Spectral conditions for almost composition operators between algebras of functions
Authors: Tonev T. 
Published: April 2014

In this article we develop a unifying theory of many years of work by a number of researchers. Especially, we establish general sufficient conditions for maps between algebras of bounded continuous functions on locally compact Hausdorff spaces to be almost composition or almost weighted composition operators, which extend the main results of many previous works on this subject. The following are some typical results. Let

T
:
A
→
B

T\colon A \to B

be a surjective map between two function algebras on locally compact Hausdorff spaces

X
X

and

Y
Y

with Choquet boundaries

δ
A
⊂
X

\delta A\subset X

and

δ
B
⊂
Y

\delta B\subset Y

. If

‖
T
f

T
g
‖
=
‖
f
g
‖

\|Tf\,Tg\|=\|fg\|

and there is an

ε
,

0
≤
ε
>
2

/

3

\varepsilon ,\,0\leq \varepsilon >2/3

, so that the peripheral spectrum

σ
π

(
T
f

T
g
)

\sigma _\pi (Tf\,Tg)

is contained in an

ε

‖
f
g
‖

\varepsilon \,\|fg\|

-neighborhood of

σ
π

(
f
g
)

\sigma _\pi (fg)

for all

f
∈
A

f\in A

and all

g
∈
A

g\in A

with

‖
g
‖
=
1

\|g\|=1

, then there is a continuous function

α
:
δ
B
→
{
±
1
}

\alpha \colon \delta B\to \{\pm 1\}

and a homeomorphism

ψ
:
δ
B
→
δ
A

\psi \colon \delta B\to \delta A

so that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue left-parenthesis upper T f right-parenthesis left-parenthesis y right-parenthesis minus alpha left-parenthesis y right-parenthesis f left-parenthesis psi left-parenthesis y right-parenthesis right-parenthesis EndAbsoluteValue less-than-or-equal-to 2 epsilon StartAbsoluteValue f left-parenthesis psi left-parenthesis y right-parenthesis right-parenthesis EndAbsoluteValue">

|

(
T
f
)
(
y
)
−
α
(
y
)

f
(
ψ
(
y
)
)

|

≤
2
ε

|

f
(
ψ
(
y
)
)

|

|(Tf)(y)-\alpha (y)\,f(\psi (y))|\leq 2\varepsilon \,|f(\psi (y))|

for each

f
∈
A

f\in A

and every

y
∈
δ
B

y\in \delta B

. Therefore,

T
T

is an almost weighted composition operator on

δ
B

\delta B

. If

‖
T
f

T
g
‖
=
‖
f
g
‖

\|Tf\,Tg\|=\|fg\|

, there are

ε
,

0
≤
ε
>
1

\varepsilon ,\,0\leq \varepsilon >1

, and

η
,
 
0
≤
η
>
1

\eta ,\ 0\leq \eta >1

, so that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d left-parenthesis sigma Subscript pi Baseline left-parenthesis upper T f upper T g right-parenthesis comma sigma Subscript pi Baseline left-parenthesis f g right-parenthesis right-parenthesis less-than-or-equal-to epsilon double-vertical-bar f g double-vertical-bar">

d
(

σ
π

(
T
f

T
g
)
,

σ
π

(
f
g
)
)
≤
ε

‖
f
g
‖

d(\sigma _\pi (Tf\,Tg),\sigma _\pi (fg))\leq \varepsilon \,\|fg\|

, while

σ
π

(
T
f
)

\sigma _\pi (Tf)

is contained in an

η
\eta

-neighborhood of

σ
π

(
f
)

\sigma _\pi (f)

for all

f
∈
A

f\in A

and all

g
∈
A

g\in A

with

‖
g
‖
=
1

\|g\|=1

. Then
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue left-parenthesis upper T f right-parenthesis left-parenthesis y right-parenthesis minus f left-parenthesis psi left-parenthesis y right-parenthesis right-parenthesis EndAbsoluteValue less-than-or-equal-to left-parenthesis epsilon plus eta right-parenthesis StartAbsoluteValue f left-parenthesis psi left-parenthesis y right-parenthesis right-parenthesis EndAbsoluteValue">

|

(
T
f
)
(
y
)
−
f
(
ψ
(
y
)
)

|

≤
(
ε
+
η
)

|

f
(
ψ
(
y
)
)

|

|(Tf)(y)-f(\psi (y))|\leq (\varepsilon +\eta )\,|f(\psi (y))|

for each

y
∈
δ
B

y\in \delta B

and every

f
∈
A

f\in A

; i.e.

T
T

is an almost composition operator on

δ
B

\delta B

. If

σ
π

(
T
f

T
g
)
⊂

σ
π

(
f
g
)

\sigma _\pi (Tf\,Tg)\subset \sigma _\pi (fg)

(
(

or

σ
π

(
f
g
)
⊂

σ
π

(
T
f

T
g
)
)

\sigma _\pi (fg)\subset \sigma _\pi (Tf\,Tg))

, and

d
(

σ
π

(
T
g
)
,

σ
π

(
g
)
)
≤
η

d(\sigma _\pi (Tg),\sigma _\pi (g))\leq \eta

for some

η
,

0
≤
η
>
1

\eta ,\,0\leq \eta >1

, for all

f
∈
A

f\in A

and all

g
∈
A

g\in A

with

‖
g
‖
=
1

\|g\|=1

, then

(
T
f
)
(
y
)
=
f
(
ψ
(
y
)
)

(Tf)(y)=f(\psi (y))

for each

f
∈
A

f\in A

and every

y
∈
δ
B

y\in \delta B

. Consequently,

T
T

is a composition operator on

δ
B

\delta B

and, therefore, an algebra isomorphism.