Abstract
We study the spectral multiplicity for the direct sum
A
⊕
B
of operators
A
and
B
on the Banach spaces
X
and
Y
. Under some domination conditions ‖
P
(
B
)‖≦
C
‖
P
(
A
)‖, in particular, ‖
B
n
‖≦
C
‖
A
n
‖,
n
≧0, we prove the addition formulas
μ
(
A
⊕
B
)=
μ
(
A
)+
μ
(
B
) for spectral multiplicities. We give valuable new applications of the main result of the author’s paper [12]. We also use the so-called Borel transformation and generalized Duhamel product in calculating the spectral multiplicity of a direct sum of the form
T
⊕
A
, where
T
is a weighted shift operator on the Wiener algebra
.