Some properties of analytic maps of operators inJ *-algebras

HereJ ^*-algebras are considered, i.e. linear spaces of operators mapping one complex Hilbert space into another, which have a kind of Jordan triple product structure. Balls are determined which contain the sets of values of functionalsf(S) (S any fixed operator) defined on the classes of (Fréchet-)holomorphic mapsf of the unit ball into the generalized upper half-plane and of the unit ball into the unit ball, respectively (see Theorems 1 and 2). Similar results were obtained for holomorphic maps of operators in the sense of functional calculus (see Theorems 3–5).     

Some properties of composition operators on entire Dirichlet series with real frequencies

In this Note we consider some problems for composition operators on a class of entire Dirichlet series with real frequencies in the complex plane whose Ritt order is zero and logarithmic orders are finite. Criteria for action and boundedness of such operators are given.     

Some properties of composition operators on Hilbert spaces of Dirichlet series

In this paper, we study some properties of composition operators on Hilbert spaces of Dirichlet series, which include the Fredholmness, Hilbert-Schmidtness, spectra, cyclic and hypercyclic phenomenons, and also answer a norm question raised by Cowen and MacCluer.     

Some properties of unbounded operators with closed range

Let H ₁, H ₂ be Hilbert spaces and T be a closed linear operator defined on a dense subspace D(T) in H ₁ and taking values in H ₂. In this article we prove the following results:

Some spectral properties of differential operators

Mathematical Notes -     

Some properties of bivariate Bernstein-Schoenberg operators

The bivariate Bernstein-Schoenberg operatorV _T of degreem, introduced in [5], is a spline approximation operator that generalizes the Bernstein polynomial operatorB _m . It is shown here that for a convex functionf,f≤V _T (f)≤B _m (f). This result is then used to show that for a twice differentiable functiong, the asymptotic error limm(V _T (g)-g) depends only on the asymptotic error for quadratic polynomials. The latter is evaluated explicitly in the special circumstances thatV _T is, in a sense, asymptotically close toB _m .     

Compactness and invariance properties of evolution operators associated with Kolmogorov operators with unbounded coefficients

In this paper we consider nonautonomous elliptic operators A with nontrivial potential term defined in I×Rd, where I is a right-halfline (possibly I=R). We prove that we can associate an evolution operator (G(t,s)) with A in the space of all bounded and continuous functions on Rd. We also study the compactness properties of the operator G(t,s). Finally, we provide sufficient conditions guaranteeing that each operator G(t,s) preserves the usual Lp-spaces and C₀(Rd).     

Boundary vs. interior conditions associated with weighted composition operators

Associated with some properties of weighted composition operators on the spaces of bounded harmonic and analytic functions on the open unit disk $\mathbb{D}$ , we obtain conditions in terms of behavior of weight functions and analytic self-maps on the interior $\mathbb{D}$ and on the boundary $\partial \mathbb{D}$ 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.     

Some spectral properties ofm-accretive differential operators

Form-accretive operatorT=T ₀+q compactness of its resolventR(z, T), zP(T) and ofR(z, T)–R(z, T ₀),zP(T)P(T ₀) under suitable assumptions onT ₀ and their perturbationq is established. This result is used in the study of spectral properties ofT andT ₀.     

Some properties of graphs of closed operators

In this paper we study some properties of graphs of closed operators in Hilbert spaces. We construct representations of von Neumann algebras induced by graphs of closed operators. We describe some classes of closed operators in terms of their characteristic matrices and study some properties of operations on graphs of closed operators.     

Some properties of operators of Abel-Poisson type

We obtain asymptotic estimates for approximations of certain classes of continuous and differentiable functions by operators $$A_{\gamma , r} (f;x) = \tfrac{1}{\pi }\int_{ - \pi }^\pi {f(x + t)} \left( {\tfrac{1}{2} + \sum\nolimits_{k = 1}^\infty {r^{k^\gamma } cor kt} } \right)dt$$ for γ=1 and 2.