This paper surveys a number of recent results obtained by C. Bereanu and the author in existence results for second order differential equations of the form (ϕ(u'))' = f(t,u,u') submitted to various boundary conditions. In the equation, ϕ: ℝ → ≤ ]-a,a[ is a homeomorphism such that ϕ(0) = 0. An important motivation is the case of the curvature operator, where ϕ(s) = s/√(1+s²). The problems are reduced to fixed point problems in suitable function space, to which Leray-Schauder...

The theory of quasimultipliers in Banach algebras is developed in order to provide a mechanism for defining the boundary values of analytic semigroups on a sector in the complex plane. Then, some methods are presented for deriving lower estimates for operators defined in terms of quasinilpotent semigroups using techniques from the theory of complex analysis.

Associated with some properties of weighted composition operators on the spaces of bounded harmonic and analytic functions on the open unit disk $$𝔻$$ , we obtain conditions in terms of behavior of weight functions and analytic self-maps on the interior $$𝔻$$ and on the boundary $$\partial 𝔻$$ respectively. We give direct proofs of the equivalence of these interior and boundary conditions. Furthermore we give another proof of the estimate for the essential norm of the difference of weighted composition operators.

A normal Banach quasi *-algebra (,) has a distinguished Banach *-algebra ${}_b$ consisting of bounded elements of . The latter *-algebra is shown to coincide with the set of elements of having finite spectral radius. If the family () of bounded invariant positive sesquilinear forms on contains sufficiently many elements then the Banach *-algebra of bounded elements can be characterized via a C*-seminorm defined by the elements of ().

We continue our study of topological partial *-algebras, focusing on the interplay between various partial multiplications. The special case of partial *-algebras of operators is examined first, in particular the link between strong and weak multiplications, on one hand, and invariant positive sesquilinear (ips) forms, on the other. Then the analysis is extended to abstract topological partial *-algebras, emphasizing the crucial role played by appropriate bounded elements, called ℳ-bounded. Finally,...

Characterizations of pairs (E,F) of complete (LF)?spaces such that every continuous linear map from E to F maps a 0?neighbourhood of E into a bounded subset of F are given. The case of sequence (LF)?spaces is also considered. These results are similar to the ones due to D. Vogt in the case E and F are Fréchet spaces. The research continues work of J. Bonet, A. Galbis, S. Önal, T. Terzioglu and D. Vogt.

Let v be a standard weight on the upper half-plane , i.e. v: → ]0,∞[ is continuous and satisfies v(w) = v(i Im w), w ∈ , v(it) ≥ v(is) if t ≥ s > 0 and $li{m}_{t\to 0}v\left(it\right)=0$. Put v₁(w) = Im wv(w), w ∈ . We characterize boundedness and surjectivity of the differentiation operator D: Hv() → Hv₁(). For example we show that D is bounded if and only if v is at most of moderate growth. We also study composition operators on Hv().

Let T be a multicyclic operator defined on some Banach space. Bounded point evaluations and analytic bounded point evaluations for T are defined to generalize the cyclic case. We extend some known results on cyclic operators to the more general setting of multicyclic operators on Banach spaces. In particular we show that if T satisfies Bishop’s property (β), then ${ℬ}_a=ℬ\setminus {\sigma }_{ap}\left(T\right)$. We introduce the concept of analytic structures and we link it to different spectral quantities. We apply this concept to retrieve...

Let $M_{m,n}$ be the space of all complex m × n matrices. The generalized unit disc in $M_{m,n}$ is >br>    $R_{m,n}=Z\in M_{m,n}:I^{\left(m\right)}-ZZ*ispositivedefinite$. Here $I^{\left(m\right)}\in M_{m,m}$ is the unit matrix. If 1 ≤ p < ∞ and α > -1, then $L_{\alpha }^p\left(R_{m,n}\right)$ is defined to be the space $L^p{R}_{m,n};{\left[det(I^{\left(m\right)}-ZZ*)\right]}^{\alpha }d{\mu }_{m,n}\left(Z\right)$, where ${\mu }_{m,n}$ is the Lebesgue measure in $M_{m,n}$, and $H_{\alpha }^p\left(R_{m,n}\right)\subset L_{\alpha }^p\left(R_{m,n}\right)$ is the subspace of holomorphic functions. In [8,9] M. M. Djrbashian and A. H. Karapetyan proved that, if $Re\beta >(\alpha +1)/p-1$ (for 1 < p < ∞) and Re β ≥ α (for p = 1), then     $f\left(\right)=T_{m,n}^{\beta }\left(f\right)\left(\right),\in R_{m,n},$where $T_{m,n}^{\beta }$ is the integral operator defined by (0.13)-(0.14). In the present paper, given 1 ≤ p <...