Hardy Space H2 Research Articles

2-Complex Symmetric Composition Operators on H2

In this paper, we study 2-complex symmetric composition operators with the conjugation J, defined by Jf(z)=(f(z¯))¯, on the Hardy space H2. More precisely, we obtain the necessary and sufficient condition for the composition operator Cϕ to be 2-complex symmetric with J when ϕ is an automorphism of D. We also characterize 2-complex symmetric with J when ϕ is a linear fractional self-map of D.

Completeness of discrete translates in the Hardy space 𝐻¹(ℝ)

We provide a characterization of discrete sets Λ ⊂ R \Lambda \subset \mathbb {R} that admit a function whose Λ \Lambda -translates are complete in the Hardy space H 1 ( R ) H^1(\mathbb {R}) . In particular, we show that such a set cannot be uniformly discrete. We then give a uniformly discrete Λ ⊂ R \Lambda \subset \mathbb {R} which admits a pair of functions such that their Λ \Lambda -translates are complete in H 1 ( R ) H^1(\mathbb {R}) .

Compensated compactness: Continuity in optimal weak topologies

For l-homogeneous linear differential operators A of constant rank, we study the implicationvj⇀vinXAvj→AvinW−lY}⇒F(vj)⇝F(v)inZ, where F is an A-quasiaffine function and ⇝ denotes an appropriate type of weak convergence. Here Z is a local L1-type space, either the space M of measures, or L1, or the Hardy space H1; X,Y are Lp-type spaces, by which we mean Lebesgue or Zygmund spaces. Our conditions for each choice of X,Y,Z are sharp. Analogous statements are also given in the case when F(v) is not a locally integrable function and it is instead defined as a distribution. In this case, we also prove Hp-bounds for the sequence (F(vj))j, for appropriate p<1, and new convergence results in the dual of Hölder spaces when (vj) is A-free and lies in a suitable negative order Sobolev space W−β,s. The choice of these Hölder spaces is sharp, as is shown by the construction of explicit counterexamples. Some of these results are new even for distributional Jacobians.

A Gaussian version of Littlewood’s theorem for random power series

We prove a Littlewood-type theorem for random analytic functions associated with not necessarily independent Gaussian processes. We show that if we randomize a function in the Hardy space H 2 ( D ) H^2(\mathbb {D}) by a Gaussian process whose covariance matrix K K induces a bounded operator on l 2 l^2 , then the resulting random function is almost surely in H p ( D ) H^p(\mathbb {D}) for any p &gt; 0 p&gt;0 . The case K = I d K=\mathrm {Id} , the identity operator, recovers Littlewood’s theorem. A new ingredient in our proof is to recast the membership problem as the boundedness of an operator. This reformulation enables us to use tools in functional analysis and is applicable to other situations.

On the essential norms of singular integral operators with constant coefficients and of the backward shift

Let X X be a rearrangement-invariant Banach function space on the unit circle T \mathbb {T} and let H [ X ] H[X] be the abstract Hardy space built upon X X . We prove that if the Cauchy singular integral operator ( H f ) ( t ) = 1 π i ∫ T f ( τ ) τ − t d τ (Hf)(t)=\frac {1}{\pi i}\int _{\mathbb {T}}\frac {f(\tau )}{\tau -t}\,d\tau is bounded on the space X X , then the norm, the essential norm, and the Hausdorff measure of non-compactness of the operator a I + b H aI+bH with a , b ∈ C a,b\in \mathbb {C} , acting on the space X X , coincide. We also show that similar equalities hold for the backward shift operator ( S f ) ( t ) = ( f ( t ) − f ^ ( 0 ) ) / t (Sf)(t)=(f(t)-\widehat {f}(0))/t on the abstract Hardy space H [ X ] H[X] . Our results extend those by Krupnik and Polonskiĭ [Funkcional. Anal. i Priloz̆en. 9 (1975), pp. 73-74] for the operator a I + b H aI+bH and by the second author [J. Funct. Anal. 280 (2021), p. 11] for the operator S S .

Nearly outer functions as extreme points in punctured Hardy spaces

The Hardy space H1 consists of the integrable functions f on the unit circle whose Fourier coefficients fˆ(k) vanish for k<0. We are concerned with H1 functions that have some additional (finitely many) holes in the spectrum, so we fix a finite set K of positive integers and consider the ‘‘punctured” Hardy spaceHK1:={f∈H1:fˆ(k)=0for all k∈K}. We then investigate the geometry of the unit ball in HK1. In particular, the extreme points of the ball are identified as those unit-norm functions in HK1 which are not too far from being outer (in the appropriate sense). This extends a theorem of de Leeuw and Rudin that deals with the classical H1 and characterizes its extreme points as outer functions. We also discuss exposed points of the unit ball in HK1.

