There are cases in which it is known that Slope(γ+z) is a singleton

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The theory of semigroups of holomorphic self-mappings of the open unit disk has been developed intensively over the last few decades (see [8] and [4] for the resent state of this theory). The local geometry of semigroup trajectories near their boundary Denjoy–Wolff point, as well as the rate of convergence to this point have attracted considerable attention. The general problem we address can be stated as follows.

•Determine the rate of convergence of hyperbolic and parabolic type semi- groups; more precisely, find the asymptotic expansion up to a term small enough.

Contreras and D´ıaz-Madrigal in [2] (see also [3, 5, 6]) considered the set Slope(γz+) of accumulation points of the function t 7→ arg (1−τ F¯ t(z)) as t→ ∞and proved that these sets do not depend onz∈∆. There are cases in which it is known that Slope(γ+z) is a singleton. The still open question is

• Whether, in general, Slope(γz+) is a singleton?

Obviously, every semigroup trajectory γz+ = {Ft(z), t≥0}, z ∈ ∆, is an analytic curve. Thus, the tangent line and the disk of curvature at each point Fs(z) exist and move as sincreases. So, one might ask

• Do tangent lines and disks of curvature have, in some sense, a limit location as s→ ∞? When is the limit curvature finite?

As it happens, in the hyperbolic case some smoothness conditions guaran- tee the affirmative answer [5]. For parabolic type semigroups whose generators have the formf(z) =a(1−z)1+α+o((1−z)1+α), it was proved that 0< α≤2 and lim

t→∞arg(1−Ft(z)) = −1

αarga (see [6] and [3]). So, in this case, all the trajectories have the same limit tangent line. In general, its existence is not known. Moreover, the finiteness of the limit curvature in the parabolic case is, in a sense, exceptional [5]. Thus, in this case a more relevant question is:

• Find the contact order of a trajectory and the limit tangent line (which is less than 2 when the limit curvature is infinite).

This problem leads to the so-called rigidity problem: finding conditions on two holomorphic mappings at a boundary point which guarantee their co- incidence. As of Burns and Krantz [1], this problem has attracted much interest

MATH. REPORTS15(65),4(2013), 523–535


(see [9] and reference therein). As a rule, rigidity problem for one-parameter semigroups is approached by looking for conditions on generators. Another approach involves semigroup asymptotics [5]. We suggest to approach the rigidity problem via contact order. Namely, we ask:

•What is the minimal contact order of trajectories required to ensure that the semigroups coincide?

All the problems above are related to the geometry of trajectories γz+ as t → ∞. Our last problem deals with a so-called backward flow invariant domain (see [7]),i.e., a subdomain Ω⊂∆ such that for eachz∈Ω, the whole trajectory γz ={Ft(z), t∈R}lies in Ω.

• Describe the structure of backward flow invariant domains. When is such a domain starlike or convex?


[1] D. Burns and S.G. Krantz,Rigidity of holomorphic mappings and a new Schwarz lemma at the boundary. J. Amer. Math. Soc.7(1994), 661–676.

[2] M.D. Contreras and S. D´ıaz-Madrigal,Analytic flows on the unit disk: angular derivatives and boundary fixed points. Pacific J. Math. 222(2005), 253–286.

[3] M. Elin, D. Khavinson, S. Reich and D. Shoikhet, Linearization models for parabolic dynamical systems via Abel’s functional equation. Ann. Acad. Sci. Fenn. Math. 35 (2010), 1–34.

[4] M. Elin and D. Shoikhet,Linearization Models for Complex Dynamical Systems. Topics in univalent functions, functions equations and semigroup theory. Birkh¨auser Basel, 2010.

[5] M. Elin and D. Shoikhet, Boundary behavior and rigidity of semigroups of holomorphic mappings. Anal. Math. Phys. 1(2011), 241–258.

[6] M. Elin, D. Shoikhet and F. Yacobzon,Linearization models for parabolic type semigroups.

J. Nonlinear Convex Anal.9(2008), 205–214.

[7] M. Elin, D. Shoikhet and L. Zalcman, A flower structure of backward flow invariant domains for semigroups. Ann. Acad. Sci. Fenn. Math. 33(2008), 3–34.

[8] D. Shoikhet,Semigroups in Geometrical Function Theory. Kluwer, Dordrecht, 2001.

