11

M. Elin

The Galilee Research Center for Applied Mathematics

of ORT Braude College

Karmiel, Israel

Definition. Let D be a domain in a complex Banach space X . A family 0 ttFS of biholomorphic self-mappings of D is said to be a one-parameter continuous semigroup on D if ,0,, stFFF stst

and xxFtt

)(lim0

for all Dx .

A family 0

, :t

S f t D X

is called a univalent

subordination chain if ,f t is biholomorphic on D ,

0, 0f t for all 0t , and , ,f s f t whenever s t .

Continuous Semigroups

33

Spirallike and starlike mappingsDefinition. Let )(XLA , Re ( ) 0A , and )bihol(D, Xf .

We say that f is an A -spirallike mapping, if for each Dx and for each 0t , the point (D))( fxfe At .

In other words, )bihol(D, Xf is A -spirallike, if the family of linear operators 0

t

Ate forms a semigroup acting on (D)f .

)bihol(D, Xf is a starlike mapping, if it is possible to choose idA , i.e., for each and for each 0t , the point (D))( fxfe t .

Equivalently, 0

tte forms a semigroup on (D)f .

In the one-dimensional case, the operator CA , 0Re .

44

In the one-dimensional settings - the well-known criteria of Nevanlinna, Study and Špaček

In multi-dimensional situations – Suffridge, Gurganus, Pfaltzgraff, Gong, …

In multi-dimensional situations – not all of the analogues hold, proofs are very complicated, examples are rather hard to construct.

Spirallike and starlike mappings

55

Since the work of Roper and Suffridge in 1995, there has been considerable interest in constructing holomorphic mappings of the unit ball in a Banach space with various geometric properties by using mappings with similar properties acting in a subspace. Such properties include convexity, starlikeness, spirallikeness, and so on. It is also of interest to extend subordination chains,  semigroups and semigroup generators.  

Extension Operators

Given two complex Banach spaces X and Y , and a family of

holomorphic mappings h on the unit ball of X , we have to construct a

family of mappings [ ]h holomorphic on the unit ball of X Y with

values in X Y such that

[ ]( ,0) : ( ),0 ,h x h x

with preserving certain (geometric) properties.

Given )Univ(h , 1: xx C

1)0(',1)0( hh ,

Roper-Suffridge extension operator

][h preserves:

convexity [K. Roper and T. J. Suffridge, 1995] starlikeness, Bloch property [I. Graham and G. Kohr, 2000] -spirallikeness [I. Graham, G. Kohr and M. Kohr, 2000]

[X. S. Liu and T. S. Liu, 2005] Loewner chains, linear-invariance [I.Graham, H.Hamada, G.Kohr, M.Kohr, 2000-2010]

they have constructed nnh CB :][ ,

1,,)('),(:),]([221 yxyyxhxhyxh nC .

7

1/2, ( )f x f x f x y

Modifications of R-S extension operator

11

2 21

, ( ) ,

, , , 1

nf

n n

f x f x J x y

x x y x y

B C C

Pfaltzgraff, Suffridge, 1997

1

, ( ) , 0,2

f x f x f x y I.Graham, G.Kohr, M.Kohr, 2000

( ), , [0,1]f x

f x f x yx

I.Graham, G.Kohr, 2000

8

1/2

11

, ( ) ,

, ( ) ,

, ( ) ,

( ),

nf

f x f x f x y

f x f x J x y

f x f x f x y

f xf x f x y

x

Modifications of R-S extension operator

, ,f x f x f x y

id

2

2

1) , 1

2) , ( ) , ,

3) ,

1 ( )4) ,

1

x

f g x g x f g x

f x

f xf x

x

- the chain rule

is invertible

99

Some notations

Let X

X , and Y

Y , be two complex Banach spaces, and let

XD , YD be the open unit balls. On the space YXZ we define a norm as follows. Let ]1,0[]1,0[: p be a continuous function which satisfies the conditions:

XY

xpyZyxD :),(

10

Main notation and notionDefinition. Let bihol(D , )XK X be closed with respect to

composition, and let ),( xf take values in )(YL be continuous

in Kf and holomorphic in Xx D . We say that ),( xf is appropriate if it satisfies the following properties:

1). YX x id),(id ;

2). ),(),())(,( xgfxgxgf , )bihol(DXg ;

3). ),( xf is invertible;

4).

X

XYL xp

xfpxf

)(),(

)( , )bihol(DXf .

Extension operator:

yxfxfyxf ),(),(),]([

1111

Extension operators for semigroups

Theorem 1. Let : D ( ) XK L Y be appropriate, and let

0t t

F K be a semigroup onDX . Then the family 0

t tF with

, ( ), , t t tF x y F x F x y forms a semigroup on ZD .

Theorem 2. Let : D ( ) XK L Y be appropriate, and let

0t t

F K be a semigroup onDX . Let 0t t

G be a semigroup of norm-

contractions on DY such that each sG commutes with , tF x .

Then the family 0

t tF with , ( ), , ( )

t t t tF x y F x F x G y

forms a semigroup on ZD .

1212

Extensions of spirallike mappingsand subordination chains

Theorem 3. Let : D ( ) XK L Y be appropriate,

and let Hol D , Xf X be A-spirallike mapping.

