arXiv subjects

Laurent W. Marcoux

Publications and source records attributed to Laurent W. Marcoux.

14 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

On products of symmetries acting on Hilbert spaces

Let $\mathcal{H}$ be a complex, separable Hilbert space (of finite or infinite dimension), and let $\mathcal{U}(\mathcal{H})$ denote the group of unitary operators on $\mathcal{H}$. A symmetry is, by definition, a unitary operator $J$ with $J^2 =I$. Denote by $\text{Sym}_k(\mathcal{H})$ the subset of $\mathcal{U}(\mathcal{H})$ consisting of those operators expressible as a product of $k$ symmetries. It is known that $\mathcal{U}(\mathcal{H}) = \text{Sym}_4(\mathcal{H})$ if $\dim \, \mathcal{H} = \infty$, while the only additional condition in finite dimensions is that the determinant be $\pm 1$. Of all the sets $\text{Sym}_k(\mathcal{H})$ with $k \in \{ 1, 2, 3, 4\}$, the case $k =3$ has been the most stubborn to characterise. Among other things, we investigate which elements of $\text{Sym}_3(\mathcal{H})$ possess exactly two eigenvalues in the setting where $\mathcal{H}$ is finite-dimensional. We also consider the problem: when is the unitary orbit of an operator $T$, i.e., the set \[ \{ U^* T U : U \in \mathcal{U}(\mathcal{H}) \} \] the same as its $\text{Sym}_k$-orbit, i.e., the set \[ \{ U^* T U: U \in \text{Sym}_k(\mathcal{H})\} ? \] Clearly, the cases of interest are when $k \le 3$.

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arXiv2025

Kaplansky's problem and unitary orbits in matrix amplifications

We study the distances between the unitary orbits of matrix amplifications of elements in certain C*-algebras. In particular, we show that the distance between unitary orbits of arbitrary elements in unital, separable, UHF-stable C*-algebras remains unchanged when amplifying to certain matrix sizes. We further exhibit examples of elements in C*-algebras where the distance between unitary orbits becomes strictly smaller after amplifying by a certain matrix size, and we demonstrate that distances between unitary orbits of amplifications are not monotone in the multiplicity of the amplifications, even in the setting of matrix algebras. Lastly, we show that topological K-theory provides obstructions in the purely infinite setting.

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arXiv2024

On commutators of square-zero Hilbert space operators

Let $\mathcal{H}$ be a complex, separable Hilbert space, and set $\mathfrak{c}($NIL$_2)=\{ MN - NM : N, M \in \mathcal{B}(\mathcal{H}), M^2 = 0 = N^2 \}$. When $\dim\, \mathcal{H}$ is finite, we characterise the set $\mathfrak{c}($NIL$_2)$ and its norm-closure CLOS$(\mathfrak{c}($NIL$_2))$. In the infinite-dimensional setting, we characterise the intersection of CLOS$(\mathfrak{c}($NIL$_2))$ with the set of biquasitriangular operators, and we exhibit an index obstruction to belonging to CLOS$(\mathfrak{c}($NIL$_2))$.

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arXiv2023

Stability relations for Hilbert space operators and a problem of Kaplansky

In his monograph on Infinite Abelian Groups, I. Kaplansky raised three ``test problems" concerning their structure and multiplicity. As noted by Azoff, these problems make sense for any category admitting a direct sum operation. Here, we are interested in the operator theoretic version of Kaplansky's second problem which asks: if $A$ and $B$ are operators on an infinite-dimensional, separable Hilbert space and $A \oplus A$ is equivalent to $B \oplus B$ in some (precise) sense, is $A$ equivalent to $B$? We examine this problem under a strengthening of the hypothesis, where a ``primitive" square root $J_2(A)$ of $A\oplus A$ is assumed to be equivalent to the corresponding square root $J_2(B)$ of $B \oplus B$. When ``equivalence" refers to similarity of operators and $A$ is a compact operator, we deduce from this stronger hypothesis that $A$ and $B$ are similar. We exhibit a counterexample (due to J. Bell) of this phenomenon in the setting of unital rings. Also, we exhibit an uncountable family $\{ U_α\}_{α\in Ω}$ of unitary operators, no two of which are unitarily equivalent, such that each $U_α$ is unitarily equivalent to $J_n(U_α)$, a ``primitive" $n^{th}$ root of $U_α\oplus U_α\oplus \cdots \oplus U_α$.

