Gives the special structure of the spectrum of bounded linear operators on a class of indecomposable Σ1e type Banach spaces;shows that there is a Σ1e type Banach space on which there is a well-bounded operator of type (B) such that the spectrum of it is the infinite countable set.

Well-bounded operators are those which possess a bounded functional calculus for the absolutely continuous functions on some compact intervals.

Shows that R(X),the class of Riesz operators,on a Σ1e type Banach space is equal to In(X),the ideal of inessential operators, so R(X) is a closed by operator norm,two-sided ideal in B(X) of co-dimension one;gives some properties of well-bounded operators on such spaces.

The theory of well bounded operators has been found many applications and formed deep connections with other areas of mathematics.

Discusses spectrum of well bounded operator of type B.

It is obtained that Iα is a bounded operator from Lp(Rn) into the Lorentz space Lq,∞(Rn).

Some necessary and sufficient conditions are given for which M_(φ) is a bounded operator from B~α to B~β_0(respectively,from B~α_0 to B~β).

This note proves that for every f∈C(\;X), the continuous functionu,u(t)=∫ t 0S(t-s)f(s) d s, t∈is a strong (classical) solution of the second inhomogeneous zero initial value problem u″=Au+f, in \, iff A is a bounded operator in X.