Complex symmetry and normality of Toeplitz composition operators on the hardy space

In this paper, we investigate the conditions under which the Toeplitz composition operator on the Hardy space H2 becomes complex symmetric with respect to a certain conjugation. We also study various normality conditions for the Toeplitz composition operator on H2.

Multiplicative property of series used in the Nevanlinna-Pick problem

In the paper we obtained substantially new sufficient condition for negativity of coefficients of power series inverse to series with positive ones. It has been proved that element-wise product of power series retains this property. In particular, it gives rise to generalization of the classical Hardy theorem about power series. These results are generalized for cases of series with multiple variables. Such results are useful in Nevanlinna – Pick theory. For example, if function k(x, y) can be represented as power series Pn≥0 an(x¯y)n, an &gt; 0, and reciprocal function 1/k(x, y) can be represented as power series Pn≥0 bn(x¯y)n such that bn &lt; 0, n &gt; 0, then k(x, y) is a reproducing kernel function for some Hilbert space of analytic functions in the unit disc D with Nevanlinna – Pick property. The reproducing kernel 1/(1 − x¯y) of the classical Hardy space H2(D) is a prime example for our theorems.

Compressions of kth-order slant Toeplitz operators to model spaces

In this paper, we consider compressions of kth-order slant Toeplitz operators to the backward shift-invariant subspaces of the classical Hardy space H2. In particular, we characterize these operators using compressed shifts and finite-rank operators of special kind.

Universal composition operators

A Hilbert space operator U is called \textit{universal} (in the sense of Rota) if every Hilbert space operator is similar to a multiple of U restricted to one of its invariant subspaces. It follows that the \textit{invariant subspace problem} for Hilbert spaces is equivalent to the statement that all minimal invariant subspaces for U are one dimensional. In this article we characterize all linear fractional composition operators Cϕf=f∘ϕ that have universal translates on both the classical Hardy spaces H2(C+) and H2(D) of the half-plane and the unit disk, respectively. The new example here is the composition operator on H2(D) with affine symbol ϕa(z)=az+(1−a) for $0

Splitting the Riesz basis condition for systems of dilated functions through Dirichlet series

Inspired by the work of Hedenmalm, Lindqvist and Seip, we consider different properties of dilations systems of a fixed function φ∈L2(0,1). More precisely, we study when the system {φ(nx)}n is a Bessel sequence, a Riesz sequence, or it satisfies the lower frame bound. We are able to characterize these properties in terms of multipliers of the Hardy space H2 of Dirichlet series and, also, in terms of Hardy spaces on the infinite polytorus. We also address the multivariate case.

On the maximal operators of weighted Marcinkiewicz type means of two-dimensional Walsh–Fourier series

Abstract Goginava proved that the maximal operator σ α , * {\sigma^{\alpha,*}} ( 0 &lt; α &lt; 1 {0&lt;\alpha&lt;1} ) of two-dimensional Marcinkiewicz type ( C , α ) {(C,\alpha)} means is bounded from the two-dimensional dyadic martingale Hardy space H p ⁢ ( G 2 ) {H_{p}(G^{2})} to the space L p ⁢ ( G 2 ) {L^{p}(G^{2})} for p &gt; 2 2 + α {p&gt;\frac{2}{2+\alpha}} . Moreover, he showed that assumption p &gt; 2 2 + α {p&gt;\frac{2}{2+\alpha}} is essential for the boundedness of the maximal operator σ α , * {\sigma^{\alpha,*}} . It was shown that at the point p 0 = 2 2 + α {p_{0}=\frac{2}{2+\alpha}} the maximal operator σ α , * {\sigma^{\alpha,*}} is bounded from the dyadic Hardy space H 2 / ( 2 + α ) ⁢ ( G 2 ) {H_{2/(2+\alpha)}(G^{2})} to the space weak- L 2 / ( 2 + α ) ⁢ ( G 2 ) {L^{2/(2+\alpha)}(G^{2})} . The main aim of this paper is to investigate the behaviour of the maximal operators of weighted Marcinkiewicz type σ α , * {\sigma^{\alpha,*}} means ( 0 &lt; α &lt; 1 {0&lt;\alpha&lt;1} ) in the endpoint case p 0 = 2 2 + α {p_{0}=\frac{2}{2+\alpha}} . In particular, the optimal condition on the weights is given which provides the boundedness from H 2 / ( 2 + α ) ⁢ ( G 2 ) {H_{2/(2+\alpha)}(G^{2})} to L 2 / ( 2 + α ) ⁢ ( G 2 ) {L^{2/(2+\alpha)}(G^{2})} . Furthermore, a strong summation theorem is stated for functions in the dyadic martingale Hardy space H 2 / ( 2 + α ) ⁢ ( G 2 ) {H_{2/(2+\alpha)}(G^{2})} .

