Volume 7, Issue 1

In this paper we apply the results obtained in [3] to establish some outcomes of the study of the behaviour of a class of linear operators, which include the Sylvester ones, acting on slice semi-regular functions. We first present a detailed study of the kernel of the linear operator ℒ f,g (when not trivial), showing that it has dimension 2 if exactly one between f and g is a zero divisor, and it has dimension 3 if both f and g are zero divisors. Afterwards, we deepen the analysis of the behaviour of the -product, giving a complete classification of the cases when the functions f v , g v and f v g v are linearly dependent and obtaining, as a by-product, a necessary and sufficient condition on the functions f and g in order their *-product is slice-preserving. At last, we give an Embry-type result which classifies the functions f and g such that for any function h commuting with f + g and f * g , we have that h commutes with f and g , too.

Let w be a semiclassical weight that is generic in Magnus’s sense, and (pn)n=0∞({p_n})_{n = 0}^\infty the corresponding sequence of orthogonal polynomials. We express the Christoffel–Darboux kernel as a sum of products of Hankel integral operators. For ψ ∈ L ∞ ( i ℝ), let W (ψ) be the Wiener-Hopf operator with symbol ψ. We give sufficient conditions on ψ such that 1/ det W (ψ) W (ψ −1 ) = det( I − Γ ϕ 1 Γ ϕ 2 ) where Γ ϕ 1 and Γ ϕ 2 are Hankel operators that are Hilbert–Schmidt. For certain, ψ Barnes’s integral leads to an expansion of this determinant in terms of the generalised hypergeometric 2m F 2m-1 . These results extend those of Basor and Chen [2], who obtained 4 F 3 likewise. We include examples where the Wiener–Hopf factors are found explicitly.

Consider averages along the prime integers ℙ given by 𝒜Nf(x)=N-1∑p∈𝕇:p≤N(logp)f(x-p).{\mathcal{A}_N}f(x) = {N^{ - 1}}\sum\limits_{p \in \mathbb{P}:p \le N} {(\log p)f(x - p).} These averages satisfy a uniform scale-free ℓ p -improving estimate. For all 1 < p < 2, there is a constant C p so that for all integer N and functions f supported on [0, N ], there holds N-1/p′‖𝒜Nf‖𝓁p′≤CpN-1/p‖f‖𝓁p.{N^{ - 1/p'}}{\left\| {{\mathcal{A}_N}f} \right\|_{\ell p'}} \le {C_p}{N^{ - 1/p}}{\left\| f \right\|_{\ell p}}. The maximal function 𝒜 * f = sup N |𝒜 N f | satisfies ( p , p ) sparse bounds for all 1 < p < 2. The latter are the natural variants of the scale-free bounds. As a corollary, 𝒜 * is bounded on ℓ p ( w ), for all weights w in the Muckenhoupt 𝒜 p class. No prior weighted inequalities for 𝒜 * were known.

We study the compactness and the hypercyclicity of Toeplitz operators Tϕ¯,b{T_{\bar \varphi ,b}} with co-analytic and bounded symbols on de Branges-Rovnyak spaces ℋ( b ). For the compactness of Tϕ¯,b{T_{\bar \varphi ,b}}, we will see that the result depends on the boundary spectrum of b . We will prove that there are non trivial compact operators of the form Tϕ¯,b{T_{\bar \varphi ,b}}, with ϕ ∈ H ∞ ∩ C (𝕋), if and only if m (σ( b ) ∩ 𝕋) = 0. We will also show that, when b is non-extreme, then Tϕ¯,b{T_{\bar \varphi ,b}} is hypercyclic if and only if ϕ is non-constant and ϕ (𝔻) ∩ 𝕋 ≠ ∅.

In this paper we study Hausdorff operators on the Bergman spaces A p (𝕌) of the upper half plane.

Paley Wiener theorem characterizes the class of functions which are Fourier transforms of ℂ ∞ functions of compact support on ℝ n by relating decay properties of those functions or distributions at infinity with analyticity of their Fourier transform. The theorem is already proved in classical case : the real case with holomorphic Fourier transform on L 2 (ℝ), the case of functions with compact support on ℝ n from Hörmander and the spherical transform on semi simple Lie groups with Gangolli theorem. Let G be a locally compact unimodular group, K a compact subgroup of G , and δ an element of unitary dual ̑K of K . In this work, we’ll give an extension of Paley-Wiener theorem with respect to δ , a class of unitary irreducible representation of K , where G is either a semi-simple Lie group or a reductive Lie group with nonempty discrete series after introducing a notion of δ -orbital integral. If δ is trivial and one dimensional, we obtain the classical Paley-Wiener theorem.

This survey shows how, for the Nevanlinna class 𝒩 of the unit disc, one can define and often characterize the analogues of well-known objects and properties related to the algebra of bounded analytic functions ℋ ∞ : interpolating sequences, Corona theorem, sets of determination, stable rank, as well as the more recent notions of Weak Embedding Property and threshold of invertibility for quotient algebras. The general rule we observe is that a given result for ℋ ∞ can be transposed to 𝒩 by replacing uniform bounds by a suitable control by positive harmonic functions. We show several instances where this rule applies, as well as some exceptions. We also briefly discuss the situation for the related Smirnov class.

Using the tools of Sz.-Nagy–Foias theory of contractions, we describe in detail the invariant subspaces of the operator S ⊕ S * , where S is the unilateral shift on a Hilbert space. This answers a question of Câmara and Ross.

In this note, we study defect operators in the case of holomorphic functions of the unit ball of ℂ n . These operators are built from weighted Bergman kernel with a holomorphic vector. We obtain a description of sub-Hilbert spaces and we give a sufficient condition so that theses spaces are the same.

We present a survey of some recent results concerning joint numerical ranges of n -tuples of Hilbert space operators, accompanied with several new observations and remarks. Thereafter, numerical ranges techniques will be applied to various problems of operator theory. In particular, we discuss problems concerning orbits of operators, diagonals of operators and their tuples, and pinching problems. Lastly, motivated by known results on the numerical radius of a single operator, we examine whether, given bounded linear operators T 1 , . . ., Tn on a Hilbert space H , there exists a unit vector x ∈ H such that | 〈 Tjx , x 〉 | is “large” for all j = 1, . . . , n .

The author has computed the bounds of the Hilbert operator on some sequence spaces [18, 19]. Through this study the author has investigated the bounds of operators on the Hilbert sequence space and the present study is a complement of those previous research.

We give a survey on approximation numbers of composition operators on the Hardy space, on the disk and on the polydisk, and add corresponding new results on their entropy numbers, revealing how they are different.

In this paper, we prove our main result that the Li-Yorke chaotic eigen set of a positive integer multiple of the backward shift operator on ℓ 2 (𝕅) is a disk in the complex plane 𝔺 and the union of such Li-Yorke chaotic eigen set’s is the whole complex plane 𝔺.