A mapping α from a normed space X into itself is called a Banach operator if there is a constant k such that 0≤ k < 1 and ‖α2(x)−α(x)‖ ≤ k‖α(x)−x‖ for all x ∈X. In this note we study some properties of Banach operators. Among other results we show that if α is a linear Banach operator on a normed space X, then N(α−1) = N((α−1)2), N(α−1)∩R(α−1)= (0) and if X is finite dimensional then X =N(α−1)⊕...

We show that if an infinite-dimensional Banach space X has a symmetric basis then there exists a bounded, linear operator R : X −→ X such that the set A = {x ∈ X : ||Rx|| → ∞} is non-empty and nowhere dense in X. Moreover, if x ∈ X \ A then some subsequence of (Rx)n=1 converges weakly to x. This answers in the negative a recent conjecture of Prǎjiturǎ. The result can be extended to any Banach s...

The object of the investigation is to study reducible $M$-ideals in Banach spaces. It is shown that if the number of $M$-ideals in a Banach space $X$ is $n(<infty)$, then the number of reducible $M$-ideals does not exceed of $frac{(n-2)(n-3)}{2}$. Moreover, given a compact metric space $X$, we obtain a general form of a reducible $M$-ideal in the space $C(X)$ of continuous functions on $X$. The...

Then Xc is the completion of {X, \\ \\c). Alternatively || ||c is the Minkowski functional of the convex hull of the unit ball. Xc has the property that any bounded linear operator L:X —> Z into a Banach space extends with preservation of norm to an operator L\XC —» Z. The Banach envelope of / (0 < p < 1) is, of course, lx. In 1969, Duren, Romberg and Shields [3] identified the dual space of H_...

It is well known that the identity is an operator with the following property: if the operator, initially defined on an n-dimensional Banach space V , can be extended to any Banach space with norm 1, then V is isometric to (n) ∞ . We show that the set of all such operators consists precisely of those with spectrum lying in the unit circle. This result answers a question raised in [5] for comple...

It is well known that the identity is an operator with the following property: if the operator, initially defined on an n-dimensional Banach space V , can be extended to any Banach space with norm 1, then V is isometric to (n) ∞ . We show that the set of all such operators consists precisely of those with spectrum lying in the unit circle. This result answers a question raised in [5] for comple...

We show that if an infinite-dimensional Banach space X has a symmetric basis then there exists a bounded, linear operator R : X −→ X such that the set A = {x ∈ X : ||R(x)|| → ∞} is non-empty and nowhere dense in X . Moreover, if x ∈ X \ A then some subsequence of (R(x)) n=1 converges weakly to x. This answers in the negative a recent conjecture of Prǎjiturǎ. The result can be extended to any Ba...

This work concerns the study of approximate boundary controllability for some nonlinear partial functional integrodifferential equations with finite delay arising in modeling materials memory, framework general Banach spaces. We give sufficient conditions that ensure system by supposing its linear undelayed part is approximately controllable, admits a resolvent operator sense Grimmer, and makin...

In this note, we show that if a Banach space X has a predual, then every bounded linear operator on X with a continuous functional calculus admits a bounded Borel functional calculus. A consequence of this is that on such a Banach space, the classes of finitely spectral and prespectral operators coincide. We also apply this result to give some sufficient conditions for an operator with an absol...