2). Con- versely, Theorem 1.6.8 will show that by finding polynomially bounded solutions to the

Proposition 1.6.1. If f (z, t) is a polynomially bounded solution of (1.1.1) such that f (z, t) = etA

Lemma 1.6.2. If Qk ∈Pk (Cn) then the following identities hold for t ∈R etAetBkQk  e−tAz k = Qk  zk (1.6.5) etAQk  e−tAz k =

Lemma 1.6.3. If Fk, Gk : [0, ∞) →Pk (Cn), k = 2, . . . , m are solutions of (1.6.2) and the limits f (z, s) := lim t→∞etA

Lemma 1.6.4. If P is a polynomial such that P (t) ≥0 for t ≥s then ˆ ∞ s P (t) e(t−s)A ∥v (z, s, t)∥n0+1 (1 −∥v (z, s, t)∥)2dt ≤ Qϵ,A,P (s) (1 −∥z∥)2 k+(A) m(A) +ϵ, ϵ > 0 where Qϵ,A,P is a polynomial of the same degree as P.

Lemma 1.6.6. If Fk : [0, ∞) →Pk (Cn), k = 2, . . . , m are polynomially bounded and the limit f (z, s) = lim t→∞etA

Corollary 1.6.7. All A-normalized polynomially bounded solutions of (1.1.1) are Loewner chains.

Theorem 1.6.8. If Fk, k = 2, . . . , n0 are polynomially bounded solutions of (1.6.2) then g (z, s) := lim t→∞etA

Theorem 3.21]). We will use the following notations σ+ (L) = {λ ∈σ (L) : Reλ > 0} σ≤(L) = {λ ∈σ (L) : Reλ ≤0} σ0 (L) = {λ ∈σ (L) : Reλ = 0} . P +, P ≤and P 0 will denote the spectral projections corresponding to σ+ (L), σ≤(L) and σ0 (L) respectively (see e.g. [DK74, p 19] ). We say that f is polynomially bounded if there exists a polynomial P such that ∥f (t)∥≤P (t), t ≥0

Theorem 1.6.11. The equation (1.1.1) always has an A-normalized polynomially bounded Loewner chain solution that is uniquely determined by the values of F ≤ k  zk, 0  , k = 2, . . . , n0, which can be prescribed arbitrarily. Furthermore, if A + ¯A is nonresonant then the solution can be chosen to be bounded.

Lemma 2.8] (cf. [GHKK08a, Lemma 2.14])

Lemma 1.6.14. Every sequence of Loewner chains {fk (z, t)} such that Dfk (0, t) = etA and e−tAfk (z, t) ≤CrP (t) , ∥z∥≤r < 1, t ≥0, where P (t) is a polynomial, has a subsequence that converges locally uniformly on Bn to a polynomially bounded Loewner chain f (z, t) for t ≥0.

Theorem 1.6.15. If F ⊂Qn0 k=2 P ≤ k  Pk (Cn)  is bounded (compact) then SF A (Bn) is normal (compact). Furthermore, given ϵ > 0 there exists a constant Cϵ,A,F such that ∥f (z)∥≤ Cϵ,A,F (1 −∥z∥)2 k+(A) m(A) +ϵ, f ∈SF A (Bn) .

Theorem 1.6.8 we have e−tAf (z, t) ≤ Qϵ,A,F (t) (1 −∥z∥)2 k+(A) m(A) +ϵ. When t = 0 the above inequality proves the fact that SF A (Bn) is normal. Furthermore, if F is also closed we can now argue by contradiction using the previous Lemma to see that SF A (Bn) is also closed.

Theorem 1.7.6. ˆSA (Bn) is compact if and only if A is nonresonant.

Proposition 1.7.10. Let A = diag (1, λ), Reλ ≥1. Define Φα,β : S (B1) →S (B2) by Φα,β (f) (z) =

Proposition 1.7.13. Let f : Bn →Cn be a holomorphic mapping and f (z) = z + ∞ X k=2 Fk  zk . Then f is A-asymptotically spirallike if and only if there exists h ∈HA (Bn) such that f (z) = lim t→∞etA

Lemma 1.7.15. Let A = diag (λ1, . . . , λn) and h ∈HA (Bn). If v = (v1, . . . , vn) is the solution of (1.7.2) then ∥vi (z, t)∥≤C            

Lemma 1.7.16. Let λ ∈C such that Reλ ≥0, a ∈C and h : [0, ∞) →C such that |h (t)| ≤C, t ≥0. If lim t→∞ ˆ t 0 esλ (h (s) + a) ds = 0 (1.7.6) then |a| ≤C.

Proposition 1.7.17. Suppose that A is normal, nonresonant and n0 = 2. Then Sa A (Bn) is a normal family. Furthermore, if f ∈Sa A (Bn) has h ∈HA (Bn) as an infinitesi- mal generator (see Proposition 1.7.13) then f can be embedded as the first element of a bounded Loewner chain with infinitesimal generator h.

Proposition 2.2.1. Let h (z) = (z1p1 (zi1) , . . . , znpn (zin)) where i1, . . . , in ∈{1, . . . , n} and pi, i = 1, . . . , n are extreme points for the class P. Then h is an extreme point for the class M (P n).

Proposition 2.2.2. Let f ∈S∗(P n) such that its infinitesimal generator is one of the extreme points from Proposition 2.2.1. Then f is an extreme point for the class S∗(P n).

Proposition 2.3.1. Let h (z) = (z1 (1 + z1) / (1 −z1) , z2 (1 + z2) / (1 −z2)) and g (z) = (az1z2 2, 0) where a ∈C. If a is small enough then h ± g ∈M.

Proposition 2.3.2. Let f (z) =  z1/ (1 −z1)2 , z2/ (1 −z2)2 and g (z) =  az1z2 2 (1 + z2)2 , 0  where a ∈C. If a is small enough then f ± g ∈S∗.

Proposition 2.3.3. If a holomorphic mapping g satisfies h ± g ∈M then g = gc for some c ∈B 2 ∩Ω. Furthermore h + gc is a mapping in M if and only if c ∈B 2 ∩Ω. In particular this proposition shows that h is not an extreme point of M. In fact now we can completely characterize the extremality in M of mappings of the form h + gc.

Corollary 2.3.4. If c ∈B2 ∩Ωthen h + gc is not an extreme point of M.

Proposition 2.3.5. If c ∈S2 ∩Ωthen h + gc is an extreme point for M.

Proposition 2.3.6. If P ∈∪∞ k=1Pk (Cn, C) and if P = 0 on Sn + then P ≡0. We will need the following consequence of the above result.

Corollary 2.3.7. If P ∈Pk (C2, C2) is such that ⟨P (z) , z⟩= 0 for z ∈S2 + then there exists Q ∈Pk−1 (C2, C) such that ⟨P (z) , z⟩= Im (z1z2) Q (z) for all z.

Lemma 2.3.8. Let z =  r1eiθ1, r2eiθ2  and r2 1 + r2 2 = 1. If θ1, θ2 →0 such that θ2/θ1 →l and l ̸= −r2 1/r2 2 then 1 sin (θ1 −θ2)Im