Javad Mashreghi

• PhD (McGill University)
• Professor (Full) at Université Laval

Using Newtonian potentials and balayages of positive Borel measures, we describe the family of superharmonically weighted Dirichlet spaces \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt}...

The family of Ces\`{a}ro operators $\sigma_n^\alpha$, $n \geq 0$ and $\alpha \in [0,1]$, consists of finite rank operators on Banach spaces of analytic functions on the open unit disc. In this work, we investigate these operators as they act on the local Dirichlet spaces $\mathcal{D}_\zeta$. It is well-established that they provide a linear approxi...

The Crouzeix ratio $\psi(A)$ 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 $\psi(A)\le 1+\sqrt{2}$, and it is conjectured that $\psi(A)\le 2$. In this note, we show that $\psi(A)\le C_N$, where $C_N$ is a constant depending only...

Given a compact convex planar domain $\Omega$ with non-empty interior, the classical Neumann's configuration constant $c_{\mathbb{R}}(\Omega)$ is the norm of the Neumann-Poincar\'e operator $K_\Omega$ acting on the space of continuous real-valued functions on the boundary $\partial \Omega$, modulo constants. We investigate the related operator norm...

The geometry of the Birkhoff polytope, i.e., the compact convex set of all \(n \times n\) doubly stochastic matrices, has been an active subject of research. While its faces, edges and facets as well as its volume have been intensely studied, other geometric characteristics such as the center and radius were left off, despite their natural uses in...

In the first of this series of two articles, we studied some geometrical aspects of the Birkhoff polytope, the compact convex set of all \(n \times n\) doubly stochastic matrices, namely the Chebyshev center, and the Chebyshev radius of the Birkhoff polytope associated with metrics induced by the operator norms from \(\ell _n^p\) to \(\ell _n^p\) f...

If U is a unitary operator on a separable complex Hilbert space H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {H}$$\end{document}, an application of the spe...

In this article we obtain an explicit formula for the Hilbert transform of log|f|,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\log |f|,$$\end{document} for the func...

Contrary to celebrated transforms such as the Fourier transform, explicit formulas for the Hilbert transform of well-known functions are rare. In this note, we present a formula for the Hilbert transform of \(\log |F/E|\), where F belongs to the Cartwright–de Branges space associated with the de Branges function E. The formula implies several other...

The geometry of the compact convex set of all n × n n\times n doubly stochastic matrices, a structure frequently referred to as the Birkhoff polytope, has been an active subject of research as of late. Geometric characteristics such as the Chebyshev center and the Chebyshev radius with respect to the operator norms from ℓ n p {\ell }_{n}^{p} to ℓ n...

The geometry of the compact convex set of all n × n doubly sto-chastic matrices, a structure frequently referred to as the Birkhoff polytope, has been an active subject of research as of late. Geometric characteristics such as the Chebyshev center and the Chebyshev radius with respect to the operator norms from ℓp to ℓp and the Schatten p-norms, bo...

The Cesàro means of Taylor polynomials n , n 0, are finite rank operators on any Banach space of analytic functions on the open unit disc. They are particularly exploited when the Taylor polyno-mials do not constitute a valid linear polynomial approximation scheme (LPAS). Notably, in local Dirichlet spaces D ⇣ , they serve as a proper LPAS. The pri...

In a celebrated paper of Marcus and Ree (1959), it was shown that if A = [a_ij] is an n × n doubly stochastic matrix, then there is a permutation σ ∈ Sn such that ∑a_ij² ≤ ∑ a_iσ(i). Erdős asked for which doubly stochastic matrices the inequality is saturated. Although Marcus and Ree provided some insight for the set of solutions, the question appe...

In the first of this series of two articles, we studied some geometrical aspects of the Birkhoff polytope, the compact convex set of all n×n doubly stochastic matrices, namely the Chebyshev center, and the Chebyshev radius of the Birkhoff polytope associated with metrics induced by the operator norms from ℓp to ℓp for 1 ≤ p ≤ ∞. In the present pape...

The geometry of the Birkhoff polytope, i.e., the compact convex set of all n × n doubly stochastic matrices, has been an active subject of research. While its faces, edges and facets as well as its volume have been intensely studied, other geometric characteristics such as the center and radius were left off, despite their natural uses in some area...

We introduce an expansion scheme in reproducing kernel Hilbert spaces, which as a special case covers the celebrated Blaschke unwinding series expansion for analytic functions. The expansion scheme is further generalized to cover Hardy spaces Hp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage...

