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Ryan O'Loughlin

Publications and source records attributed to Ryan O'Loughlin.

At least 19 recordsLinked to original sources
arXiv2026

On the abstract approach to spectral constants: a proof of the Clouâtre--Ostermann--Ransford conjecture

Clouâtre, Ostermann, and Ransford formulated an abstract version of Crouzeix's conjecture involving a bounded unital homomorphism from a uniform algebra into matrices and a unital antilinear map. They conjectured that contractivity of the associated symmetrised map forces the homomorphism to have norm at most two. We prove this conjecture, in fact without requiring the antilinear map to be contractive and for homomorphisms into the bounded operators on a Hilbert space. The proof combines positivity of real parts, an operator-valued Herglotz theorem, and the perturbation lemma of Lorist and Schwenninger used in the recent proof of Crouzeix's conjecture.

math.FA↗
arXiv2026

Semialgebraic Dimension and Truncated Toeplitz Models for Complex Symmetric Matrices

We answer negatively a finite-dimensional unitary-model question for complex symmetric operators. More precisely, we show that, for every \(n\geq 10\), not every \(n\times n\) symmetric matrix is unitarily equivalent to a direct sum of truncated Toeplitz operators. In order to do this, we first use semialgebraic dimension, a tool from real algebraic geometry, to prove a general theorem showing that, if \(\mathcal X\) is a semialgebraic family of complex symmetric matrices, then the set of complex symmetric matrices which are unitarily equivalent to an element of \(\mathcal X\) is semialgebraic and has dimension at most $\dim_{\mathbb R}\mathcal X+\frac{n(n-1)}2.$ We then apply this theorem to show that when $n\geq 10$ there exist irreducible symmetric $n \times n$ matrices which are not unitarily equivalent to a truncated Toeplitz operator. Although unitary equivalence is too restrictive, we prove that every finite-dimensional complex symmetric operator is complex-orthogonally equivalent to a coanalytic truncated Toeplitz operator. We also answer positively a refined representation question by showing that whenever a symmetric matrix is unitarily equivalent to a truncated Toeplitz operator, it is a matrix representation of that operator with respect to a conjugation-invariant orthonormal basis.

math.FA↗
arXiv2026

A Counterexample to the Gau--Wang--Wu Conjecture on Partial Isometries

We disprove the conjecture of Gau, Wang and Wu that the numerical range of a finite-dimensional partial isometry, when circular, must be centred at the origin. More precisely, we prove that there is an \(\varepsilon>0\) such that, for every \(a\in(0,\varepsilon)\), one can find a rank-four partial isometry \(V_a\in\M_6(\R)\) satisfying $$ W(V_a)=\{ζ\in\C:|ζ-a|\leq 3/4\}. $$

math.FA↗
arXiv2026

Sharp spectral constants for scaled $q$-numerical ranges

For $ n \geq 2$, $A\in M_n(\mathbb C)$ and $0<|q|\leq 1$, let $Ω_q(A)=q^{-1}W_q(A)$ be the scaled $q$-numerical range. We prove that for every $γ\geq 1$, \[ Ω_{η(γ)}(A) =\bigcup_{κ(S)\leqγ}W(S^{-1}AS), \qquad η(γ)=\frac{2}{γ+γ^{-1}}, \] where $κ(S)=\|S\|\,\|S^{-1}\|$. As a consequence, we prove the sharp inequality \[ \|p(A)\|\leq \max\!\left\{1,\frac{2|q|}{1+\sqrt{1-|q|^2}}\right\} \max_{z\inΩ_q(A)}|p(z)|, \] for all polynomials $p$.

math.FA↗
arXiv2026

Describing the Numerical Range and $C$-Numerical Range of Matrices via Their Unitary Orbits and the Joukowsky Transform

We generalise the Elliptical Range Theorem to characterise the numerical range of matrices belonging to a subspace of the space of \(3 \times 3\) matrices. Using Specht's Theorem, which characterizes when two matrices are unitarily equivalent, we then provide a novel proof of the Elliptical Range Theorem. Finally, we give an explicit description of the $C$-numerical range for $2 \times 2$ matrices and for rank-one matrices.

math.FA↗
arXiv2026

A Schwarz-Jack lemma, circularly symmetric domains and numerical ranges

We prove a Schwarz-Jack lemma for holomorphic functions on the unit disk with the property that their maximum modulus on each circle about the origin is attained at a point on the positive real axis. With the help of this result, we establish monotonicity and convexity properties of conformal maps of circularly symmetric and bi-circularly symmetric domains. As an application, we give a new proof of Crouzeix's theorem that the numerical range of any $2\times 2$ matrix is a $2$-spectral set for the matrix. Unlike other proofs, our approach does not depend on the explicit formula for the conformal mapping of an ellipse onto the unit disk.

