SURJECTIVE ISOMETRIES ON $C^{1}[0,1]$ WITH RESPECT TO SEVERAL NORMS (Researches on isometries from various viewpoints)

SURJECTIVE ISOMETRIES ON

_C^{1}[0

,1

]

WITH RESPECT TO

SEVERAL NORMS TAKESHI MIURA

ABSTRACT. Let

_C^{1}[0

,1

]

beacomplexlinearspaceof allcontinuouslydifferentiablecom‐

plex valued functions onthe unit interval

_[0

,1

]

. Wegive acharacterization ofsurjec‐

tive, not _{necessarily linear,} isometries on

C^{1}[0

,1

]

with respecttothefollowing norms:

\Vert f\Vert_{ $\Sigma$}=\Vert f\Vert_{\infty}+\Vert f'\Vert_{\infty}, \displaystyle \Vert f\Vert_{C}=\sup\{|f(t)|+|f'(t)| : t\in[0, 1]\}

and

_{\Vert f\Vert_{ $\sigma$}=|f(0)|+\Vert f'\Vert_{\infty}}

for

_{f\in C^{1}[0}

,1

]

,respectively.

1. INTRODUCTION

Let M and N be real or

complex

normed linear _spaces with norms

\Vert\cdot\Vert_{M}

and

\Vert\cdot\Vert_{N},

respectively.

We say thata

_mapping

T:M\rightarrow N is an

_isometry

if and

only

if

\Vert T(a)-T(b)\Vert_{N}=\Vert a-b\Vert_{M} (a, b\in M)

.

It should be

_emphasized

that we never assume

linearity

of isometries unless otherwise

stated. Let X bea

_compact

Hausdorff_spaceand

C(X)

the Banach_spaceof allcontinuous

complex

valued functions on X with the _supremum norm

\Vert\cdot\Vert_{\infty}

. Denote

by

C_{\mathrm{R}}(X)

the

real Banach _space ofall continuous real valued functions on X. Banach

[1,

Theorem 3

in

_Chapter

_XI]

_proved

that if T:

C_{\mathbb{R}}(X)

\rightarrow

C_{\mathbb{R}}(Y)

is a

surjective

isometry

and if X

and Y are

_compact

metric _spaces, then there exist acontinuous function u:Y\rightarrow

\{\pm 1\}

and a

homeomorphism

_$\varphi$: Y \rightarrow X such that

T(f)(y)

=

T(0)(y)+u(y)f( $\varphi$(y))

for all

f

\in

C_{\mathrm{R}}(X)

_{and y} \in Y. Stone

[18,

Theorem

83]

generalized

the result

by

Banach for

compact

Hausdorff spaces X and Y. On the other

hand,

the so‐called Banach‐Stone

theoremstatesthat if T:

C(X)\rightarrow C(Y)

isa

surjective

complex

linear

isometry,

then there

exist acontinuousfunctionu : Y\rightarrow \mathbb{C} with

|u(y)|

=1 for

y\in Y

and a

homeomorphism

$\varphi$:Y\rightarrow X

such that

_{T(f)(y)=u(y)f( $\varphi$(y))}

for all

_{f\in C(X)}

and

_{y\in Y.}

Let

_C^{1}[0

,1

]

be the Banachspaceofall

continuously

differentiable

complex

valued func‐

tionsonthe unit interval

[0

,1

]

with thenorm

\displaystyle \Vert f\Vert_{C}=\sup\{|f(t)|+|f'(t)| : t\in [0, 1]\}

for

f

\in

C^{1}[0

,1

]

. Cambern

[4,

Theorem

1.5]

gave a characterization for

surjective complex

linear isometries from

_C^{1}[0

,1

]

onto

itself;

tobe more

explicit,

if T:

C^{1}[0, 1]\rightarrow C^{1}[0

,1

]

is a

surjective complex

linear

isometry,

then there exists c \in \mathbb{C} with

_|c|

= 1 such that

T(f)(t)=cf(t)

for all

_{f\in C^{1}[0}

,1

]

and t\in

[0

,1

]

, or

T(f)(t)=cf(1-t)

forall

f\in C^{1}[0

,1

]

and t\in

_[0

,1

]

. Theresult

by

Cambem hasbeenextendedin various

directions;

Pathak

[16,

Theorem

_2.5]

described

_{surjective complex}

linear isometries between thè Banachspaceof 2000 Mathematics_{Subject Classification.} 46\mathrm{J}10.

all n times

continuously

differentiable functions. Rao and

Roy

[17,

Theorem

4.1]

con‐

sidered

_{surjective complex}

linear isometries on

C^{1}[0

,1

]

with thenorm

\Vert f\Vert_{\infty}+\Vert f'\Vert_{\infty}

for

f

\in

C^{1}[0

,1

]

. Jarosz and Pathak

[9,

Theorem

3]

gave ascheme to

verify

that

surjective

complex

linear isometries are

given

by

homeomorphisms.

