On partial isometries with circular numerical range

In their LAMA 2016 paper Gau, Wang and Wu conjectured that a partial isometry A acting on ℂ n cannot have a circular numerical range with a non-zero center, and proved this conjecture for n ≤ 4. We prove it for operators with rank A = n − 1 and any n. The proof is based on the unitary similarity of A to a compressed shift operator SB generated by a finite Blaschke product B. We then use the description of the numerical range of SB as intersection of Poncelet polygons, a special representation of Blaschke products related to boundary interpolation, and an explicit formula for the barycenter of the vertices of Poncelet polygons involving elliptic functions.

The dual of the space of bounded operators on a Banach space

Given Banach spaces X and Y, we ask about the dual space of the 𝒧(X, Y). This paper surveys results on tensor products of Banach spaces with the main objective of describing the dual of spaces of bounded operators. In several cases and under a variety of assumptions on X and Y, the answer can best be given as the projective tensor product of X ** and Y *.

Continuous embedding between P-de Branges spaces

In this paper we study the continuity of the embedding operator ℓ : ℋ p (E) ↪ ℋ q (E) when 0 < p < q ⩽ ∞. The necessary and sufficient condition has already been described in [10] if p > 1. In this work, we address the problem when p = 1, using a new approach, but asking some additional hypothesis about the Hermite-Biehler function E. We give also a different proof for the case p > 1.

On (m, P)-expansive operators: products, perturbation by nilpotents, Drazin invertibility

A generalisation of m-expansive Hilbert space operators T ∈ B(ℋ) [18, 20] to Banach space operators T ∈ B(𝒳) is obtained by defining that a pair of operators A, B ∈ B(𝒳) is (m, P)-expansive for some operator P ∈ B(𝒳) if Δ A,B m (P)= ( I - L A R B ) m ( P ) = ∑ j = 0 m ( - 1 ) j ( j m ) {\left( {I - {L_A}{R_B}} \right)^m}\left( P \right) = \sum\nolimits_{j = 0}^m {{{\left( { - 1} \right)}^j}\left( {_j^m} \right)} AjPBj ≤0; LA(X) = AX and RB(X)=XB. Unlike m-isometric and m-left invertible operators, commuting products and perturbations by commuting nilpotents of (m, I)-expansive operators do not result in expansive operators: using elementary algebraic properties of the left and right multiplication operators, a sufficient condition is proved. For Drazin invertible A and B ∈ B(ℋ), with Drazin inverses Ad and Bd, a sufficient condition proving (Ad, Bd) ^ (A, B) is (m − 1, P)-isometric (resp., (m − 1, P)-contractive) for m even (resp., m odd) is given, and a Banach space analogue of this result is proved.

Cesàro and Abel ergodic theorems for integrated semigroups

Let {S(t)}t ≥ 0 be an integrated semigroup of bounded linear operators on the Banach space 𝒳 into itself and let A be their generator. In this paper, we consider some necessary and sufficient conditions for the Cesàro mean and the Abel average of S(t) converge uniformly on ℬ(𝒳). More precisely, we show that the Abel average of S(t) converges uniformly if and only if 𝒳 = ℛ(A) ⊕ 𝒩(A), if and only if ℛ(Ak) is closed for some integer k and ∥ λ 2 R(λ, A) ∥ → 0 as λ→ 0+, where ℛ(A), 𝒩(A) and R(λ, A), be the range, the kernel, the resolvent function of A, respectively. Furthermore, we prove that if S(t)/t 2 → 0 as t → 1, then the Cesàro mean of S(t) converges uniformly if and only if the Abel average of S(t) is also converges uniformly.

Besov-type spaces for the κ-Hankel wavelet transform on the real line

In this paper, we shall introduce functions spaces as subspaces of Lp κ (ℝ) that we call Besov-κ-Hankel spaces and extend the concept of κ-Hankel wavelet transform in Lp κ(ℝ) space. Subsequently we will characterize the Besov-κ-Hankel space by using κ-Hankel wavelet coefficients.

Fractional Cesàro Matrix and its Associated Sequence Space

In this research, we introduce a new fractional Cesàro matrix and investigate the topological properties of the sequence space associated with this matrix.We also introduce a fractional Gamma matrix aswell and obtain some factorizations for the Hilbert operator based on Cesàro and Gamma matrices. The results of these factorizations are two new inequalities one ofwhich is a generalized version of thewell-known Hilbert’s inequality. There are also some challenging problems that authors share at the end of the manuscript and invite the researcher for trying to solve them.

The Numerical Range of C* ψ Cφ and Cφ C* ψ

In this paper we investigate the numerical range of C* bφ m Caφ n and Caφ n C* bφ m on the Hardy space where φ is an inner function fixing the origin and a and b are points in the open unit disc. In the case when |a| = |b| = 1 we characterize the numerical range of these operators by constructing lacunary polynomials of unit norm whose image under the quadratic form incrementally foliate the numerical range. In the case when a and b are small we show numerical range of both operators is equal to the numerical range of the operator restricted to a 3-dimensional subspace.

M-hypercyclicity of C 0-semigroup and Svep of its generator

Let 𝒯 = (Tt ) t ≥0 be a C 0-semigroup on a separable infinite dimensional Banach space X, with generator A. In this paper, we study the relationship between the single valued extension property of generator A, and the M-hypercyclicity of the C 0-semigroup. Specifically, we prove that if A does not have the single valued extension property at λ ∈ iℝ, then there exists a closed subspace M of X, such that the C 0-semigroup 𝒯 is M-hypercyclic. As a corollary, we get certain conditions of the generator A, for the C 0-semigroup to be M-hypercyclic.

Conformable differential operators for meromorphically multivalent functions

We define a conformable diff-integral operator for a class of meromorphically multivalent functions. We show that this conformable operator adheres to the semigroup property. We then use the subordination properties to prove inclusion conditions, sufficienrt inclusion conditions and convolution properties for this class of conformable operators.