arXiv subjects

Jonathan R. Partington

Publications and source records attributed to Jonathan R. Partington.

At least 19 recordsLinked to original sources

arXiv2026

Equivalence after extension for general Toeplitz operators

This paper reviews the subject of equivalence after extension in the context of Toeplitz operators, truncated Toeplitz operators, and further generalizations including dual and multiband truncated Toeplitz operators. Paired operators are shown to play a key role here. Properties under investigation by these methods include the invertibility and Fredholm properties of generalized Toeplitz operators, with new results showing how the kernels of such operators can often be fully described given only limited information.

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arXiv2026

Frame constructions associated with operator orbits

This paper studies frames in Hilbert spaces generated by the orbits of (in)-finitely many vectors under a single operator, presenting new results on multiplication operators and operators composed of Jordan blocks, which generalizes existing results of Cabrelli, Molter, Paternostro and Philipp by means of techniques which deal with weighted interpolation, weighted composition operators, and Beurling--Lax theory related to shifts of infinity multiplicity. Likewise, we discuss Carleson frames and give counterexamples to a recent conjecture of Aldroubi, Cabrelli, Krishtal and Molter.

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arXiv2025

On the relation between distances and seminorms on Fr\'echet spaces, with application to isometries

A study is made of linear isometries on Fr\'echet spaces for which the metric is given in terms of a sequence of seminorms. This establishes sufficient conditions on the growth of the function that defines the metric in terms of the seminorms to ensure that a linear operator preserving the metric also preserves each of these seminorms. As an application, characterizations are given of the isometries on various spaces including those of holomorphic functions on complex domains and continuous functions on open sets, extending the Banach--Stone theorem to surjective and nonsurjective cases.

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arXiv2025

Hardy operators: In the footsteps of Brown, Halmos, and Shields

This paper discusses the two classical Hardy operators $\mathcal{H}_{1}$ on $L^2(0, 1)$ and $\mathcal{H}_{\infty}$ on $L^2(0, \infty)$ initially studied by Brown, Halmos and Shields. Particular emphasis is given to the construction of explicit cyclic and $*$-cyclic vectors in conjunction with a characterization of their invariant and reducing subspaces. We also provide a complete description of the frame vectors for $I - \mathcal{H}_{1}^{*}$.

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arXiv2024

Linear isometries on the annulus: description and spectral properties

We give a complete characterisation of the linear isometries of ${\rm Hol}(\Omega)$, where $\Omega$ is the half-plane, the complex plane or an annulus centered at 0 and symmetric to the unit circle. Moreover, we introduce new techniques to describe the holomorphic maps on the annulus that preserve the unit circle, and we finish by proving results about the spectra of the linear isometries on the annulus.

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arXiv2024

Paired kernels and truncated Toeplitz operators

This paper considers paired operators in the context of the Lebesgue Hilbert space $L^2$ on the unit circle and its subspace, the Hardy space $H^2$. The kernels of such operators, together with their analytic projections, which are generalizations of Toeplitz kernels, are studied. Inclusion relations between such kernels are considered in detail, and the results are applied to describing the kernels of finite-rank asymmetric truncated Toeplitz operators.

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arXiv2024

Kernels of paired operators and their adjoints

We review the basic properties of paired operators and their adjoints, the transposed paired operators, with particular reference to commutation relations, and we study the properties of their kernels, bringing out their similarities and also, somewhat surprisingly, their stark differences. Various notions expressing different invariance properties are also reviewed and we extend to paired operators some known invariance results.

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arXiv2024

Invariance and near invariance for non-cyclic shift semigroups

We characterise the subspaces of $H^2(\mathbb D)$ that are invariant under the semigroups generated by two higher-order shifts $S^m $ and $S^{n}$. Complete descriptions are obtained for both invariant and nearly invariant subspaces associated with these operators and their adjoints. The approach is based on vector-valued Hardy spaces, the Beurling--Lax theorem, and matrix-valued inner functions. Finally we apply these results to non-cyclic shift semigroups and to Toeplitz operators induced by finite Blaschke products.

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arXiv2024

Rhaly operators: more on generalized Ces\`aro operators

Rhaly operators, as generalizations of the Ces\`aro operator, are studied from the standpoint of view of spectral theory and invariant subspaces, extending previous results by Rhaly and Leibowitz to a framework where generalized Ces\`aro operators arise naturally.

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arXiv2024

Composition operators between Toeplitz kernels

Recently, it was shown that the image of a Toeplitz kernel of dimension greater than $1$ under composition by an inner function is nearly $S^*$-invariant if and only if the inner function is an automorphism. Building on this, we determine the minimal Toeplitz kernel containing the image of a Toeplitz kernel under a composition operator with a general inner symbol, and extend this to weighted composition operators. Specifically, the corresponding cases for minimal model spaces are also given, thereby extending known work on the action of composition operators on model spaces. Finally, we use the equivalences between Toeplitz kernels to derive the explicit maximal vectors for several Toeplitz kernels, with symbols expressed in terms of composition operators and inner functions.

