Symmetric Invariant Subspaces of Complexifications of Linear Operators
We prove the existence of the invariant subspaces of some operators in a real Banach space. For example, linear isometries have invariant subspaces
math.FA↗
arXiv subjects
K. V. Storozhuk
Publications and source records attributed to K. V. Storozhuk.
Obstructions for uniform stability of C_0-semigroup
Let T be a C_0-semigroup on X with generator A. We prove that if the abscissa of uniform boundedness of the resolvent A is non-negative then for each a non-decreasing function h:[0,\infty] -> [0,\infty], there are x' in X' and x in X such that integral from 0 to \infty of h(|< x',T(t)x)>| is equal to \infty. If Sp(A) contained a number from iR then such x may be taken in D(A^\infty).
math.FA↗
Slowly Changing Vectors and the Asymptotic Finite-Dimensionality of an Operator Semigroup
Let $T:X\to X$ be a linear power bounded operator on Banach space. Let $X_0$ is a subspace of vectors tending to zero under iterating of $T$. We prove that if $X_0$ is not equal to $X$ then there exists $λ$ in Sp(T) such that, for every $ε>0$, there is $x$ such that $|Tx-λx|<ε$ but $|T^nx|>1-ε$ for all $n$. The technique we develop enables us to establish that if $X$ is reflexive and there exists a compactum $K$ in $X$ such that for every norm-one $x\in X$ $ρ\{T^nx, K\}<α(T)<1$ for some $n=n_1, n_2,...$ then $codim(X_0)<\infty$. The results hold also for a one-parameter semigroup.
math.FA↗