Volume 9, Issue 1

A recent work derived expressions for the induced p -norm of a special class of circulant matrices A ( n , a , b ) ∈ ℝ n × n , with the diagonal entries equal to a ∈ ℝ and the off-diagonal entries equal to b ≥ 0. We provide shorter proofs for all the results therein using Fourier analysis. The key observation is that a circulant matrix is diagonalized by a DFT matrix. The results comprise an exact expression for ǁ A ǁ p , 1 ≤ p ≤ ∞, where A = A ( n , a , b ), a ≥ 0 and for ǁ A ǁ 2 where A = A ( n , − a , b ), a ≥ 0; for the other p -norms of A ( n , − a , b ), 2 < p < ∞, upper and lower bounds are derived.

The aim of this paper is to carry out an analysis of five trascendental operators acting on the space of slice regular functions, namely *-exponential, *-sine and *-cosine and their hyperbolic analogues. The first three of them were introduced by Colombo, Sabadini and Struppa and some features of *-exponential were investigated in a previous paper by Altavilla and the author. We show how exp * ( f ), sin * ( f ), cos * ( f ), sinh * ( f ) and cosh * ( f ) can be written in terms of the real and the vector part of the function f and we examine the relation between cos * and cosh * when the domain Ω is product and when it is slice. In particular we prove that when Ω is slice, then cos * ( f ) = cosh * ( f * I ) holds if and only if f is ℂ I preserving, while in the case Ω is product there is a much larger family of slice regular functions for which the above relation holds.

In this survey, we present several results related to characterizing the surjective isometries on Banach sequence spaces. Our survey includes full proofs of these characterizations for the classical spaces as well as more recent results for combinatorial Banach spaces and Tsirelson-type spaces. Along the way, we pose many open problems related to the structure of the group of surjective isometries for various Banach spaces.

For p ∈ (1, ∞) \ {2}, some properties of the space ℳ p of multipliers on ℓ p A are derived. In particular, the failure of the weak parallelogram laws and the Pythagorean inequalities is demonstrated for ℳ p . It is also shown that the extremal multipliers on the ℓ p A spaces are exactly the monomials, in stark contrast to the p = 2 case.

This paper explores the long journey from projective tensor products of a pair of Banach spaces, passing through the definition of nuclear operators still on the realm of projective tensor products, to the of notion of trace-class operators on a Hilbert space, and shows how and why these concepts (nuclear and trace-class operators, that is) agree in the end.

In this paper, we improve some numerical radius inequalities for Hilbert space operators under suitable condition. We also compare our results with some known results. As application of our result, we obtain an operator inequality.

Previously, spectra of certain weighted composition operators W ѱ, φ on H 2 were determined under one of two hypotheses: either φ converges under iteration to the Denjoy-Wolff point uniformly on all of 𝔻 rather than simply on compact subsets, or φ is “essentially linear fractional.” We show that if φ is a quadratic self-map of 𝔻 of parabolic type, then the spectrum of W ѱ, φ can be found when these maps exhibit both of the aforementioned properties, and we determine which symbols do so.

Let ( X , 𝒜, μ ) be a σ−finite measure space. A transformation ϕ : X → X is non-singular if μ ∘ ϕ −1 is absolutely continuous with respect with μ . For this non-singular transformation, the composition operator C ϕ : 𝒟( C ϕ ) → L 2 ( μ ) is defined by C ϕ f = f ∘ ϕ , f ∈ 𝒟( C ϕ ). For a fixed positive integer n ≥ 2, basic properties of product C ϕn · · · C ϕ 1 in L 2 ( μ ) are presented in Section 2, including the boundedness and adjoint. Under the assistance of these properties, normality and quasinormality of specific bounded C ϕn · · · C ϕ 1 in L 2 ( μ ) are characterized in Section 3 and 4 respectively, where C ϕ 1 , C ϕ 2 , · · ·, C ϕn are all densely defined.

Over the past several years, optimal polynomial approximants (OPAs) have been studied in many different function spaces. In these settings, numerous papers have been devoted to studying the properties of their zeros. In this paper, we introduce the notion of optimal polynomial approximant in the space L p , 1 ≤ p ≤ ∞. We begin our treatment by showing existence and uniqueness for 1 < p < ∞. For the extreme cases of p = 1 and p = ∞, we show that uniqueness does not necessarily hold. We continue our development by elaborating on the special case of L 2 . Here, we create a test to determine whether or not a given 1st degree OPA is zero-free in ̄𝔻. Afterward, we shed light on an orthogonality condition in L p . This allows us to study OPAs in L p with the additional tools from the L 2 setting. Throughout this paper, we focus many of our discussions on the zeros of OPAs. In particular, we show that if 1 < p < ∞, f ∈ H p , and f (0) ≠ 0, then there exists a disk, centered at the origin, in which all the associated OPAs are zero-free. Toward the end of this paper, we use the orthogonality condition to compute the coefficients of some OPAs in L p . To inspire further research in the general theory, we pose several open questions throughout our discussions.

