W. Żelazko

Topological Simplicity of a Certain LF-Algebra

We show that the LF-algebra considered by Akkar and Nacir in [1] is topologically simple.

Operators on Locally Convex Spaces

This paper is essentially a survey of some previous author’s results, but it contains also some new concepts and theorems. In the second section we discuss the (already known) concepts of an operator algebra on a locally convex space and of a topologizable operator (see definitions below). We also introduce and study here the (new) concept of jointly topologizable operators. In the third section we discuss some questions open for all infinitely dimensional Banach spaces, such as the problem of existence of closed proper invariant and hyperinvariant subspaces for all continuous endomorphisms of a given space, and the Fell-Doran Problem in the representation theory of algebras. These problems can be solved in positive for the locally convex space (s) of all numerical sequences. The novelty here is the Theorem 3.2 stating that all continuous endomorphisms of the complex space (s), which are not scalar multiples of the identity operator, have closed proper hyperinvariant subspaces.

Concerning closed invariant subspaces for endomorphisms of the space (s)

We show that every (continuous) endomorphism of the complex space (s) has a proper closed invariant subspace which is either of dimension one or of codimension one. We give also a related result for the real space (s).

Concerning Strong Generation of L (ε)

Denote by ε the locally convex space of entire functions in one complex variable, endowed with the compact-open topology, and by L(ε) — the algebra of all continuous linear maps from ε into itself. Our main result states that L(ε) is strongly generated by the semigroup T(s), s ≥ 0, of translations

Generation of B(X) two commutative subalgebras - results and open problems

(T_{(s)}x)(\zeta)\ =\ x(\zeta\ +\ s),\ x\ \epsilon\ \varepsilon,\ \zeta\ \epsilon\ {\cal C}

A characterization of maximal ideals in commutative Banach algebras

and by the operator Tz, given by

A characterization of multiplicative linear functionals in complex Banach algebras

T_zx=zx, where\ z(\zeta)=\zeta

On joint spectra of commuting families of operators

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Metric generalizations of Banach algebras

Let X be a real or complex Banach space and let τ be a topology on the algebra B(X) of all endomorphisms of X. For a non-void subset S ⊂ B(X), let alg(S) denote the smallest subalgebra of B(X) which contains S, that is, the set of all linear combinations of finite products of elements of S, and let algτ (S) denote its τ -closure. We shall also consider the uniform (norm) topology on B(X), which we denote by u, and the strong topology, which we denote by s. We say that B(X) is algebraically generated by S if B(X) = alg(S), uniformly generated by S if B(X) = algu(S), and strongly generated by S if B(X) = algs(S). Our general problem is whether B(X) can be generated by two commutative subalgebras A1 and A2, i.e. by S = A1 ∪ A2. More specifically, we consider