[9] D. Shoikhet, Another look at the Burns–Krantz theorem. J. Anal. Math. 105 (2008), 19–42.

ORT Braude College, P.O. Box 78, 21982 Karmiel,

Israel ORT Braude College, P.O. Box 78, 21982 Karmiel,

Israel ORT Braude College, P.O. Box 78, 21982 Karmiel,




A normal operator N is a bounded operator in a Hilbert space such that it commutes with its adjoint, N N = NN. It is a remarkable fact that this simple algebraic condition is strong enough to ensure that a normal operator is, when the ambiental Hilbert space is transformed by an isometric isomorphism, similar to the multiplication by a function on anL2 space. Thus, when considered as a single operator, a normal operator has the best spectral theory one can dream of. However, given two normal operatorsAandB on the same Hilbert space H, it is known that, in general, the productsAB and BA may not be normal, for example,A=

2 1 1 1

and B = 0 1

1 0

. However, as a consequence of the Fuglede-Putnam Theorem (cf. [2] and [8]) if, in addition, A and B commute, then AB is normal. But there is a simple and striking example showing that the general picture transcends the commutativity: if both A and B are two arbitrary unitary operators (isometric isomorphisms) on the same Hilbert space, then both AB and BAare unitary operators, and hence, normal.

The problem that we want to address is the following:

Problem. Given a Hilbert space H characterise those pairs of bounded normal operators A andB on H such the operator AB is normal as well.

If H is finite dimensional this problem was solved by F.R. Gantmaher and M.G. Krein in 1930 [3], as follows:

Theorem 1. Assume that His a finite dimensional Hilbert space and let A, B∈ B(H) two normal operators. The following assertions are equivalent:

(a) AB is normal.

(b) BAis normal.

(c) There exists an orthogonal decomposition

H=R1⊕ R2⊕ · · · ⊕ Rk,

such that for each j = 1, . . . , k, the subspaces reduce both A and B and the compressed operatorsPRjA|Rj andPRjB|Rj are multiples of unitary operators in Rj.

This theorem was rediscovered in 1948 by N.A. Wiegmann in [10] and generalised to compact operators one year later by the same author in [11]. In


1953, I. Kaplansky [6] gave an example of two normal operators A and B on the same Hilbert space for which AB is normal but BA is not. An attempt to understand this asymmetry was made in 1991 by F. Kittaneh in [7] who showed that, given two normal operators A and B on the same Hilbert space such that at least one of the operators A, B, AB, and BA is compact, then AB is compact if and only ifBAis compact. Finally, in the noncompact case, a description of those pairs of normal operatorsAandB such that both of the operatorsABandBAare normal was obtained in [4] by means of the Spectral Multiplicity Theorem in terms of von Neumann’s direct integral representation, cf. [1, 5, 9].

As a conclusion, the problem is open in the nonsymmetric case, that is, only the product AB is assumed to be normal and, of course, without any compactness assumptions of any involved operators.


[1] J. Dixmier,Von Neumann Algebras.North Holland, Amsterdam–New York–Oxford 1981.

[2] B. Fuglede, A commutativity theorem for normal operators. Proc. Nat. Acad. Sci.

USA36(1950), 35–40.

[3] F.R. Gantmaher and M.G. Krein, O normalnyh operatorah v ermitovom prostranstve.

Izv. fiz-mat ob-vapri Kazanskom Univ. 1(1929-1930),3, 71–84.

[4] A. Gheondea, When are the products of normal operators normal?. Bull. Math. Soc.

Sci. Math. Roumanie (N.S.)52(2009), 129–150.

[5] R.V. Kadison and J.R. Ringrose, Fundamentals of the Theory of Operator Algebras, Vol. 1, Elementary Theory, Vol. 2, Advanced Theory, Amer. Math. Soc., Providence R.I. 1997.

[6] I. Kaplansky,Products of normal operators. Duke Math. J.20(1953), 257–260.

[7] F. Kittaneh, On the normality of operator products. Linear Multilinear Algebra 30 (1991), 1–4.

[8] C.R. Putnam,On normal operators in Hilbert space. Amer. J. Math. 73(1951), 357–362.

[9] J. von Neumann,On rings of operators. Reduction theory. Ann. of Math. (2)50(1949) 401–485.

