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Spectrum of an bounded linear operator and invariant subspaces
Last modified: 2014-01-17
Abstract
Let $X_1$ and $X_2$ be a closed invariant subspaces of a linear
bounded operator $T\in B(X)$, where $X$ is a Banach space.
In the talk we will give conditions for invertibility of $T\in B(X)$ in respect of invertibility of its
restriction to invariant subspaces in two cases:
CASE I: $X=X_1\oplus X_2$. In this case $\sigma (T)=\sigma
(T_{|X_1})\cup \sigma (T_{|X_2})$.
CASE II: $X=X_1+ X_2$. In this case $X_1\cap X_2\ne\{ 0\} $ and
the relation of invertibility of $T$
trough of invertibility of $T_{X_1}$ and $T_{X_2}$ start to be
more complicate. We need to involve one more invariant subspace
$X_1\cap X_2$ and the restriction of $T$ on it.
bounded operator $T\in B(X)$, where $X$ is a Banach space.
In the talk we will give conditions for invertibility of $T\in B(X)$ in respect of invertibility of its
restriction to invariant subspaces in two cases:
CASE I: $X=X_1\oplus X_2$. In this case $\sigma (T)=\sigma
(T_{|X_1})\cup \sigma (T_{|X_2})$.
CASE II: $X=X_1+ X_2$. In this case $X_1\cap X_2\ne\{ 0\} $ and
the relation of invertibility of $T$
trough of invertibility of $T_{X_1}$ and $T_{X_2}$ start to be
more complicate. We need to involve one more invariant subspace
$X_1\cap X_2$ and the restriction of $T$ on it.
Keywords
Invariant subspace, Spectrum of operator