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arXiv2022

Around the closures of the set of commutators and the set of differences of idempotent elements of $\mathcal{B}(\mathcal{H})$

We describe the norm-closures of the set $\mathfrak{C}_{\mathfrak{E}}$ of commutators of idempotent operators and the set $\mathfrak{E} - \mathfrak{E}$ of differences of idempotent operators acting on a finite-dimensional complex Hilbert space, as well as characterising the intersection of the closures of these sets with the set $\mathcal{K}(\mathcal{H})$ of compact operators acting on an infinite-dimensional, separable Hilbert space. Finally, we characterise the closures of the set $\mathfrak{C}_{\mathfrak{P}}$ of commutators of orthogonal projections and the set $\mathfrak{P} - \mathfrak{P}$ of differences of orthogonal projections acting on an arbitrary complex Hilbert space.

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arXiv2021

Matrix Algebras with a Certain Compression Property I

An algebra $\mathcal{A}$ of $n\times n$ complex matrices is said to be \textit{idempotent compressible} if $E\mathcal{A}E$ is an algebra for all idempotents $E\in\mathbb{M}_n(\mathbb{C})$. Analogously, $\mathcal{A}$ is said to be \textit{projection compressible} if $P\mathcal{A}P$ is an algebra for all orthogonal projections $P$ in $\mathbb{M}_n(\mathbb{C})$. In this paper we construct several examples of unital algebras that admit these properties. In addition, a complete classification of the unital idempotent compressible subalgebras of $\mathbb{M}_3(\mathbb{C})$ is obtained up to similarity and transposition. It is shown that in this setting, the two notions of compressibility agree: a unital subalgebra of $\mathbb{M}_3(\mathbb{C})$ is projection compressible if and only if it is idempotent compressible. Our findings are extended to algebras of arbitrary size in the sequel to this paper.

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arXiv2019

Operators which are polynomially isometric to a normal operator

Let $\mathcal{H}$ be a complex, separable Hilbert space and $\mathcal{B}(\mathcal{H})$ denote the algebra of all bounded linear operators acting on $\mathcal{H}$. Given a unitarily-invariant norm $\| \cdot \|_u$ on $\mathcal{B}(\mathcal{H})$ and two linear operators $A$ and $B$ in $\mathcal{B}(\mathcal{H})$, we shall say that $A$ and $B$ are \emph{polynomially isometric relative to} $\| \cdot \|_u$ if $\| p(A) \|_u = \| p(B) \|_u$ for all polynomials $p$. In this paper, we examine to what extent an operator $A$ being polynomially isometric to a normal operator $N$ implies that $A$ is itself normal. More explicitly, we first show that if $\| \cdot \|_u$ is any unitarily-invariant norm on $\mathbb{M}_n(\mathbb{C})$, if $A, N \in \mathbb{M}_n(\mathbb{C})$ are polynomially isometric and $N$ is normal, then $A$ is normal. We then extend this result to the infinite-dimensional setting by showing that if $A, N \in \mathcal{B}(\mathcal{H})$ are polynomially isometric relative to the operator norm and $N$ is a normal operator whose spectrum neither disconnects the plane nor has interior, then $A$ is normal, while if the spectrum of $N$ is not of this form, then there always exists a non-normal operator $B$ such that $B$ and $N$ are polynomially isometric. Finally, we show that if $A$ and $N$ are compact operators with $N$ normal, and if $A$ and $N$ are polynomially isometric with respect to the $(c,p)$-norm studied by Chan, Li and Tu, then $A$ is again normal.

