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Discussion on Matrices Fixed Nullity in Complement Problem of Operator Matrices
Let H and K be separable infinite-dimensional Hilbert spaces, and let A ∈ B ( H ) and B ∈ B ( K ) be given operators. We denote by M C the operator acting on H ⊕ K of the form M C = A C 0 B . In this paper, some necessary and sufficient conditions are obtained for M C to be a Fredholm operator with...
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| Published in: | Complex analysis and operator theory 2024-05, Vol.18 (4), Article 98 |
|---|---|
| Main Authors: | , , |
| Format: | Article |
| Language: | English |
| Subjects: | |
| Citations: | Items that this one cites |
| Online Access: | Get full text |
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| Summary: | Let
H
and
K
be separable infinite-dimensional Hilbert spaces, and let
A
∈
B
(
H
)
and
B
∈
B
(
K
)
be given operators. We denote by
M
C
the operator acting on
H
⊕
K
of the form
M
C
=
A
C
0
B
. In this paper, some necessary and sufficient conditions are obtained for
M
C
to be a Fredholm operator with
n
(
M
C
)
>
0
and
ind
(
M
C
)
<
0
for some left invertible or invertible operator
C
∈
B
(
K
,
H
)
. Meanwhile, for the nullity of
M
C
, we discuss the relationship between
n
(
M
C
)
and
n
(
A
) by different method. As the application of above results, the weak properties of Weyl’s theorem for upper triangular operator matrices are explored. |
|---|---|
| ISSN: | 1661-8254 1661-8262 |
| DOI: | 10.1007/s11785-024-01542-0 |