Let A be a bounded, injective and self-adjoint linear operator on a complex separable Hilbert space. We prove that there is a pure isometry, V , so that AV > 0 and A is Hankel with respect to V , i.e. V * A = AV , if and only if A is not invertible. The isometry V can be chosen to be isomorphic to N ∈ ℕ ∪ {+∞} copies of the unilateral shift if A has spectral multiplicity at most N . We further show that the set of all isometries, V , so that A is Hankel with respect to V , are in bijection with the set of all closed, symmetric restrictions of A −1 .

Given Banach space operators A i , B i ( i = 1, 2), let δ i denote (the generalised derivation) δ i ( X ) = ( L Ai − R Bi )( X ) = A i X − XB i . If 0 ∈ σ a ( B i ), i = 1, 2, and if Δδ1,δ2n(I)=(Lδ1Rδ1-I)n(I)=0 \Delta _{{\delta _1},\delta 2}^n\left( I \right) = {\left( {{L_{{\delta _1}}}{R_{{\delta _1}}} - I} \right)^n}\left( I \right) = 0 , then ΔA1,A2n(I)=0 \Delta _{{A_1},A2}^n\left( I \right) = 0 . For Hilbert space pairs ( A , B ) such that 0 ∈ σ a ( B * ) and Δδ*,δn(I)=0(i.e., δ is n-isometric) \Delta _{{\delta ^*},\delta }^n\left( I \right) = 0\left( {i.e.,\,\delta \,is\,n - isometric} \right) , where δ = δ A , B and δ * = δ A * , B *, this implies ΔA*,An(I)=0 \Delta _{{A^*},A}^n\left( I \right) = 0 (and hence there exists a positivie integer m ≤ n such that A is strictly m -isometric). If Δδ*,δn(I)=0 \Delta _{{\delta ^*},\delta }^n\left( I \right) = 0 , then there exists a scalar λ such that 0 ∈ σ a (( B − λ I ) * ) and, given δ is strictly n -isometric, there exists a positive integer m ≤ n such that A − λ I is strictly m -isometric. Furthermore, there exist decompositions ℋ = ℋ 1 ⊕ ℋ 2 and ℋ = ℋ 11 ⊕ ℋ 22 of ℋ and t i -nilpotent operators N i ( i = 1, 2) such that either A − λI = αI + N 1 and B − λI = (0 I | ℋ 1 ⊕ 2 e it I | ℋ2 ) + N 2 , or, A − λI = αI + N 1 , α = e it , 0 ≤ t < 2 π , and B − λI = (0 I |ℋ 11 ⊕ 2 e it I |ℋ 22 ) + N 2 , or, A − λ = ( α 1 I |ℋ 1 ⊕ α 2 I |ℋ 2 ) + N 1 and B λ = (0I|ℋ 11 ⊕ μI |ℋ 22 ) + N 2 , where μ = e it | μ |, 0 ≤ t < 2 π , 0 < | μ | < 2, α1=eit| μ |+i4-| μ |22 {\alpha _1} = {e^{it}}{{\left| \mu \right| + i\sqrt {4 - {{\left| \mu \right|}^2}} } \over 2} and α2=eit| μ |-i4-| μ |22 {\alpha _2} = {e^{it}}{{\left| \mu \right| - i\sqrt {4 - {{\left| \mu \right|}^2}} } \over 2} .

We study the cyclic, supercyclic and hypercyclic properties of a composition operator C ϕ on the Segal-Bargmann space ℋ(ℰ), where ϕ ( z ) = Az + b , A is a bounded linear operator on ℰ, b ∈ ℰ with || A || ⩽ 1 and A * b belongs to the range of ( I – A * A )½. Specifically, under some conditions on the symbol ϕ we show that if C ϕ is cyclic then A* is cyclic but the converse need not be true. We also show that if C ϕ * is cyclic then A is cyclic. Further we show that there is no supercyclic composition operator on the space ℋ(ℰ) for certain class of symbols ϕ .

In this article we continue our investigation of the Paatero space. We prove that the intersection of every Paatero class V ( k ) with every Hardy space H p is closed in that Hp and associate singular continuous measures to elements of V ( k ). There have been no examples in the literature of functions in V ( k ) with zeros in the unit disk other than the one at the origin. We close this gap in the literature. We derive a representation of the measure associated to a function in V ( k ) for functions whose derivatives are rational, or algebraic, or transcendental functions in the unit disk.Finally, we consider the notion of regulated domains, introduced by Pommerenke and show that there are regulated domains whose boundary is not of bounded boundary rotation.

Operators of type f → ψf ◦ φ acting on function spaces are called weighted composition operators. If the weight function ψ is the constant function 1, then they are called composition operators. We consider weighted composition operators acting on Hardy–Smirnov spaces and prove that their unitarily invariant properties are reducible to the study of weighted composition operators on the classical Hardy space over a disc. We give examples of such results, for instance proving that Forelli’s theorem saying that the isometries of non–Hilbert Hardy spaces over the unit disc need to be special weighted composition operators extends to all non–Hilbert Hardy–Smirnov spaces. A thorough study of boundedness of weighted composition operators is performed.