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arXiv2024

Cyclic nearly invariant subspaces for semigroups of isometries

In this paper, the structure of the nearly invariant subspaces for discrete semigroups generated by several (even infinitely many) automorphisms of the unit disc is described. As part of this work, the near $S^*$-invariance property of the image space $C_φ({\rm ker\, } T)$ is explored for composition operators $C_φ$, induced by inner functions $φ$, and Toeplitz operators $T$. After that, the analysis of nearly invariant subspaces for strongly continuous multiplication semigroups of isometries is developed with a study of cyclic subspaces generated by a single Hardy class function. These are characterised in terms of model spaces in all cases when the outer factor is a product of an invertible function and a rational (not necessarily invertible) function. Techniques used include the theory of Toeplitz kernels and reproducing kernels.

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arXiv2024

Paired kernels and their applications

This paper considers paired operators in the context of the Lebesgue Hilbert space on the unit circle and its subspace, the Hardy space $H^2$. The kernels of such operators, together with their analytic projections, which are generalizations of Toeplitz kernels, are studied. Results on near-invariance properties, representations, and inclusion relations for these kernels are obtained. The existence of a minimal Toeplitz kernel containing any projected paired kernel and, more generally, any nearly $S^*$-invariant subspace of $H^2$, is derived. The results are applied to describing the kernels of finite-rank asymmetric truncated Toeplitz operators.

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arXiv2023

Paired operators and paired kernels

This paper is concerned with paired operators in the context of the Lebesgue Hilbert space on the unit circle and its subspace, the Hardy space. By considering when such operators commute, generalizations of the Brown--Halmos results for Toeplitz operators are derived. Further, the kernels of such operators are described, giving results on invariant and nearly-invariant subspaces, together with a generalization of Coburn's theorem on Toeplitz kernels.

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arXiv2023

Invariant subspaces of the Cesaro operator

This paper explores various classes of invariant subspaces of the classical Cesàro operator $C$ on the Hardy space $H^2$. We provide a new characterization of the finite co-dimensional $C$-invariant subspaces, based on earlier work of the first two authors, and determine exactly which model spaces are $C$-invariant subspaces. We also describe the $C$-invariant subspaces contained in model spaces and establish that they are all cyclic. Along the way, we re-examine an associated Hilbert space of analytic functions on the unit disk developed by Kriete and Trutt. We also make a connection between the adjoint of the Cesàro operator and certain composition operators on $H^2$ which have universal translates in the sense of Rota.

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arXiv2022

Admissibility of retarded diagonal systems with one-dimensional input space

We investigate infinite-time admissibility of a control operator $B$ in a Hilbert space state-delayed dynamical system setting of the form $\dot{z}(t)=Az(t)+A_1 z(t-τ)+Bu(t)$, where $A$ generates a diagonal $C_0$-semigroup, $A_1\in\mathcal{L}(X)$ is also diagonal and $u\in L^2(0,\infty;\mathbb{C})$. Our approach is based on the Laplace embedding between $L^2$ and the Hardy space $H^2(\mathbb{C}_+)$. The results are expressed in terms of the eigenvalues of $A$ and $A_1$ and the sequence representing the control operator.

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arXiv2022

Insights on the Cesàro operator: shift semigroups and invariant subspaces

A closed subspace is invariant under the Cesàro operator $\mathcal{C}$ on the classical Hardy space $H^2(\mathbb D)$ if and only if its orthogonal complement is invariant under the $C_0$-semigroup of composition operators induced by the affine maps $φ_t(z)= e^{-t}z + 1 - e^{-t}$ for $t\geq 0$ and $z\in \mathbb D$. The corresponding result also holds in the Hardy spaces $H^p(\mathbb D)$ for $1<p<\infty$. Moreover, in the Hilbert space setting, by linking the invariant subspaces of $\mathcal{C}$ to the lattice of the closed invariant subspaces of the standard right-shift semigroup acting on a particular weighted $L^2$-space on the line, we exhibit a large class of non-trivial closed invariant subspaces and provide a complete characterization of the finite codimensional ones, establishing, in particular, the limits of such an approach towards describing the lattice of all invariant subspaces of $\mathcal{C}$. Finally, we present a functional calculus argument which allows us to extend a recent result by Mashreghi, Ptak and Ross regarding the square root of $\mathcal{C}$, and discuss its invariant subspaces.

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arXiv2022

Multiplication by a finite Blaschke product on weighted Bergman spaces: commutant and reducing subspaces

We provide a characterization of the commutant of analytic Toeplitz operators $T_B$ induced by finite Blachke products $B$ acting on weighted Bergman spaces which, as a particular instance, yields the case $B(z)=z^n$ on the Bergman space solved recently by by Abkar, Cao and Zhu. Moreover, it extends previous results by Cowen and Wahl in this context and applies to other Banach spaces of analytic functions such as Hardy spaces $H^p$ for $1<p<\infty$. Finally, we apply this approach to study the reducing subspaces of $T_{B}$ in weighted Bergman spaces.

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arXiv2021

Laplace-Carleson embeddings and infinity-norm admissibility

A full characterization of the boundedness of Laplace--Carleson embeddings on $L^\infty$ is provided, in terms of the Carleson intensity of the respective measure and of a suitable weighted Berezin transform of the measure. Moreover, boundedness results, and in some cases full characterizations of boundedness, are proved for a large class of Orlicz spaces. These findings are crucial for characterizing admissibility of control operators for linear diagonal semigroup systems in a variety of contexts. A particular focus is laid on essentially bounded inputs.

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