[10] N.A. Wiegmann,Normal products of matrices. Duke Math. J.15(1948), 633–638.

[11] N.A. Wiegmann,A note on infinite normal matrices. Duke Math. J.16(1949), 535–538.

Bilkent University, Department of Mathematics,

06800 Bilkent, Ankara, Turkey,

“Simion Stoilow” Institute of Mathematics of the Romanian Academy,

1-764, 014700 Bucharest, Romania



In what follows, n≥1 is an integer. Leth·,·istand for the inner product in the complex n-dimensional space Cn given by hz, wi =z1w1+· · ·+znwn, where z = (z1, . . . , zn) and w= (w1, . . . , wn) are coordinate representation of z, w ∈Cnin the standard base{e1, . . . , en}ofCn. The inner product induces the Euclidean norm|z|=hz, zi1/2. Denote by B the unit ball{z ∈Cn :|z|<

1}. We let v be the volume measure in Cn, normalized so that v(B) = 1.

We will also consider a class of weighted volume measures on B: for α >−1 we define a finite measure vα on B by dvα(z) =cα(1− |z|2)αdv(z), where cα is a normalizing constant so that vα(B) = 1. One can easily calculate that cα = n+αn

. When 1 ≤ p < ∞, let Lp stand for the Lebesgue space of all measurable functions inB which modulus with the exponent pis integrable in the unit ball; forp=∞let it be the space of all essentially bounded measurable functions in the unit ball. Denote byk · kp the norm onLp(for all 1≤p≤ ∞).

A Bi-holomorphic mapping of B onto itself (a bi–holomorphic automor- phism) has the following form

ϕa(w) = a−hw,ai|a|2 a−(1− |a|2)1/2(w−hw,ai|a|2 a)

1− hw, ai for a∈B,

up to unitary transformations; for a = 0, we set ϕa = −IdB. Observe that ϕa(0) =a. Since ϕa is involutive,i.e.,ϕa◦ϕa= IdB, we also have ϕa(a) = 0.

By Aut(B) = {U ◦ϕa :a ∈ B, U ∈ U }, where U is the group of all unitary transformations of the space Cn, is denoted the group of all bi–holomorphic automorphisms of the unit ball.

For a holomorphic function f(z) = f(z1, . . . , zn) with ∇f(z) we denote the complex gradient ∇f(z) =


∂z1 , . . . ,∂f∂z(z)


. The Bloch space Bcontains all functionsf holomorphic inB for which the semi–normkfkβ = supz∈B(1−

|z|2)|∇f(z)| is finite. One can obtain a true norm by adding |f(0)|, more precisely, in the following waykfkB =|f(0)|+kfkβ, f ∈ B. It is well known that B is a Banach space with the above norm. The standard reference for Bloch space of the unit disc is [1]. For the high dimensions case we refer to [8, 9, 11]. There are several equivalent ways to introduce the Bloch space in the unit ball of Cn. The preceding one is natural and straightforward but the norm defined in that way is not invariant with respect to the group Aut(B).


The following Bloch norm has this property. For f ∈ H(B), we define the invariant gradient |∇f(z)|, where ˜˜ ∇f(z) = ∇(f ◦ϕz)(0), and where ϕz is an automorphism of the unit ball such that ϕz(0) = z. This norm is invariant w.r.t. automorphisms of the unit ball. Namely,|∇(f˜ ◦ϕ)|=|( ˜∇f)◦ϕ|for all ϕ ∈Aut(B). The Bloch space B contains all holomorphic functions f in the ball B for which kfkβ˜ := supz∈B|∇f(z)|˜ < ∞ (cf. [11, Theorem 3.4] or [8]).

For n = 1 we have |∇f˜ (z)| = (1− |z|2)|∇f(z)|, but for n > 1 this is no longer true. Notice that k · kβ˜ is also a semi–norm. We obtain a norm by kfkB˜=|f(0)|+kfkβ˜, f ∈ B.

For α >−1 the Bergman projection operatorPα is defined by Pαg(z) =



Kα(z, w)g(w) dvα(w), g∈Lp,

where Kα(z, w) = (1−hz,wi)1 n+1+α, z, w ∈ B is the weighted Bergman kernel.

