توصيفگر ها :
ضرب آرنز , گروه فشرده موضعي , ضربگر , عملگر تاوبري , عملگر هم ـ تاوبري , فردهلم با شاخص صفر
چكيده انگليسي :
• Cely, L., Galego, E. M. and González, M., Tauberian convolution operators acting on L1(G), J. Math. Anal.
Appl., 446 (2017), 299–306.
• Caly, L., Calego, E. M. and Gonzalez, M., Convolution operators on group algebras which are tauberian or
cotauberian, J. Math. Anal. Appl., 465 (2018), 309-317.
Let X and Y are complex Banach spaces and T : X −→ Y is a (continuous linear) operator from X to Y . Anoperator T : X −→ Y is called tauberian if the second conjugate T∗∗ : X∗∗ −→ Y∗∗ satisfies(T∗∗)−1(Y ) ⊂ X and T is called cotauberian if its conjugate T
∗is tauberian. In 1976, Kalton and Wilansky [15] mentioned such definitions, but Wilansky called tauberian the conservative matrices that sum no bounded divergent sequence [25].
Tauberian operators were introduced to investigate a problem in summability theory. Since that introduction, they have made a deep impact on the isomorphic theory of Banach spaces. In fact, these operators have been useful inseveral contexts of Banach space theory that have no apparent or obvious connections. In [8], the authors have tried to find out the reason and how this category of describe operators and point out their importance. Let us recall that
an operator T : X −→ Y is said to be upper semi-Fredholm if it has closed range and finite dimensional kernel, we denote by R(T) and N(T)the range and the kernel of T respectively. In 1976, Yang extends the theory of Fredholm operators to the case of tauberian operators with closed range [26]. In 1974, Davis, Figiel, Johnson and Pelczynski obtain their famous factorization for weakly compact operators, which is the main source of examples of tauberian
operators [6]. For basic results on Fredholm theory and tauberian operators we refer to [1, 8, 10] respectively. Given
a Banach algebra A, a map T : A −→ A is a multiplier of A if
T(xy) = x(T(y)) = (T(x))y (x, y ∈ A).
The concept of a multiplier first appears in harmonic analysis in connection with the theory of summability for Fourier series, multipliers have also appeared in a variety of other contexts, among which we mention the general theory of Banach algebras, representation theory for Banach algebras and the study of Banach modules, the theory of singular
integrals, interpolation theory, stochastic processes. Also, multipliers on Banach algebra without order is Reviewed
[16]. Wendel famous theorem states, Let G be a locally compact Abelian group. Then the space of multipliers for
L1(G) is isometrically isomorphic to M(G). The multipliers for the group algebra L1(G) have been investigated
by Edwards, Helson and Wendel. If G be a locally compact abelian group and µ be a complex Borel measure on G.
Our goal in this thesis is to study convoltion operator that’s mean, multipliers on group algebras L1(G) and M(G).
In fact, we will give a characterization for this class of operators. We show that every cotauberian convolution operator Tµ is tauberian. Finally, we specify, the relationship between tauberian and cotauberian operators acting on
L1(G) with the tauberian and cotauberian operators acting on M(G).
In chapter 1, we introduce basic definitions and concepts.
In chapter 2, we introduce tauberian and cotauberian operators and we give examples of them, also introduce upper semi-Fordhelm operators and lower semi-Fordhelm operators. In fact, the upper semi-Fordhelm operators are
tauberian operators and lower semi-Fordhelm operators are cotauberian operators. At the end, we explain residuum operator
