15.—The Measure of Non-compactness of Some Linear Integral Operators

1973 ◽  
Vol 71 (2) ◽  
pp. 167-179
Author(s):  
C. A. Stuart
Keyword(s):  
Compact Operator ◽  
Essential Spectrum ◽  
Differential Operators ◽  
Integral Operators ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Resolvent Set ◽  
Linear Integral ◽  
Necessary And Sufficient ◽  
Linear Integral Operators

SynopsisThe measure of non-compactness of linear integral operators on the half-line [0, ∞) of a special type is studied. In particular, a necessary and sufficient condition is established for an operator of this type to define a compact operator from L2(0, ∞) into itself. These results are then used to discuss the spectrum of second-order differential operators. A necessary and sufficient condition for the spectrum to be discrete is established together with estimates for the distance of a point in the resolvent set from the essential spectrum.

NOETHER IDENTITIES OF A DIFFERENTIAL OPERATOR: THE KOSZUL–TATE COMPLEX

2005 ◽  
Vol 02 (05) ◽  
pp. 873-886 ◽  
Author(s):  
G. SARDANASHVILY
Keyword(s):  
Differential Operator ◽  
Fiber Bundle ◽  
Differential Operators ◽  
Chain Complex ◽  
Boundary Operator ◽  
Lagrangian System ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

Given a generic Lagrangian system, its Euler–Lagrange operator obeys Noether identities which need not be independent, but satisfy first-stage Noether identities, and so on. This construction is generalized to arbitrary differential operators on a smooth fiber bundle. Namely, if a certain necessary and sufficient condition holds, one can associate to a differential operator the exact chain complex with the boundary operator whose nilpotency restarts all the Noether identities characterizing the degeneracy of an original differential operator.

scholarly journals On Approximation of the Perturbed Inclusion

2007 ◽  
Vol 14 (2) ◽  
pp. 253-267
Author(s):  
Alexander I. Bulgakov ◽  
Anna A. Grigorenko ◽  
Anatoliy I. Korobko
Keyword(s):  
Stable Process ◽  
Approximate Solutions ◽  
Continuous Functions ◽  
The Other ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Space Of Continuous Functions ◽  
Linear Integral ◽  
The Right ◽  
Necessary And Sufficient

Abstract The paper is concerned with the so-called perturbed inclusion in the space of continuous functions. The right-hand side of the inclusion is represented by an algebraic sum of the values of two multi-valued maps, one of which consists of compacts and the other is not necessarily closed-valued and is a composition of a linear integral operator and multimap convex-valued with respect to switching. For such an inclusion it is proved that approximation in the space of summable functions of the values of a multimap convex-valued with respect to switching is not always a stable process. The necessary and sufficient condition for the closure of the set of approximate solutions to converge to the closure of the set of solutions for perturbed inclusion is derived.

scholarly journals Microlocal regularity on step two nilpotent Lie groups

1988 ◽  
Vol 31 (1) ◽  
pp. 25-39
Author(s):  
Kenneth G. Miller
Keyword(s):  
Differential Operators ◽  
Pseudodifferential Operators ◽  
Partial Differential Operator ◽  
Irreducible Unitary Representation ◽  
Nilpotent Lie Groups ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Partial Differential ◽  
Partial Differential Operators ◽  
Necessary And Sufficient

A necessary and sufficient condition for a homogeneous left invariant partial differential operator P on a nilpotent Lie group G to be hypoelliptic is that π(P) be injective in π for every nontrivial irreducible unitary representation π of G. This was conjectured by Rockland in [18], where it was also proved in the case of the Heisenberg group. The necessity of the condition in the general case was proved by Beals [2] and the sufficiency by Helffer and Nourrigat [4]. In this paper we present a microlocal version of this theorem when G is step two nilpotent. The operator may be homogeneous with respect to any family of dilations on G, not just the natural dilations. We may also consider pseudodifferential operators as well as partial differential operators.

scholarly journals Some applications of generalized Ruscheweyh derivatives involving a general fractional derivative operator to a class of analytic functions with negative coefficients II

2020 ◽  
Vol 28 (1) ◽  
pp. 85-103
Author(s):  
Waggas Galib Atshan ◽  
S. R. Kulkarni
Keyword(s):  
Fractional Derivative ◽  
Linear Combination ◽  
Analytic Functions ◽  
Univalent Functions ◽  
Integral Operators ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Derivative Operator ◽  
Necessary And Sufficient

AbstractIn this paper, we study a class of univalent functions f as defined by making use of the generalized Ruscheweyh derivatives involving a general fractional derivative operator, satisfying{\mathop{\rm Re}\nolimits} \left\{{{{z\left({{\bf{J}}_1^{\lambda,\mu}f\left(z \right)} \right)'} \over {\left({1 - \gamma} \right){\bf{J}}_1^{\lambda,\mu}f\left(z \right) + \gamma {z^2}\left({{\bf{J}}_1^{\lambda,\mu}f\left(z \right)} \right)''}}} \right\} > \beta.A necessary and sufficient condition for a function to be in the class A_\gamma ^{\lambda,\mu,\nu}\left({n,\beta} \right) is obtained. Also, our paper includes linear combination, integral operators and we introduce the subclass A_{\gamma,{c_m}}^{\lambda,\mu,\nu}\left({1,\beta} \right) consisting of functions with negative and fixed finitely many coefficients. We study some interesting properties of A_{\gamma,{c_m}}^{\lambda,\mu,\nu}\left({1,\beta} \right).

Characterizations of nuclear pseudo-differential operators on ℤ with some applications

2018 ◽  
Vol 13 (4) ◽  
pp. 33
Author(s):  
Majid JamalpourBirgani
Keyword(s):  
Differential Operators ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Bounded Operators ◽  
Nuclear Operators ◽  
Pseudo Differential Operators ◽  
Necessary And Sufficient ◽  
Adjoint Operators

In this paper, we give necessary and sufficient conditions on the symbols σ, such that the corresponding pseudo-differential operators Tσ from Lp1(ℤ) into Lp2(ℤ), 1 ≤ p1,p2 < ∞, be nuclear. We show that the adjoint operators of the nuclear pseudo-differential operators from Lp′2(ℤ) into Lp′1(ℤ) are nuclear and present a necessary and sufficient condition on the symbols of the nuclear pseudo-differential operators from L2(ℤ) into L2(ℤ) to be self-adjoint. As applications, We get the symbol of the product of the nuclear operators with the bounded operators, and a necessary and sufficient condition on the symbols of nuclear operators is given to be normal.

scholarly journals Note on norm of an m-linear integral-type operator between weighted-type spaces

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Stevo Stević
Keyword(s):  
Type Operator ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Integral Type ◽  
Type Spaces ◽  
Linear Integral ◽  
Necessary And Sufficient ◽  
Weighted Type

AbstractWe find a necessary and sufficient condition for the boundedness of an m-linear integral-type operator between weighted-type spaces of functions, and calculate norm of the operator, complementing some results by L. Grafakos and his collaborators. We also present an inequality which explains a detail in the proof of the boundedness of the linear integral-type operator on $L^{p}({\mathbb {R}}^{n})$ L p ( R n ) space.

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