Bergman type projections are central operators when dealing with questions related to analytic function spaces. One wants to prove that Bergman projec- tions are bounded and the exact operator norm of the operator is often difficult to obtain. By the Forelli–Rudin theorem [3], the operatorPα:Lp →Lp∩H(B) is bounded if and only if α >1/p−1; here 1≤p <∞ and H(B) is the space of holomorphic functions in the unit ball. In the same paper, they obtain the norm of Pα for p= 1 and p= 2. On the other hand, for n= 1, the Bergman projectionPα :L→ Bis bounded and onto (see [10]). Forn >1 the operator Pα:L→ B is surjective; this can be seen from [11, Theorem 3.4].

The β( ˜β)−norm and B( ˜B)−norm of the Bergman projectionPα:L→ B( ˜B) are

kPαkβ( ˜β) = sup


kPαgkβ( ˜β), kPαkB( ˜B) = sup


kPαgkB( ˜B),

respectively. Note that for every α > −1 we have kPαkB( ˜B) ≤ 1 +kPαkβ( ˜β). From the proof of [11, Theorem 3.4] we find out thatkPαgkβ˜≤Ckgk, where C is a positive constant. The later implies that Pα is a bounded operator.

It is convenient to introduceθ=n+ 1 +α. DenoteCα = Γ(θ+1)

Γ2(θ2+12), where Γ is Euler’s Gamma function. In our recent paper [4], we prove the following two results.

1. We find the exact norm of Pα w.r.t. β−Bloch seminorm. We prove that for the Bergman projection Pα holds kPαkβ = Cα (Theorem 1.2 in [4]). This generalizes the main result from the recent paper of Per¨al¨a [5]

in two directions. Namely, for α = 0 we put P = P0 and we have



presents the main result in [5] (Theorem 2.5). As an immediate corollary, we have the following norm estimates of the Bergman projection Cα ≤ kPαkB ≤ 1 +Cα. However, one can derive kPαkB = 1 +Cα, as proved in [6] quite recently (see Corollary 3.1).

2. We estimate the ˜β−Bloch seminorm of the Bergman projection Pα. In order to formulate this result, assume thatn >1 and define the following function on the real line:

`(t) =θ Z


|(1−w1) cost+w2sint|

|w1−1|θ dvα(w).

For α > −1 we have `(π/2) = π2`(0) = π2Cα. For the ˜β−seminorm of the Bergman projection Pα holds kPαkβ˜ = ˜Cα := max0≤t≤π/2`(t) and


2Cα ≤ kPαkβ˜


2 Cα.

Conjecture 1. In connection with our second result, we conjecture that kPαkβ˜= ˜Cα= π2Cα and kPαkB˜= 1 +π2Cα.

Note than in order to prove this conjecture it is enough to establish that

`(t) is increasing in 0≤t≤ π2.


[1] J.M. Anderson, J. Clunie and Ch. Pomerenke,On Bloch functions and normal functions.

J. Reine Angew. Math. 270(1974), 12–37.

[2] S. Axler and K. Zhu,Boundary behavior of derivatives of analytic functions. Michigan Math. J.39(1992), 129–143.

[3] F. Forelli and W. Rudin,Projections on spaces of holomorphic functions in balls. Indiana Univ. Math. J.24(1974), 593–602.

[4] D. Kalaj and M. Markovi´c,Norm of the Bergman projection. to appear in Math. Scand, arXiv:1203.6009.

[5] A. Per¨al¨a, On the optimal constant for the Bergman projection onto the Bloch space.

Ann. Acad. Sci. Fenn. Math. 37(2012), 245–249.

[6] A. Per¨al¨a,Bloch space and the norm of the Bergman projection. Ann. Acad. Sci. Fenn.

Math. 38(2013), 849–853.

[7] W. Rudin,Function Theory in the Unit Ball ofCn. Springer–Verlag, New York, 1980.

[8] R.M. Timoney,Bloch functions in several complex variables I. Bull. Lond. Math. Soc.

12(1980), 241–267.

[9] R.M. Timoney,Bloch functions in several complex variables II. J. Reine Angew. Math.

319(1980), 1–22.


[10] K. Zhu,Operator Theory in Function Spaces. Marcel Dekker Inc., New York, 1990.

[11] K. Zhu,Spaces of Holomorphic Functions in the Unit Ball. Springer–Verlag, New York, 2005.