We establish three Hardy-type inequalities in which the arithmetic means of a sequence of non-negative real numbers are replaced by the weighted means over nested subsets of the sequence. This work originated from some delicate calculations previously performed for evaluating the norm of infinite L-matrices.

In this article, we present some recent results related to the calculation of the induced p-norm of n times n circulant matrices A(n, a, b) with diagonal entries equal to a in R and off-diagonal entries equal to b in R. For circulant matrices with nonnegative entries, an explicit formula for the induced p-norm (p between 1 and infinity) is given, w...

The objective of the present paper is to establish three Hardy-type inequalities in which the arithmetic mean over a sequence of non-negative real numbers is replaced by some weighted arithmetic mean over some nested subsets of the given sequence of numbers. One of these inequalities stems from a calculation in a paper of Bouthat and Mashreghi on s...

In a celebrated paper of Marcus and Ree (1959), it was shown that if $A=[a_{ij}]$ is an $n \times n$ doubly stochastic matrix, then there is a permutation $\sigma \in S_n$ such that $\sum_{i,j=1}^{n} a_{i,j}^{2} \leq \sum_{i=1}^{n} a_{i,\sigma(i)}$. Erd\H{o}s asked for which doubly stochastic matrices the inequality is saturated. Although Marcus an...

The objective of the present paper is to establish three Hardy-type inequalities in which the arithmetic mean over a sequence of non-negative real numbers is replaced by some weighted arithmetic mean over some nested subsets of the given sequence of numbers. One of these inequalities stems from a calculation in a paper of Bouthat and Mashreghi on s...

In an otherwise instructive 2012 article, Szilard provided a flawed argument purportedly establishing that the left (resp. right) Riemann sum of f(x) = 1/1+x^2 with respect to the uniform partition of [0,1] into n equal intervals is monotonically decreasing (resp. increasing) relative to n. A few years later, D. Borwein, J. M. Borwein and B. Sims d...

Operator Theory by Example is aimed at graduate students just getting started in operator theory. Rather than discuss the subject in the abstract, this book covers the subject through twenty instructive examples of a wide variety of operators. For each operator, we discuss its norm, spectrum, commutant, invariant subspaces, and interesting properti...

We formalize the observation that the same summability methods converge in a Banach space $X$ and its dual $X^*$. At the same time we determine conditions under which these methods converge in the weak and weak*-topologies on $X$ and $X^*$ respectively. We also derive a general limitation theorem, which yields a necessary condition for the converge...

We introduce the notion of an asymptotically equicontinuous sequence of linear operators, and use it to prove the following result. If $X,Y$ are topological vector spaces, if $T_n,T:X\to Y$ are continuous linear maps, and if $D$ is a dense subset of $X$, then the following statements are equivalent: (i) $T_nx\to Tx$ for all $x\in X$, and (ii) $T_n...

Operator Theory by Example is aimed at graduate students just getting started in operator theory. Rather than discuss the subject in the abstract, this book covers the subject through twenty instructive examples of a wide variety of operators. For each operator, we discuss its norm, spectrum, commutant, invariant subspaces, and interesting properti...

Operator Theory by Example is aimed at graduate students just getting started in operator theory. Rather than discuss the subject in the abstract, this book covers the subject through twenty instructive examples of a wide variety of operators. For each operator, we discuss its norm, spectrum, commutant, invariant subspaces, and interesting properti...

Operator Theory by Example is aimed at graduate students just getting started in operator theory. Rather than discuss the subject in the abstract, this book covers the subject through twenty instructive examples of a wide variety of operators. For each operator, we discuss its norm, spectrum, commutant, invariant subspaces, and interesting properti...

Operator Theory by Example is aimed at graduate students just getting started in operator theory. Rather than discuss the subject in the abstract, this book covers the subject through twenty instructive examples of a wide variety of operators. For each operator, we discuss its norm, spectrum, commutant, invariant subspaces, and interesting properti...

Operator Theory by Example is aimed at graduate students just getting started in operator theory. Rather than discuss the subject in the abstract, this book covers the subject through twenty instructive examples of a wide variety of operators. For each operator, we discuss its norm, spectrum, commutant, invariant subspaces, and interesting properti...

Operator Theory by Example is aimed at graduate students just getting started in operator theory. Rather than discuss the subject in the abstract, this book covers the subject through twenty instructive examples of a wide variety of operators. For each operator, we discuss its norm, spectrum, commutant, invariant subspaces, and interesting properti...

