Toeplitz Operators whose Symbols Are Borel Measures

In this paper, we are concerned with Toeplitz operators whose symbols are complex Borel measures. When a complex Borel measure μ on the unit circle is given, we give a formal definition of a Toeplitz operator T μ with symbol μ , as an unbounded linear operator on the Hardy space. We then study various properties of T μ . Among them, there is a theorem that the domain of T μ is represented by a trichotomy. Also, it was shown that if the domain of T μ contains at least one polynomial, then T μ is densely defined. In addition, we give evidence for the conjecture that T μ with a singular measure μ reduces to a trivial linear operator.

The structure of 𝓐-free measures revisited

We refine a recent result on the structure of measures satisfying a linear partial differential equation 𝓐μ = σ, μ, σ are Radon measures, considering the measure μ(x) = g(x)dx + μus(x̃)(μs(x̄) + dx̄) where x = (x̃,x̄) ∈ ℝk × ℝd−k, μus is a uniformly singular measure in x̃0 and the measure μs is a singular measure. We proved that for μus-a.e. x̃0 the range of the Radon-Nykodim derivative $\begin{array}{} \tilde{f}(\tilde{{\bf x}}_0) = \frac{d \mu_{us}}{d | \mu_{us}|}(\tilde{{\bf x}}_0) \end{array}$ is in the set ∩ξ̃∈P̃𝓚erAP̃(ξ) and, if μs is different to zero, for μs-a.e. x̄0 the range of the Radon-Nykodim derivative $\begin{array}{} \bar{f}(\bar{{\bf x}}_0) = \frac{d \mu_{s}}{d | \mu_{s}|}(\bar{{\bf x}}_0) \end{array}$ is in the set ∪ξ̄∈P̄ 𝓚erAP̄(ξ) where P̃ × P̄ = P is a manifold determined by the main symbol AP = AP̃ ⋅ AP̄ of the operator 𝓐.

Integral Representation of Functions of Bounded Variation

Functions of bounded variations form important transition between absolute continuous and singular functions. With Bainov’s introduction of impulsive differential equations having solutions of bounded variation, this class of functions had eventually entered into the theory of differential equations. However, the determination of existence of solutions is still problematic because the solutions of differential equations is usually at least absolute continuous which is disrupted by the solutions of bounded variations. As it is known, if f:[a,bλ]→Rn is of bounded variation then f is the sum of an absolute continuous function fa and a singular function fs where the total variation of fs generates a singular measure τ and fs is absolute continuous with respect to τ. In this paper we prove that a function of bounded variation f has two representations: one is f which was described with an absolute continuous part with respect to the Lebesgue measure λ, while in the other an integral with respect to τ forms the absolute continuous part and t(τ) defines the singular measure. Both representations are obtained as parameter transformation images of an absolute continuous function on total variation domain [a,bν].

Long-time behavior of a stochastic predator–prey model with modified Leslie–Gower and Holling-type II schemes

In this paper, we consider a stochastic predator–prey model with modified Leslie–Gower and Holling-type II schemes. We analyze long-time behavior of densities of the distributions of the solution. We prove that the densities can converge in [Formula: see text] to an invariant density or can converge weakly to a singular measure under appropriate conditions.