University of Montenegro,

Faculty of Natural Sciences and Mathematics, Cetinjski put b.b. 81000 Podgorica,

Montenegro University of Montenegro,

Faculty of Natural Sciences and Mathematics, Cetinjski put b.b. 81000 Podgorica,





For a non-constant analytic function f on the unit disk D = {z ∈ C :

|z|<1},we consider the quantities M(f) = sup


|f0(z)| and m(f) = inf



Note that M(f) is a positive number (possibly +∞) whereas m(f) is a finite nonnegative number. F. John proved the following result.

Theorem 1 (John (1969)). There exists a number γ ∈ [π/2, log(97 + 56√

3)] with the following property: if a non-constant analytic functionf onD satisfies the condition M(f)≤eγm(f),then f is univalent on D.

The largest possible number γ with the property in the theorem is called the (logarithmic) John constant and will be denoted by γ1. Yamashita (1978) improved John’s result by showing that γ1 ≤π. Gevirtz (1989) further proved that γ1 ≤ λπ and conjectured that γ1 = λπ, where λ = 0.6278. . . is the number determined by a transcendental equation. It seems that there is no further progress concerning the John constant.

We may consider a similar problem for zf0(z)/f(z) instead of f0(z) for an analytic functionf on Dwithf(0) = 0, f0(0) = 1.We will denote byAthe class of such functions. Let

L(f) = sup


zf0(z) f(z)

and l(f) = inf


zf0(z) f(z)

for f ∈ A. Here, the value of zf0(z)/f(z) at z = 0 will be understood as limz→0zf0(z)/f(z) = 1 as usual. Note that 0 ≤l(f) ≤ 1 ≤L(f) ≤ +∞.We may now formulate the problem as follows.

Problem 1. Find the largest number δ1>0 with the following property:

If a function f ∈ A satisfies the condition L(f) ≤eδ1l(f) then f is univalent on D.

Since the value 1 plays a special role in the study of zf0(z)/f(z), the following problem is also natural to consider.

Problem 2. Find the largest number δ0>0 with the following property:

If a function f ∈ Asatisfies the condition e−δ0/2 <|zf0(z)/f(z)|< eδ0/2 onD, then f is univalent on D.


Obviously,δ1 ≤δ0≤2δ1.Yong Chan Kim and the author observed in [1]

that π/6 = 0.523· · · ≤δ0 ≤π= 3.14. . . . Moreover, they improved it in [2] as in the following:

Theorem 2. π

3 = 1.04719· · ·< δ0 < 5π

7 = 2.24399. . . . Theorem 3. 7π

25 = 0.87964· · ·< δ1< 5π

7 = 2.24399. . . .

We remark that the above results are not optimal. Indeed, more elabo- rative numerical computations would yield slightly better bounds. Determina- tions of δ0 and δ1 are challenging open problems.

To give an upper bound, we should construct a non-univalent function satisfying the condition in Problems 1 or 2. The function Fa∈ A determined by the differential equation zFa0(z)/Fa(z) =

1−iz 1+iz


is a candidate for an extremal one, whereais a positive constant andiis the imaginary unit √


As is easily seen, L(Fa)/l(Fa) =eπa.

We can pose open problems, which may be easier than the above ones. Let abe the supremum of the numbersasuch thatFais univalent onD.Likewise, let a be the infimum of the numbers ofa such thatFa is not univalent on D. Obviously,δ0 ≤πa ≤πa.The upper bounds in Theorems 2 and 3 are indeed obtained by showing that a<5/7.

Problem 3. Do the following relations hold true?

(1) a=a, (2) δ0 =πa, (3) δ01.


[1] Y.C. Kim and T. Sugawa, On univalence of the power deformation z(f(z)z )c. Chinese Ann. Math. Ser. B.