Operator Theory by Example is aimed at graduate students just getting started in operator theory. Rather than discuss the subject in the abstract, this book covers the subject through twenty instructive examples of a wide variety of operators. For each operator, we discuss its norm, spectrum, commutant, invariant subspaces, and interesting properti...

Operator Theory by Example is aimed at graduate students just getting started in operator theory. Rather than discuss the subject in the abstract, this book covers the subject through twenty instructive examples of a wide variety of operators. For each operator, we discuss its norm, spectrum, commutant, invariant subspaces, and interesting properti...

Operator Theory by Example is aimed at graduate students just getting started in operator theory. Rather than discuss the subject in the abstract, this book covers the subject through twenty instructive examples of a wide variety of operators. For each operator, we discuss its norm, spectrum, commutant, invariant subspaces, and interesting properti...

Operator Theory by Example is aimed at graduate students just getting started in operator theory. Rather than discuss the subject in the abstract, this book covers the subject through twenty instructive examples of a wide variety of operators. For each operator, we discuss its norm, spectrum, commutant, invariant subspaces, and interesting properti...

Operator Theory by Example is aimed at graduate students just getting started in operator theory. Rather than discuss the subject in the abstract, this book covers the subject through twenty instructive examples of a wide variety of operators. For each operator, we discuss its norm, spectrum, commutant, invariant subspaces, and interesting properti...

Operator Theory by Example is aimed at graduate students just getting started in operator theory. Rather than discuss the subject in the abstract, this book covers the subject through twenty instructive examples of a wide variety of operators. For each operator, we discuss its norm, spectrum, commutant, invariant subspaces, and interesting properti...

Operator Theory by Example is aimed at graduate students just getting started in operator theory. Rather than discuss the subject in the abstract, this book covers the subject through twenty instructive examples of a wide variety of operators. For each operator, we discuss its norm, spectrum, commutant, invariant subspaces, and interesting properti...

Operator Theory by Example is aimed at graduate students just getting started in operator theory. Rather than discuss the subject in the abstract, this book covers the subject through twenty instructive examples of a wide variety of operators. For each operator, we discuss its norm, spectrum, commutant, invariant subspaces, and interesting properti...

Operator Theory by Example is aimed at graduate students just getting started in operator theory. Rather than discuss the subject in the abstract, this book covers the subject through twenty instructive examples of a wide variety of operators. For each operator, we discuss its norm, spectrum, commutant, invariant subspaces, and interesting properti...

Operator Theory by Example is aimed at graduate students just getting started in operator theory. Rather than discuss the subject in the abstract, this book covers the subject through twenty instructive examples of a wide variety of operators. For each operator, we discuss its norm, spectrum, commutant, invariant subspaces, and interesting properti...

Operator Theory by Example is aimed at graduate students just getting started in operator theory. Rather than discuss the subject in the abstract, this book covers the subject through twenty instructive examples of a wide variety of operators. For each operator, we discuss its norm, spectrum, commutant, invariant subspaces, and interesting properti...

Operator Theory by Example is aimed at graduate students just getting started in operator theory. Rather than discuss the subject in the abstract, this book covers the subject through twenty instructive examples of a wide variety of operators. For each operator, we discuss its norm, spectrum, commutant, invariant subspaces, and interesting properti...

Operator Theory by Example is aimed at graduate students just getting started in operator theory. Rather than discuss the subject in the abstract, this book covers the subject through twenty instructive examples of a wide variety of operators. For each operator, we discuss its norm, spectrum, commutant, invariant subspaces, and interesting properti...

Operator Theory by Example is aimed at graduate students just getting started in operator theory. Rather than discuss the subject in the abstract, this book covers the subject through twenty instructive examples of a wide variety of operators. For each operator, we discuss its norm, spectrum, commutant, invariant subspaces, and interesting properti...

We show that if u is a compactly supported distribution on the complex plane such that, for every pair of entire functions f, g, $$\begin{aligned} \langle u,f{\overline{g}}\rangle =\langle u,f\rangle \langle u,{\overline{g}}\rangle , \end{aligned}$$then u is supported at a single point. As an application, we complete the classification of all weigh...

The partial Taylor sums \(S_n\), \(n \ge 0\), are finite rank operators on any Banach space of analytic functions on the open unit disc. In the classical setting of disc algebra \({\mathcal {A}}\), the precise value of \(\Vert S_n\Vert _{{\mathcal {A}} \rightarrow {\mathcal {A}}}\) is not known. These numbers are referred as the Lebesgue constants...