math.FA↗

arXiv2019

Normal operators with highly incompatible off-diagonal corners

Let $\mathcal{H}$ be a complex, separable Hilbert space, and $\mathcal{B}(\mathcal{H})$ denote the set of all bounded linear operators on $\mathcal{H}$. Given an orthogonal projection $P \in \mathcal{B}(\mathcal{H})$ and an operator $D \in \mathcal{B}(\mathcal{H})$, we may write $D=\begin{bmatrix} D_1& D_2 D_3 & D_4 \end{bmatrix}$ relative to the decomposition $\mathcal{H} = \mathrm{ran}\, P \oplus \mathrm{ran}\, (I-P)$. In this paper we study the question: for which non-negative integers $j, k$ can we find a normal operator $D$ and an orthogonal projection $P$ such that $\mathrm{rank}\, D_2 = j$ and $\mathrm{rank}\, D_3 = k$? Complete results are obtained in the case where $\mathrm{dim}\, \mathcal{H} < \infty$, and partial results are obtained in the infinite-dimensional setting.

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arXiv2017

Residual finite dimensionality and representations of amenable operator algebras

We consider a version of a famous open problem formulated by Kadison, asking whether bounded representations of operator algebras are automatically completely bounded. We investigate this question in the context of amenable operator algebras, and we provide an affirmative answer for representations whose range is residually finite-dimensional. Furthermore, we show that weak-${}^*$ closed, amenable, residually finite-dimensional operator algebras are similar to $C^*$-algebras, and in particular have the property that all their bounded representations are completely bounded. We prove our results for operator algebras having the so-called total reduction property, which is known to be weaker than amenability.

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arXiv2016

Compact ideals and rigidity of representations for amenable operator algebras

We examine rigidity phenomena for representations of amenable operator algebras which have an ideal of compact operators. We establish that a generalized version of Kadison's conjecture on completely bounded homomorphisms holds for the algebra if the associated quotient is abelian. We also prove that injective completely bounded representations of the algebra are similar to complete isometries. The main motivating example for these investigations is the recent construction of Choi, Farah and Ozawa of an amenable operator algebra that is not similar to a $C^*$-algebra, and we show how it fits into our framework. All of our results hold in the presence of the total reduction property, a property weaker than amenability.

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arXiv2014

On selfadjoint extensions of semigroups of partial isometries

Let $\mathcal S$ be a semigroup of partial isometries acting on a complex, infinite-dimensional, separable Hilbert space. In this paper we seek criteria which will guarantee that the selfadjoint semigroup $\mathcal T$ generated by $\mathcal S$ consists of partial isometries as well. Amongst other things, we show that this is the case when the set of final projections of elements of $\mathcal S$ generates an abelian von Neumann algebra of uniform finite multiplicity.

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arXiv2013

Abelian, amenable operator algebras are similar to C*-algebras

Suppose that H is a complex Hilbert space and that B(H) denotes the bounded linear operators on H. We show that every abelian, amenable operator algebra is similar to a C*-algebra. We do this by showing that if A is an abelian subalgebra of B(H) with the property that given any bounded representation $\varrho: A \to B(H_\varrho)$ of A on a Hilbert space $H_\varrho$, every invariant subspace of $\varrho(A)$ is topologically complemented by another invariant subspace of $\varrho(A)$, then A is similar to an abelian $C^*$-algebra.

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arXiv2012

On Almost-Invariant Subspaces and Approximate Commutation

A closed subspace of a Banach space $\cX$ is almost-invariant for a collection $\cS$ of bounded linear operators on $\cX$ if for each $T \in \cS$ there exists a finite-dimensional subspace $\cF_T$ of $\cX$ such that $T \cY \subseteq \cY + \cF_T$. In this paper, we study the existence of almost-invariant subspaces of infinite dimension and codimension for various classes of Banach and Hilbert space operators. We also examine the structure of operators which admit a maximal commuting family of almost-invariant subspaces. In particular, we prove that if $T$ is an operator on a separable Hilbert space and if $TP-PT$ has finite rank for all projections $P$ in a given maximal abelian self-adjoint algebra $\fM$ then $T=M+F$ where $M\in\fM$ and $F$ is of finite rank.

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