[2] Y.C. Kim and T. Sugawa, Univalence criteria and analogues of the John constant. to appear in Bull. Austr. Math. Soc. (DOI:

Tohoku University,

Graduate School of Information Sciences, Aoba-ku, Sendai 980-8579




We denote by H2 and H the standard Hardy spaces on the unit disc D. An interpolation sequence inH is a sequence of points (zj)j∈N inDwith the property that for every sequence of bounded complex numbers (wj)j∈N

there exists a functionφ∈Hsuch thatφ(zj) =wj. The celebrated Carleson Theorem gives a necessary and sufficient condition for this to happen; this is

(1) inf







One of the proofs of this theorem uses the fact that condition (1) can be shown to be equivalent to the pair of conditions


zj −zn 1−z¯jzn





|f(zj)|2(1− |zj|2)<∞ ,∀f ∈H2. (3)

Another, more recent, interpolation theorem refers to theDirichlet space.

This is the space D of functions f(z) = P

n=0anzn analytic in D, with the property that



(n+ 1)|an|2 <∞,

where the square of the norm is defined to be the quantity on the left.

In this case, the interpolation condition requires from the function φ to belong to a proper subclass ofH, that ofmultipliers; these are characterized by the fact that φf ∈ Dwhenever f ∈ D. It has been shown by Sundberg and Marshall (building on previous work by Stegenga) that a sequence (zj)j∈N in Dis interpolating if and only if there areγ, C >0 such that the following two conditions are satisfied:


zj −zn



≤(1− |zn|2)γ, n6=j, (4)




log 1

1− |zj|2 −1

<∞ ,∀f ∈ D.


One can see that conditions (3) and (5) are not explicit; they are usually translated in a concrete description by using the so-called Carleson mesures, but we will not pursue this direction.


It turns out that there is a more abstract frame in which we can state these results as particular cases of a general situation. Namely, a reproducing kernel Hilbert space (RKHS) is a Hilbert spaceHof functions defined on a set Λ such that the evaluations f 7→ f(λ) are continuous. According to Riesz’s Theorem, they can be represented by scalar products with functions in H:

f(λ)hf, kλi;

kλ are called reproducing vectors.

A special class of RKHS is formed byPick spaces, which have the property (I am simplifying slightly) that there is another RKHS H0 whose reproducing vectors are given by

k0λ(µ) = 1− 1 kλ(µ).

Pick spaces are in fact precisely the spaces on which the analogue of the classical Nevanlinna–Pick theorem is true.

For general RKHS the interpolation property refers, similar to the case of the Dirichlet space, to multipliers. Thus, an interpolation sequence for an RKHS H is a sequence (λj)j∈N in Λ such that for any bounded sequence of complex numbers (aj) there exists a multiplierφ withφ(λj) =aj.

It turns out that both H2 and D are Pick spaces, and the above condi- tions are particular cases of general ones. Namely, (2) and (4) say that the normalized reproducing vectors are separated; that is, there exists δ <1 such that

(6) |h kλj

kkλjk, kλn

kkλnki| ≤δ whenever n6=j, while (3) and (5) say that

(7) X


|f(zj)|2kkλjk−1 <∞ ,∀f ∈ H.

It can be shown that (6) and (7) are necessary conditions for an interpo- lation sequence. So, we are left with the question:

Open problem. Are (6) and (7) sufficient conditions for an interpola- tion sequence in a general Pick space?

The problem appears as a conjecture in [1], Question 9.49. If true, it would unify in a Hilbert space framework several independent results in func- tion theory. It is known to be true for other particular spaces of functions as well; the proofs always use hard analysis. On the other hand, Bøe proves it for a large class of RKHS, using purely Hilbert space methods; however, it



Finally, let us note that the conjecture is equivalent to the following matricial finite-dimensional statement.

For any δ < 1, M > 0, there exists M1 > 0 with the following property:

SupposeN, n∈N,ζij ∈C,i= 1, . . . , n,j= 1, . . . , N, satisfy





ij|2 1−











for all i6=k, and the matrix A:= (aik)ni,k=1 defined by

aik =







satisfies kAk ≤ M (the operator norm). ThenA is invertible and kA−1k ≤M1.

Note that M1 depends on δ and M, but should be independent of N and n.

Rather than give precise references for all statements above, we refer to Seip’s book [2] for the history of the problem and precise attributions.


[1] J. Agler and J.E. McCarthy,Pick Interpolation and Hilbert Function Spaces. AMS, 2002.

[2] K. Seip, Interplation and Sampling in Spaces of Analytic Functions. AMS, 2004.

“Simion Stoilow” Institute of Mathematics of the Romanian Academy,

P.O. Box 1-764, Bucharest 014700,





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