scholarly journals Linear Operators and Equations with Partial Integrals

2019 ◽  
Vol 65 (3) ◽  
pp. 390-433
Author(s):  
A S Kalitvin ◽  
V A Kalitvin
Keyword(s):  
Spectral Radius ◽  
Spectral Properties ◽  
Volterra Operator ◽  
Linear Equations ◽  
Continuous Functions ◽  
Linear Operators ◽  
Fredholm Alternative ◽  
Fredholm Equations ◽  
Spaces Of Continuous Functions ◽  
Banach Ideal Spaces

We consider linear operators and equations with partial integrals in Banach ideal spaces, spaces of vector functions, and spaces of continuous functions. We study the action, regularity, duality, algebras, Fredholm properties, invertibility, and spectral properties of such operators. We describe principal properties of linear equations with partial integrals. We show that such equations are essentially different compared to usual integral equations. We obtain conditions for the Fredholm alternative, conditions for zero spectral radius of the Volterra operator with partial integrals, and construct resolvents of invertible equations. We discuss Volterra-Fredholm equations with partial integrals and consider problems leading to linear equations with partial integrals.

scholarly journals BOUNDED LINEAR OPERATORS ON SPACES IN NORMED DUALITY

2007 ◽  
Vol 49 (1) ◽  
pp. 145-154
Author(s):  
BRUCE A. BARNES
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Integral Operators ◽  
Continuous Functions ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Closed Range ◽  
Bounded Linear ◽  
Spaces Of Continuous Functions

Abstract.LetTbe a bounded linear operator on a Banach spaceW, assumeWandYare in normed duality, and assume thatThas adjointT†relative toY. In this paper, conditions are given that imply that for all λ≠0, λ−Tand λ −T†maintain important standard operator relationships. For example, under the conditions given, λ −Thas closed range if, and only if, λ −T†has closed range.These general results are shown to apply to certain classes of integral operators acting on spaces of continuous functions.

scholarly journals Positive linear operators and the approximation of continuous functions on locally compact abelian groups

1980 ◽  
Vol 30 (2) ◽  
pp. 180-186 ◽  
Author(s):  
Walter R. Bloom ◽  
Joseph F. Sussich
Keyword(s):  
Continuous Functions ◽  
Linear Operators ◽  
Positive Linear Operators ◽  
Locally Compact Abelian Group ◽  
Periodic Functions ◽  
Locally Compact ◽  
Character Group ◽  
Spaces Of Continuous Functions ◽  
Compact Abelian Groups ◽  
Approximation Of Continuous Functions

AbstractIn 1953 P. P. Korovkin proved that if (Tn) is a sequence of positive linear operators defined on the space C of continuous real 2π-periodic functions and limn→rTnf = f uniformly for f = 1, cos and sin. then limn→rTnf = f uniformly for all f∈C. We extend this result to spaces of continuous functions defined on a locally compact abelian group G, with the test family {1, cos, sin} replaced by a set of generators of the character group of G.

scholarly journals A spectral radius problem connected with weak compactness

1993 ◽  
Vol 35 (1) ◽  
pp. 85-94 ◽  
Author(s):  
Hans-Olav Tylli
Keyword(s):  
Asymptotic Behaviour ◽  
Banach Spaces ◽  
Spectral Radius ◽  
Spectral Properties ◽  
Weak Compactness ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Radius Problem

The asymptotic behaviour has been determined for several natural geometric or topological quantities related to (degrees of) compactness of bounded linear operators on Banach spaces; see for instance [24], [25] and [17]. This paper complements these results by studying the spectral properties of some quantities related to weak compactness.

scholarly journals A Korovkin type theorem for weighted spaces of continuous functions

1997 ◽  
Vol 55 (2) ◽  
pp. 239-248 ◽  
Author(s):  
Walter Roth
Keyword(s):  
Type Theorem ◽  
Weighted Approximation ◽  
Continuous Functions ◽  
Linear Operators ◽  
Weighted Spaces ◽  
Korovkin Type Theorem ◽  
Korovkin Type Approximation Theorem ◽  
Spaces Of Continuous Functions ◽  
Locally Convex Topologies ◽  
Locally Convex

