Let H 1 and H 2 be Hilbert spaces, let T ∈ B ( H...

Fletcher Hays

Fletcher Hays

Answered

2022-06-22

Let H 1 and H 2 be Hilbert spaces, let T B ( H 1 , H 2 ). Suppose that Ker T is finite-dimensional and that Im T is closed in H 2 . Prove that Ker ( T + K ) is finite-dimensional for each K K ( H 1 , H 2 ).

1. Define Hilbert space as direct sum of two complemented subspaces
2. For compact operator use Hilbert Schmidt decomposition

Main idea is to prove that intersection of Spectrum of such $T and compact K is finite

Answer & Explanation

Stevinivm

Stevinivm

Expert

2022-06-23Added 18 answers

By assumptions the operator T :   ker T I m T is invertible. Therefore for any w ker T we have
T w c w
for a constant c > 0.

Assume by contradiction, that ker ( T + K ) is infinite dimensional. Then there exists an infinite orthonormal system v n in ker ( T + K ) . As K is compact, then K v n 0. Thus T v n 0. We have
v n = w n + u n , w n ker T ,   u n ker T
and
T v n = T u n c u n
hence u n 0. The sequence w n has an accumulation point, therefore the sequence v n has an accumulation point as well. This leads to a contradiction.

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