Recent results have shown that any closed operator A commuting with the backwards shift S∗ restricted to K u := H 2 ⊖ uH, where u is an inner function, can be realized as a Nevanlinna function of S∗ u := S ∗| K2 u , A = φ(S∗ u), where φ belongs to a certain class of Nevanlinna functions which depend on u. In this paper this result is generalized to show that given any contraction T of class C0(...

The paper is a short survey around the author’s recent works on topics related to complex Monge-Ampère equations and strictly pseudoconvex pseudo-Hermitian manifolds. 1. Invariant differential operators In complex analysis of one variable, the fact that the invariant property for Laplace operator under holomorphic change of coordinates plays an important role. Namely, Let φ : D1 → D2 be a holom...

The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find some unknown parameters of the model. The equation for the error of the optimal solution is derived through the statistical errors of the input data (background, observation, and model errors). A numerical algorithm is developed to construct an a posteriori covariance...

We characterize the symbols φ for which there exists a weight w such that weighted composition operator MwCφ is compact on Bergman space Bα2. also bounded but not compact. investigate when Hilbert-Schmidt

Let ∞ and c be the Banach spaces of bounded and convergent sequence x = (xk) with the usual supremum norm. Let σ be a one-to-one mapping of N, the set of positive integers, into itself and T : ∞ → ∞ a linear operator defined by Tx = (Txk)= (xσ(k)). An element φ ∈ ′ ∞, the conjugate space of ∞, is called an invariant mean or a σ-mean if and only if (i) φ(x) ≥ 0 when the sequence x = (xk) has xk ...

Let φ : ℝ(n) × [0, ∞)→[0, ∞) be a Musielak-Orlicz function and A an expansive dilation. In this paper, the authors introduce the anisotropic Hardy space of Musielak-Orlicz type, H(A)(φ)(ℝ(n)), via the grand maximal function. The authors then obtain some real-variable characterizations of H(A)(φ)(ℝ(n)) in terms of the radial, the nontangential, and the tangential maximal functions, which general...

Studying elliptic measure data problem with strongly nonlinear operator whose growth is described by the means of fully anisotropic N-function, we prove uniqueness for a broad class measures. In order to provide it, framework capacities in Orlicz–Sobolev spaces developed and capacitary characterization bounded given. Moreover, give an example Young function Φ, such that | ξ p ≲ Φ ( ) log α 1 + ...