On the construction of the orthonormal wavelet in the Hardy space H2(ℝ)

We are concerned with the orthonormal wavelet [Formula: see text] in the Hardy space [Formula: see text] which is a closed subspace of [Formula: see text] without negative frequency components. It is well known that there does not exist an [Formula: see text]-wavelet such that [Formula: see text] is continuous on [Formula: see text] and satisfies [Formula: see text] for some [Formula: see text]. The aim of this paper is to find a critical decay rate in the existing [Formula: see text]-wavelet under the condition that [Formula: see text] is continuous on [Formula: see text]. Moreover, we also construct a concrete [Formula: see text]-wavelet having infinite vanishing moments.

Doubly commuting mixed invariant subspaces in the polydisc

We obtain a complete characterization for doubly commuting mixed invariant subspaces of the Hardy space over the unit polydisc. We say a closed subspace Q of H2(Dn) is mixed invariant if Mzj(Q)⊆Q for 1≤j≤k and Mzj⁎(Q)⊆Q, k+1≤j≤n for some integer k∈{1,2,…,n−1}. We prove that a mixed invariant subspace Q of H2(Dn) is doubly commuting if and only ifQ=ΘH2(Dk)⊗Qθ1⊗⋯⊗Qθn−k, where Θ∈H∞(Dk) is some inner function and Qθj is either a Jordan block H2(D)⊖θjH2(D) for some inner function θj or the Hardy space H2(D). Furthermore, an explicit representation for the commutant of an n-tuple of doubly commuting shifts as well as a representation for the commutant of a doubly commuting tuple of shifts and co-shifts are obtained. Finally, we discuss some concrete examples of mixed invariant subspaces.

Weighted composition operators: isometries and asymptotic behaviour

This paper studies the behaviour of iterates of weigh\-ted composition operators acting on spaces of analytic functions, with particular emphasis on the Hardy space H2. Questions relating to uniform, strong and weak convergence are resolved in many cases. Connected to this is the question when a weighted composition operator is an isometry, and new results are given in the case of the Hardy and Bergman spaces.

Extended eigenvalues of composition operators

A complex scalar λ is said to be an extended eigenvalue of a bounded linear operator A on a complex Hilbert space if there is a nonzero operator X such that AX=λXA. The results in this paper provide a full solution to the problem of computing the extended eigenvalues for those composition operators Cφ induced on the Hardy space H2(D) by linear fractional transformations φ of the unit disk.

On the equivalence between weak BMO and the space of derivatives of the Zygmund class

Abstract In this paper, we will discuss the space of functions of weak bounded mean oscillation. In particular, we will show that this space is the dual space of the special atom space, whose dual space was already known to be the space of derivative of functions (in the sense of distribution) belonging to the Zygmund class of functions. We show, in particular, that this proves that the Hardy space H 1 {H}^{1} strictly contains the special atom space.

The F-composition operator on Hardy space ℍ2

The purpose of this paper is to introduce a new operator on Hardy space ℍ2 called the F-composition operator which is generalization of composition operator. Some basic properties of this operator are given in this paper.

Idempotent, model, and Toeplitz operators attaining their norms

We study idempotent, model, and Toeplitz operators that attain the norm. Notably, we prove that if Q is a backward shift invariant subspace of the Hardy space H2(D), then the model operator SQ attains its norm. Here SQ=PQMz|Q, the compression of the shift Mz on the Hardy space H2(D) to Q.

Quasi-inner functions and local factors

We introduce the notion of quasi-inner function and show that the product u=ρ∞∏ρv of m+1 ratios of local L-factors ρv(z)=γv(z)/γv(1−z) over a finite set F of places of Q inclusive of the archimedean place is quasi-inner on the left of the critical line ℜ(z)=12 in the following sense. The off diagonal part u21 of the matrix of the multiplication by u in the orthogonal decomposition of the Hilbert space L2 of square integrable functions on the critical line into the Hardy space H2 and its orthogonal complement is a compact operator. When interpreted on the unit disk, the quasi-inner condition means that the associated Haenkel matrix is compact. We show that none of the individual non-archimedean ratios ρv is quasi-inner and, in order to prove our main result we use Gauss multiplication theorem to factor the archimedean ratio ρ∞ into a product of m quasi-inner functions whose product with each ρv retains the property to be quasi-inner. Finally we prove that Sonin's space is simply the kernel of the diagonal part u22 for the quasi-inner function u=ρ∞, and when u(F)=∏v∈Fρv the kernels of the u(F)22 form an inductive system of infinite dimensional spaces which are the semi-local analogues of (classical) Sonin's spaces.