We introduce a sequence of Hankel style operators $H^k$, k = 1,2,3,..., which act on the Bergman space of the unit disk. These operators are intermediate between the classical big and small Hankel operators. We study the boundedness and Schatten-von Neumann properties of the $H^k$ and show, among other things, that $H^k$ are cut-off at 1/k. Recall that the big Hankel operator is cut-off at 1 and the small Hankel operator at 0.

We study the representation of orthogonally additive mappings acting on Hilbert C*-modules and Hilbert H*-modules. One of our main results shows that every continuous orthogonally additive mapping f from a Hilbert module W over 𝓚(𝓗) or 𝓗𝓢(𝓗) to a complex normed space is of the form f(x) = T(x) + Φ(⟨x,x⟩) for all x ∈ W, where T is a continuous additive mapping, and Φ is a continuous linear mapping.

On donne des conditions larges sur un champ de normes symplectiques sur ${\mathbf{R}}^{2n}$ pour que les opérateurs d’ordre zéro associés opèrent sur $L^2({\mathbf{R}}^n)$; les éléments de cet espace se laissent alors écrire comme somme d’états propres, de niveau d’énergie bornée, de la famille d’oscillateurs harmoniques associée.

We survey results concerning the L2 boundedness of oscillatory and Fourier integral operators and discuss applications. The article does not intend to give a broad overview; it mainly focuses on topics related to the work of the authors.[Proceedings of the 6th International Conference on Harmonic Analysis and Partial Differential Equations, El Escorial (Madrid), 2002].

Ostrowski-Kantorovich theorem of Halley method for solving nonlinear operator equations in Banach spaces is presented. The complete expression of an upper bound for the method is given based on the initial information. Also some properties of $S$-order of convergence and sufficient asymptotic error bound will be discussed.

Some Ostrowski’s type inequalities for the Riemann-Stieltjes integral ${\int }_a^{b}f\left(e^{it}\right)du\left(t\right)$ of continuous complex valued integrands $f:𝒞\left(0,1\right)\to ℂ$ defined on the complex unit circle $𝒞\left(0,1\right)$ and various subclasses of integrators $u:\left[a,b\right]\subseteq \left[0,2\pi \right]\to ℂ$ of bounded variation are given. Natural applications for functions of unitary operators in Hilbert spaces are provided as well.

An exact criterion is derived for an operator valued weight function $W(e^{is},e^{it})$ on the torus to have a factorization $W(e^{is},e^{it})=\Phi (e^{is},e^{it})*\Phi (e^{is},e^{it})$, where the operator valued Fourier coefficients of Φ vanish outside of the Helson-Lowdenslager halfplane $\Lambda =(m,n)\in {\mathbb{Z}}^2:m\ge 1\cup (0,n):n\ge 0$, and Φ is “outer” in a related sense. The criterion is expressed in terms of a regularity condition on the weighted space $L^2\left(W\right)$ of vector valued functions on the torus. A logarithmic integrability test is also provided. The factor Φ is explicitly constructed in terms of Toeplitz operators...