Suppose that Ate f K and ( ) C L Y such that , , At Cte f x e f x .

Then the mapping f is A -spirallike, where 0

0

AA

B C

with any

accretive operator B which commutes with C and , f x .

Theorem 4 (I.Graham-H.Hamada-G.Kohr). Let ,f t be a univalent

subordination chain. Suppose that ,Ate f s K and ( ) C L Y such that

, , , ,C t sAt Ase f s x e e f s x .

Then the family , ,F t defined by , , .,At AtF x y t e e f t is a

univalent subordination chain.

1313

Extensions of spirallike mappingsExample. Let CnX with an arbitrary norm, Y be a complex Banach space;

1

1, ( ) nff x y J x y

Then the mapping f is A -spirallike, where

0

2trace(A)0 id

( 1) Y

A

AB

r n

for any accretive operator B .

D = 2, : 1 , 1

r

X Yx y Z X Y x y r

Let A be a diagonal matrix, and let f be A-spirallike.

1414

Extensions of spirallike mappingsCY

For any point ](D)[),( 00 fwz , the image ](D)[ f contains the set

0,,:),( 0

)Re(trace

0 twewzezwz r

AtAt

1515

Extensions of spirallike mappings

Example. Let X be a complex Hilbert space, Y be a complex Banach space;

2, : 1

r

X Yx y Z X Y x y D =

Let A be an accretive operator, * A , and let f be A-spirallike with respect

to a boundary point ( ) 0 f such that ( ), 0 f x .

2

( ),,

1 ,

rf xf x y y

x

Then the mapping f is 0

20 id

Y

A

Br

-spirallike for any accretive operator B .

16

, ( ), ,f x y f x f x y

Extension operator:

? Perturbation of the first coordinate:

ˆ [ ]( , ) ( ) ( , , ), ,h x y f x Q f x y f x y

Extreme Points, Support PointsTheorem 5 (I.Graham-H.Hamada-G.Kohr). Let F ⊆ K be a nonempty compact set. Then Φ(exF) ⊆ exΦ(F) and Φ(suppF) ⊆ suppΦ(F).

Further question

Perturbation of the first coordinate:

2 21

ˆ [ ]( , ) ( ) '( ) ( ), '( ) ,

, , 1,n

h x y h x h x Q y h x y

x y x y

C C[J. R. Jr. Muir, 2005]

where Q is a homogeneous polynomial of degree 2.

Muir’s extension operator

ˆ [ ]h preserves starlikeness of h whenever 4

1)(sup

1

yQ

y

Suffridge’s criterion of starlikeness:

a (locally) biholomorphic mapping Hol , nH Β C

normalized by (0) 0, (0) ,H DH I is starlike if and only if

1Re ( ) ( ), 0DH x H x x

Geometric explanationˆ [ ]h is starlike ˆ( , ) [ ]( ),nz w h B

ˆ( , ) : ( , ) [ ]( )t t ntG z w e z e w h B

[ ]( , ) : ( ), '( ) ,ˆ [ ] ,

ˆ [ ]( , ) ( ) '( ) ( ), '( )

h x y h x h x yh

h x y h x h x Q y h x y

where ( , ) ( ),z w z Q w w is the automorphism of the space

][][ˆ hh is starlike

nnt hhG BB ][][

tt GF 1: is a semigroup on nh B][

2( , ) ( ) ( ),t t t ttF z w e z e e Q w e w

Geometric explanation

2

( , ) : ( , )

( , ) ( ) ( ),

t tt

t t t tt

G z w e z e w

F z w e z e e Q w e w

act on the image of [ ]h

nB

x

y

yxhxhyxh )('),(),]([

z

w )]([ nh B

),( wz

ze t

we t

Theorem 1. Let ),( CUnivh . For [0,1) and 0x define

the set 2

00

21)('1)(':: xxhxxhx .

Then the image )( h covers the open disk of radius

2

00 1)('4

1xxh

centered at )( 0xh .

Theorem 2. Let ),( CUnivh , 0 1 be such that

)()( hh . For 0x define as above.

Then the image )( h covers the open disk of radius

2

00 1)('4

xxh

centered at )( 0xh

Covering results

2 , , 0t te e t

),( CUnivh means univalent non-normalized

By the Koebe 1/4 Theorem: Let ),( CUnivh . Then the image ( )h

covers the open disk of radius 2

0 0

1'( ) 1

4h x x centered at )( 0xh .

Theorem 3 Let C:h be a -spirallike function and

YH CB : be defined by ))('),((:),(1

yxhxhyxH r .

Roper-Suffridge type operator

Suppose that )(YLB generates a semigroup of strict contractions.

Then for each point )(),( 00 BHwz and 0t ,

)(, 00 BHweeze Bttr

tt

whenever 2

11 1)('4

1xxh

eR

rtB

tt

with

01

1 zehx t .

Theorem 4 Let C:h be -spirallike and CYQ :

be a homogeneous polynomial of degree Nr .

Spiralikeness for Muir’s type operator

Then the mapping :H YB C defined by

1

( , ) : ( ( ) '( ) ( ), '( ) )rH x y h x h x Q y h x y

is

Yr

id0

0

-spirallike for each C , 0Re ,

whenever Re

4

1)(sup

1

yQ

y.

Moreover, this bound is sharp.

2323

Thank you for your attention!