In the conformable fractional calculus, TαTβ≠TβTα and IαIβ≠IβIα, where Tα and Iα are conformable fractional differential and integral operators, respectively. Also, Tβ≠Tnα and Iβ≠Inα, where β=nα for ...some n∈N. In this work, we introduce a modification to the definition of conformable fractional derivative and conformable fractional integral. Our definition characterizes by the realization of the commutative property between the differential operators and the integral operators. The higher derivatives in the sense of the modified conformable fractional derivative are coinciding with the sequential derivatives. Finally, we present the solutions of the Cauchy-Euler and linear (with constant coefficients) fractional differential equations to show our new approach.
In this paper, we introduce and study the diskcyclicity and disk transitivity of a set of operators. We establish a diskcyclicity criterion and give the relationship between this criterion and the ...diskcyclicity. As applications, we study the diskcyclicty of
C
0
-semigroups and
C
-regularized groups. We show that a diskcyclic
C
0
-semigroup exists on a complex topological vector space
X
if and only if dim(
X
) = 1 or dim(
X
) = ∞ and we prove that diskcyclicity and disk transitivity of
C
0
-semigroups (resp
C
-regularized groups) are equivalent.
The goal of the paper is to obtain analogs of the sampling theorems and of the Riesz–Boas interpolation formulas which are relevant to the discrete Hilbert and Kak–Hilbert transforms in
$l^{2}$
.
In this article we present a straightforward generalisation to groups with operators of a number of invariants well-known in the theory of modules, having a special bearing on phenomena of ...semisimplicity. We examine the behaviour of the generalised invariants in relation to the fundamental construction of restricted direct sums and in particular in the context of semisimple groups with operators. We also consider a particular type of morphisms between groups with operators, which naturally preserves the generalised invariants in question and thus shows itself to be an adequate notion for their study in the more general frame considered.
We show that the discrete Hilbert transform and the discrete Kak–Hilbert transform are infinitesimal generators of one-parameter groups of operators in
${{\ell }^{2}}$
.
M. sova ((10)) proved that the infinitesimal generator of all uniformly continuous cosine family, of operators in Banach space, is a bounded operator. We show by counter-example that the result ...mentioned above is not true in general on Fréchet spaces, and we prove that the infinitesimal generator of an uniformly continuous cosine family of operators in a class of Fréchet spaces (quojection) is necessarily continuous.
In this paper, Hardy's uncertainty principle and unique continuation properties of Schrödinger equations with operator potentials in Hilbert space-valued $L^{2}$ classes are obtained. Since the ...Hilbert space $H$ and linear operators are arbitrary, by choosing the appropriate spaces and operators we obtain numerous classes of Schrödinger type equations and its finite and infinite many systems which occur in a wide variety of physical systems.
We present a realization for some
K
-functionals associated with Jacobi expansions in terms of generalized Jacobi–Weierstrass operators. Fractional powers of the operators as well as results ...concerning simultaneous approximation and Nikolskii–Stechkin type inequalities are also considered.
We discuss some relations between the local existence of analytic selections of eigenvectors (LSP
=
“NON-SVEP”) for an operator in Banach space and some chaoticity properties of linear dynamical ...system (with discrete or continuous time) generated by this operator. Our main goal is to prove the existence of a strong connection of some results known for many years in the local spectral theory to some important problems (which seem to be not solved so far) in the linear chaos theory. We also find a simple particular solution of the problem formulated in “Eigenvectors Selection Conjecture” (Banasiak and Moszyński in Discrete Contin Dyn Syst A 20(3):577–587,
2008
, Conjecture 4.3, p. 585) and we formulate a new convenient spectral criterion for linear chaos. To make the assumptions more clear we introduce some special parts of the point spectrum of a closed operator, including the right-inversion point spectrum. Using this new criterion we prove chaoticity of a large class of super-upper-triangular operators in
l
p
and
c
0
spaces and also of some strongly continuous semi-groups generated by such operators.
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