arXiv subjects

Bamdad R. Yahaghi

Publications and source records attributed to Bamdad R. Yahaghi.

11 recordsLinked to original sources

arXiv2026

Fully non-zero matrices and generators of full algebras of operators

In a wide variety of fields, every non-scalar matrix $M$ is similar to a matrix all of whose entries are non-zero. We give extensions of this result, including some in the division ring context. We then apply these results to the question of which pairs of operators generate the algebra of all operators on finite- and infinite-dimensional spaces. In the case of complex, separable infinite-dimensional Hilbert space, this is intimately related to the unsolved Invariant Subspace Problem. Finally, linear independence of subsets of arbitrary TVS's and generators of arbitrary TA's are shown to be stable in the sense of the theorems presented.

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arXiv2025

Spectrum in alternative topological algebras and a new look at old theorems

In this paper, we consider real and complex algebras as well as algebras over general fields. In Section 2, we revisit and prove several results on (quadratic) algebras over general fields. As an example, we demonstrate that a quadratic algebra over a field of characteristic not $2$ is flexible if and only if it is proper--a concept introduced in this paper. In Section 3, we show how to develop the spectral theory in the context of complex (resp. real) one-sided alternative topological algebras. As an application of the existence of spectrum, we prove the existence of nontrivial hyperinvariant linear manifolds for nonscalar (resp. nonquadratic) continuous linear operators acting on complex (resp. real) Fréchet spaces. Along the way, spectral theory is used to prove several topological counterparts of the well-known theorems of Frobenius, Hurwitz, Gelfand-Mazur, and Zorn. This is done, for example, in the context of left (resp. right) alternative topological algebras whose duals separate their elements. In Section 4, we consider real and complex algebras in various topological settings and reconsider and prove several results. For instance, it is shown that given a $ 1 < k \in \mathbb{N}$, on any locally complex algebra, there exists a unique nonzero vector space norm, say, $\|.\|$, satisfying the identity $\|a^k\| = \|a\|^k$ on the algebra. In Section 5, among other things, we revisit and slightly strengthen the celebrated theorems of Frobenius, Zorn, Gelfand-Mazur, and Hurwitz, and also give slight extensions of their topological counterparts, e.g., theorems of Albert, Kaplansky, and Urbanik-Wright to name a few, in several settings.

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arXiv2024

Extensions of the fundamental theorem of algebra

In this paper motivated by the celebrated fundamental theorem of algebra and its standard proof utilizing Liouville's Theorem, we prove the fundamental theorem of algebra type results for both commutative and noncommutative polynomials in the setting of left (resp. right) alternative topological complex algebras whose topological duals separates their elements and that of such real algebras whose centers contain certain copies of complex numbers. An application of one of the main results of the paper is the existence of eigenvalues for matrices with entries from arbitrary finite-dimensional complex algebras. We also prove the existence of right eigenvalues for matrices with entries from finite-dimensional associative real algebras that contain copies of the complex numbers.

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arXiv2022

On the existence of left and right eigenvalues

In this note, we consider arbitrary finite-dimensional real algebras containing a copy of complex numbers. It is proved that matrices with entries from an arbitrary finite-dimensional real algebra containing a square root of negative one in its left (resp. right) associate set have left (resp. right) eigenvalues. A quick consequence of our main result is the existence of left and right eigenvalues for matrices with entries from finite-dimensional alternatives algebras containing a copy of complex numbers, e.g., octonions, and more generally matrices with entries from the real Cayley-Dickson algebras.

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arXiv2018

Burnside's theorem in the setting of general fields

We extend a well-known theorem of Burnside in the setting of general fields as follows: for a general field $F$ the matrix algebra $M_n(F)$ is the only algebra in $M_n(F)$ which is spanned by an irreducible semigroup of triangularizable matrices. In other words, for a semigroup of triangularizable matrices with entries from a general field irreducibility is equivalent to absolute irreducibility. As a consequence of our result we prove a stronger version of a theorem of Janez Bernik.

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arXiv2018

Burnside type theorems in real and quaternion settings

In this note, we consider irreducible semigroups of real, complex, and quaternionic matrices with real spectra. We prove Burnside type theorems in the settings of reals and quaternions. First, we prove that an irreducible semigroup of triangularizable matrices in $M_n(\mathbb{R})$ contains a vector space basis for $M_n(\mathbb{R})$. In other words, $M_n(\mathbb{R})$ is the only irreducible subalgebra of itself that is spanned by an irreducible semigroup of triangularizable matrices in $M_n(\mathbb{R})$. Next, we use this result to show that, up to similarity, $M_n(\mathbb{R})$ is the only irreducible $\mathbb{R}$-algebra in $M_n(\mathbb{H})$ that is spanned by an irreducible semigroup of matrices in $M_n(\mathbb{H})$ with real spectra. Some consequences of our mains results are presented.

