Let H be a separable Hilbert space and let B(H) be the set of all bounded operators acting on H. Given T∈ B(H), we show that T has a proper invariant subspace, i.e., there exists a proper Hilbert subspace L⊂H such that T(L)⊆L. This problem has only been solved for special cases so far and in this article we try to offer a solution that can take care of the most general cases.
The preliminary definitions and notations can be found in the thesis of Jonathan Noel “The invariant subspace problem”, Department of Mathematics and Statistic, Thompson River University, 2011.
Given T as in the above, without loss generality, we can assume that 
is one to one and onto, otherwise T will have an proper invariant subspace. So if 
is any vector in 
then the subspace 
which is spanned by the elements of the form 
will be equal to 
, because 
is invariant under the action of 
. Thus any element 
can be written as a norm convergent sum, 
, where 
is a complex number. Before we propose the general solution, we consider the following special case. Suppose we have 
, defined by 
with 
Lemma 1.1. If 
for some vector 
, then 
has a proper invariant subspace.
Proof. Let 
be the roots of the polynomial 
Hence we have 
Then at least one of the factors, say 
, is not invertible. This implies that either 
has a proper invariant subspace, or there exits an integer 
with 
which implies the existence of a one dimensional invariant subspace for 
The case of 
, with 
, is much more complex and Lemma 1 is not working for this case. At this point we assume that 
, and 
. Next, let us pick an element 
, with 
. Now it is well known that a subset 
of 
is compact if every sequence in 
has a bounded subsequence with its limits included in 
. Next, consider the following set, 
Subsequently, define the following set 
to be the subset of the closure of convex hall of 
consisting of all elements of the form 
, with 
.
It is easy using the above comment to check that 
is a convex and compact subset of 
Hence by the Brower fixed point theorem there exists an element 
in 
for which 
If 
is different from zero then we have a non trivial invariant subspace and we are done. Otherwise, we have to consider the case where 
This means we have 
But assuming that 
does not have a non trivial invariant subspace implies that for every 
where, by the above, each 
and 
.
Next, let us denote 
. Then continue the above argument for each 
If for any integer 
the subspace generated by the set 
, will not span 
, then this implies that 
has a non trivial invariant subspace, otherwise the Brower fixed point acting over the compact convex hall of the above set implies the existence of a fixed point under the action of 
, in which case we have the following infinite dimensional polynomial 
with 
.
On the other hand, if we have a nontrivial fixed point then the proof is complete. As we mentioned in the above if 
does not have nontrivial invariant subspace we get that the polynomial 
converges in norm topology to zero. But considering the fact that 
and 
therefore the sets 
have a following limit point 
with 
and this will implies contradiction since as 
tends to infinity this will force 
to be equal to zero. This proves the final theorem which is the main focus of this article.
Theorem 1.2. Every Bounded Operator acting on a separable Hilbert space has a proper invariant subspace.
| [1] | R. Abraham, J.E. Mardsen, T.Ratiu. “Manifolds, Tensor Analysis and applications”. | ||
| In article | |||
| [2] | Jonathan Noel, “The invariant subspace problem”, Thesis, Department of Mathematics and Statistic, Thompson River University, 2011. | ||
| In article | |||
Published with license by Science and Education Publishing, Copyright © 2022 Bahman Mashood
This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit
https://creativecommons.org/licenses/by/4.0/
| [1] | R. Abraham, J.E. Mardsen, T.Ratiu. “Manifolds, Tensor Analysis and applications”. | ||
| In article | |||
| [2] | Jonathan Noel, “The invariant subspace problem”, Thesis, Department of Mathematics and Statistic, Thompson River University, 2011. | ||
| In article | |||