Analytic functions, despite exhibiting very satisfactory behavior inside their domains of definition, e.g., being infinitely differentiable and having power series representations, could be quite wild and show bizarre performance at the boundary points. Not being continuous at a single boundary point is the simplest of such phenomena. However, much...

We show that there exists a de Branges–Rovnyak space H(b)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {H}}(b)$$\end{document} on the unit disk containing a...

We establish the following Hilbert‐space analog of the Gleason–Kahane–Żelazko theorem. If H${\mathcal {H}}$ is a reproducing kernel Hilbert space with a normalized complete Pick kernel, and if Λ$\Lambda$ is a linear functional on H${\mathcal {H}}$ such that Λ(1)=1$\Lambda (1)=1$ and Λ(f)≠0$\Lambda (f)\ne 0$ for all cyclic functions f∈H$f\in {\mathc...

The Hilbert L-matrix As=[a_{ij}(s)], where a_{ij}(s)=1/(max⁡{i,j}+s) with i,j≥0, was introduced in [3]. As a surprising property, we showed that its 2-norm is constant for s≥s0, where the critical point s_0 is unknown but relies in the interval (1/4,1/2). In this note, using some delicate calculations we sharpen this result by improving the upper a...

We explore the relation between Hadamard products of two entire functions in the weighted Fock space \(\mathrm {F}^p_\gamma \) and their integral Gaussian means. By introducing the key auxiliary function \(\kappa _{\gamma ,r}\), we show that the growth of the Gaussian means of the Hadamard product \(f*g*\kappa _{\gamma ,r}\) is controlled by the gr...

The classical theorems of Mittag-Leffler and Weierstrass show that when (λn)n⩾1 is a sequence of distinct points in the open unit disk D, with no accumulation points in D, and (wn)n⩾1 is any sequence of complex numbers, there is an analytic function φ on D for which φ(λn)=wn. A celebrated theorem of Carleson [2] characterizes when, for a bounded se...

In this note, we study the induced p-norm of circulant matrices A(n,±a,b), acting as operators on the Euclidean space Rn. For circulant matrices whose entries are nonnegative real numbers, in particular for A(n,a,b), we provide an explicit formula for the p-norm, 1≤p≤∞. The calculation for A(n,−a,b) is more complex. The 2-norm is precisely determin...

In this paper we give complete descriptions of the set of square roots of certain classical operators, often providing specific formulas. The classical operators included in this discussion are the square of the unilateral shift, the Volterra operator, certain compressed shifts, the unilateral shift plus its adjoint, the Hilbert matrix, and the Ces...

In this note we study the induced $p$-norm of circulant matrices $A(n,\pm a, b)$, acting as operators on the Euclidean space $\mathbb{R}^n$. For circulant matrices whose entries are nonnegative real numbers, in particular for $A(n,a,b)$, we provide an explicit formula for the $p$-norm, $1 \leq p \leq \infty$. The calculation for $A(n,-a,b)$ is more...

The $L$-matrix $A_s=[1/(n+s)]$ was introduced in \cite{MRtmp}. As a surprising property, we showed that its 2-norm is constant for $s \geq s_0$, where the critical point $s_0$ is unknown but relies in the interval $(1/4,1/2)$. In this note, using some delicate calculations we sharpen this result by improving the upper and lower bounds of the interv...

We show that an L-matrices A = [a_n], with lacunary coefficients (a_n) is a bounded operator on ℓ2 , provided that (a_n) satisfy an explicit decay rate. Moreover, by a concrete example, we see that the decay restriction is optimal. The extension to operators on p spaces, for p > 1, is also discussed.

We construct a Hilbert holomorphic function space H on the unit disk such that the polynomials are dense in H, but the odd polynomials are not dense in the odd functions in H. As a consequence, there exists a function f in H that lies outside the closed linear span of its Taylor partial sums \(s_n(f)\), so it cannot be approximated by any triangula...

We show that if $u$ is a compactly supported distribution on the complex plane such that, for every pair of entire functions $f,g$, \[ \langle u,f\overline{g}\rangle=\langle u,f\rangle\langle u,\overline{g}\rangle, \] then $u$ is supported at a single point. As an application, we complete the classification of all weighted Dirichlet spaces on the u...