math.CV↗
arXiv2026

$q$-Numerical Ranges and Spectral Sets

We study spectral constants for convex domains $Ω$ containing the spectrum of an operator. We extend the Crouzeix--Palencia framework by obtaining bounds depending on a parameter $γ$ and relating these bounds to geometric properties of $Ω$ and the numerical range $W(A)$. We generalise the proof that the numerical range is a $1+\sqrt{2}$-spectral set to scaled $q$-numerical ranges. We also propose a generalisation of Crouzeix's Conjecture in the context of $q$-numerical ranges.

math.FA↗
arXiv2025

Double-layer potentials, configuration constants and applications to numerical ranges

Given a compact convex planar domain $Ω$ with non-empty interior, the classical Neumann's configuration constant $c_{\mathbb{R}}(Ω)$ is the norm of the Neumann-Poincaré operator $K_Ω$ acting on the space of continuous real-valued functions on the boundary $\partial Ω$, modulo constants. We investigate the related operator norm $c_{\mathbb{C}}(Ω)$ of $K_Ω$ on the corresponding space of complex-valued functions, and the norm $a(Ω)$ on the subspace of analytic functions. This change requires introduction of techniques much different from the ones used in the classical setting. We prove the equality $c_{\mathbb{R}}(Ω) = c_{\mathbb{C}}(Ω)$, the analytic Neumann-type inequality $a(Ω) < 1$, and provide various estimates for these quantities expressed in terms of the geometry of $Ω$. We apply our results to estimates for the holomorphic functional calculus of operators on Hilbert space of the type $\|p(T)\| \leq K \sup_{z \in Ω} |p(z)|$, where $p$ is a polynomial and $Ω$ is a domain containing the numerical range of the operator $T$. Among other results, we show that the well-known Crouzeix-Palencia bound $K \leq 1 + \sqrt{2}$ can be improved to $K \leq 1 + \sqrt{1 + a(Ω)}$. In the case that $Ω$ is an ellipse, this leads to an estimate of $K$ in terms of the eccentricity of the ellipse.

math.FA↗
arXiv2025

Bounded Toeplitz Products on the Hardy Space

A Toeplitz operator on the Hardy space of the unit circle is bounded if and only if its symbol is bounded. For two Toeplitz operators, there are no known function-theoretic conditions for their symbols, which are equivalent to the product of the Toeplitz operators being bounded. In this paper, we provide a solution to this problem, by showing under certain assumptions that the product of two Toeplitz operators is bounded if and only if the product of their symbols is bounded.

math.FA↗
arXiv2024

Relating Superconducting Optoelectronic Networks to Classical Neurodynamics

The circuits comprising superconducting optoelectronic synapses, dendrites, and neurons are described by numerically cumbersome and formally opaque coupled differential equations. Reference 1 showed that a phenomenological model of superconducting loop neurons eliminates the need to solve the Josephson circuit equations that describe synapses and dendrites. The initial goal of the model was to decrease the time required for simulations, yet an additional benefit of the model was increased transparency of the underlying neural circuit operations and conceptual clarity regarding the connection of loop neurons to other physical systems. Whereas the original model simplified the treatment of the Josephson-junction dynamics, essentially by only considering low-pass versions of the dendritic outputs, the model resorted to an awkward treatment of spikes generated by semiconductor transmitter circuits that required explicitly checking for threshold crossings and distinct treatment of time steps wherein somatic threshold is reached. Here we extend that model to simplify the treatment of spikes coming from somas, again making use of the fact that in neural systems the downstream recipients of spike events almost always perform low-pass filtering. We provide comparisons between the first and second phenomenological models, quantifying the accuracy of the additional approximations. We identify regions of circuit parameter space in which the extended model works well and regions where it works poorly. For some circuit parameters it is possible to represent the downstream dendritic response to a single spike as well as coincidences or sequences of spikes, indicating the model is not simply a reduction to rate coding. The governing equations are shown to be nearly identical to those ubiquitous in the neuroscience literature for modeling leaky-integrator dendrites and neurons.

cs.NE↗
arXiv2024

On the Crouzeix ratio for $N\times N$ matrices

The Crouzeix ratio $ψ(A)$ of an $N\times N$ complex matrix $A$ is the supremum of $\|p(A)\|$ taken over all polynomials $p$ such that $|p|\le 1$ on the numerical range of $A$. It is known that $ψ(A)\le 1+\sqrt{2}$, and it is conjectured that $ψ(A)\le 2$. In this note, we show that $ψ(A)\le C_N$, where $C_N$ is a constant depending only on $N$ and satisfying $C_N<1+\sqrt{2}$. The proof is based on a study of the continuity properties of the map $A\mapsto ψ(A)$.