Botelho and Jamison

[2,

Theo‐

rem

3.5]

investigated

surjective

complex

linear isometries between

C^{1}([0,1], E)

, where E

denotes afinite dimensional Hilbert _space. We refer the reader to

_{[6, 7]}

for a _survey of

the

_study

ofisometrieson variousfunction_spaces.

The purpose ofthis paper is to describe

_surjective

isometries on

C^{1}[0

,1

]

without as‐

suming

lineairity

of_maps. In

_fact,

the

_following

is the main theorem of this_paper, which

extends the result

_by

Rao and

_Roy

_[17,

Theorem

_4.1]:

2. MAIN RESULTS

Theorem 2.1. Let T:

C^{1}[0, 1]f\rightarrow C^{1}[0

_,1

]

be a

surjective isometry,

which need not be

linear,

with

_respect

to the norm

\Vert f\Vert_{ $\Sigma$}

=

\Vert f\Vert_{\infty}+\Vert f'\Vert_{\infty}

. Then there exists a constant

c\in \mathbb{C} with

_|c|=1

such that

T(f)(t)=T(0)(t)+cf(t)

(\forall f\in C^{1}[0,1]

,

\forall t\in[0,1

or

T(f)(t)=T(0)(t)+cf(1-t)

(\forall f\in C^{1}[0,1],

\forall t\in[0,1

or

T(f)(t)=T(0)(t)+\overline{cf(t)}

(\forall f\in C^{1}[0,1],

\forall t\in[0,1

or

T(f)(t)=T(0)(t)+\overline{cf(1-t)} (\forall f\in C^{1}[0,1], \forall t\in[0,1

where

-denotes the

_complex

_conjugate.

Conversely,

each

of

the above _maps is a

surjective

isometry

on

C^{1}[0

,1

]

with

respect

to

\Vert\cdot\Vert_{ $\Sigma$}

, where

T(0)

bs an

arbitrary

element

of

C^{1}[0

,1

].

The

_following

result is a

special

caseof

[2,

Theorem

3.5]

by

Botelho and

Jamison;

in

fact, they

consider

_surjective

linear isometries on

C^{1}([0,1], H)

with

_respect

to the norm

\displaystyle \sup\{\Vert f(t)\Vert_{H}+\Vert f'(t)\Vert_{H}

: t\in

_[0

,1 where H denotes afinite dimensional Hilbert space.

We can

identify

C^{1}[0

,1

]

with

C^{1}([0,1],\mathbb{R}^{2})

. If

T_{0}

is a

surjective

real linear

isometry

on

C^{1}[0

,1

]

, then we may

regard

T_{0}

as a

surjective

linear

isometry

on

C^{1}([0,1],\mathbb{R}^{2})

.

Thus,

T_{0}

is characterized

_by

_[2,

Theorem

_3.5].

On the other

_hand,

we can _provethe

following

result as a

corollary

to Theorem 2.1.

Corollary

2.2. Let T:

C^{1}[0, 1]\rightarrow C^{1}[0

_,1

]

be a

surjective isometry,

which need not be

linear,

with

_respect

to the norm

\displaystyle \Vert f\Vert_{C}=\sup\{|f(t)|+|f'(t)|

:

t\in[0

_,1 Then there exists a constantc\in \mathbb{C} with

|c|=1

such that

T(f)(t)=T(0)(t)+cf(t)

(\forall f\in C^{1}[0,1],

\forall t\in[0,1

or

T(f)(t)=T(0)(t)+cf(1-t)

(\forall f\in C^{1}[0,1]

,

\forall t\in[0,1

or

T(f)(t)=T(0)(t)+\overline{cf(t)}

(\forall f\in C^{1}[0,1]

,

\forall t\in[0,1

or

Conversely,

each

_of

the above _maps is a

surjective isometry

on

C^{1}[0

,1

]

with

respect

to

||\cdot\Vert_{C}

, where

T(0)

is an

arbitrary

element

of

C^{1}[0

,1

].