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arXiv2017

Theorems of Burnside and Wedderburn revisited

We approach celebrated theorems of Burnside and Wedderburn via simultaneous triangularization. First, for a general field $F$, we prove that $M_n(F)$ is the only irrreducible subalgebra of triangularizable matrices in $M_n(F)$ provided such a subalgebra exists. This provides a slight generalization of a well-known theorem of Burnside. Next, for a given $n > 1$, we characterize all fields $F$ such that Burnside's Theorem holds in $M_n(F)$, i.e., $M_n(F)$ is the only irreducible subalgebra of itself. In fact, for a subfield $F$ of the center of a division ring $D$, our simple proof of the aforementioned extension of Burnside's Theorem can be adjusted to establish a Burnside type theorem for irreducible $F$-algebras of triangularizable matrices in $M_n(D)$ with inner eigenvalues in $F$, namely such subalgebras of $M_n(D)$ are similar to $M_n(F)$. We use Burnside's theorem to present a simple proof of a theorem due to Wedderburn. Then, we use our Burnside type theorem to prove an extension of Wedderburn's Theorem as follows: A subalgebra of a semi-simple left Artinian $F$-algebra is nilpotent iff the algebra, as a vector space over the field $F$, is spanned by its nilpotent members and that the minimal polynomials of all of its members split into linear factors over $F$. We conclude with an application of Wedderburn's Theorem.

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arXiv2016

An extension of a theorem of Kaplansky

A theorem of Kaplansky asserts that a semigroup of matrices with entries from a field whose members all have singleton spectra is triangularizable. Indeed, Kaplansky's Theorem unifies well-known theorems of Kolchin and Levitzki on simultaneous triangularizability of semigroups of unipotent and nilpotent matrices, respectively. First, we present a new and simple proof of Kaplansky's Theorem over fields of characteristic zero. Next, we show that this proof can be adjusted to show that the counterpart of Kolchin's Theorem over division rings of characteristic zero implies that of Kaplansky's Theorem over such division rings. Also, we give a generalization of Kaplansky's Theorem over general fields. We show that this extension of Kaplansky's Theorem holds over a division ring $Δ$ provided the counterpart of Kaplansky's Theorem holds over $Δ$.

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arXiv2015

A theorem of Kaplansky revisited

We present a new and simple proof of a theorem due to Kaplansky which unifies theorems of Kolchin and Levitzki on triangularizability of semigroups of matrices. We also give two different extensions of the theorem. As a consequence, we prove the counterpart of Kolchin's Theorem for finite groups of unipotent matrices over division rings. We also show that the counterpart of Kolchin's Theorem over division rings of characteristic zero implies that of Kaplansky's Theorem over such division rings.

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arXiv2015

On modules of linear transformations

Let $D$ be a division ring, $\mathcal V$ and $ \mathcal W$ vector spaces over $D$, and ${\mathcal L(\mathcal V,\mathcal W)}$ the ${\mathcal L(\mathcal W)}$-${\mathcal L(\mathcal V)}$ bimodule of all linear transformations from $\mathcal V$ into $\mathcal W$. We prove some basic results about certain submodules of $\mathcal L(\mathcal V, \mathcal W)$. For instance, we show, among other results, that a right submodule (resp. left submodule) of ${\mathcal L(\mathcal V,\mathcal W)}$ is finitely generated whenever its image (resp. coimage) is finite-dimensional.

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arXiv2015

On nest modules of matrices over division rings

Let $ m , n \in \mathbb{N}$, $D$ be a division ring, and $M_{m \times n}(D)$ denote the bimodule of all $m \times n$ matrices with entries from $D$. First, we characterize one-sided submodules of $M_{m \times n}(D)$ in terms of left row reduced echelon or right column reduced echelon matrices with entries from $D$. Next, we introduce the notion of a nest module of matrices with entries from $D$. We then characterize submodules of nest modules of matrices over $D$ in terms of certain finite sequences of left row reduced echelon or right column reduced echelon matrices with entries from $D$. We use this result to characterize principal submodules of nest modules. We also describe subbimodules of nest modules of matrices. As a consequence, we characterize (one-sided) ideals of nest algebras of matrices over division rings.

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