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