Theorem 2.3. Let T:

C^{1}[0, 1]\rightarrow C^{1}[0

_,1

]

be a

_surjective

_isometry,

which need not be

linear,

with

_respect

to the norm

\Vert f\Vert_{ $\sigma$}=|f(0)|+\Vert f'\Vert_{\infty}

. Then there existaconstantc\in \mathbb{C}

with

_|c|

= 1

, a continuous unimodular

function $\beta$

:

[0

,1

]

\rightarrow \mathbb{C} and a

homeomorphism

$\rho$:

[0, 1]\rightarrow[0

_,1

]

such that

T_{0}(f)(t)=cf(0)+\displaystyle \int_{0}^{t} $\beta$(s)f'( $\rho$(s))ds

(\forall f\in C^{1}[0,1]

,

\forall t\in[0,1

or

T_{0}(f)(t)=c\displaystyle \overline{f(0)}+\int_{0}^{t} $\beta$(s)f'( $\rho$(s))ds

(\forall f\in C^{1}[0,1]

,

\forall t\in[0,1

or

T_{0}(f)(t)=cf(0)+\displaystyle \int_{0}^{t} $\beta$(s)\overline{f'(p(s))}ds

(\forall f\in C^{1}[0,1]

,

\forall t\in[0,1

or

T_{0}(f)(t)=c\displaystyle \overline{f(0)}+\int_{0}^{t} $\beta$(s)\overline{f'( $\rho$(s))}ds (\forall f\in C^{1}[0,1], \forall t\in[0,1

where

_{T_{0}(f)(t)=T(f)(t)-T(0)(t)}

.

Conversely,

each

_of

the above maps is a

surjective isometw

on

C^{1}[0

,1

]

with

respect

to

\Vert\cdot\Vert_{ $\sigma$}

, where

T(0)

is an

arbitrary

element

of

C^{1}[0

,1

].

A

_key

of

_proofs

ofthe main results isa

significant

result relatedtoisometries_proven

_by

Mazur and Ulam. The Mazur‐Ulamtheorem

_[13]

statesthat if T isa

surjective

isometry

between normed linear spaces, then

T-T(0)

is real

linear; consequently

T-T(0)

is

a

_surjective,

real linear

_isometry.

Väisälä

[19]

_gave a

simple proof

of the Mazur‐Ulam

theorem. Theorem 2.1 states that

_surjective

real linear

_isometry

_T-T(0)

on

C^{1}[0

,1

]

is

the sameas

complex

linear one_{up to}the

complex conjugate;

similar resultswere _proven

for function

_algebras

_[5,

_8,

_14]

and forfunction_spacesunder additional

_assumptions

_[12].

Onthe other

_hand,

real linear isometries are

_quite

different from

complex

linearones in

general;

such an

elementary example

is

given

in

[12,

Example

6.2].

A characterization

is obtained in

_[15]

in order that

_surjective

real linear isometrieson function _spaces with

respect

tothesupremumnormbeofthe canonical

form,

that

is,

acombinationof

weighted

composition operators

and the

_{complex conjugate.}

_Surjective,

non‐canonical isometries

are

investigated

in

[10].

Let

_C^{1}[0

,1

]

be the Banach space of all

continuously

differentiable

complex

valued func‐

tionsonthe unit interval

[0

,1

]

with

respect..to

the

following

norms:,

\Vert f\Vert_{ $\Sigma$}=\Vert f\Vert_{\infty}+\Vert f'\Vert_{\infty}

,

\Vert f\Vert_{ $\sigma$}=|f(0)|+\Vert f'\Vert_{\infty}

and

\displaystyle \Vert f\Vert_{C}=\sup\{|f(t)|+|f'(t)| : t\in[0, 1]\}

for

_{f\in C^{1}[0}

,1

]

, where

\Vert\cdot\Vert_{\infty}

denotes thesupremumnorm on

[0

,1

]

. Let

$\Gamma$=\{z\in \mathbb{C}

:

|z|=

1\}

be the unit circle in the

_{complex plane}

\mathbb{C}, andset

X_{ $\Sigma$}=[0, 1]\times [0

,1

]

\times $\Gamma$,

with the

_{product topology.}

Define

(1)

\tilde{f}(r, s, z)=f(r)+zf'(s)

for

_f

\in

C^{1}[0

,1

]

and

(r, s, z)

\in

X_{ $\Sigma$}

; thus

\tilde{f}(r, s, z)

