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Science and Education Publishing (SciEP) is a multidisciplinary open access publisher of Journals covering the fields of Science, Technology and Medicine. All SciEP publications are subject to high-quality peer review, editorial and production processes and are freely available online.

Welcome to the April, 2013 edition of SciEP newsletter, providing monthly summaries of articles on science & education related topics of interest to scientist and professionals. The mission of our journals is to provide an open access forum for publishing new findings on issues pertinent to scientific and academic field. We also invite you to submit original research papers, review articles, short reports, or case studies are considered for publication in our Multidisciplinary Journals.

American Journal of Public Health Research
DOI: 10.12691/ajphr-1-1-3

The Implementation of a Tele-Homecare System with Service Oriented Architecture and HL7 Message Transmission Standard
Ching-Sung Wang

American Journal of Medical and Biological Research
DOI: 10.12691/ajmbr-1-1-2

The Influence of Ethanol on Pyruvate Kinases Activity in Vivo, in Vitro, in Silico
Lelevich Sergey Vladimirovich, Khrustalev Vladislav Victorovich, Barkovsky Eugene Victorovich, Shedogubova Tatyana Aleksandrovna

American Journal of Medical Sciences and Medicine
DOI: 10.12691/ajmsm-1-1-1

Clinical and Histological Features in Wegener Granulomatosis
Li Cavoli Gioacchino, Passantino Rita, Bono Luisa, Tortorici Calogera, Ferrantelli Angelo, Giammarresi Carlo, Rotolo Ugo

American Journal of Infectious Diseases and Microbiology
DOI: 10.12691/ajidm-1-1-2

The Curse of Lymphatic Filariasis: Would the Continual Use of Diethylcarbamazine Eliminate this Scourge in Papua New Guinea?
Anne Lanham, Lillian Mwanri

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A Short Proof of Von Neumann’s Conjecture

Bahman Mashood

66 Shipely Avenue, Daly City, CA 94015, US

Abstract: 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.

Keywords: Hilbert space, bounded operator, invariant subspace, polynomial, orthogonal basis

Received September 22, 2022; Revised October 27, 2022; Accepted November 08, 2022

Citation: Bahman Mashood, A Short Proof of Von Neumann’s Conjecture. American Journal of Mathematical Analysis, 10 (1). 1-2. January 2022. doi:10.12691/ajma-10-1-1

1. Main Arguments

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.

References

[1]   | R. Abraham, J.E. Mardsen, T.Ratiu. “Manifolds, Tensor Analysis and applications”.
[2]   | Jonathan Noel, “The invariant subspace problem”, Thesis, Department of Mathematics and Statistic, Thompson River University, 2011.