The Pommiez operator (∆ f )(z) = ( f (z)− f (0))/z is considered in the space (G) of the holomorphic functions in an arbitrary finite Runge domain G. A new proof of a representation formula of Linchuk of the commutant of ∆ in (G) is given. The main result is a representation formula of the commutant of the Pommiez operator in an arbitrary invariant hyperplane of (G). It uses an explicit convolu...

Abstract: The main aim of this paper is to prove that the maximal operator of a subsequence of the (one-dimensional) logarithmic means of Vilenkin-Fourier series is of weak type (1,1). Moreover, we prove that the maximal operator of the logarithmic means of quadratical partial sums of double Vilenkin-Fourier series is of weak type (1,1), provided that the supremum in the maximal operator is tak...

Let A be a linear bounded operator in a Hilbert space H, N(A) and R(A) its null-space and range, and A∗ its adjoint. The operator A is called Fredholm iff dim N(A) = dim N(A∗) := n < ∞ and R(A) and R(A∗) are closed subspaces of H. A simple and short proof is given of the following known result: A is Fredholm iff A = B + F , where B is an isomorphism and F is a finite-rank operator. The proof co...

In general, a symmetric operator does not have a unique spectral function. If T is a symmetric operator and F(t) is a spectral function for T, by the theory of Naimark (see Akhiezer and Glazman [1]) there exists a Hilbert space H of which H is a subspace and a self-adjoint operator T onif with resolution of the identity E(t) such that T + is an extension of T and F(t) = P£(<) where P is the pro...

In this work, an iterative method based on a matrix form of LSQR algorithm is constructed for solving the linear operator equation A(X) = B and the minimum Frobenius norm residual problem ||A(X)−B||F where X ∈ S := {X ∈ Rn×n | X = G(X)}, F is the linear operator from Rn×n onto Rr×s, G is a linear selfconjugate involution operator and B ∈ Rr×s. Numerical examples are given to verify the efficien...

The purpose of this paper is to prove existence and uniqueness theorem for solving an operator equation of the form F(x)+ G(x) = 0, where F is a Gâteaux differentiable operator and G is a Lipschitzian operator defined on an open convex subset of a Banach space. Our result extends and improves the previously known results in recent literature. 2010 Mathematics Subject Classification: 49M15, 65K10

We review the construction of minimally bosonized supersymmetric quantum mechanics and its relation to hidden supersymmetries in pure parabosonic (parafermionic) systems. The simplest N = 1 supersymmetric quantum mechanical system is the superoscillator [1] given by the Hamiltonian H = 1 2 {b, b−}+ 1 2 [f, f−]. (1) In addition to H , the system is characterized by the two conserved integrals of...

Let Dn be the unit polydisc of Cn. The class of all holomorphic functions with domain Dn will be denoted by H(Dn). Let φ be a holomorphic self-map of Dn, the composition operator Cφ induced by φ is defined by (Cφ f )(z) = f (φ(z)) for z ∈Dn and f ∈H(Dn). If, in addition, ψ is a holomorphic function defined on Dn, the weighted composition operator ψCφ induced by ψ and φ is defined by ψCφ(z) = ψ(...

(with DF = 1 for F = P). As explained in [2], essential properties required on the operators DF are ”locality” and ”computability”. At each face F of P, the operator DF should depend only of the translation class modulo Z of the normal cone No(P, F ) to P at a generic point of F (if F is of codimension k, the normal cone is an affine cone of dimension k). In particular if P is an integral polyt...

In the paper titled as “Jackson-type inequality on the sphere” 2004 , Ditzian introduced a spherical nonconvolution operator Ot,r , which played an important role in the proof of the wellknown Jackson inequality for spherical harmonics. In this paper, we give the lower bound of approximation by this operator. Namely, we prove that there are constants C1 and C2 such that C1ω2r f, t p ≤ ‖Ot,rf − ...