=

f(0)+zf'(s)

if

(r, s, z)

\in

X_{ $\sigma$}

, and

\tilde{f}(r, s, z)=f(s)+zf'(s)

if

_{(r, s, z)\in X_{c}}

. The function

\tilde{f}

is continuouson

X_{ $\Sigma$}

. Let

C(K)

be

the Banach_spaceof all continuous

_complex

valued functionson a

_compact

Hausdorff_space

Kwith

_respect

to_{the supremum}norm

\Vert\cdot\Vert_{\infty}

. We define

A_{ $\Sigma$}=\{\tilde{f}\in C(X_{ $\Sigma$})

:

f\in C^{1}[0

,1

A_{ $\sigma$}=A_{ $\Sigma$}|_{X_{ $\sigma$}}

and

_{A_{C}=A_{ $\Sigma$}|_{X_{c}}}

. Let

(A, X) \in\{(A_{ $\Sigma$}, X_{ $\Sigma$})

,

(A_{ $\sigma$}, X_{ $\sigma$})

,

(A_{C},

X_{c}

Then A is

anormedlinear

subspace

of

C(X)

. Let 1 \in

C^{1}[0

,1

]

be the constant function suchthat

1(t)=1

for all t\in

_[0

,1

]

.

By

(1),,

we see that A has constant function

\overline{1}

. Notice that A

separates

points

of X in thesense that

for.

each

pair

ofdistinct

points

_{x_{1}, x_{2}} \in X there

exists

\tilde{f}\in A

such that

_{\tilde{f}(x_{1})}

_\neq

_{\tilde{f}(x_{2})}

. The

correspondence

f\mapsto

\tilde{f}

is a

complex

linear

isometry

from

_(C^{1}[0,1]

, onto

(A, \Vert\cdot\Vert_{\infty})

; where -

\Vert f\Vert

=

\Vert f\Vert_{ $\Sigma$}

if

A=A_{ $\Sigma$},

\Vert f\Vert

=

\Vert f\Vert_{ $\sigma$}

if

_{A=A_{ $\sigma$}}

and

_{\Vert f\Vert =\Vert f\Vert_{C}}

if

_{A=A_{C}}

. Notethat

if=i\tilde{f}

for

f\in C^{1}[0

_,1

]

. We denote

by

A^{*} the

_complex

dual _spaceof

_{(A, \Vert\cdot\Vert_{\infty})}

. Let

$\delta$_{x}

: A\rightarrow \mathbb{C}be the

point

evaluationdefined

as

$\delta$_{x}(\tilde{f}) =\tilde{f}(x)

for

\tilde{f}\in A

andx\in X. We see that the set of all extreme

points

ofthe

unit ball of A^{*} is

_{\{ $\lambda \delta$_{x} : x\in X, $\lambda$\in $\Gamma$\}.}

Let T:

C^{1}[0, 1]\rightarrow C^{1}[0

_,1

]

be a

surjective isometry.

Define a

mapping T_{0}

:

C^{1}[0, 1]\rightarrow

C^{1}[0

,1

]

as

T_{0}

=

T-T(0)

.

By

the Mazur‐Ulam

theorem,

T_{0}

is a

surjective,

real linear

isometry

from

_C^{1}[0

,1

]

ontoitself. We define S:A\rightarrow Aas

(2)

S(\tilde{f})=\overline{T_{0}(f)} (\tilde{f}\in A)

.

Since

_f\mapsto

\tilde{f}

is a

_surjective

_isometry

from

C^{1}[0

,1

]

onto A, it is a

bijection,

and thus S

is well defined. As

_f\mapsto

\tilde{f}

is a

surjective complex

linear

isometry,

S is a

surjective

real

linear

_isometry

on A. We define a

mapping S_{*}:A^{*}\rightarrow A^{*}

as

(3)

S_{*}( $\eta$)(\tilde{f})={\rm Re} $\eta$(S(\tilde{f}))-i{\rm Re} $\eta$(S(i\tilde{f}))

for

_{$\eta$\in A^{*}}

and

\tilde{f}\in A

. It is routinetocheck that the

mapping

S_{*}

isa

surjective

real linear

isometry

with

_respect

tothe

_operator

norm onA^{*}

(cf. [15,

Proposition

1

Proof of Theorem

_2.1,

_Corollary

2.2 and Theorem 2.3are

given

in

[11].

In

fact,

Kawa‐

mura, Koshimizu and the authorofthispaper

generalize

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