## Octav Olteanu: Invariant Subspaces and Invariant Balls of Bounded Linear Operators, p. 261-271

### Abstract:

Let be a complex Banach space, the algebra of all bounded operators acting on , the Fréchet space of all entire functions from the complex plane into , endowed with the topology of uniform convergence on the compact subsets in . Let be a subset of such that the product of any two elements from is still an element of . For , one denotes by the linear continuous operator from into defined by , , . One proves that there exist common invariant subspaces of all operators in . In particular, there exist common invariant subspaces for all operators in . One describes the elements of a dense subspace of such an invariant subspace. On the other side, a differential equation with operator - valued solution is discussed, in the Hilbert space setting. A related example is given, where an invariant subspace appears as a kernel of a differential operator. This subspace stands for a hyperinvariant subspace related to an arbitrary operator in . Finally, the invariance of the unit ball in some spaces in terms of polynomials is discussed. To this end, polynomial approximation on unbounded subsets is applied.

Key Words: invariant subspace problem, invariant balls, bounded operators, sandwich type theorems for operators, approximation

2010 Mathematics Subject Classification: Primary 47A15,
Secondary 41A10, 47A63