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Toeplitz Operators with Homogeneous Symbols on Polyharmonic Spaces
Ist Teil von
Complex analysis and operator theory, 2021-09, Vol.15 (6), Article 107
Ort / Verlag
Cham: Springer International Publishing
Erscheinungsjahr
2021
Link zum Volltext
Quelle
SpringerLink (Online service)
Beschreibungen/Notizen
We describe
C
∗
-algebras generated by Toeplitz operators with homogeneous symbols acting on polyharmonic Bergman spaces of the upper half-plane
Π
. The symbols considered here have finite limits at the points 0 and
π
. Under these conditions on the family of symbols, a Toeplitz operator acting on the true polyharmonic space
H
(
n
)
2
(
Π
)
is unitarily equivalent to a
2
×
2
matrix-valued function defined on
R
¯
. The
C
∗
-algebra generated by these matrix-valued functions turns out to be isomorphic to the algebra
C
:
=
f
=
(
f
ij
)
∈
M
2
(
C
(
R
¯
)
)
:
f
(
±
∞
)
is diagonal
,
f
11
(
±
∞
)
=
f
22
(
∓
∞
)
.
Besides, we prove that the
C
∗
-algebra generated by Toeplitz operators with homogeneous symbols, acting on the polyharmonic Bergman space
H
n
2
(
Π
)
, is isomorphic to the
C
∗
-subalgebra of
M
2
n
(
C
(
R
¯
)
)
consisting of all matrix-valued functions
f
=
(
f
ij
)
such that
f
(
-
∞
)
=
λ
1
I
0
I
0
I
λ
2
I
,
f
(
+
∞
)
=
λ
2
I
0
I
0
I
λ
1
I
,
λ
1
,
λ
2
∈
C
,
where
I
is the
n
×
n
identity matrix.