We prove a Korovkin type approximation theorem for positive linear operators on weighted spaces of continuous real-valued functions on a compact Hausdorff space X. These spaces comprise a variety of subspaces of C (X) with suitable locally convex topologies and were introduced by Nachbin 1967 and Prolla 1977. Some early Korovkin type results on the weighted approximation of real-valued functions in one and several variables with a single weight function are due to Gadzhiev 1976 and 1980.

scholarly journals Jointly separating maps between vector-valued function spaces

2020 ◽  
Vol 70 (3) ◽  
pp. 707-718
Author(s):  
Ziba Pourghobadi ◽  
Masoumeh Najafi Tavani ◽  
Fereshteh Sady
Keyword(s):  
Banach Algebras ◽  
Function Algebra ◽  
Complex Banach Space ◽  
Continuous Functions ◽  
Linear Operators ◽  
Hausdorff Spaces ◽  
Absolutely Continuous Functions ◽  
Spaces Of Continuous Functions ◽  
Vector Valued Function ◽  
Vector Valued

AbstractLet X and Y be compact Hausdorff spaces, E be a real or complex Banach space and F be a real or complex locally convex topological vector space. In this paper we study a pair of linear operators S, T : A(X, E) → C(Y, F) from a subspace A(X, E) of C(X, E) to C(Y, F), which are jointly separating, in the sense that Tf and Sg have disjoint cozeros whenever f and g have disjoint cozeros. We characterize the general form of such maps between certain classes of vector-valued (as well as scalar-valued) spaces of continuous functions including spaces of vector-valued Lipschitz functions, absolutely continuous functions and continuously differentiable functions. The results can be applied to a pair T : A(X) → C(Y) and S : A(X, E) → C(Y, F) of linear operators, where A(X) is a regular Banach function algebra on X, such that f ⋅ g = 0 implies Tf ⋅ Sg = 0, for all f ∈ A(X) and g ∈ A(X, E). If T and S are jointly separating bijections between Banach algebras of scalar-valued functions of this class, then they induce a homeomorphism between X and Y and, furthermore, T−1 and S−1 are also jointly separating maps.

scholarly journals BOUNDARY-CHAOTIC BEHAVIOUR OF CONTINUOUS FUNCTIONS UNDER THE ACTION OF OPERATORS

2003 ◽  
Vol 46 (2) ◽  
pp. 451-464
Author(s):  
Luis Bernal-González ◽  
María del Carmen Calderón-Moreno
Keyword(s):  
Composition Operators ◽  
Primary 47B38 ◽  
Holomorphic Functions ◽  
Continuous Functions ◽  
Linear Operators ◽  
Multiplication Operators ◽  
Spaces Of Holomorphic Functions ◽  
Spaces Of Continuous Functions ◽  
Left And Right ◽  
Secondary 30D40

AbstractIn this paper we introduce two classes of operators on spaces of continuous functions with values in $F$-spaces under the action of which many functions behave chaotically near the boundary. Several examples—including onto linear operators, left and right composition operators, multiplication operators, and operators with pointwise dense range or with some stability property—are given. This new theory extends one recently developed on spaces of holomorphic functions.AMS 2000 Mathematics subject classification: Primary 47B38. Secondary 30D40; 46E10; 54D45

On the Principal Eigenvalue of Disconjugate BVPs with L1-Coefficients

2002 ◽  
Vol 2 (1) ◽  
Author(s):  
Guy Degla
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Principal Eigenvalue ◽  
Continuous Functions ◽  
Linear Operators ◽  
The Maximum Principle ◽  
Multipoint Boundary ◽  
Spaces Of Continuous Functions ◽  
Multipoint Boundary Value Problems ◽  
Ordered Banach Spaces

AbstractEstablishing some general results on the principal eigenvalues for a class of multipoint boundary value problems, we prove existence and uniqueness theorems for positive solutions of nonlinear multipoint boundary value problems. Our argument is based mainly on the maximum principle [4] (highlighting new ordered Banach spaces of continuous functions) and on the continuity of the spectral radius mapping for compact linear operators.

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