We refine a result of [J. E. McCarthy, Common range of co-analytic Toeplitz operators, J. Amer. Math. Soc. 3 1990, 4, 793–799] and explore the common range of the co-analytic Toeplitz operators on a model space. The tools used to do this also yield information about when one can interpolate with an outer function.

We present a maximum principle for metrics with negative curvature. This principle is essentially a reformulation of the Minda–Schober proof of Ahlfors’s theorem, which by itself is a version of Schwarz’s lemma in differential geometry language. This maximum principle leads to the concept of extremal metrics.

We give an elementary proof of an analogue of Fejér’s theorem in weighted Dirichlet spaces with superharmonic weights. This provides a simple way of seeing that polynomials are dense in such spaces.

We show that there exists a de Branges-Rovnyak space $\mathcal{H}(b)$ on the unit disk containing a function $f$ with the following property: even though $f$ can be approximated by polynomials in $\mathcal{H}(b)$, neither the Taylor partial sums of $f$ nor their Ces\`aro, Abel, Borel or logarithmic means converge to $f$ in $\mathcal{H}(b)$. A key t...

Evaluating the norm of infinite matrices, as operators acting on the sequence space ℓ2, is not an easy task. For a few celebrated matrices, e.g., the Hilbert matrix and the Cesàro matrix, the precise value of the norm is known. But, for many other important cases we use estimated values of norm. In this note, we study the norm of L-matrices A = [a_...

We show that, in every weighted Dirichlet space on the unit disk with superharmonic weight, the Taylor series of a function in the space is (C,α)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{...

Using the fractional derivatives Dt of Hardy-Littlewood and auxiliary functions ξγ,r and ηγ,r, we study the growth of the Hadamard product f⁎g in the weighted Bergman space Aγq(D). We precisely evaluate the growth of dilations of Dα+β(f⁎g⁎ξγ,r), whenever Dαf∈Aγp(D) and Dβg∈Aγq(D). The main result has numerous special cases which are interesting in...

We show that, if $f$ is an outer function and $a\in[0,1)$, then the set of functions $\{\log |(f\circ\psi)^*|: \psi:\mathcal{D}\to\mathcal{D} \text{ holomorphic}, |\psi(0)|\le a\}$ is uniformly integrable on the unit circle. As an application, we derive a simple proof of the fact that, if $f$ is outer and $\phi:\mathcal{D}\to\mathcal{D}$ is holomor...

The Gleason-Kahane-\.Zelazko theorem states that a linear functional on a Banach algebra that is non-zero on invertible elements is necessarily a scalar multiple of a character. Recently this theorem has been extended to certain Banach function spaces that are not algebras. In this article we present a brief survey of these extensions.

We give an elementary proof of an analogue of Fej\'er's theorem in weighted Dirichlet spaces with superharmonic weights. This provides a simple way of seeing that polynomials are dense in such spaces.

Let $X$ be a Banach holomorphic function space on the unit disk. A linear polynomial approximation scheme for $X$ is a sequence of bounded linear operators $T_n:X\to X$ with the property that, for each $f\in X$, the functions $T_n(f)$ are polynomials converging to $f$ in the norm of the space. We completely characterize those spaces $X$ that admit...

The Hadamard product of two power series is obtained by multiplying them coefficientwise. In this paper we characterize those power series that act as Hadamard multipliers on all weighted Dirichlet spaces on the disk with superharmonic weights, and we obtain sharp estimates on the corresponding multiplier norms. Applications include an analogue of...

We establish the following Hilbert-space analogue of the Gleason-Kahane-\.Zelazko theorem. If $\mathcal{H}$ is a reproducing kernel Hilbert space with a normalized complete Pick kernel, and if $\Lambda$ is a linear functional on $\mathcal{H}$ such that $\Lambda(1)=1$ and $\Lambda(f)\ne0$ for all cyclic functions $f\in\mathcal{H}$, then $\Lambda$ is...

The classical theorems of Mittag-Leffler and Weierstrass show that when $\{\lambda_n\}$ is a sequence of distinct points in the open unit disk $\D$, with no accumulation points in $\D$, and $\{w_n\}$ is any sequence of complex numbers, there is an analytic function $\phi$ on $\D$ for which $\phi(\lambda_n) = w_n$. A celebrated theorem of Carleson \...

We show that, in every weighted Dirichlet space on the unit disk with superharmonic weight, the Taylor series of a function in the space is $(C,\alpha)$-summable to the function in the norm of the space, provided that $\alpha>1/2$. We further show that the constant $1/2$ is sharp, in marked contrast with the classical case of the disk algebra.