math.FA↗
arXiv2024

Szegő Limit Theorem for Truncated Toeplitz Operators

We discuss generalizations of the Szegő Limit Theorem to truncated Toeplitz operators. In particular, we consider compressions of Toeplitz operators to an increasing sequence of finite dimensional model spaces. We present two theorems. The first is a new variant of the Szegő Limit Theorem in this setting. The second relates to the variant given by Strouse-Timotin-Zarrabi in 2017, and characterizes the sequences for which that result holds.

math.FA↗
arXiv2023

Crouzeix's conjecture for classes of matrices

For a matrix $A$ which satisfies Crouzeix's conjecture, we construct several classes of matrices from $A$ for which the conjecture will also hold. We discover a new link between cyclicity and Crouzeix's conjecture, which shows that Crouzeix's Conjecture holds in full generality if and only if it holds for the differentiation operator on a class of analytic functions. We pose several open questions, which if proved, will prove Crouzeix's conjecture. We also begin an investigation into Crouzeix's conjecture for symmetric matrices and in the case of $3 \times 3$ matrices, we show Crouzeix's conjecture holds for symmetric matrices if and only if it holds for analytic truncated Toeplitz operators.

math.FA↗
arXiv2023

Symmetric matrix representations of truncated Toeplitz operators on finite dimensional spaces

In this paper, we study matrix representations of truncated Toeplitz operators with respect to orthonormal bases which are invariant under a canonical conjugation map. In particular, we determine necessary and sufficient conditions for when a 3-by-3 symmetric matrix is the matrix representation of a truncated Toeplitz operator with respect to a given conjugation invariant orthonormal basis. We specialise our result to the case when the conjugation invariant orthonormal basis is a modified Clark basis. As a corollary to this specialisation, we answer a previously stated open conjecture in the negative, and show that not every unitary equivalence between a complex symmetric matrix and a truncated Toeplitz operator arises from a modified Clark basis representation. Finally, we show that a given 3-by-3 symmetric matrix is the matrix representation of a truncated Toeplitz operator with respect to a conjugation invariant orthonormal basis if and only if a specified system of polynomial equations is satisfied with a real solution.

math.FA↗
arXiv2022

Matrix-valued truncated Toeplitz operators: unbounded symbols, kernels and equivalence after extension

This paper studies matrix-valued truncated Toeplitz operators, which are a vectorial generalisation of truncated Toeplitz operators. It is demonstrated that, although there exist matrix-valued truncated Toeplitz operators without a matrix symbol in $L^p$ for any $p \in (2, \infty ]$, there is a wide class of matrix-valued truncated Toeplitz operators which possess a matrix symbol in $L^p$ for some $p \in (2, \infty ]$. In the case when the matrix-valued truncated Toeplitz operator has a symbol in $L^p$ for some $p \in (2, \infty ]$, an approach is developed which bypasses some of the technical difficulties which arise when dealing with problems concerning matrix-valued truncated Toeplitz operators with unbounded symbols. Using this new approach, two new notable results are obtained. The kernel of the matrix-valued truncated Toeplitz operator is expressed as an isometric image of an $S^*$-invariant subspace. Also, a Toeplitz operator is constructed which is equivalent after extension to the matrix-valued truncated Toeplitz operator. In a different yet overlapping vein, it is also shown that multidimensional analogues of the truncated Wiener-Hopf operators are unitarily equivalent to certain matrix-valued truncated Toeplitz operators.

math.FA↗
arXiv2021

Symbols of compact truncated Toeplitz operators

This paper characterises the dual of the model space $K_I^1$, where $I$ is an inner function, intersected with the shifted Hardy space, $z H^1$. With this duality result, it is then shown that every bounded truncated Toeplitz operator on the model space $K_I^2$ has a bounded symbol if and only if every compact truncated Toeplitz operator on $K_I^2$ has a symbol which is of the form $I$ multiplied by a continuous function.

math.FA↗
arXiv2021

Dual-band general Toeplitz operators

We relate dual-band general Toeplitz operators to block truncated Toeplitz operators and, via equivalence after extension, with Toeplitz operators with $4 \times 4$ matrix symbols. We discuss their norm, their kernel, Fredhomlness, invertibility and spectral properties in various situations, focusing on the spectral properties of the dual-band shift, which turns out to be considerably complex, leading to new and nontrivial connections with the boundary behaviour of the associated inner function.

math.FA↗