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We investigate the boundedness and compactness properties of integral operators, Volterra operators, and generalized Volterra operators between the Dales–Davie algebras. Additionally, we study the analogous properties of these operators when applied to the Lipschitz versions of Dales–Davie algebras.

We consider a class of Volterra cubic stochastic operators. We describe the set of fixed points, the invariant sets and construct several Lyapunov functions to use them in the study of the asymptotical behavior of the given Volterra cubic stochastic operators. A complete description of the set of limit points is given, and we show that such operators have the ergodic property.

Integral equations frequently appear in many mechanics problems. Several of them are grouped based on the location of an unknown function or the integration interval. Here we have a boundary problem that will be rearranged into Volterra integral equation. From this equation, the integral operator (namely the Volterra operator) will be developed by determining the kernel. Then the eigenvalues of this operator will be sought.

In this expository article, we discuss the evaluation and estimation of the operator norms of various functions of the Volterra operator.

This paper introduces an efficient numerical algorithm for solving a significant class of linear and nonlinear time-fractional partial differential equation governed by Fredholm–Volterra operator in the sense of Robin conditions. A direct approach based on the normalized orthonormal function systems that fitted from the Gram–Schmidt orthogonalization process is utilized to transcribe the problem under study into appropriate Hilbert space. Some functional analysis theories such as upper error bound and convergence behavior under some assumptions which give the hypothetical premise of the proposed calculation are likewise talked about. Mathematical properties of the numerical results obtained are analyzed as well as general features of many numerical solutions have been identified. At long last, the used outcomes demonstrate that the present calculation and mimicked toughening give a decent planning procedure to such models.

It is a classical theorem of Sarason that an analytic function of bounded mean oscillation (BMOA) is of vanishing mean oscillation if and only if its rotations converge in norm to the original function as the angle of the rotation tends to zero. In a series of two papers, Blasco et al. have raised the problem of characterizing all semigroups of holomorphic functions (_t) that can replace the semigroup of rotations in Sarason’s theorem. We give a complete answer to this question, in terms of a logarithmic vanishing oscillation condition on the infinitesimal generator of the semigroup (_t) . In addition, we confirm the conjecture of Blasco et al. that all such semigroups are elliptic. We also investigate the analogous question for the Bloch and the little Bloch spaces, and surprisingly enough, we find that the semigroups for which the Bloch version of Sarason’s theorem holds are exactly the same as in the BMOA case.

For every α ∈ ( 0 , + ∞ ) and p , q ∈ ( 1 , + ∞ ) let T α be the operator L p [ 0 , 1 ] → L q [ 0 , 1 ] defined via the equality ( T α f ) ( x ) : = ∫ 0 x α f ( y ) d y . We study the norms of T α ∗ for every p , q . In the case p = q we further study its spectrum, point spectrum, eigenfunctions, and the norms of its iterates. Moreover, for the case p = q = 2 we determine the point spectrum and eigenfunctions for T α ∗ T α ∗ , where T α ∗ is the adjoint operator.

We study the real and imaginary parts of the powers of the Volterra operator on L 2 [ 0 , 1 ] , specifically their eigenvalues, their norms and their numerical ranges.

For a quasinilpotent operator T on a separable Hilbert space H , Douglas and Yang define k x = lim sup λ → 0 ln ‖ ( λ - T ) - 1 x ‖ ln ‖ ( λ - T ) - 1 ‖ for each nonzero vector x , and call Λ ( T ) = { k x : x ≠ 0 } the power set of T . In this paper, we prove that Λ ( T ) is right closed, that is, sup σ ∈ Λ ( T ) for each nonempty subset σ of Λ ( T ) . Moreover, for any right closed subset σ of [0, 1] containing 1, we show that there exists a quasinilpotent operator T with Λ ( T ) = σ . Finally, we prove that the power set of V , the Volterra operator on L 2 [ 0 , 1 ] , is (0, 1].