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On the structure of fixed point sets of some compact maps in the Fréchet space

Zbyněk Kubáček (1993)

Mathematica Bohemica

The aim of this note is 1. to show that some results (concerning the structure of the solution set of equations (18) and (21)) obtained by Czarnowski and Pruszko in [6] can be proved in a rather different way making use of a simle generalization of a theorem proved by Vidossich in [8]; and 2. to use a slight modification of the “main theorem” of Aronszajn from [1] applying methods analogous to the above mentioned idea of Vidossich to prove the fact that the solution set of the equation (24), (25)...

Order intervals in C ( K ) . Compactness, coincidence of topologies, metrizability

Zbigniew Lipecki (2022)

Commentationes Mathematicae Universitatis Carolinae

Let K be a compact space and let C ( K ) be the Banach lattice of real-valued continuous functions on K . We establish eleven conditions equivalent to the strong compactness of the order interval [ 0 , x ] in C ( K ) , including the following ones: (i) { x > 0 } consists of isolated points of K ; (ii) [ 0 , x ] is pointwise compact; (iii) [ 0 , x ] is weakly compact; (iv) the strong topology and that of pointwise convergence coincide on [ 0 , x ] ; (v) the strong and weak topologies coincide on [ 0 , x ] . Moreover, the weak topology and that of pointwise convergence...

Regular vector lattices of continuous functions and Korovkin-type theorems-Part II

Francesco Altomare, Mirella Cappelletti Montano (2006)

Studia Mathematica

By applying the results of the first part of the paper, we establish some Korovkin-type theorems for continuous positive linear operators in the setting of regular vector lattices of continuous functions. Moreover, we present simple methods to construct Korovkin subspaces for finitely defined operators and for the identity operator and we determine those classes of operators which admit finite-dimensional Korovkin subspaces. Finally, we give a Korovkin-type theorem for continuous positive projections....

Regular vector lattices of continuous functions and Korovkin-type theorems-Part I

Francesco Altomare, Mirella Cappelletti Montano (2005)

Studia Mathematica

We introduce and study a new class of locally convex vector lattices of continuous functions on a locally compact Hausdorff space, which we call regular vector lattices. We investigate some general properties of these spaces and of the subspaces of so-called generalized affine functions. Moreover, we present some Korovkin-type theorems for continuous positive linear operators; in particular, we study Korovkin subspaces for finitely defined operators, for the identity operator and for positive...

Structural aspects of truncated archimedean vector lattices: good sequences, simple elements

Richard N. Ball (2021)

Commentationes Mathematicae Universitatis Carolinae

The truncation operation facilitates the articulation and analysis of several aspects of the structure of archimedean vector lattices; we investigate two such aspects in this article. We refer to archimedean vector lattices equipped with a truncation as truncs. In the first part of the article we review the basic definitions, state the (pointed) Yosida representation theorem for truncs, and then prove a representation theorem which subsumes and extends the (pointfree) Madden representation theorem....

The band generated by homomorphisms on Banach lattices.

David C. Carothers, William A. Feldman (1998)

Extracta Mathematicae

This paper will consider the closure of the set of operators which may be expressed as a sum of lattice homomorphisms whose range is contained in a Dedekind complete Banch lattice.

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