We construct a Hilbert holomorphic function space $H$ on the unit disk such that the polynomials are dense in $H$, but the odd polynomials are not dense in the odd functions in $H$. As a consequence, there exists a function $f$ in $H$ that lies outside the closed linear span of its Taylor partial sums $s_n(f)$, so it cannot be approximated by any t...

It is well known that if \(f\in H^1\) and \(g\in H^q\), where \(1\le q<\infty \), then the integral means of order q of their Hadamard product \(f*g\) satisfy \(M_q(r,f*g)\le \Vert f\Vert _{H^1}\Vert g\Vert _{H^q}\), uniformly for each \(0<r<1\), and consequently \(\Vert f*g\Vert _{H^q}\le \Vert f\Vert _{H^1}\Vert g\Vert _{H^q}\). In this note, we...

The Hadamard product of two power series is obtained by multiplying them coefficientwise. In this paper we characterize those power series that act as Hadamard multipliers on all weighted Dirichlet spaces on the disk with superharmonic weights, and we obtain sharp estimates on the corresponding multiplier norms. Applications include an analogue of...

Let X be a Banach holomorphic function space on the unit disk. A linear polynomial approximation scheme for X is a sequence of bounded linear operators Tn:X→X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\od...

In Beurling’s approach to inner functions for the shift operator S on the Hardy space H 2, a function f is inner when f ⊥ S n f for all \(n \geqslant 1\). Inspired by this approach, this paper develops a notion of an inner vector x for any operator T on a Hilbert space, via the analogous condition x ⊥ T n x for all \(n \geqslant 1\). We study these...

We study the compactness of the composition operator on de Branges–Rovnyak spaces. Inspired by a paper by Lyubarskii–Malinnikova on model spaces, we give some necessary and some sufficient conditions for compactness. In the paper of Lyubarskii-Malinnikova, the key point is some Bernstein inequality on model spaces due to Cohn (and based on a deep i...

We give a simple proof of the fact that a finite measure $\mu$ on the unit disk is a Carleson measure for the Dirichlet space if it satisfies the Carleson one-box condition $\mu(S(I))=O(\phi(|I|))$, where $\phi:(0,2\pi]\to(0,\infty)$ is an increasing function such that $\int_0^{2\pi}(\phi(x)/x)\,dx<\infty$. We further show that the integral conditi...

We show that, for many holomorphic function spaces on the unit disk, a continuous endomorphism that sends inner functions to inner functions is necessarily a weighted composition operator.

In most classical holomorphic function spaces on the unit disk in which the polynomials are dense, a function $f$ can be approximated in norm by its dilates $f_r(z):=f(rz)~(r<1)$, in other words, $\lim_{r\to1^-}\|f_r-f\|=0$. We construct a de Branges-Rovnyak space ${\mathcal H}(b)$ in which the polynomials are dense, and a function $f\in{\mathcal H...

We show that every linear functional on the Dirichlet space that is non-zero on nowhere-vanishing functions is necessarily a multiple of a point evaluation. Continuity of the functional is not assumed. As an application, we obtain a characterization of weighted composition operators on the Dirichlet space as being exactly those linear maps that sen...

We show that, for many holomorphic function spaces on the unit disk, a continuous endomorphism that sends inner functions to inner functions is necessarily a weighted composition operator.

In this paper we survey and bring together several approaches to obtaining inner functions for Toeplitz operators. These approaches include the classical definition, the Wold decomposition, the operator-valued Poisson Integral, and Clark measures. We then extend these notions somewhat to inner functions on model spaces. Along the way we present som...

We show that, if f is an outer function and a ∈ [0, 1), then the set of functions (log |(f o ψ)*| :ψ: D → D holomorphic, |ψ(0)| ≤ a) is uniformly integrable on the unit circle. As an application, we derive a simple proof of the fact that, if f is outer and ϕ: D → D is holomorphic, then f o ϕ is outer.

In this paper we explore the notion of inner function in a broader context of operator theory.

The (scalar-valued) Darlington synthesis problem from electrical network theory asks the following question. Given a ∈ H∞ , do there exist b, c, d ∈ H∞ such that the matrix-valued analytic function $$\displaystyle U = \begin {bmatrix} a & -b \\ c & d \end {bmatrix} $$ is unitary almost everywhere on \(\mathbb {T}\)?

This chapter will cover some basic facts about the Schur class.