Volume 4, Issue 1

In this note, we improve the idea of the Tsallis entropy in a complex domain. This improvement is contingent on the fractional operator in a complex domain (type Alexander). We clarify some new classes of analytic functions, which are planned in view of the geometry function theory. This category of entropy is called fractional entropy; accordingly, we demand them fractional entropic geometry classes. Other geometric properties are established in the sequel. Our exhibition is supported by the Maxwell Lemma and Jack Lemma.

Let Ω be an open simply connected proper subset of the complex plane and φ an analytic self map of Ω. If f is in the Hardy-Smirnov space defined on Ω, then the operator that takes f to f º φ is a composition operator. We show that for any Ω, analytic self maps that induce bounded Hermitian composition operators are of the form Φ(w) = aw + b where a is a real number. For ceratin Ω, we completely describe values of a and b that induce bounded Hermitian composition operators.

In this paper we consider the map L defined on the Bergman space La2(𝔺+)$L_a^2({{\rm\mathbb{C}}_{\rm{ + }}})$ of the right half plane ℂ + by (Lf)(w)=πM′(w)∫𝔺+(fM′)(s)|bw(s)|2dA˜(s)$(Lf)(w) = \pi M'(w)\int\limits_{{{\rm\mathbb{C}}_{\rm{ + }}}} {\left( {{f \over {M'}}} \right)} (s){\left| {{b_w}(s)} \right|^2}d\tilde A(s)$ where bw¯(s)=1π1+w1+w2Rew(s+w)2${b_{\bar w}}(s) = {1 \over {\sqrt \pi }}{{1 + w} \over {1 + w}}{{2{\mathop{Re}\nolimits} w} \over {{{(s + w)}^2}}}$, s ∈ ℂ + and Ms=1-s1+s$Ms = {{1 - s} \over {1 + s}}$. We show that L commutes with the weighted composition operators Wa , a ∈ 𝔻 defined on La2(𝔺+)$L_a^2({{\rm\mathbb{C}}_{\rm{ + }}})$, as Waf=(f∘ta)M′M′∘ta${W_a}f = (f \circ {t_a}){{M'} \over {M' \circ {t_a}}}$, f∈La2(𝔺+)$f \in L_a^2(\mathbb{C_ + })$ . Here ta(s)=-ids+(1-c)(1+c)s+id$${t_a}(s) = {{ - ids + (1 - c)} \over {(1 + c)s + id}}, if a = c + id ∈ 𝔻 c , d ∈ ℝ. For a ∈ 𝔻, define Va:La2(𝔺+)→La2(𝔺+)${V_a}:L_a^2({{\mathbb{C}}_{\rm{ + }}}) \to L_a^2({{\mathbb{C}}_{\rm{ + }}})$by ( V a g)(s) = ( g ∘ t a )(s)la(s) where la(s)=1-|a|2((1+c)s+id)2$la(s) = {{1 - {{\left| a \right|}^2}} \over {{{((1 + c)s + id)}^2}}}$.We look at the action of the class of unitary operators Va , a ∈ 𝔻 on the linear operator L . We establish that Lˆ = L where L⌢=∫𝔻VaLVadA(a)$\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over L} = \int\limits_{\mathbb{D}} {{V_a}L{V_a}dA(a)}$ and dA is the area measure on 𝔻. In fact the map L satisfies the averaging condition L˜(w1)=∫DL˜(ta¯(w1))dA(a),for all w1∈C+$$\tilde L({w_1}) = \int\limits_{\mathbb{D}} {\tilde L({t_{\bar a}}({w_1}))dA(a),{\rm{for all }}{w_1} \in {{\rm{C}}_{\rm{ + }}}}$$ where L˜(w1)=〈Lbw¯1,bw¯1〉$\tilde L({w_1}) = \left\langle {L{b_{{{\bar w}_1}}},{b_{{{\bar w}_1}}}} \right\rangle$.

Some trace inequalities of Shisha-Mond type for operators in Hilbert spaces are provided. Applications in connection to Grüss inequality and for convex functions of selfadjoint operators are also given.

We consider the infinite Jacobi block matrices in the completely indeterminate case, i. e. such that the deficiency indices of the corresponding Jacobi operators are maximal. For such matrices, some criteria of complete indeterminacy are established. These criteria are similar to several known criteria of indeterminacy of the Hamburger moment problem in terms of the corresponding scalar Jacobi matrices and the related systems of orthogonal polynomials.

Quantitative versions of weighted estimates obtained by F. Ruiz and J.L. Torrea [30, 31] for the operator are obtained. As a consequence, some sufficient conditions for the boundedness of Min the two weight setting in the spirit of the results obtained by C. Pérez and E. Rela [26] and very recently by M. Lacey and S. Spencer [17] for the Hardy-Littlewood maximal operator are derived. As a byproduct some new quantitative estimates for the Poisson integral are obtained.

Using the Kato-Rosenblum theorem, we describe the absolutely continuous spectrum of a class of weighted integral Hankel operators in L 2 (ℝ + ). These self-adjoint operators generalise the explicitly diagonalisable operator with the integral kernel s α t α (s + t) -1-2α , where α > -1/2. Our analysis can be considered as an extension of J. Howland’s 1992 paper which dealt with the unweighted case, corresponding to α = 0.

This paper is selective survey on the space l A p and its multipliers. It also includes some connections of multipliers to Birkhoff-James orthogonality

Being expected as a Banach space substitute of the orthogonal projections on Hilbert spaces, generalized n-circular projections also extend the notion of generalized bicontractive projections on JB*-triples. In this paper, we study some geometric properties of JB*-triples related to them. In particular, we provide some structure theorems of generalized n-circular projections on an often mentioned special case of JB*-triples, i.e., Hilbert C*-modules over abelian C*-algebras C 0 (Ω).

Wiener-Hopf factorisation plays an important role in the theory of Toeplitz operators. We consider here Toeplitz operators in the Hardy spaces H p of the upper half-plane and we review how their Fredholm properties can be studied in terms of a Wiener-Hopf factorisation of their symbols, obtaining necessary and sufficient conditions for the operator to be Fredholm or invertible, as well as formulae for their inverses or one-sided inverses when these exist. The results are applied to a class of singular integral equations in L −1 (ℝ)